From f355a72c300e96075c9188427abb9921bd3d82e8 Mon Sep 17 00:00:00 2001 From: LehengChen Date: Sun, 28 Jun 2026 19:20:39 +0800 Subject: [PATCH 1/2] Add Poincare TeX Astrolabe extractor --- tools/poincare_tex_extract.py | 2971 +++++++++++++++++++++++++++++++++ 1 file changed, 2971 insertions(+) create mode 100644 tools/poincare_tex_extract.py diff --git a/tools/poincare_tex_extract.py b/tools/poincare_tex_extract.py new file mode 100644 index 00000000..edf3241b --- /dev/null +++ b/tools/poincare_tex_extract.py @@ -0,0 +1,2971 @@ +#!/usr/bin/env python3 +"""Build the Poincare Challenge Astrolabe store from Morgan--Tian TeX. + +This is intentionally project-specific. The generic ``tex2mdx.py`` is good for +first-pass reading docs, but it drops labels and references. Here the TeX source +is the authority for statement boundaries and labels; the generated MDX is a +structured source format that can be re-extracted deterministically. + +Outputs: + projects/poincare-conjecture/.astrolabe/docs-src/*.mdx + readable MDX with ``astrolabe:begin/end`` statement markers + projects/poincare-conjecture/.astrolabe/docs/*.mdx + reading docs with ``\\entryblock{hash}`` + projects/poincare-conjecture/.astrolabe/atoms/*.md + projects/poincare-conjecture/.astrolabe/edges/*.md +""" + +from __future__ import annotations + +import argparse +import json +import os +import re +import shutil +import sys +import tarfile +import tempfile +import unicodedata +import urllib.request +from collections import Counter, defaultdict +from dataclasses import dataclass, field +from pathlib import Path + +from astrolabe_store import AstrolabeStorage, validate_store +from tex2mdx import ( + collect_macros, + convert_font_switches, + convert_inline, + convert_lists, + strip_comments, + strip_definitions, + tidy, + _balanced, +) + + +DEFAULT_PROJECT = Path("projects/poincare-conjecture") +DEFAULT_SRC = Path("/tmp/poincare-arxiv-src") +ARXIV_EPRINT = "https://arxiv.org/e-print/math/0607607" + +ENV_SORT = { + "thm": "theorem", + "cor": "corollary", + "lem": "lemma", + "prop": "proposition", + "claim": "claim", + "defn": "definition", + "exam": "example", + "conj": "conjecture", + "ex": "exercise", + "rem": "remark", + "assumption": "assumption", + "addendum": "addendum", +} + +SORT_LABEL = { + "theorem": "Theorem", + "corollary": "Corollary", + "lemma": "Lemma", + "proposition": "Proposition", + "claim": "Claim", + "definition": "Definition", + "example": "Example", + "conjecture": "Conjecture", + "exercise": "Exercise", + "remark": "Remark", + "assumption": "Assumption", + "addendum": "Addendum", +} + +STATEMENT_ENV_RE = re.compile( + r"\\begin\{(" + "|".join(re.escape(k) for k in ENV_SORT) + r")\}" + r"(?:\[([^\]]*)\])?(.*?)\\end\{\1\}", + re.S, +) +PROOF_RE = re.compile(r"\\begin\{proof\}(.*?)\\end\{proof\}", re.S) +PROOF_TOKEN_RE = re.compile(r"\\(begin|end)\{proof\}") +LABEL_RE = re.compile(r"\\label\{([^{}]+)\}") +REF_RE = re.compile(r"\\(?:ref|eqref)\{([^{}]+)\}") +CITE_RE = re.compile(r"\\cite[tp]?\*?(?:\[[^\]]*\])?\s*\{([^{}]+)\}") +DISPLAY_START_RE = re.compile(r"\$\$|\\\[|\\begin\{(?:equation|eqnarray|align|gather|multline)\*?\}") +DISPLAY_BLOCK_ENVS = ["align", "align*", "gather", "gather*", "multline", "multline*", "eqnarray", "eqnarray*"] +TEXT_MACROS = { + r"\f1": r"\frac", + r"\g1": r"\Sigma", + r"\01": r"\Omega", + r"\c1": r"\gamma", + r"\o1": r"\omega", + r"\d1": r"\delta", + r"\e1": r"\epsilon", + r"\l1": r"\Lambda", + r"\m1": r"\Theta", + r"\t1": r"\theta", + r"\v1": r"\varphi", + r"\w1": r"\wedge", + r"\lemin": r"\le \min", + r"\betacos": r"\beta\cos", + r"\cdotexp": r"\cdot \exp", + r"\setminusint": r"\setminus \operatorname{int}", + r"\lbrack": "[", + r"\rbrack": "]", +} +OUTER_FONT_SWITCHES = { + "bf": "**", + "bfseries": "**", + "it": "*", + "itshape": "*", + "em": "*", + "sl": "*", + "slshape": "*", + "tt": "`", + "ttfamily": "`", + "sc": "", + "scshape": "", + "sf": "", + "sffamily": "", + "rm": "", + "rmfamily": "", +} +GENERATED_SENTINEL = "" +LEGACY_GENERATED_DOC_NAMES = { + "01-intro.mdx", + "02-prelim.mdx", + "03-flowbasics.mdx", + "04-maxprin.mdx", + "05-converge2.mdx", + "06-newcompar.mdx", + "07-newcomp2.mdx", + "08-noncoll.mdx", + "09-temp2kappa.mdx", + "10-bddcurvbdddist.mdx", + "11-singlimit2.mdx", + "12-stdsoln.mdx", + "13-surgery.mdx", + "14-energy1.mdx", + "15-canonnbhd.mdx", +} +LOCAL_ANAPHORA_RE = re.compile( + r"(?i)\b(this|previous|preceding|following)\s+" + r"(theorem|lemma|proposition|corollary|claim|definition|remark)\b" +) +RESOLVABLE_ANAPHORA_RE = re.compile( + r"(?i)\b(previous|preceding|following)\s+" + r"(theorem|lemma|proposition|corollary|claim|definition|remark|example)\b" +) +PROSE_DEPENDENCY_SIGNAL_RE = re.compile( + r"(?i)\b(" + r"follows? (?:immediately |directly )?from|" + r"by|from|using|applying|apply|combining|together with|" + r"in view of|as a consequence of|according to|implies?|" + r"we conclude|we deduce|we obtain|this proves|completes the proof|" + r"proved above|established above|leads? (?:immediately )?to" + r")\b" +) +PROSE_DEPENDENCY_NEGATIVE_RE = re.compile( + r"(?i)\b(" + r"will|shall|later|below|next|in section|following section|" + r"chapter|appendix|see|cf\.|compare|recall|called|denote|" + r"is proved in|are proved in|was proved in|were proved in|" + r"we prove|we shall prove|will prove" + r")\b" +) +PROSE_NEXT_STATEMENT_SOURCE_RE = re.compile( + r"(?i)\b(" + r"the following is|" + r"following\s+(?:theorem|lemma|proposition|corollary|claim|statement|result)|" + r"leads? (?:immediately )?to (?:the )?following" + r")\b" +) +THEOREM_LIKE_SORTS = {"theorem", "lemma", "proposition", "corollary", "claim"} +TERM_STOP_WORDS = { + "appendix", + "case", + "central", + "chapter", + "condition", + "conditions", + "constant", + "curvature", + "definition", + "direction", + "domain", + "equation", + "equivalent", + "example", + "exists", + "family", + "flow", + "function", + "generate", + "length", + "lemma", + "manifold", + "metric", + "minimizing", + "neighborhood", + "point", + "preserves", + "property", + "proposition", + "remark", + "result", + "section", + "set", + "solution", + "space", + "strong", + "structure", + "system", + "theorem", + "time", + "within", +} +TERM_STOP_PHRASES = { + "boundary contained", + "cap in whose core contains", + "component", + "converges in the gromov hausdorff sense", + "surgery operation at time", +} +TERM_BAD_INTERNAL_WORDS = { + "contains", + "converges", + "satisfies", +} +TERM_SINGLE_ALLOWLIST = { + "convex", + "geodesic", + "neck", + "soliton", + "worldline", +} +TERM_BAD_BOUNDARY_WORDS = { + "a", + "an", + "and", + "at", + "by", + "for", + "from", + "if", + "in", + "is", + "of", + "on", + "or", + "the", + "to", + "with", +} +MAX_DEFINITION_TERM_USES_PER_SOURCE = 3 +MAX_DEFINITION_TERM_USES_PER_TERM = 30 +SECTION_COMMAND_RE = re.compile(r"\\(section|subsection|subsubsection)\*?\s*") +EDGE_LAYER_METADATA = { + "explicit": { + "evidence_type": "tex-reference", + "confidence": 1.0, + "review_status": "accepted", + "inference": "explicit", + "kind": "reference", + "scope": "statement-or-proof", + }, + "proof_containment_dependency": { + "evidence_type": "proof-contained-theorem-like-statement", + "confidence": 0.85, + "review_status": "accepted", + "inference": "inferred", + "kind": "dependency", + "scope": "proof-containment", + }, + "prose_dependency": { + "evidence_type": "inferential-prose-reference", + "confidence": 0.8, + "review_status": "accepted", + "inference": "inferred", + "kind": "dependency", + "scope": "prose", + }, + "local_anaphora": { + "evidence_type": "local-anaphora-reference", + "confidence": 0.72, + "review_status": "accepted", + "inference": "inferred", + "kind": "dependency", + "scope": "local-context", + }, + "definition_term": { + "evidence_type": "same-chapter-definition-term-match", + "confidence": 0.45, + "review_status": "candidate", + "inference": "weak", + "kind": "definition-use", + "scope": "same-chapter-term-match", + }, + "prose_mention": { + "evidence_type": "prose-mention-nearest-previous-statement", + "confidence": 0.35, + "review_status": "candidate", + "inference": "weak", + "kind": "mention", + "scope": "prose", + }, + "proof_contains": { + "evidence_type": "proof-containment", + "confidence": 0.9, + "review_status": "accepted", + "inference": "structural", + "kind": "containment", + "scope": "proof", + }, + "section_sequence": { + "evidence_type": "section-reading-order-adjacency", + "confidence": 0.25, + "review_status": "accepted", + "inference": "navigational", + "kind": "sequence", + "scope": "section", + }, + "chapter_sequence": { + "evidence_type": "chapter-reading-order-adjacency", + "confidence": 0.2, + "review_status": "accepted", + "inference": "navigational", + "kind": "sequence", + "scope": "chapter", + }, +} + + +@dataclass +class Statement: + chapter: int + segment_key: str + file_stem: str + index: int + env: str + sort: str + raw_body: str + start: int + end: int + body_start: int + body_end: int + opt_title: str = "" + labels: list[str] = field(default_factory=list) + hash: str = "" + + @property + def mtref(self) -> str: + return f"{self.chapter}.{self.index}" + + +@dataclass +class ChapterSegment: + key: str + order: int + chapter: int + file_stem: str + title_raw: str + start: int + end: int + + +@dataclass +class ProofSpan: + file_stem: str + start: int + body_start: int + body_end: int + end: int + depth: int + owner_label: str = "" + + @property + def body_range(self) -> tuple[int, int]: + return self.body_start, self.body_end + + +@dataclass(frozen=True) +class DefinitionTerm: + term: str + source: str + raw_start: int + raw_end: int + + +def canon(record: dict) -> str: + return json.dumps(record, sort_keys=True, ensure_ascii=False) + + +def norm_label(label: str) -> str: + """Normalize TeX labels split across source lines.""" + return re.sub(r"\s+", " ", label.strip()) + + +def safe_extract_tar(tf: tarfile.TarFile, dest: Path) -> None: + """Extract an arXiv tarball without allowing path traversal.""" + root = dest.resolve() + for member in tf.getmembers(): + target = (root / member.name).resolve() + if target != root and root not in target.parents: + raise RuntimeError(f"Refusing to extract unsafe tar member: {member.name}") + tf.extractall(root) + + +def ensure_source(src: Path) -> Path: + """Use an existing source dir, or download/extract arXiv e-print to it.""" + if (src / "booktemplate.tex").exists(): + return src + src.mkdir(parents=True, exist_ok=True) + with tempfile.NamedTemporaryFile(suffix=".tar.gz", delete=False) as tmp: + tmp_path = Path(tmp.name) + try: + print(f"downloading {ARXIV_EPRINT} -> {tmp_path}") + urllib.request.urlretrieve(ARXIV_EPRINT, tmp_path) + with tarfile.open(tmp_path, "r:*") as tf: + safe_extract_tar(tf, src) + finally: + try: + tmp_path.unlink() + except OSError: + pass + if not (src / "booktemplate.tex").exists(): + raise FileNotFoundError(f"booktemplate.tex not found in {src}") + return src + + +def include_order(src: Path) -> list[str]: + root = strip_comments((src / "booktemplate.tex").read_text(encoding="utf-8", errors="replace")) + return [m.group(1).strip().removesuffix(".tex") for m in re.finditer(r"\\include\{([^{}]+)\}", root)] + + +def source_texts(src: Path, order: list[str]) -> dict[str, str]: + return { + stem: strip_comments((src / f"{stem}.tex").read_text(encoding="utf-8", errors="replace")) + for stem in order + } + + +def read_group_at(s: str, i: int) -> tuple[str, int] | None: + while i < len(s) and s[i].isspace(): + i += 1 + if i >= len(s) or s[i] != "{": + return None + return _balanced(s, i) + + +def remove_balanced_macro(s: str, name: str, keep: str = "") -> str: + """Remove ``\name{...}`` occurrences with balanced braces.""" + out: list[str] = [] + i = 0 + pat = re.compile(r"\\" + re.escape(name) + r"\s*") + while True: + m = pat.search(s, i) + if not m: + out.append(s[i:]) + break + out.append(s[i:m.start()]) + g = read_group_at(s, m.end()) + if not g: + out.append(m.group(0)) + i = m.end() + continue + body, j = g + out.append(keep.format(body=body) if keep else "") + i = j + return "".join(out) + + +def chapter_commands(text: str) -> list[tuple[int, int, bool, str]]: + """Return ``(start, title_end, starred, title_raw)`` for each chapter.""" + found: list[tuple[int, int, bool, str]] = [] + for m in re.finditer(r"\\chapter(\*)?\s*", text): + g = read_group_at(text, m.end()) + if not g: + continue + title, j = g + found.append((m.start(), j, bool(m.group(1)), title)) + return found + + +def chapter_segments(texts: dict[str, str], order: list[str]) -> list[ChapterSegment]: + """Split included files by actual TeX chapter boundaries. + + Morgan--Tian has multi-chapter include files (notably ``prelim`` and + ``surgery``). The theorem counter is reset by ``\chapter``, so statement + identities and reader docs have to follow these boundaries, not filenames. + """ + segments: list[ChapterSegment] = [] + chapter = 0 + order_index = 0 + for stem in order: + text = texts[stem] + commands = chapter_commands(text) + if not commands: + continue + for i, (start, _title_end, starred, title_raw) in enumerate(commands): + if not starred: + chapter += 1 + number = 0 if starred and not segments else chapter + end = commands[i + 1][0] if i + 1 < len(commands) else len(text) + order_index += 1 + segments.append( + ChapterSegment( + key=f"{stem}:{i}", + order=order_index, + chapter=number, + file_stem=stem, + title_raw=title_raw, + start=start, + end=end, + ) + ) + return segments + + +def strip_labels(s: str) -> str: + return LABEL_RE.sub("", s) + + +def statement_labels(raw_body: str) -> list[str]: + """Labels before the first display-math block are statement labels. + + Morgan--Tian often labels equations inside theorem environments. Those + labels should stay textual references for now, not edges to the enclosing + statement. + """ + first_display = DISPLAY_START_RE.search(raw_body) + limit = first_display.start() if first_display else len(raw_body) + return [norm_label(m.group(1)) for m in LABEL_RE.finditer(raw_body) if m.start() <= limit] + + +def collect_aux_label_texts(texts: dict[str, str], segments: list[ChapterSegment]) -> dict[str, str]: + """Best-effort labels for refs that are not statement atoms. + + These stay textual on purpose. Equation/figure/section labels are not atoms + in this pass, but rendering "Equation (2.7)" is much better than leaking + "Equation (conemetric)". + """ + out: dict[str, str] = {} + for seg in segments: + text = texts[seg.file_stem][seg.start:seg.end] + eq_no = 0 + for m in re.finditer( + r"\\begin\{(equation|eqnarray|align|gather|multline)\*?\}(.*?)\\end\{\1\*?\}", + text, + re.S, + ): + if "*" in m.group(0).split("}", 1)[0]: + continue + eq_no += 1 + for lab in LABEL_RE.findall(m.group(2)): + out.setdefault(norm_label(lab), f"{seg.chapter}.{eq_no}") + + fig_no = 0 + for m in re.finditer(r"\\begin\{figure\*?\}(.*?)\\end\{figure\*?\}", text, re.S): + fig_no += 1 + for lab in LABEL_RE.findall(m.group(1)): + out.setdefault(norm_label(lab), f"{seg.chapter}.{fig_no}") + + section_no = 0 + subsection_no = 0 + section_pat = re.compile(r"\\(section|subsection|subsubsection)\*?\s*") + for m in section_pat.finditer(text): + g = read_group_at(text, m.end()) + if not g: + continue + _title, j = g + if m.group(1) == "section": + section_no += 1 + subsection_no = 0 + number = f"{seg.chapter}.{section_no}" + elif m.group(1) == "subsection": + subsection_no += 1 + number = f"{seg.chapter}.{section_no}.{subsection_no}" + else: + number = f"{seg.chapter}.{section_no}.{subsection_no}" + trailer = text[j:j + 160] + for lab in LABEL_RE.findall(trailer): + out.setdefault(norm_label(lab), number) + + # Chapter labels usually appear immediately after the chapter command. + chap = chapter_commands(text) + if chap: + trailer = text[chap[0][1]:chap[0][1] + 160] + for lab in LABEL_RE.findall(trailer): + out.setdefault(norm_label(lab), str(seg.chapter)) + return out + + +def clean_citations(s: str) -> str: + def repl(m: re.Match) -> str: + keys = ", ".join(k.strip() for k in m.group(1).split(",") if k.strip()) + return f"[{keys}]" if keys else "" + + return CITE_RE.sub(repl, s) + + +def normalize_title( + s: str, + label2hash: dict[str, str] | None = None, + label2text: dict[str, str] | None = None, +) -> str: + if label2hash is not None: + s = linkify_refs(s, label2hash, label2text) + s = strip_labels(clean_citations(s)) + s = convert_text_inline(s) + # TeX titles frequently wrap refs in \protect{...}; grouping braces are + # unsafe in MDX text and add no semantics once refs are converted. + prev = None + while prev != s: + prev = s + s = re.sub(r"\{(\\entryref\{[0-9a-f]+\})\}", r"\1", s) + s = re.sub(r"\s+", " ", s) + return s.strip() + + +def linkify_refs( + s: str, + label2hash: dict[str, str], + label2text: dict[str, str] | None = None, +) -> str: + # Consume the common "Kind~\ref{label}" form at once so the rendered link + # says "Theorem 5.4" rather than "Theorem Theorem 5.4". + kinds = ( + "Theorem|Lemma|Proposition|Corollary|Claim|Definition|Remark|Example|" + "Conjecture|Exercise|Assumption|Addendum" + ) + pat = re.compile(rf"\b({kinds})~?\\(?:ref|eqref)\{{([^{{}}]+)\}}") + + def kind_ref(m: re.Match) -> str: + lab = norm_label(m.group(2)) + h = label2hash.get(lab) + if h: + return f"\\entryref{{{h}}}" + text = label2text.get(lab) if label2text else None + return f"{m.group(1)} {text or lab}" + + s = pat.sub(kind_ref, s) + + def bare_ref(m: re.Match) -> str: + lab = norm_label(m.group(1)) + h = label2hash.get(lab) + if h: + return f"\\entryref{{{h}}}" + text = label2text.get(lab) if label2text else None + return text or lab + + return REF_RE.sub(bare_ref, s) + + +def replace_section_commands( + s: str, + chapter: int, + label2hash: dict[str, str] | None = None, + label2text: dict[str, str] | None = None, +) -> str: + out: list[str] = [] + i = 0 + pat = re.compile(r"\\(chapter|section|subsection|subsubsection)\*?") + while True: + m = pat.search(s, i) + if not m: + out.append(s[i:]) + break + g = read_group_at(s, m.end()) + if not g: + out.append(s[i:m.end()]) + i = m.end() + continue + body, j = g + out.append(s[i:m.start()]) + title = normalize_title(body, label2hash, label2text) + cmd = m.group(1) + if cmd == "chapter": + out.append(f"\n# Chapter {chapter} -- {title}\n") + elif cmd == "section": + out.append(f"\n## {title}\n") + elif cmd == "subsection": + out.append(f"\n### {title}\n") + else: + out.append(f"\n#### {title}\n") + i = j + return "".join(out) + + +def convert_quote_envs(s: str) -> str: + def repl(m: re.Match) -> str: + body = m.group(1).strip() + return "\n\n" + "\n".join("> " + line for line in body.splitlines()) + "\n\n" + + return re.sub(r"\\begin\{quote\}(.*?)\\end\{quote\}", repl, s, flags=re.S) + + +def convert_proof_markers(s: str) -> str: + s = re.sub(r"\\begin\{proof\}\s*", "\n\n**Proof.** ", s) + return re.sub(r"\\end\{proof\}", "\n\n", s) + + +def cleanup_display_math_body(body: str) -> str: + """Normalize old TeX text boxes that contain inline math delimiters.""" + def mbox_math_repl(m: re.Match) -> str: + prefix = m.group(1) + math = m.group(2) + suffix = m.group(3) + out = "" + if prefix: + out += rf"\text{{{prefix}}}" + out += math + if suffix: + out += rf"\text{{{suffix}}}" + return out + + return re.sub(r"\\mbox\{([^{}$]*?)\$([^$]+)\$([^{}$]*?)\}", mbox_math_repl, body) + + +def convert_math(s: str) -> str: + """Convert display math without reprocessing generated aligned blocks.""" + s = re.sub(r"\\\[(.*?)\\\]", lambda m: f"\n$$\n{cleanup_display_math_body(m.group(1).strip())}\n$$\n", s, flags=re.S) + s = re.sub( + r"\\begin\{equation\*?\}(.*?)\\end\{equation\*?\}", + lambda m: f"\n$$\n{cleanup_display_math_body(m.group(1).strip())}\n$$\n", + s, + flags=re.S, + ) + for env in DISPLAY_BLOCK_ENVS: + s = re.sub( + rf"\\begin\{{{re.escape(env)}\}}(.*?)\\end\{{{re.escape(env)}\}}", + lambda m: f"\n$$\n\\begin{{aligned}}\n{cleanup_display_math_body(m.group(1).strip())}\n\\end{{aligned}}\n$$\n", + s, + flags=re.S, + ) + + def bare_display(m: re.Match) -> str: + body = cleanup_display_math_body(m.group(1).strip()) + return f"\n\n$$\n{body}\n$$\n\n" if body else "\n\n" + + return re.sub(r"\$\$(.*?)\$\$", bare_display, s, flags=re.S) + + +def normalize_markdown_math_boundaries(s: str) -> str: + s = re.sub(r"(\*\*(?:Theorem|Corollary|Lemma|Proposition|Claim|Definition|Example|Conjecture|Exercise|Remark|Assumption|Addendum)\.?\*\*)\s+\$\$", r"\1\n\n$$", s) + s = re.sub(r"\$\$\s+(\*\*(?:Theorem|Corollary|Lemma|Proposition|Claim|Definition|Example|Conjecture|Exercise|Remark|Assumption|Addendum)\.?\*\*)", r"$$\n\n\1", s) + return s + + +def drop_display_environments(s: str) -> str: + for n in ["figure", "figure*", "table", "table*", "wrapfigure", "titlepage", "abstract"]: + s = re.sub(rf"\\begin\{{{re.escape(n)}\}}.*?\\end\{{{re.escape(n)}\}}", "", s, flags=re.S) + return s + + +def cleanup_latex_scaffolding(s: str) -> str: + s = remove_balanced_macro(s, "protect", keep="{body}") + s = remove_balanced_macro(s, "lefteqn", keep="{body}") + s = re.sub(r"\\protect\s*", "", s) + s = re.sub(r"\{(\\entryref\{[0-9a-f]+\})\}", r"\1", s) + s = re.sub(r"\\(?:maketitle|tableofcontents|newpage|clearpage|bigskip|medskip|smallskip|noindent|par)\b", "", s) + s = re.sub(r"\\(?:pagestyle|thispagestyle|bibliographystyle|bibliography|includeonly)\{[^{}]*\}", "", s) + s = re.sub(r"\\(?:printindex|makeindex)\b", "", s) + # Old TeX spacing commands that should not survive in prose. + s = re.sub(r"\\[,;:!]\s*", " ", s) + s = s.replace("~", " ") + return s + + +def expand_text_macros(s: str) -> str: + for old, new in TEXT_MACROS.items(): + s = s.replace(old, new) + s = re.sub(r"\\root\s+([^\\{}\s]+)\s*\\of\s*\{([^{}]+)\}", r"\\sqrt[\1]{\2}", s) + return s + + +def convert_legacy_bold_in_math(s: str) -> str: + """Convert old TeX font switches inside math to KaTeX-safe commands.""" + def repl(m: re.Match) -> str: + prefix = m.group(1) + body = m.group(2).strip() + if not body: + return prefix + cmd = r"\mathbf" if prefix in (r"\bf", r"\textbf") else r"\mathit" + return f"{cmd}{{{body}}}" + + # Braced forms inside math, e.g. ${\bf t}$. + s = re.sub(r"\{(\\(?:bf|it|rm|textbf|textit))\s+([^{}]+)\}", repl, s) + # Simple switch forms up to the next obvious math delimiter. + s = re.sub(r"\\bf\s+([A-Za-z0-9]+)", r"\\mathbf{\1}", s) + s = re.sub(r"\\it\s+([A-Za-z0-9]+)", r"\\mathit{\1}", s) + return re.sub(r"\\rm\s+", "", s) + + +def math_ranges(s: str) -> list[tuple[int, int]]: + """Return best-effort markdown math ranges so prose transforms can avoid them.""" + ranges: list[tuple[int, int]] = [] + i = 0 + while i < len(s): + starts = [(s.find(token, i), token) for token in ("$$", "$") if s.find(token, i) != -1] + starts = [(pos, token) for pos, token in starts if pos >= 0] + if not starts: + break + pos, token = min(starts, key=lambda item: item[0]) + j = s.find(token, pos + len(token)) + if j == -1: + break + ranges.append((pos, j + len(token))) + i = j + len(token) + return ranges + + +def position_in_ranges(pos: int, ranges: list[tuple[int, int]]) -> bool: + return any(start <= pos < end for start, end in ranges) + + +def convert_outer_font_switch_groups(s: str) -> str: + """Convert balanced text font groups that may span inline math.""" + names = "|".join(sorted(OUTER_FONT_SWITCHES, key=len, reverse=True)) + pat = re.compile(r"\{\\(" + names + r")\b[ \t\n]*") + ranges = math_ranges(s) + out: list[str] = [] + i = 0 + while True: + m = pat.search(s, i) + if not m: + out.append(s[i:]) + break + if position_in_ranges(m.start(), ranges): + out.append(s[i:m.end()]) + i = m.end() + continue + out.append(s[i:m.start()]) + kind = m.group(1) + inner, end = _balanced(s, m.start()) + content = convert_outer_font_switch_groups(inner[len("\\" + kind):].strip()) + mark = OUTER_FONT_SWITCHES[kind] + out.append(f"{mark}{content}{mark}" if mark and content else content) + i = end + return "".join(out) + + +TEX_ACCENT_COMBINING = { + "'": "\u0301", + "`": "\u0300", + "^": "\u0302", + "\"": "\u0308", + "~": "\u0303", + "=": "\u0304", + ".": "\u0307", + "u": "\u0306", + "v": "\u030c", + "H": "\u030b", + "r": "\u030a", + "c": "\u0327", + "d": "\u0323", + "b": "\u0331", + "k": "\u0328", +} +TEX_ACCENT_RE = re.compile( + r"\\(?P['`\"^~=.uvHrcdbk])\s*(?:\{(?P\\i|\\j|[A-Za-z])\}|(?P[A-Za-z]))" +) +PROSE_SYMBOL_MACROS = { + r"\S": "Section", + r"\P": "Paragraph", + r"\LaTeX": "LaTeX", + r"\TeX": "TeX", +} + + +def tex_accent_repl(m: re.Match) -> str: + accent = m.group("accent") + char = m.group("grouped") or m.group("bare") or "" + if char == r"\i": + char = "i" + elif char == r"\j": + char = "j" + mark = TEX_ACCENT_COMBINING.get(accent, "") + return unicodedata.normalize("NFC", char + mark) if mark else char + + +def cleanup_prose_tex_commands(s: str) -> str: + """Clean TeX commands that occur in prose, leaving math spans untouched.""" + s = re.sub(r"`([^`'\n]{1,120})'", r"'\1'", s) + s = s.replace("``", '"').replace("''", '"') + s = TEX_ACCENT_RE.sub(tex_accent_repl, s) + for old, new in PROSE_SYMBOL_MACROS.items(): + s = re.sub(re.escape(old) + r"\b", new, s) + s = re.sub(r"\\(?:quad|qquad)\b", " ", s) + s = re.sub(r"\\[ ,;:!]\s*", " ", s) + return re.sub(r"[ \t]{2,}", " ", s) + + +def move_footnotes_after_following_font_group(s: str) -> str: + """Keep inline footnotes from splitting an immediately following font group.""" + out: list[str] = [] + i = 0 + pat = re.compile(r"\\footnote\s*") + font_pat = re.compile(r"\{\\(" + "|".join(sorted(OUTER_FONT_SWITCHES, key=len, reverse=True)) + r")\b") + while True: + m = pat.search(s, i) + if not m: + out.append(s[i:]) + break + g = read_group_at(s, m.end()) + if not g: + out.append(s[i:m.end()]) + i = m.end() + continue + footnote_end = g[1] + j = footnote_end + while j < len(s) and s[j].isspace(): + j += 1 + if j < len(s) and s[j] == "{" and font_pat.match(s, j): + _font_inner, font_end = _balanced(s, j) + out.append(s[i:m.start()]) + out.append(s[j:font_end]) + out.append(s[m.start():footnote_end]) + i = font_end + continue + out.append(s[i:footnote_end]) + i = footnote_end + return "".join(out) + + +def split_math_spans(s: str) -> list[tuple[bool, str]]: + """Best-effort split into math and prose spans for markdown-only transforms.""" + spans: list[tuple[bool, str]] = [] + i = 0 + while i < len(s): + starts = [(s.find(token, i), token) for token in ("$$", "$") if s.find(token, i) != -1] + starts = [(pos, token) for pos, token in starts if pos >= 0] + if not starts: + spans.append((False, s[i:])) + break + pos, token = min(starts, key=lambda item: item[0]) + if pos > i: + spans.append((False, s[i:pos])) + j = s.find(token, pos + len(token)) + if j == -1: + spans.append((False, s[pos:])) + break + spans.append((True, s[pos:j + len(token)])) + i = j + len(token) + return spans + + +def convert_text_inline(s: str) -> str: + s = expand_text_macros(s) + s = convert_outer_font_switch_groups(s) + out: list[str] = [] + for is_math, part in split_math_spans(s): + if is_math: + out.append(convert_legacy_bold_in_math(part)) + else: + part = convert_inline(part) + part = convert_font_switches(part) + part = re.sub(r"\\(?:em|it|rm|bf)\b\s*", "", part) + part = cleanup_prose_tex_commands(part) + out.append(part) + rendered = "".join(out) + rendered = re.sub(r"\*\*\s*\*\*", " ", rendered) + return cleanup_latex_scaffolding(rendered) + + +def convert_fragment( + raw: str, + chapter: int, + label2hash: dict[str, str], + label2text: dict[str, str] | None = None, + *, + statement: bool = False, +) -> str: + s = strip_comments(raw) + s = drop_display_environments(s) + s = strip_definitions(s) + s = remove_balanced_macro(s, "label") + s = remove_balanced_macro(s, "index") + s = move_footnotes_after_following_font_group(s) + s = remove_balanced_macro(s, "footnote", keep=" ({body})") + s = linkify_refs(s, label2hash, label2text) + s = clean_citations(s) + if not statement: + s = replace_section_commands(s, chapter, label2hash, label2text) + s = convert_proof_markers(s) + s = convert_quote_envs(s) + s = convert_lists(s) + s = convert_math(s) + # tex2mdx can create empty display wrappers around aligned blocks when the + # source mixed $$ with old eqnarray syntax; remove the harmless wrappers. + s = re.sub(r"\$\$\s*\$\$\s*(?=\n\\begin\{aligned\})", "", s, flags=re.S) + s = convert_text_inline(s) + s = normalize_markdown_math_boundaries(s) + s = re.sub(r"\n{3,}", "\n\n", s) + return tidy(s) + + +def atom_record(st: Statement, label2text: dict[str, str]) -> dict: + # Keep atom notes hash-local: embedding target hashes via label2hash would + # create content-address cycles for mutually referencing statements. The + # dependency graph stores those links; generated docs-src prose can still + # render clickable entryrefs where it is not part of an atom hash. + notes = convert_fragment(st.raw_body, st.chapter, {}, label2text, statement=True) + rec = { + "sort": st.sort, + "source": "tex", + "src": "morgan-tian", + "mtref": st.mtref, + "chapter": st.chapter, + "tex_file": st.file_stem, + "generator": "tools/poincare_tex_extract.py", + "title": normalize_title(st.opt_title, {}, label2text) if st.opt_title else "", + "labels": st.labels, + "notes": notes, + } + if st.labels: + rec["label"] = st.labels[0] + if not rec["title"]: + rec.pop("title") + return rec + + +def parse_statements( + texts: dict[str, str], + segments: list[ChapterSegment], + storage: AstrolabeStorage, +) -> tuple[dict[str, str], dict[str, str], dict[str, list[Statement]]]: + label2hash: dict[str, str] = {} + label2text: dict[str, str] = collect_aux_label_texts(texts, segments) + by_segment: dict[str, list[Statement]] = {} + + for seg in segments: + text = texts[seg.file_stem] + items: list[Statement] = [] + for idx, m in enumerate(STATEMENT_ENV_RE.finditer(text, seg.start, seg.end), 1): + st = Statement( + chapter=seg.chapter, + segment_key=seg.key, + file_stem=seg.file_stem, + index=idx, + env=m.group(1), + sort=ENV_SORT[m.group(1)], + opt_title=(m.group(2) or "").strip(), + raw_body=m.group(3), + start=m.start(), + end=m.end(), + body_start=m.start(3), + body_end=m.end(3), + labels=statement_labels(m.group(3)), + ) + items.append(st) + by_segment[seg.key] = items + for items in by_segment.values(): + for st in items: + for lab in st.labels: + label2text[lab] = st.mtref + for items in by_segment.values(): + for st in items: + rec = atom_record(st, label2text) + st.hash = storage._compute_hash(["__self__"], canon(rec)) + for lab in st.labels: + label2hash[lab] = st.hash + return label2hash, label2text, by_segment + + +def statements_by_file( + order: list[str], + by_segment: dict[str, list[Statement]], +) -> dict[str, list[Statement]]: + out: dict[str, list[Statement]] = {stem: [] for stem in order} + for items in by_segment.values(): + for st in items: + out.setdefault(st.file_stem, []).append(st) + return {stem: sorted(items, key=lambda st: st.start) for stem, items in out.items()} + + +def all_statements(by_segment: dict[str, list[Statement]]) -> list[Statement]: + return [st for items in by_segment.values() for st in items] + + +def statement_inner_label_owners(by_segment: dict[str, list[Statement]]) -> dict[str, Statement]: + """Map equation/figure/other labels inside statements back to the statement. + + Statement labels are the leading labels already exposed through + ``label2hash``. Later labels usually name equations inside the statement; + references to those labels are useful dependencies on the enclosing + statement, but they should not become standalone atoms in this pass. + """ + owners: dict[str, Statement] = {} + statement_labels_set = {lab for st in all_statements(by_segment) for lab in st.labels} + for st in all_statements(by_segment): + for m in LABEL_RE.finditer(st.raw_body): + lab = norm_label(m.group(1)) + if lab in statement_labels_set: + continue + owners.setdefault(lab, st) + return owners + + +def proof_owner_hash( + proof: ProofSpan, + statements: list[Statement], + label2hash: dict[str, str], +) -> str | None: + owner_hash = label2hash.get(proof.owner_label) if proof.owner_label else None + if owner_hash: + return owner_hash + prev = previous_statement(statements, proof.start) + return prev.hash if prev else None + + +def proof_owner_label(raw_body: str) -> str: + """Return a label from leading proof headings such as ``(Of Lemma~\ref{x})``. + + Only the leading heading counts. Prose like "For a proof of Theorem ..." + later in a proof body is a reference, not an owner declaration. + """ + prefix = raw_body[:500].lstrip() + prefix = re.sub(r"^\{?\\(?:em|it|rm)\b\s*", "", prefix) + prefix = prefix.lstrip(" \t\r\n({[") + pat = re.compile( + r"(?i)^(?:proof\s+)?of\s+" + r"(?:(?:the|a)\s+)?" + r"(?:(?:Theorem|Lemma|Proposition|Corollary|Claim|Definition|Remark|" + r"Example|Conjecture|Exercise|Assumption|Addendum)\s*)?" + r"~?\s*(?:\\(?:ref|eqref)\{([^{}]+)\}|\\protect\{\\(?:ref|eqref)\{([^{}]+)\}\})" + ) + m = pat.search(prefix) + return norm_label(next((g for g in m.groups() if g), "")) if m else "" + + +def parse_proof_spans(texts: dict[str, str], order: list[str]) -> dict[str, list[ProofSpan]]: + """Parse proof environments with a stack so nested proofs stay well scoped.""" + spans: dict[str, list[ProofSpan]] = {stem: [] for stem in order} + for stem in order: + stack: list[tuple[int, int, int]] = [] + text = texts[stem] + for m in PROOF_TOKEN_RE.finditer(text): + if m.group(1) == "begin": + stack.append((m.start(), m.end(), len(stack) + 1)) + continue + if not stack: + continue + start, body_start, depth = stack.pop() + body_end = m.start() + spans[stem].append( + ProofSpan( + file_stem=stem, + start=start, + body_start=body_start, + body_end=body_end, + end=m.end(), + depth=depth, + owner_label=proof_owner_label(text[body_start:body_end]), + ) + ) + spans[stem].sort(key=lambda p: p.start) + return spans + + +def previous_statement(statements: list[Statement], pos: int) -> Statement | None: + prev = None + for st in statements: + if st.start < pos: + prev = st + else: + break + return prev + + +def next_statement(statements: list[Statement], pos: int) -> Statement | None: + for st in statements: + if st.start > pos: + return st + return None + + +def statement_at(statements: list[Statement], pos: int) -> Statement | None: + for st in statements: + if st.start <= pos < st.end: + return st + if st.start > pos: + break + return None + + +def proof_at(proofs: list[ProofSpan], pos: int) -> ProofSpan | None: + matches = [p for p in proofs if p.body_start <= pos < p.body_end] + return max(matches, key=lambda p: p.body_start) if matches else None + + +def nearby_statement_by_sort( + statements: list[Statement], + pos: int, + sort: str, + *, + direction: str, +) -> Statement | None: + if direction == "following": + for st in statements: + if st.start > pos and st.sort == sort: + return st + return None + for st in reversed(statements): + if st.start < pos and st.sort == sort: + return st + return None + + +def reference_sentence(text: str, start: int, end: int) -> str: + """Small context window for deciding whether a prose ref is inferential.""" + _a, _b, sentence = reference_sentence_span(text, start, end) + return sentence + + +def reference_sentence_span(text: str, start: int, end: int) -> tuple[int, int, str]: + """Return ``(start, end, normalized_text)`` for the sentence around a ref.""" + before = [ + text.rfind("\n\n", 0, start), + text.rfind(". ", 0, start), + text.rfind("; ", 0, start), + ] + a = max(before) + a = 0 if a < 0 else a + 1 + after = [ + pos for pos in ( + text.find(". ", end), + text.find("; ", end), + text.find("\n\n", end), + ) + if pos != -1 + ] + b = min(after) if after else min(len(text), end + 320) + return a, b, " ".join(text[a:b].split()) + + +def refs_in_fragment(fragment: str, label2hash: dict[str, str]) -> list[str]: + hashes: list[str] = [] + for lab in REF_RE.findall(fragment): + h = label2hash.get(norm_label(lab)) + if h and h not in hashes: + hashes.append(h) + return hashes + + +def refs_in_fragment_with_spans( + fragment: str, + label2hash: dict[str, str], + *, + base_offset: int = 0, +) -> list[dict]: + refs: list[dict] = [] + for m in REF_RE.finditer(fragment): + lab = norm_label(m.group(1)) + h = label2hash.get(lab) + if h: + refs.append({ + "hash": h, + "label": lab, + "start": base_offset + m.start(), + "end": base_offset + m.end(), + }) + return refs + + +def prose_dependency_clause_spans(sentence: str) -> list[tuple[int, int, str]]: + """Split a context window so inferential refs do not cross sentence boundaries.""" + spans: list[tuple[int, int, str]] = [] + start = 0 + for m in re.finditer(r"(?<=[.!?])\s+(?=[A-Z\\])|;\s+", sentence): + end = m.start() + if sentence[start:end].strip(): + spans.append((start, end, sentence[start:end])) + start = m.end() + if sentence[start:].strip(): + spans.append((start, len(sentence), sentence[start:])) + return spans + + +def normalize_match_text(s: str) -> str: + s = re.sub(r"\$[^$]*\$", " ", s) + s = re.sub(r"\\[A-Za-z]+", " ", s) + s = re.sub(r"[^A-Za-z0-9 -]+", " ", s) + s = re.sub(r"[-]+", " ", s) + return re.sub(r"\s+", " ", s).strip().lower() + + +TERM_HIDDEN_MACROS = { + "cite", + "citep", + "citet", + "eqref", + "footnote", + "index", + "label", + "pageref", + "ref", +} + + +def skip_optional_bracket(s: str, i: int) -> int: + while i < len(s) and s[i].isspace(): + i += 1 + if i >= len(s) or s[i] != "[": + return i + depth = 0 + while i < len(s): + if s[i] == "\\": + i += 2 + continue + if s[i] == "[": + depth += 1 + elif s[i] == "]": + depth -= 1 + if depth == 0: + return i + 1 + i += 1 + return len(s) + + +def skip_macro_arguments(s: str, i: int) -> int: + i = skip_optional_bracket(s, i) + while True: + while i < len(s) and s[i].isspace(): + i += 1 + if i >= len(s) or s[i] != "{": + return i + _inner, i = _balanced(s, i) + + +def normalize_match_text_with_map(raw: str) -> tuple[str, list[int]]: + """Normalize raw TeX for term matching, including hidden macro arguments.""" + return normalize_visible_match_text_with_map(raw, skip_hidden_macros=False) + + +def normalize_visible_match_text_with_map(raw: str, *, skip_hidden_macros: bool = True) -> tuple[str, list[int]]: + """Normalize visible raw TeX and map normalized chars back to raw offsets.""" + chars: list[str] = [] + positions: list[int] = [] + + def emit_space(pos: int) -> None: + if chars and chars[-1] != " ": + chars.append(" ") + positions.append(pos) + + i = 0 + while i < len(raw): + ch = raw[i] + if ch == "$": + j = raw.find("$", i + 1) + emit_space(i) + i = len(raw) if j == -1 else j + 1 + continue + if ch == "\\": + if i + 1 < len(raw) and raw[i + 1].isalpha(): + command_start = i + i += 1 + command_name_start = i + while i < len(raw) and raw[i].isalpha(): + i += 1 + command = raw[command_name_start:i] + if skip_hidden_macros and command in TERM_HIDDEN_MACROS: + emit_space(command_start) + i = skip_macro_arguments(raw, i) + continue + if command in {"quad", "qquad"}: + emit_space(command_start) + continue + continue + i += 1 + if i >= len(raw): + break + ch = raw[i] + if ch.isalnum(): + chars.append(ch.lower()) + positions.append(i) + else: + emit_space(i) + i += 1 + + while chars and chars[0] == " ": + chars.pop(0) + positions.pop(0) + while chars and chars[-1] == " ": + chars.pop() + positions.pop() + return "".join(chars), positions + + +def find_term_span_in_raw(raw: str, term: str) -> tuple[int, int] | None: + normalized, positions = normalize_visible_match_text_with_map(raw) + if not normalized: + return None + m = term_match_pattern(term).search(normalized) + if not m: + return None + return positions[m.start()], positions[m.end() - 1] + 1 + + +def find_specific_term_span_in_raw(raw: str, term: str, covering_terms: list[str]) -> tuple[int, int] | None: + normalized, positions = normalize_visible_match_text_with_map(raw) + if not normalized: + return None + covered: list[tuple[int, int]] = [] + for covering_term in covering_terms: + if covering_term == term: + continue + for m in term_match_pattern(covering_term).finditer(normalized): + covered.append((m.start(), m.end())) + for m in term_match_pattern(term).finditer(normalized): + if any(start <= m.start() and m.end() <= end for start, end in covered): + continue + return positions[m.start()], positions[m.end() - 1] + 1 + return None + + +def term_span_record(st: Statement, raw_start: int, raw_end: int) -> dict: + return absolute_statement_span(st, raw_start, raw_end) + + +def clean_definition_term(raw: str) -> str: + term = normalize_match_text(raw) + words = term.split() + while words and words[0] in {"a", "an", "the"}: + words.pop(0) + while words and words[0] in {"dimensional"}: + words.pop(0) + while words and words[-1] in TERM_BAD_BOUNDARY_WORDS: + words.pop() + term = " ".join(words) + if not term: + return "" + if len(words) > 6 or sum(ch.isalpha() for ch in term) < 4: + return "" + if words[0] in TERM_BAD_BOUNDARY_WORDS or words[-1] in TERM_BAD_BOUNDARY_WORDS: + return "" + if any(word in TERM_BAD_INTERNAL_WORDS for word in words): + return "" + if len(words) == 1 and words[0] not in TERM_SINGLE_ALLOWLIST: + return "" + if term in TERM_STOP_WORDS or term in TERM_STOP_PHRASES: + return "" + return term + + +def definition_terms(st: Statement, label2text: dict[str, str]) -> list[DefinitionTerm]: + if st.sort != "definition": + return [] + rendered = convert_fragment(st.raw_body, st.chapter, {}, label2text, statement=True) + candidates: list[tuple[str, str]] = [] + if st.opt_title: + candidates.append((normalize_title(st.opt_title, {}, label2text), "title")) + for m in re.finditer(r"\*([^*]{3,180})\*", rendered[:1800]): + candidates.append((m.group(1), "italic")) + plain = normalize_match_text(rendered[:1000]) + fallback_patterns = [ + ("called-if", r"\bis called (?:a|an|the)?\s*([a-z][a-z0-9 ]{3,70})\s+if\b"), + ("said-to-be-if", r"\bis said to be (?:a|an|the)?\s*([a-z][a-z0-9 ]{3,70})\s+if\b"), + ("we-call", r"\bwe call [a-z0-9 ]{0,80}? (?:a|an|the)?\s*([a-z][a-z0-9 ]{3,70})"), + ] + for source, pat in fallback_patterns: + candidates.extend((m.group(1), source) for m in re.finditer(pat, plain)) + + terms: list[DefinitionTerm] = [] + seen: set[str] = set() + for candidate, source in candidates: + term = clean_definition_term(candidate) + if not term or term in seen: + continue + span = find_term_span_in_raw(st.raw_body, term) + if not span: + continue + seen.add(term) + terms.append(DefinitionTerm(term=term, source=source, raw_start=span[0], raw_end=span[1])) + return sorted(terms, key=lambda t: (len(t.term.split()), len(t.term)), reverse=True)[:4] + + +def term_match_pattern(term: str) -> re.Pattern: + return re.compile(r"(? tuple[dict[tuple[str, str], set[str]], dict[tuple[str, str], dict]]: + """Parse explicit prose patterns where both dependent and dependency are refs.""" + edges: dict[tuple[str, str], set[str]] = {} + details: dict[tuple[str, str], dict] = {} + patterns = [ + ( + re.compile(r"\b(?:deduce|derive|obtain|conclude|prove)\b(?P.*?)\bfrom\b(?P.*)", re.I | re.S), + "prose-deduce-from", + ), + ( + re.compile(r"(?P.*?)\bfollows? (?:immediately |directly )?from\b(?P.*)", re.I | re.S), + "prose-follows-from", + ), + ( + re.compile( + r"(?P.*?)\b(?:is|are|was|were) " + r"(?:an? )?(?:immediate |direct )?consequence of\b(?P.*)", + re.I | re.S, + ), + "prose-consequence-of", + ), + ] + for clause_start, _clause_end, clause in prose_dependency_clause_spans(sentence): + for pat, via in patterns: + m = pat.search(clause) + if not m: + continue + source_refs = refs_in_fragment_with_spans( + m.group("src"), + label2hash, + base_offset=sentence_start + clause_start + m.start("src"), + ) + target_refs = refs_in_fragment_with_spans( + m.group("dst"), + label2hash, + base_offset=sentence_start + clause_start + m.start("dst"), + ) + for source_ref in source_refs: + for target_ref in target_refs: + src_hash = source_ref["hash"] + dst_hash = target_ref["hash"] + if src_hash == dst_hash: + continue + pair = (src_hash, dst_hash) + edges.setdefault(pair, set()).add(via) + add_edge_detail(details, pair, "proseTriggers", { + "via": [via], + "label": target_ref["label"], + "sourceLabel": source_ref["label"], + "sentence": " ".join(sentence.split()), + "sentenceSpan": tex_span_record(file_stem, sentence_start, sentence_start + len(sentence)), + "sourceRefSpan": tex_span_record(file_stem, source_ref["start"], source_ref["end"]), + "refSpan": tex_span_record(file_stem, target_ref["start"], target_ref["end"]), + "sourceStrategy": "refs-in-inferential-sentence", + }) + return edges, details + + +def merge_edge_maps(*maps: dict[tuple[str, str], set[str]]) -> dict[tuple[str, str], set[str]]: + merged: dict[tuple[str, str], set[str]] = {} + for edge_map in maps: + for pair, via_set in edge_map.items(): + merged.setdefault(pair, set()).update(via_set) + return merged + + +def edge_map_difference( + edge_map: dict[tuple[str, str], set[str]], + existing_pairs: set[tuple[str, str]], +) -> dict[tuple[str, str], set[str]]: + return {pair: set(via_set) for pair, via_set in edge_map.items() if pair not in existing_pairs} + + +def detail_map_difference( + details: dict[tuple[str, str], dict], + edges: dict[tuple[str, str], set[str]], +) -> dict[tuple[str, str], dict]: + return {pair: details[pair] for pair in edges if pair in details} + + +def tex_span_record(file_stem: str, start: int, end: int) -> dict: + return { + "file": f"{file_stem}.tex", + "start": start, + "end": end, + "coordinateSpace": "comment-stripped-tex", + "sourceTransform": "tex2mdx.strip_comments", + } + + +def proof_span_record(proof: ProofSpan) -> dict: + return { + "file": f"{proof.file_stem}.tex", + "start": proof.start, + "bodyStart": proof.body_start, + "bodyEnd": proof.body_end, + "end": proof.end, + "depth": proof.depth, + "ownerLabel": proof.owner_label, + "coordinateSpace": "comment-stripped-tex", + "sourceTransform": "tex2mdx.strip_comments", + } + + +def add_edge_detail( + details: dict[tuple[str, str], dict], + pair: tuple[str, str], + key: str, + value: dict, +) -> None: + bucket = details.setdefault(pair, {}).setdefault(key, []) + if value not in bucket: + bucket.append(value) + + +def absolute_statement_span(st: Statement, raw_start: int, raw_end: int) -> dict: + return tex_span_record(st.file_stem, st.body_start + raw_start, st.body_start + raw_end) + + +def statement_location(st: Statement) -> dict: + """Compact, stable source pointer for reviewing generated graph edges.""" + return { + "file": f"{st.file_stem}.tex", + "chapter": st.chapter, + "mtref": st.mtref, + "sort": st.sort, + "label": st.labels[0] if st.labels else "", + "span": { + "start": st.start, + "end": st.end, + "coordinateSpace": "comment-stripped-tex", + "sourceTransform": "tex2mdx.strip_comments", + }, + } + + +def edge_metadata_factory( + by_segment: dict[str, list[Statement]], + *, + evidence_type: str, + confidence: float, + review_status: str, + inference: str, + kind: str, + scope: str, + evidence_extra_by_pair: dict[tuple[str, str], dict] | None = None, +): + """Return per-edge metadata without changing the collector edge-map shape.""" + hash2statement = {st.hash: st for st in all_statements(by_segment)} + + def metadata(src_hash: str, dst_hash: str, via_set: set[str]) -> dict: + src = hash2statement.get(src_hash) + dst = hash2statement.get(dst_hash) + evidence: dict = { + "type": evidence_type, + "via": sorted(via_set), + } + if evidence_extra_by_pair: + evidence.update(evidence_extra_by_pair.get((src_hash, dst_hash), {})) + if src: + evidence["sourceStatement"] = statement_location(src) + if dst: + evidence["targetStatement"] = statement_location(dst) + return { + "confidence": confidence, + "reviewStatus": review_status, + "inference": inference, + "kind": kind, + "scope": scope, + "evidence": evidence, + } + + return metadata + + +def slice_without_ranges(text: str, start: int, end: int, ranges: list[tuple[int, int]]) -> str: + out: list[str] = [] + cursor = start + for a, b in sorted(ranges): + a = max(a, start) + b = min(b, end) + if b <= cursor or a >= end: + continue + if a > cursor: + out.append(text[cursor:a]) + cursor = max(cursor, b) + if cursor < end: + out.append(text[cursor:end]) + return "".join(out) + + +def proof_dependency_body( + text: str, + proof: ProofSpan, + statements: list[Statement], + proof_spans: list[ProofSpan], +) -> str: + """Proof text with nested statements/proofs removed to avoid double counting.""" + ignored: list[tuple[int, int]] = [ + (st.start, st.end) + for st in statements + if proof.body_start <= st.start and st.end <= proof.body_end + ] + ignored.extend( + (child.start, child.end) + for child in proof_spans + if proof.start < child.start and child.end <= proof.body_end + ) + return slice_without_ranges(text, proof.body_start, proof.body_end, ignored) + + +def write_macros(src: Path, project: Path) -> None: + ast = project / ".astrolabe" + texts = [ + p.read_text(encoding="utf-8", errors="replace") + for p in sorted(src.glob("*.tex")) + sorted(src.glob("*.sty")) + sorted(src.glob("*.cls")) + ] + macros = collect_macros(texts) + (ast / "katex-macros.json").write_text( + json.dumps(macros, ensure_ascii=False, indent=2, sort_keys=True) + "\n", + encoding="utf-8", + ) + + +def purge_old(storage: AstrolabeStorage) -> None: + def rec(h: str) -> dict: + try: + return json.loads(storage.data[h]["record"]) + except Exception: + return {} + + old_atoms = { + h for h, e in storage.data.items() + if len(e["ref"]) == 1 and rec(h).get("src") == "morgan-tian" + } + old_edges = { + h for h, e in storage.data.items() + if len(e["ref"]) > 1 + and rec(h).get("src") == "morgan-tian" + and rec(h).get("source") == "tex" + and (rec(h).get("rel") == "references" or rec(h).get("generator") == "tools/poincare_tex_extract.py") + } + foreign_edges = { + h for h, e in storage.data.items() + if len(e["ref"]) > 1 and h not in old_edges and any(r in old_atoms for r in e["ref"]) + } + if foreign_edges: + sample = ", ".join(sorted(foreign_edges)[:5]) + raise RuntimeError( + "Refusing to purge Morgan--Tian atoms because non-generated edges " + f"still reference them: {sample}" + ) + for h in old_atoms | old_edges: + storage.data.pop(h, None) + + +def register_atoms( + storage: AstrolabeStorage, + by_segment: dict[str, list[Statement]], + label2text: dict[str, str], +) -> None: + for items in by_segment.values(): + for st in items: + rec = atom_record(st, label2text) + storage.data[st.hash] = {"ref": [st.hash], "record": canon(rec)} + + +def collect_dependency_edges( + texts: dict[str, str], + order: list[str], + by_segment: dict[str, list[Statement]], + label2hash: dict[str, str], + aux_label2statement: dict[str, Statement] | None = None, +) -> tuple[dict[tuple[str, str], set[str]], dict[tuple[str, str], dict]]: + edges: dict[tuple[str, str], set[str]] = {} + details: dict[tuple[str, str], dict] = {} + aux_label2statement = aux_label2statement or {} + stmt_by_start = statements_by_file(order, by_segment) + proof_by_file = parse_proof_spans(texts, order) + + for stem in order: + text = texts[stem] + statements = stmt_by_start[stem] + proofs = proof_by_file[stem] + + for st in statements: + for m in REF_RE.finditer(st.raw_body): + lab = norm_label(m.group(1)) + h = label2hash.get(lab) + via = "statement" + if not h and lab in aux_label2statement: + h = aux_label2statement[lab].hash + via = "statement-aux-label-proxy" + if h and h != st.hash: + pair = (st.hash, h) + edges.setdefault(pair, set()).add(via) + add_edge_detail(details, pair, "refTriggers", { + "via": via, + "label": lab, + "context": "statement", + "sourceSpan": absolute_statement_span(st, m.start(), m.end()), + }) + + # Proof references are dependencies of the proved statement. Prefer an + # explicit "Proof of Lemma \ref{...}" heading; otherwise use the nearest + # preceding statement, matching ordinary TeX theorem/proof layout. + for proof in proofs: + owner_hash = proof_owner_hash(proof, statements, label2hash) + if owner_hash is None: + continue + ignored: list[tuple[int, int]] = [ + (st.start, st.end) + for st in statements + if proof.body_start <= st.start and st.end <= proof.body_end + ] + ignored.extend( + (child.start, child.end) + for child in proofs + if proof.start < child.start and child.end <= proof.body_end + ) + for m in REF_RE.finditer(text, proof.body_start, proof.body_end): + if any(a <= m.start() < b for a, b in ignored): + continue + lab = norm_label(m.group(1)) + h = label2hash.get(lab) + via = "proof" + if not h and lab in aux_label2statement: + h = aux_label2statement[lab].hash + via = "proof-aux-label-proxy" + if h and h != owner_hash: + pair = (owner_hash, h) + edges.setdefault(pair, set()).add(via) + add_edge_detail(details, pair, "refTriggers", { + "via": via, + "label": lab, + "context": "proof", + "sourceSpan": tex_span_record(stem, m.start(), m.end()), + "proofSpan": proof_span_record(proof), + }) + return edges, details + + +def collect_proof_containment_dependency_edges( + structural_edges: dict[tuple[str, str], set[str]], + structural_details: dict[tuple[str, str], dict], + by_segment: dict[str, list[Statement]], +) -> tuple[dict[tuple[str, str], set[str]], dict[tuple[str, str], dict]]: + """Semantic dependencies from a theorem to theorem-like statements in its proof.""" + edges: dict[tuple[str, str], set[str]] = {} + details: dict[tuple[str, str], dict] = {} + hash2statement = {st.hash: st for st in all_statements(by_segment)} + for pair in structural_edges: + _src_hash, dst_hash = pair + target = hash2statement.get(dst_hash) + if target and target.sort in THEOREM_LIKE_SORTS: + edges.setdefault(pair, set()).add("proof-contained-statement") + if pair in structural_details: + details[pair] = { + "proofSpan": structural_details[pair].get("proofSpan"), + "containmentReason": "target theorem-like statement occurs inside source proof", + } + return edges, details + + +def collect_local_anaphora_edges( + texts: dict[str, str], + order: list[str], + by_segment: dict[str, list[Statement]], + label2hash: dict[str, str], +) -> tuple[dict[tuple[str, str], set[str]], dict[tuple[str, str], dict]]: + """Resolve local references such as "previous lemma" in proofs/statements.""" + edges: dict[tuple[str, str], set[str]] = {} + details: dict[tuple[str, str], dict] = {} + stmt_by_start = statements_by_file(order, by_segment) + proof_by_file = parse_proof_spans(texts, order) + + for stem in order: + text = texts[stem] + statements = stmt_by_start[stem] + proofs = proof_by_file[stem] + for m in RESOLVABLE_ANAPHORA_RE.finditer(text): + direction = m.group(1).lower() + sort = m.group(2).lower() + current = statement_at(statements, m.start()) + via = "local-anaphora-statement" + if current: + owner_hash = current.hash + trigger_context = "statement" + trigger_detail: dict = {} + else: + proof = proof_at(proofs, m.start()) + if not proof: + continue + owner_hash = proof_owner_hash(proof, statements, label2hash) + via = "local-anaphora-proof" + trigger_context = "proof" + trigger_detail = {"proofSpan": proof_span_record(proof)} + if not owner_hash: + continue + target = nearby_statement_by_sort( + statements, + m.start(), + sort, + direction="following" if direction == "following" else "previous", + ) + if target and target.hash != owner_hash: + pair = (owner_hash, target.hash) + edges.setdefault(pair, set()).add(via) + trigger = { + "via": via, + "phrase": m.group(0), + "direction": direction, + "sort": sort, + "context": trigger_context, + "sourceSpan": tex_span_record(stem, m.start(), m.end()), + } + trigger.update(trigger_detail) + add_edge_detail(details, pair, "anaphoraTriggers", trigger) + return edges, details + + +def collect_definition_term_edges( + by_segment: dict[str, list[Statement]], + label2text: dict[str, str], + existing_pairs: set[tuple[str, str]], +) -> tuple[dict[tuple[str, str], set[str]], dict[tuple[str, str], dict]]: + """Low-confidence same-chapter edges from later term use to definitions.""" + statements = all_statements(by_segment) + position = {st.hash: i for i, st in enumerate(statements)} + degree: Counter[str] = Counter() + for src_hash, dst_hash in existing_pairs: + degree[src_hash] += 1 + degree[dst_hash] += 1 + + definitions = [ + (st, definition_terms(st, label2text)) + for st in statements + if st.sort == "definition" + ] + definitions = [(st, terms) for st, terms in definitions if terms] + term_owners_by_chapter: dict[tuple[int, str], set[str]] = defaultdict(set) + for definition, terms in definitions: + for term in terms: + term_owners_by_chapter[(definition.chapter, term.term)].add(definition.hash) + ambiguous_terms = { + key for key, owners in term_owners_by_chapter.items() if len(owners) > 1 + } + terms_by_chapter: dict[int, set[str]] = defaultdict(set) + for definition, terms in definitions: + for term in terms: + terms_by_chapter[definition.chapter].add(term.term) + covering_terms_by_chapter: dict[tuple[int, str], list[str]] = {} + for chapter, terms in terms_by_chapter.items(): + for term in terms: + covering_terms_by_chapter[(chapter, term)] = sorted( + ( + other for other in terms + if other != term and term_match_pattern(term).search(other) + ), + key=lambda value: (len(value.split()), len(value)), + reverse=True, + ) + + edges: dict[tuple[str, str], set[str]] = {} + evidence_details: dict[tuple[str, str], dict] = {} + uses_by_source: Counter[str] = Counter() + uses_by_term: Counter[tuple[str, str]] = Counter() + for definition, terms in definitions: + for source in statements: + if source.hash == definition.hash: + continue + if source.chapter != definition.chapter: + continue + if position[source.hash] <= position[definition.hash]: + continue + if degree[source.hash] > 0 and degree[definition.hash] > 0: + continue + if uses_by_source[source.hash] >= MAX_DEFINITION_TERM_USES_PER_SOURCE: + continue + for term in terms: + if (definition.chapter, term.term) in ambiguous_terms: + continue + term_key = (definition.hash, term.term) + if uses_by_term[term_key] >= MAX_DEFINITION_TERM_USES_PER_TERM: + continue + covering_terms = covering_terms_by_chapter[(definition.chapter, term.term)] + source_span = find_specific_term_span_in_raw(source.raw_body, term.term, covering_terms) + if not source_span: + continue + pair = (source.hash, definition.hash) + if pair in existing_pairs or pair in edges: + break + source_degree = degree[source.hash] + target_degree = degree[definition.hash] + edges.setdefault(pair, set()).add(f"definition-term:{term.term}") + evidence_details[pair] = { + "term": term.term, + "termSource": term.source, + "selectionReason": "semantic-connectivity-backfill", + "endpointSemanticDegreeBeforeTermBackfill": { + "source": source_degree, + "target": target_degree, + }, + "termAmbiguity": "unique-in-chapter", + "termSpecificity": ( + "not-covered-by-longer-defined-term" + if covering_terms else "no-longer-defined-term" + ), + "sourceMatchSpan": term_span_record(source, source_span[0], source_span[1]), + "targetTermSpan": term_span_record(definition, term.raw_start, term.raw_end), + } + degree[source.hash] += 1 + degree[definition.hash] += 1 + uses_by_source[source.hash] += 1 + uses_by_term[term_key] += 1 + break + return edges, evidence_details + + +def collect_structural_edges( + texts: dict[str, str], + order: list[str], + by_segment: dict[str, list[Statement]], + label2hash: dict[str, str], +) -> tuple[dict[tuple[str, str], set[str]], dict[tuple[str, str], dict]]: + """Structural containment edges from proof owners to nested statements.""" + edges: dict[tuple[str, str], set[str]] = {} + details: dict[tuple[str, str], dict] = {} + stmt_by_start = statements_by_file(order, by_segment) + proof_by_file = parse_proof_spans(texts, order) + for stem in order: + statements = stmt_by_start[stem] + proofs = proof_by_file[stem] + for st in statements: + containers = [ + proof + for proof in proofs + if proof.body_start <= st.start and st.end <= proof.body_end + ] + if not containers: + continue + proof = max(containers, key=lambda p: p.body_start) + owner_hash = proof_owner_hash(proof, statements, label2hash) + if owner_hash and owner_hash != st.hash: + pair = (owner_hash, st.hash) + edges.setdefault(pair, set()).add("proof-contains") + details[pair] = { + "proofSpan": proof_span_record(proof), + "containmentDepth": proof.depth, + "targetContainedSpan": tex_span_record(st.file_stem, st.start, st.end), + } + return edges, details + + +def collect_section_sequence_edges( + texts: dict[str, str], + order: list[str], + segments: list[ChapterSegment], + by_segment: dict[str, list[Statement]], +) -> dict[tuple[str, str], set[str]]: + """Reading-order structural edges between adjacent statements in each section.""" + edges: dict[tuple[str, str], set[str]] = {} + stmt_by_start = statements_by_file(order, by_segment) + for seg in segments: + text = texts[seg.file_stem] + cuts: list[int] = [seg.start] + pos = seg.start + while True: + m = SECTION_COMMAND_RE.search(text, pos, seg.end) + if not m: + break + cuts.append(m.start()) + g = read_group_at(text, m.end()) + pos = g[1] if g else m.end() + cuts = sorted(set(cuts)) + for i, start in enumerate(cuts): + end = cuts[i + 1] if i + 1 < len(cuts) else seg.end + local = [ + st for st in stmt_by_start[seg.file_stem] + if st.segment_key == seg.key and start <= st.start < end + ] + for prev, nxt in zip(local, local[1:]): + if prev.hash != nxt.hash: + edges.setdefault((prev.hash, nxt.hash), set()).add("section-sequence") + return edges + + +def collect_chapter_sequence_edges( + by_segment: dict[str, list[Statement]], +) -> dict[tuple[str, str], set[str]]: + """Reading-order structural edges between adjacent statements in a chapter.""" + edges: dict[tuple[str, str], set[str]] = {} + for statements in by_segment.values(): + local = sorted(statements, key=lambda st: st.start) + for prev, nxt in zip(local, local[1:]): + if prev.hash != nxt.hash: + edges.setdefault((prev.hash, nxt.hash), set()).add("chapter-sequence") + return edges + + +def collect_prose_dependency_edges( + texts: dict[str, str], + order: list[str], + segments: list[ChapterSegment], + by_segment: dict[str, list[Statement]], + label2hash: dict[str, str], +) -> tuple[dict[tuple[str, str], set[str]], dict[tuple[str, str], dict]]: + """Conservative semantic dependencies from inferential prose references.""" + edges: dict[tuple[str, str], set[str]] = {} + details: dict[tuple[str, str], dict] = {} + stmt_by_start = statements_by_file(order, by_segment) + proof_by_file = parse_proof_spans(texts, order) + segments_by_file: dict[str, list[ChapterSegment]] = defaultdict(list) + for seg in segments: + segments_by_file[seg.file_stem].append(seg) + + for stem in order: + text = texts[stem] + statements = stmt_by_start[stem] + ignored = [(st.start, st.end) for st in statements] + ignored.extend((p.start, p.end) for p in proof_by_file[stem]) + ignored.sort() + processed_sentences: set[tuple[int, int]] = set() + + for m in REF_RE.finditer(text): + lab = norm_label(m.group(1)) + dst_hash = label2hash.get(lab) + if not dst_hash: + continue + if any(a <= m.start() < b for a, b in ignored): + continue + line_start = text.rfind("\n", 0, m.start()) + 1 + line_end = text.find("\n", m.end()) + line = text[line_start:line_end if line_end != -1 else len(text)] + if re.search(r"\\(?:chapter|section|subsection|subsubsection)\*?\s*\{", line): + continue + + sent_start, sent_end, sentence = reference_sentence_span(text, m.start(), m.end()) + sentence_key = (sent_start, sent_end) + if sentence_key in processed_sentences: + continue + processed_sentences.add(sentence_key) + + raw_sentence = text[sent_start:sent_end] + explicit_pairs, explicit_details = explicit_prose_dependency_pairs( + raw_sentence, + label2hash, + file_stem=stem, + sentence_start=sent_start, + ) + edges = merge_edge_maps(edges, explicit_pairs) + for pair, detail in explicit_details.items(): + for trigger in detail.get("proseTriggers", []): + add_edge_detail(details, pair, "proseTriggers", trigger) + + if not PROSE_NEXT_STATEMENT_SOURCE_RE.search(sentence): + continue + if not PROSE_DEPENDENCY_SIGNAL_RE.search(sentence): + continue + if PROSE_DEPENDENCY_NEGATIVE_RE.search(sentence): + continue + + seg = next((s for s in segments_by_file[stem] if s.start <= m.start() < s.end), None) + if not seg: + continue + local_statements = [st for st in statements if st.segment_key == seg.key] + owner = next_statement(local_statements, m.end()) + if not owner or owner.start - m.end() > 1400: + continue + for ref_hash in refs_in_fragment(sentence, label2hash): + if owner.hash != ref_hash: + pair = (owner.hash, ref_hash) + edges.setdefault(pair, set()).add("prose-dependency-following") + add_edge_detail(details, pair, "proseTriggers", { + "via": ["prose-dependency-following"], + "label": lab, + "sentence": sentence, + "sentenceSpan": tex_span_record(stem, sent_start, sent_end), + "refSpan": tex_span_record(stem, m.start(), m.end()), + "sourceStrategy": "following-statement-after-inferential-prose", + }) + return edges, details + + +def collect_prose_mention_edges( + texts: dict[str, str], + order: list[str], + segments: list[ChapterSegment], + by_segment: dict[str, list[Statement]], + label2hash: dict[str, str], +) -> tuple[dict[tuple[str, str], set[str]], dict[tuple[str, str], dict]]: + """Weak prose mention edges attached to the nearest prior statement. + + These are intentionally not semantic dependencies. They make roadmap and + expository references discoverable in optional graph layers without changing + the default dependency graph. + """ + edges: dict[tuple[str, str], set[str]] = {} + details: dict[tuple[str, str], dict] = {} + stmt_by_start = statements_by_file(order, by_segment) + proof_by_file = parse_proof_spans(texts, order) + segments_by_file: dict[str, list[ChapterSegment]] = defaultdict(list) + for seg in segments: + segments_by_file[seg.file_stem].append(seg) + + for stem in order: + text = texts[stem] + statements = stmt_by_start[stem] + ignored = [(st.start, st.end) for st in statements] + ignored.extend((p.start, p.end) for p in proof_by_file[stem]) + ignored.sort() + + for m in REF_RE.finditer(text): + lab = norm_label(m.group(1)) + dst_hash = label2hash.get(lab) + if not dst_hash: + continue + if any(a <= m.start() < b for a, b in ignored): + continue + line_start = text.rfind("\n", 0, m.start()) + 1 + line_end = text.find("\n", m.end()) + line = text[line_start:line_end if line_end != -1 else len(text)] + if re.search(r"\\(?:chapter|section|subsection|subsubsection)\*?\s*\{", line): + continue + + seg = next((s for s in segments_by_file[stem] if s.start <= m.start() < s.end), None) + if not seg: + continue + local_statements = [st for st in statements if st.segment_key == seg.key] + owner = previous_statement(local_statements, m.start()) + if owner and owner.hash != dst_hash: + pair = (owner.hash, dst_hash) + edges.setdefault(pair, set()).add("prose") + sent_start, sent_end, sentence = reference_sentence_span(text, m.start(), m.end()) + add_edge_detail(details, pair, "mentionTriggers", { + "via": "prose", + "label": lab, + "sentence": sentence, + "sentenceSpan": tex_span_record(stem, sent_start, sent_end), + "refSpan": tex_span_record(stem, m.start(), m.end()), + "sourceStrategy": "nearest-previous-statement", + }) + return edges, details + + +def register_edges( + storage: AstrolabeStorage, + edges: dict[tuple[str, str], set[str]], + *, + rel: str = "references", + edge_class: str = "semantic", + notes: str = "Morgan--Tian cross-reference.", + extra: dict | None = None, + per_edge_extra=None, +) -> None: + for (src_hash, dst_hash), via_set in sorted(edges.items()): + via = sorted(via_set) + rec = { + "sort": "(morgan-tian, morgan-tian)", + "source": "tex", + "src": "morgan-tian", + "rel": rel, + "edgeClass": edge_class, + "generator": "tools/poincare_tex_extract.py", + "via": via, + "notes": notes, + } + if extra: + rec.update(extra) + if per_edge_extra: + rec.update(per_edge_extra(src_hash, dst_hash, via_set)) + record = canon(rec) + hid = storage._compute_hash([src_hash, dst_hash], record) + storage.data.setdefault(hid, {"ref": [src_hash, dst_hash], "record": record}) + + +def validate_generated_span(span: object, context: str) -> None: + if not isinstance(span, dict): + raise ValueError(f"{context}: missing span") + if not isinstance(span.get("start"), int) or not isinstance(span.get("end"), int): + raise ValueError(f"{context}: span start/end must be integers") + if span["start"] >= span["end"]: + raise ValueError(f"{context}: span start must be before end") + if span.get("coordinateSpace") != "comment-stripped-tex": + raise ValueError(f"{context}: span coordinateSpace must be comment-stripped-tex") + if span.get("sourceTransform") != "tex2mdx.strip_comments": + raise ValueError(f"{context}: span sourceTransform must be tex2mdx.strip_comments") + + +def validate_generated_edge_metadata(data: dict) -> None: + required = { + "confidence", + "edgeClass", + "evidence", + "generator", + "inference", + "kind", + "rel", + "reviewStatus", + "scope", + "source", + "src", + "via", + } + definition_term_required = { + "endpointSemanticDegreeBeforeTermBackfill", + "selectionReason", + "sourceMatchSpan", + "targetTermSpan", + "term", + "termAmbiguity", + "termSpecificity", + "termSource", + } + + def validate_trigger_list(evidence: dict, key: str, span_keys: tuple[str, ...], context: str) -> None: + triggers = evidence.get(key) + if not isinstance(triggers, list) or not triggers: + raise ValueError(f"{context}: missing evidence.{key}") + for idx, trigger in enumerate(triggers): + if not isinstance(trigger, dict): + raise ValueError(f"{context}: evidence.{key}[{idx}] must be a mapping") + for span_key in span_keys: + if span_key not in trigger: + raise ValueError(f"{context}: missing evidence.{key}[{idx}].{span_key}") + validate_generated_span(trigger[span_key], f"{context} evidence.{key}[{idx}].{span_key}") + + for h, entry in data.items(): + ref = entry["ref"] + if len(ref) <= 1: + continue + rec = json.loads(entry["record"]) + if rec.get("generator") != "tools/poincare_tex_extract.py": + continue + missing = sorted(required - set(rec)) + if missing: + raise ValueError(f"Generated edge {h}: missing metadata fields {missing}") + evidence = rec.get("evidence") + if not isinstance(evidence, dict): + raise ValueError(f"Generated edge {h}: evidence must be a mapping") + if evidence.get("via") != rec.get("via"): + raise ValueError(f"Generated edge {h}: evidence.via and record via differ") + for side in ("sourceStatement", "targetStatement"): + statement = evidence.get(side) + if not isinstance(statement, dict): + raise ValueError(f"Generated edge {h}: missing evidence.{side}") + validate_generated_span(statement.get("span"), f"Generated edge {h} evidence.{side}") + if rec.get("kind") == "definition-use": + missing_detail = sorted(definition_term_required - set(evidence)) + if missing_detail: + raise ValueError( + f"Generated definition-use edge {h}: missing evidence fields {missing_detail}" + ) + validate_generated_span(evidence.get("sourceMatchSpan"), f"Generated edge {h} sourceMatchSpan") + validate_generated_span(evidence.get("targetTermSpan"), f"Generated edge {h} targetTermSpan") + elif rec.get("kind") == "reference": + validate_trigger_list(evidence, "refTriggers", ("sourceSpan",), f"Generated edge {h}") + elif rec.get("kind") == "mention": + validate_trigger_list(evidence, "mentionTriggers", ("sentenceSpan", "refSpan"), f"Generated edge {h}") + elif rec.get("kind") == "containment": + validate_generated_span(evidence.get("proofSpan"), f"Generated edge {h} proofSpan") + validate_generated_span(evidence.get("targetContainedSpan"), f"Generated edge {h} targetContainedSpan") + elif rec.get("kind") == "dependency": + evidence_type = evidence.get("type") + if evidence_type == "proof-contained-theorem-like-statement": + validate_generated_span(evidence.get("proofSpan"), f"Generated edge {h} proofSpan") + elif evidence_type == "local-anaphora-reference": + validate_trigger_list(evidence, "anaphoraTriggers", ("sourceSpan",), f"Generated edge {h}") + elif evidence_type == "inferential-prose-reference": + validate_trigger_list(evidence, "proseTriggers", ("sentenceSpan", "refSpan"), f"Generated edge {h}") + + +RAW_REF_RESIDUE_RE = re.compile(r"\\(?:label|ref|eqref|cite)\{") +FONT_SWITCH_RESIDUE_RE = re.compile(r"\{\\(?:em|it|sl|bf|rm|tt|sc)\b|\\(?:em|it|sl|bf|rm|tt|sc)\b") +FRAGMENTED_FONT_MATH_RE = re.compile(r"\*[A-Za-z][^*\n]{0,80}\*\$") +TRAILING_FONT_BRACE_RE = re.compile( + r"\b[A-Za-z][A-Za-z -]{2,80}\}\s+(?:if|is|are|consists|centered|from|in|of)\b" +) +PROSE_TEX_COMMAND_RESIDUE_RE = re.compile(r"\\(?!entryref\{|entryblock\{)[A-Za-z]+\*?\b") + + +def validate_generated_text_quality(data: dict, project: Path) -> None: + """Fail fast on TeX residues that break generated MDX/atom readability.""" + texts: list[tuple[str, str]] = [] + for h, entry in data.items(): + if len(entry["ref"]) != 1: + continue + rec = json.loads(entry["record"]) + if rec.get("generator") == "tools/poincare_tex_extract.py": + texts.append((f"atom {h}", rec.get("notes", ""))) + + ast = project / ".astrolabe" + for directory in ("docs-src", "docs"): + for path in sorted((ast / directory).glob("*.mdx")): + text = path.read_text(encoding="utf-8") + if GENERATED_SENTINEL in text: + texts.append((str(path.relative_to(project)), text)) + + for context, text in texts: + if RAW_REF_RESIDUE_RE.search(text): + raise ValueError(f"{context}: raw TeX ref/label/cite residue") + if FRAGMENTED_FONT_MATH_RE.search(text): + raise ValueError(f"{context}: fragmented font switch around inline math") + for is_math, part in split_math_spans(text): + if is_math: + if part.startswith("$$") and "$" in part[2:-2]: + raise ValueError(f"{context}: nested inline dollar in display math") + continue + if FONT_SWITCH_RESIDUE_RE.search(part): + raise ValueError(f"{context}: raw TeX font switch residue") + if TRAILING_FONT_BRACE_RE.search(part): + raise ValueError(f"{context}: trailing TeX font group brace residue") + if PROSE_TEX_COMMAND_RESIDUE_RE.search(part): + raise ValueError(f"{context}: raw prose TeX command residue") + + +def isolated_count(hashes: list[str], edges: set[tuple[str, str]]) -> int: + degree: Counter[str] = Counter() + for src_hash, dst_hash in edges: + degree[src_hash] += 1 + degree[dst_hash] += 1 + return sum(1 for h in hashes if degree[h] == 0) + + +def component_sizes(hashes: list[str], edges: set[tuple[str, str]]) -> list[int]: + adj: dict[str, set[str]] = defaultdict(set) + for src_hash, dst_hash in edges: + adj[src_hash].add(dst_hash) + adj[dst_hash].add(src_hash) + seen: set[str] = set() + sizes: list[int] = [] + for h in hashes: + if h in seen: + continue + stack = [h] + seen.add(h) + n = 0 + while stack: + cur = stack.pop() + n += 1 + for nxt in adj[cur]: + if nxt not in seen: + seen.add(nxt) + stack.append(nxt) + sizes.append(n) + return sorted(sizes, reverse=True) + + +def graph_audit( + texts: dict[str, str], + order: list[str], + by_segment: dict[str, list[Statement]], + label2hash: dict[str, str], + label2text: dict[str, str], + edges: dict[tuple[str, str], set[str]], + mention_edges: dict[tuple[str, str], set[str]] | None = None, + structural_edges: dict[tuple[str, str], set[str]] | None = None, + semantic_edge_layers: dict[str, dict[tuple[str, str], set[str]]] | None = None, + structural_edge_layers: dict[str, dict[tuple[str, str], set[str]]] | None = None, +) -> dict: + statements = all_statements(by_segment) + stmt_hashes = [st.hash for st in statements] + stmt_hash_set = set(stmt_hashes) + hash2stmt = {st.hash: st for st in statements} + stmt_by_start = statements_by_file(order, by_segment) + proof_by_file = parse_proof_spans(texts, order) + edge_pairs = set(edges) + mention_pairs = set(mention_edges or {}) + structural_pairs = set(structural_edges or {}) + semantic_layer_pairs = { + name: set(layer_edges) + for name, layer_edges in (semantic_edge_layers or {"semantic": edges}).items() + } + structural_layer_pairs = { + name: set(layer_edges) + for name, layer_edges in (structural_edge_layers or {"structural": structural_edges or {}}).items() + } + explicit_pairs = semantic_layer_pairs.get("explicit", set()) + proof_contains_pairs = structural_layer_pairs.get("proof_contains", structural_pairs) + section_sequence_pairs = structural_layer_pairs.get("section_sequence", set()) + chapter_sequence_pairs = structural_layer_pairs.get("chapter_sequence", set()) + all_edge_pairs = edge_pairs | mention_pairs | structural_pairs + + degree: Counter[str] = Counter() + in_degree: Counter[str] = Counter() + out_degree: Counter[str] = Counter() + for src_hash, dst_hash in edge_pairs: + out_degree[src_hash] += 1 + in_degree[dst_hash] += 1 + degree[src_hash] += 1 + degree[dst_hash] += 1 + + isolated = [st for st in statements if degree[st.hash] == 0] + labelled = [st for st in statements if st.labels] + unlabelled = [st for st in statements if not st.labels] + + via_counts: Counter[str] = Counter() + via_target_sorts: dict[str, Counter[str]] = defaultdict(Counter) + for via_set in edges.values(): + via_counts.update(via_set) + for (_src_hash, dst_hash), via_set in edges.items(): + dst = hash2stmt.get(dst_hash) + if not dst: + continue + for via in via_set: + via_target_sorts[via][dst.sort] += 1 + + semantic_layer_stats: dict[str, dict[str, int]] = {} + cumulative_pairs: set[tuple[str, str]] = set() + for name, pairs in semantic_layer_pairs.items(): + cumulative_pairs |= pairs + semantic_layer_stats[name] = { + "edges": len(pairs), + "new_vs_explicit": len(pairs - explicit_pairs) if explicit_pairs else 0, + "overlap_explicit": len(pairs & explicit_pairs) if explicit_pairs else 0, + "isolated_cumulative": isolated_count(stmt_hashes, cumulative_pairs), + } + structural_layer_stats = { + name: { + "edges": len(pairs), + "new_vs_semantic": len(pairs - edge_pairs), + } + for name, pairs in structural_layer_pairs.items() + } + + statement_label_set = set(label2hash) + aux_label_set = set(label2text) - statement_label_set + aux_label_owners = statement_inner_label_owners(by_segment) + aux_proxy_pairs = { + pair for pair, via_set in edges.items() + if any("aux-label-proxy" in via for via in via_set) + } + ref_context: Counter[str] = Counter() + unresolved: Counter[str] = Counter() + prose_statement_refs: Counter[str] = Counter() + prose_pairs: set[tuple[str, str]] = set() + local_anaphora: Counter[str] = Counter() + nested_statement_hashes: set[str] = set() + + explicit_owner_mismatches: list[dict[str, str]] = [] + proof_spans = [p for spans in proof_by_file.values() for p in spans] + for proof in proof_spans: + if not proof.owner_label: + continue + owner_hash = label2hash.get(proof.owner_label) + prev = previous_statement(stmt_by_start[proof.file_stem], proof.start) + if owner_hash and prev and owner_hash != prev.hash: + owner = hash2stmt.get(owner_hash) + explicit_owner_mismatches.append({ + "file": proof.file_stem, + "owner_label": proof.owner_label, + "owner": owner.mtref if owner else owner_hash, + "nearest_previous": prev.mtref, + }) + + for stem in order: + text = texts[stem] + statements_in_file = stmt_by_start[stem] + proofs = proof_by_file[stem] + for st in statements_in_file: + if any(p.body_start <= st.start and st.end <= p.body_end for p in proofs): + nested_statement_hashes.add(st.hash) + for m in LOCAL_ANAPHORA_RE.finditer(text): + local_anaphora[f"{m.group(1).lower()} {m.group(2).lower()}"] += 1 + for m in REF_RE.finditer(text): + pos = m.start() + lab = norm_label(m.group(1)) + target = "statement" if lab in statement_label_set else "aux" if lab in aux_label_set else "unresolved" + context = "prose" + for st in statements_in_file: + if st.start <= pos < st.end: + context = "statement" + break + if st.start > pos: + break + if context == "prose": + proof = next((p for p in reversed(proofs) if p.body_start <= pos < p.body_end), None) + if proof: + context = "proof" + ref_context[f"{context}:{target}"] += 1 + if target == "unresolved": + unresolved[lab] += 1 + if context == "prose" and target == "statement": + prose_statement_refs[lab] += 1 + prev = previous_statement(statements_in_file, pos) + dst_hash = label2hash[lab] + if prev and prev.hash != dst_hash: + prose_pairs.add((prev.hash, dst_hash)) + + combined_with_prose = edge_pairs | prose_pairs + combined_with_mentions = edge_pairs | mention_pairs + sizes = component_sizes(stmt_hashes, edge_pairs) + all_sizes = component_sizes(stmt_hashes, all_edge_pairs) + top_connected = sorted( + statements, + key=lambda st: degree[st.hash], + reverse=True, + )[:10] + + return { + "statements": len(statements), + "edges": len(edge_pairs), + "semantic_edges": len(edge_pairs), + "prose_mention_edges": len(mention_pairs), + "structural_edges": len(structural_pairs), + "semantic_edge_layers": semantic_layer_stats, + "structural_edge_layers": structural_layer_stats, + "edge_layer_metadata": EDGE_LAYER_METADATA, + "all_statement_edges": len(all_edge_pairs), + "semantic_plus_mentions_edges": len(combined_with_mentions), + "isolated": len(isolated), + "isolated_with_explicit_semantic": isolated_count(stmt_hashes, explicit_pairs) if explicit_pairs else len(isolated), + "isolated_with_mentions": isolated_count(stmt_hashes, combined_with_mentions), + "isolated_with_all_edge_classes": isolated_count(stmt_hashes, all_edge_pairs), + "isolated_percent": round(len(isolated) / len(statements), 4) if statements else 0, + "no_incoming": sum(1 for h in stmt_hashes if in_degree[h] == 0), + "no_outgoing": sum(1 for h in stmt_hashes if out_degree[h] == 0), + "sorts": dict(sorted(Counter(st.sort for st in statements).items())), + "isolated_by_sort": dict(sorted(Counter(st.sort for st in isolated).items())), + "isolated_by_chapter": dict(sorted(Counter(str(st.chapter) for st in isolated).items(), key=lambda kv: int(kv[0]))), + "labelled": len(labelled), + "unlabelled": len(unlabelled), + "isolated_labelled": sum(1 for st in labelled if degree[st.hash] == 0), + "isolated_unlabelled": sum(1 for st in unlabelled if degree[st.hash] == 0), + "components": len(sizes), + "singleton_components": sum(1 for size in sizes if size == 1), + "largest_components": sizes[:10], + "components_all_edge_classes": len(all_sizes), + "singleton_components_all_edge_classes": sum(1 for size in all_sizes if size == 1), + "largest_components_all_edge_classes": all_sizes[:10], + "via_counts": dict(sorted(via_counts.items())), + "via_target_sorts": { + via: dict(sorted(counts.items())) + for via, counts in sorted(via_target_sorts.items()) + }, + "proof_spans": len(proof_spans), + "proof_max_depth": max((p.depth for p in proof_spans), default=0), + "explicit_proof_owners": sum(1 for p in proof_spans if p.owner_label), + "explicit_owner_mismatches": explicit_owner_mismatches[:20], + "explicit_owner_mismatch_count": len(explicit_owner_mismatches), + "nested_statement_count": len(nested_statement_hashes), + "nested_statement_isolated_semantic": sum(1 for h in nested_statement_hashes if degree[h] == 0), + "nested_contains_edges": len(proof_contains_pairs), + "section_sequence_edges": len(section_sequence_pairs), + "chapter_sequence_edges": len(chapter_sequence_pairs), + "inner_label_owner_count": len(aux_label_owners), + "aux_label_proxy_edges": len(aux_proxy_pairs), + "ref_context": dict(sorted(ref_context.items())), + "unresolved_refs": dict(unresolved.most_common(20)), + "prose_statement_refs": sum(prose_statement_refs.values()), + "top_prose_statement_refs": dict(prose_statement_refs.most_common(20)), + "simulated_prose_new_pairs": len(prose_pairs - edge_pairs), + "simulated_isolated_with_prose": isolated_count(stmt_hashes, combined_with_prose), + "local_anaphora_hits": sum(local_anaphora.values()), + "local_anaphora": dict(local_anaphora.most_common(20)), + "top_connected": [ + { + "degree": degree[st.hash], + "sort": st.sort, + "mtref": st.mtref, + "label": st.labels[0] if st.labels else "", + "title": normalize_title(st.opt_title, {}, label2text) if st.opt_title else "", + } + for st in top_connected + if st.hash in stmt_hash_set + ], + } + + +def slugify_title( + title_raw: str, + fallback: str, + label2hash: dict[str, str] | None = None, + label2text: dict[str, str] | None = None, +) -> str: + title = normalize_title(title_raw, label2hash, label2text) + title = re.sub(r"\\entryref\{[0-9a-f]+\}", "", title) + title = re.sub(r"\\(?:ref|eqref)\{[^{}]*\}", "", title) + title = re.sub(r"\\([A-Za-z]+)", r"\1", title) + title = re.sub(r"[$\\{}]", " ", title) + slug = re.sub(r"[^A-Za-z0-9]+", "-", title).strip("-").lower() + return slug[:72].strip("-") or fallback + + +def is_generated_doc(path: Path) -> bool: + try: + sample = path.read_text(encoding="utf-8", errors="replace")[:4096] + except OSError: + return False + return GENERATED_SENTINEL in sample + + +def is_legacy_generated_doc(path: Path) -> bool: + """Legacy first-pass Poincare docs predate the generator sentinel.""" + return path.name in LEGACY_GENERATED_DOC_NAMES + + +def marker_attr(value: str) -> str: + return ( + value.replace("&", "&") + .replace('"', """) + .replace("<", "<") + .replace(">", ">") + .replace("--", "- -") + ) + + +def generate_docs( + texts: dict[str, str], + project: Path, + segments: list[ChapterSegment], + by_segment: dict[str, list[Statement]], + label2hash: dict[str, str], + label2text: dict[str, str], +) -> None: + ast = project / ".astrolabe" + docs_src_dir = ast / "docs-src" + docs_dir = ast / "docs" + docs_src_dir.mkdir(parents=True, exist_ok=True) + docs_dir.mkdir(parents=True, exist_ok=True) + + # Preserve the project index; all numbered chapter docs are regenerated. + for d in (docs_src_dir, docs_dir): + for p in d.glob("[0-9][0-9]-*.mdx"): + if p.name != "00-index.mdx": + if not is_generated_doc(p) and not is_legacy_generated_doc(p): + raise RuntimeError(f"Refusing to overwrite non-generated doc: {p}") + p.unlink() + + for seg in segments: + text = texts[seg.file_stem] + items = by_segment[seg.key] + src_out: list[str] = [GENERATED_SENTINEL + "\n\n"] + doc_out: list[str] = [GENERATED_SENTINEL + "\n\n"] + last = seg.start + + for st in items: + before = convert_fragment(text[last:st.start], seg.chapter, label2hash, label2text) + src_out.append(before) + doc_out.append(before) + + label_attr = st.labels[0] if st.labels else "" + title_attr = normalize_title(st.opt_title, label2hash, label2text) if st.opt_title else "" + attrs = [ + f'kind="{st.sort}"', + 'src="morgan-tian"', + f'mtref="{st.mtref}"', + ] + if label_attr: + attrs.append(f'label="{marker_attr(label_attr)}"') + if title_attr: + attrs.append(f'title="{marker_attr(title_attr)}"') + body = convert_fragment(st.raw_body, seg.chapter, label2hash, label2text, statement=True) + head_sep = "\n\n" if body.lstrip().startswith("$$") else " " + src_out.append( + "\n" + f"\n" + f"**{SORT_LABEL[st.sort]}.**{head_sep}{body}" + "\n\n" + ) + doc_out.append(f"\n\\entryblock{{{st.hash}}}\n\n") + last = st.end + + tail = convert_fragment(text[last:seg.end], seg.chapter, label2hash, label2text) + src_out.append(tail) + doc_out.append(tail) + + filename = f"{seg.order:02d}-{slugify_title(seg.title_raw, seg.file_stem, label2hash, label2text)}.mdx" + docs_src_dir.joinpath(filename).write_text(tidy("".join(src_out)), encoding="utf-8") + docs_dir.joinpath(filename).write_text(tidy("".join(doc_out)), encoding="utf-8") + + +def main() -> None: + ap = argparse.ArgumentParser() + ap.add_argument("src", nargs="?", default=str(DEFAULT_SRC), help="Morgan--Tian arXiv source directory") + ap.add_argument("--project", default=str(DEFAULT_PROJECT), help="project directory") + ap.add_argument("--report", action="store_true", help="print a graph extraction audit") + ap.add_argument("--report-json", default="", help="write the graph extraction audit to this JSON file") + args = ap.parse_args() + + src = ensure_source(Path(os.path.expanduser(args.src)).resolve()) + project = Path(args.project).resolve() + storage = AstrolabeStorage(str(project)) + order = include_order(src) + texts = source_texts(src, order) + segments = chapter_segments(texts, order) + + purge_old(storage) + label2hash, label2text, by_segment = parse_statements(texts, segments, storage) + register_atoms(storage, by_segment, label2text) + aux_label2statement = statement_inner_label_owners(by_segment) + explicit_edges, explicit_details = collect_dependency_edges(texts, order, by_segment, label2hash, aux_label2statement) + proof_structural_edges, proof_structural_details = collect_structural_edges(texts, order, by_segment, label2hash) + proof_dependency_candidates, proof_dependency_details = collect_proof_containment_dependency_edges( + proof_structural_edges, + proof_structural_details, + by_segment, + ) + proof_dependency_edges = edge_map_difference(proof_dependency_candidates, set(explicit_edges)) + proof_dependency_details = detail_map_difference(proof_dependency_details, proof_dependency_edges) + semantic_so_far = merge_edge_maps(explicit_edges, proof_dependency_edges) + prose_dependency_candidates, prose_dependency_details = collect_prose_dependency_edges(texts, order, segments, by_segment, label2hash) + prose_dependency_edges = edge_map_difference(prose_dependency_candidates, set(semantic_so_far)) + prose_dependency_details = detail_map_difference(prose_dependency_details, prose_dependency_edges) + semantic_so_far = merge_edge_maps(semantic_so_far, prose_dependency_edges) + anaphora_candidates, anaphora_details = collect_local_anaphora_edges(texts, order, by_segment, label2hash) + anaphora_edges = edge_map_difference(anaphora_candidates, set(semantic_so_far)) + anaphora_details = detail_map_difference(anaphora_details, anaphora_edges) + edges = merge_edge_maps(semantic_so_far, anaphora_edges) + definition_term_candidates, definition_term_details = collect_definition_term_edges(by_segment, label2text, set(edges)) + definition_term_edges = edge_map_difference(definition_term_candidates, set(edges)) + definition_term_details = { + pair: detail for pair, detail in definition_term_details.items() + if pair in definition_term_edges + } + edges = merge_edge_maps(edges, definition_term_edges) + prose_edges, prose_details = collect_prose_mention_edges(texts, order, segments, by_segment, label2hash) + section_sequence_edges = collect_section_sequence_edges(texts, order, segments, by_segment) + chapter_sequence_edges = edge_map_difference( + collect_chapter_sequence_edges(by_segment), + set(section_sequence_edges), + ) + structural_edges = merge_edge_maps(proof_structural_edges, section_sequence_edges, chapter_sequence_edges) + + def edge_meta(layer: str): + return edge_metadata_factory(by_segment, **EDGE_LAYER_METADATA[layer]) + + explicit_meta = edge_metadata_factory( + by_segment, + **EDGE_LAYER_METADATA["explicit"], + evidence_extra_by_pair=explicit_details, + ) + proof_dependency_meta = edge_metadata_factory( + by_segment, + **EDGE_LAYER_METADATA["proof_containment_dependency"], + evidence_extra_by_pair=proof_dependency_details, + ) + prose_dependency_meta = edge_metadata_factory( + by_segment, + **EDGE_LAYER_METADATA["prose_dependency"], + evidence_extra_by_pair=prose_dependency_details, + ) + anaphora_meta = edge_metadata_factory( + by_segment, + **EDGE_LAYER_METADATA["local_anaphora"], + evidence_extra_by_pair=anaphora_details, + ) + definition_term_meta = edge_metadata_factory( + by_segment, + **EDGE_LAYER_METADATA["definition_term"], + evidence_extra_by_pair=definition_term_details, + ) + prose_mention_meta = edge_metadata_factory( + by_segment, + **EDGE_LAYER_METADATA["prose_mention"], + evidence_extra_by_pair=prose_details, + ) + proof_structural_meta = edge_metadata_factory( + by_segment, + **EDGE_LAYER_METADATA["proof_contains"], + evidence_extra_by_pair=proof_structural_details, + ) + section_sequence_meta = edge_meta("section_sequence") + chapter_sequence_meta = edge_meta("chapter_sequence") + register_edges(storage, explicit_edges, per_edge_extra=explicit_meta) + register_edges( + storage, + proof_dependency_edges, + rel="depends", + edge_class="semantic", + notes="Morgan--Tian proof-contained statement dependency.", + per_edge_extra=proof_dependency_meta, + ) + register_edges( + storage, + prose_dependency_edges, + rel="depends", + edge_class="semantic", + notes="Morgan--Tian high-confidence prose dependency.", + per_edge_extra=prose_dependency_meta, + ) + register_edges( + storage, + anaphora_edges, + rel="depends", + edge_class="semantic", + notes="Morgan--Tian local anaphora dependency.", + per_edge_extra=anaphora_meta, + ) + register_edges( + storage, + definition_term_edges, + rel="uses", + edge_class="semantic", + notes=( + "Morgan--Tian same-chapter definition term match. " + "This is a low-confidence candidate edge for review." + ), + per_edge_extra=definition_term_meta, + ) + register_edges( + storage, + prose_edges, + rel="mentions", + edge_class="prose", + notes="Morgan--Tian prose mention. The source is the nearest preceding statement in the same chapter segment.", + per_edge_extra=prose_mention_meta, + ) + register_edges( + storage, + proof_structural_edges, + rel="contains", + edge_class="structural", + notes="Morgan--Tian proof containment. The source is the statement whose proof contains the target statement.", + per_edge_extra=proof_structural_meta, + ) + register_edges( + storage, + section_sequence_edges, + rel="in-section", + edge_class="structural", + notes="Morgan--Tian reading-order adjacency within a section.", + per_edge_extra=section_sequence_meta, + ) + register_edges( + storage, + chapter_sequence_edges, + rel="in-chapter", + edge_class="structural", + notes="Morgan--Tian reading-order adjacency across sections in a chapter.", + per_edge_extra=chapter_sequence_meta, + ) + generate_docs(texts, project, segments, by_segment, label2hash, label2text) + write_macros(src, project) + audit = graph_audit( + texts, + order, + by_segment, + label2hash, + label2text, + edges, + prose_edges, + structural_edges, + semantic_edge_layers={ + "explicit": explicit_edges, + "proof_containment_dependency": proof_dependency_edges, + "prose_dependency": prose_dependency_edges, + "local_anaphora": anaphora_edges, + "definition_term": definition_term_edges, + }, + structural_edge_layers={ + "proof_contains": proof_structural_edges, + "section_sequence": section_sequence_edges, + "chapter_sequence": chapter_sequence_edges, + }, + ) + + validate_store(storage.data) + validate_generated_edge_metadata(storage.data) + validate_generated_text_quality(storage.data, project) + storage._save() + n_atoms = sum(1 for e in storage.data.values() if len(e["ref"]) == 1) + n_edges = sum(1 for e in storage.data.values() if len(e["ref"]) > 1) + n_statements = sum(len(v) for v in by_segment.values()) + print(f"registered {n_statements} Morgan--Tian statements") + print(f"store: {n_atoms} atoms, {n_edges} edges") + print(f"docs: {len(segments)} chapter docs in {project / '.astrolabe'}") + if args.report: + print(json.dumps(audit, ensure_ascii=False, indent=2, sort_keys=True)) + if args.report_json: + Path(args.report_json).write_text( + json.dumps(audit, ensure_ascii=False, indent=2, sort_keys=True) + "\n", + encoding="utf-8", + ) + + +if __name__ == "__main__": + main() From 12250273e64fe2d6e5ea911d8f2053a99e962413 Mon Sep 17 00:00:00 2001 From: LehengChen Date: Sun, 28 Jun 2026 19:20:53 +0800 Subject: [PATCH 2/2] Regenerate Poincare Astrolabe data --- .../.astrolabe/atoms/002f3496c54c.md | 21 + .../.astrolabe/atoms/00499f9fd446.md | 23 + .../.astrolabe/atoms/006e42ef5bb5.md | 17 + .../.astrolabe/atoms/00e5410e6aa1.md | 17 + .../.astrolabe/atoms/01028c93347d.md | 18 + .../.astrolabe/atoms/012eefbbc44a.md | 18 + .../.astrolabe/atoms/016cf36686b2.md | 18 + .../.astrolabe/atoms/01a829e22cf6.md | 21 + .../.astrolabe/atoms/01f0d3baa28f.md | 23 + .../.astrolabe/atoms/028d3b5182b5.md | 18 + .../.astrolabe/atoms/033786a89c6d.md | 24 + .../.astrolabe/atoms/035b5e6ca144.md | 17 + .../.astrolabe/atoms/03957890f08f.md | 16 + .../.astrolabe/atoms/03c26a36f415.md | 18 + .../.astrolabe/atoms/0425e2f3cfe1.md | 56 + .../.astrolabe/atoms/04518fa1ac2f.md | 32 + .../.astrolabe/atoms/04dd3c3ea1c8.md | 27 + 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projects/poincare-conjecture/.astrolabe/edges/ff578c6a5d22.md create mode 100644 projects/poincare-conjecture/.astrolabe/edges/ff649155c213.md create mode 100644 projects/poincare-conjecture/.astrolabe/edges/ff92d49ef581.md create mode 100644 projects/poincare-conjecture/.astrolabe/edges/ffe39eb07c39.md create mode 100644 projects/poincare-conjecture/.astrolabe/edges/fffd833792e7.md diff --git a/projects/poincare-conjecture/.astrolabe/atoms/002f3496c54c.md b/projects/poincare-conjecture/.astrolabe/atoms/002f3496c54c.md new file mode 100644 index 00000000..cc219c49 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/002f3496c54c.md @@ -0,0 +1,21 @@ +--- +chapter: 11 +generator: tools/poincare_tex_extract.py +label: distch +labels: +- distch +mtref: '11.12' +ref: +- 002f3496c54c +sort: claim +source: tex +src: morgan-tian +tex_file: singlimit2 +--- +Let $Q$ be an upper bound on $R(x,0)$ for all $x\in +M_\infty$. Then for any points $x,y\in M_\infty$ and any $t\in +(-T,0]$ we have + +$$ +d_{t}(x,y)\le d_0(x,y)+16\sqrt{\frac{Q}{3}}T. +$$ diff --git a/projects/poincare-conjecture/.astrolabe/atoms/00499f9fd446.md b/projects/poincare-conjecture/.astrolabe/atoms/00499f9fd446.md new file mode 100644 index 00000000..1b1597c3 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/00499f9fd446.md @@ -0,0 +1,23 @@ +--- +chapter: 14 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '14.14' +ref: +- 00499f9fd446 +sort: definition +source: tex +src: morgan-tian +tex_file: surgery +--- +Let $({\mathcal M},G)$ be a Ricci flow with surgery, and let $x$ be +a point of space-time. Set $t=\mathbf{t}(x)$. For any $r>0$ and $\Delta +t>0$ we say that the *backward parabolic neighborhood* +$P(x,t,r,-\Delta t)$ *exists* in ${\mathcal M}$ if there is an +embedding $B(x,t,r)\times (t-\Delta t,t]\to {\mathcal M}$ compatible +with time and the vector field. Similarly, we say +that the *forward parabolic neighborhood* $P(x,t,r,\Delta t)$ +*exists* in ${\mathcal M}$ if there is an embedding +$B(x,t,r)\times [t,t+\Delta t)\to {\mathcal M}$ compatible with time +and the vector field. A *parabolic neighborhood* is either a +forward or backward parabolic neighborhood. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/006e42ef5bb5.md b/projects/poincare-conjecture/.astrolabe/atoms/006e42ef5bb5.md new file mode 100644 index 00000000..25bfa0da --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/006e42ef5bb5.md @@ -0,0 +1,17 @@ +--- +chapter: 2 +generator: tools/poincare_tex_extract.py +label: line +labels: +- line +mtref: '2.14' +ref: +- 006e42ef5bb5 +sort: lemma +source: tex +src: morgan-tian +tex_file: prelim +--- +Any complete Riemannian manifold $X$ of non-negative Ricci curvature +containing a minimizing line is isometric to a product $N\times \Ar$ +for some Riemannian manifold $N$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/00e5410e6aa1.md b/projects/poincare-conjecture/.astrolabe/atoms/00e5410e6aa1.md new file mode 100644 index 00000000..61f5f0fc --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/00e5410e6aa1.md @@ -0,0 +1,17 @@ +--- +chapter: 5 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '5.16' +ref: +- 00e5410e6aa1 +sort: corollary +source: tex +src: morgan-tian +tex_file: converge2 +--- +Suppose that $(U,g(t)),\ 0\le t0$. If $R(p,T)=0$ for some $p\in +U$, then $(U,g(t))$ is flat for every $t\in [0,T]$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/01a829e22cf6.md b/projects/poincare-conjecture/.astrolabe/atoms/01a829e22cf6.md new file mode 100644 index 00000000..921a7bb1 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/01a829e22cf6.md @@ -0,0 +1,21 @@ +--- +chapter: 3 +generator: tools/poincare_tex_extract.py +label: corI.8.3 +labels: +- corI.8.3 +mtref: '3.26' +ref: +- 01a829e22cf6 +sort: corollary +source: tex +src: morgan-tian +tex_file: flowbasics +--- +Let $(M,g(t)),\ a\le t\le b$, be a Ricci flow with $(M,g(t))$ complete for every $t +\in [0,T)$. Fix a positive function $K(t)$, and suppose that $\mathit{Ric}_{g(t)}(x,t)\le (n-1)K(t)$ for all $x\in M$ and all $t\in [a,b]$. Let +$x_0,x_1$ be two points of $M$. Then + +$$ +d_a(x_0,x_1)\le d_b(x_0,x_1)+4(n-1)\int_a^b\sqrt{\frac{2K(t)}{3}}dt. +$$ diff --git a/projects/poincare-conjecture/.astrolabe/atoms/01f0d3baa28f.md b/projects/poincare-conjecture/.astrolabe/atoms/01f0d3baa28f.md new file mode 100644 index 00000000..f423c97b --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/01f0d3baa28f.md @@ -0,0 +1,23 @@ +--- +chapter: 11 +generator: tools/poincare_tex_extract.py +label: 2epshorns +labels: +- 2epshorns +mtref: '11.30' +ref: +- 01f0d3baa28f +sort: lemma +source: tex +src: morgan-tian +tex_file: singlimit2 +--- +Let $({\mathcal M},G)$ be a generalized $3$-dimensional Ricci flow +defined for $0\le t0$ be +fixed. Then there is $N$ depending only on $X$ and $\zeta$ such the +conclusion of the previous claim holds for every $c\in X$ and every +$n\ge N$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/033786a89c6d.md b/projects/poincare-conjecture/.astrolabe/atoms/033786a89c6d.md new file mode 100644 index 00000000..80d85d8b --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/033786a89c6d.md @@ -0,0 +1,24 @@ +--- +chapter: 6 +generator: tools/poincare_tex_extract.py +label: DL +labels: +- DL +mtref: '6.29' +ref: +- 033786a89c6d +sort: corollary +source: tex +src: morgan-tian +tex_file: newcompar +--- +At any $q\in {\mathcal U}_x(\tau)$ we have + +$$ +\nabla L_x^{\tau}(q)=2\sqrt{\tau}X(\tau) +$$ + +where $X(\tau)$ is the horizontal component of $\gamma'(\tau)$, +where $\gamma$ is the unique minimizing ${\mathcal L}$-geodesic +connecting $x$ to $q$. (See Fig. 0.2 in the +Introduction.) diff --git a/projects/poincare-conjecture/.astrolabe/atoms/035b5e6ca144.md b/projects/poincare-conjecture/.astrolabe/atoms/035b5e6ca144.md new file mode 100644 index 00000000..0f2de0ea --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/035b5e6ca144.md @@ -0,0 +1,17 @@ +--- +chapter: 19 +generator: tools/poincare_tex_extract.py +label: S2timesI +labels: +- S2timesI +mtref: '19.13' +ref: +- 035b5e6ca144 +sort: lemma +source: tex +src: morgan-tian +tex_file: canonnbhd +--- +The union $U$ of the $N_i$ in a finite or infinite chain of +$\epsilon$-necks is diffeomorphic to $S^2\times (0,1)$. In +particular, it is an $\epsilon$-tube. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/03957890f08f.md b/projects/poincare-conjecture/.astrolabe/atoms/03957890f08f.md new file mode 100644 index 00000000..88299d1e --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/03957890f08f.md @@ -0,0 +1,16 @@ +--- +chapter: 9 +generator: tools/poincare_tex_extract.py +label: noncompsol +labels: +- noncompsol +mtref: '9.46' +ref: +- 03957890f08f +sort: proposition +source: tex +src: morgan-tian +tex_file: temp2kappa +--- +There is no two- or three-dimensional Ricci flow satisfying the hypotheses of +Theorem 9.42 with $(M,g)$ non-compact and of positive curvature. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/03c26a36f415.md b/projects/poincare-conjecture/.astrolabe/atoms/03c26a36f415.md new file mode 100644 index 00000000..80fc8ffb --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/03c26a36f415.md @@ -0,0 +1,18 @@ +--- +chapter: 5 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '5.4' +ref: +- 03c26a36f415 +sort: remark +source: tex +src: morgan-tian +tex_file: converge2 +--- +Conditions (1) and (2a) in the definition above also appear in the definition +in the case of complete limits. It is Condition (2b) that is extra in this +incomplete case. It says that once $k$ is sufficiently large then the image +$\varphi_\ell(V_k)$ contains all points satisfying two conditions: they are at +most a given bounded distance from $x_\ell$, and also they are at least a fixed +distance from the boundary of $U_\ell$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/0425e2f3cfe1.md b/projects/poincare-conjecture/.astrolabe/atoms/0425e2f3cfe1.md new file mode 100644 index 00000000..b2d3d066 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/0425e2f3cfe1.md @@ -0,0 +1,56 @@ +--- +chapter: 9 +generator: tools/poincare_tex_extract.py +label: epstube +labels: +- epstube +mtref: '9.81' +ref: +- 0425e2f3cfe1 +sort: definition +source: tex +src: morgan-tian +tex_file: temp2kappa +--- +*An $\epsilon$-tube ${\mathcal T}$* in +a Riemannian $3$-manifold $M$ is a submanifold diffeomorphic to the +product of $S^2$ with a non-degenerate interval with the following +properties: + +- **(1)** Each boundary component $S$ of ${\mathcal T}$ is the central +$2$-sphere of an $\epsilon$-neck $N(S)$ in $M$. +- **(2)** ${\mathcal T}$ +is a union of $\epsilon$-necks and the closed half $\epsilon$-necks +whose boundary sphere is a component of $\partial{\mathcal T}$. +Furthermore, the central $2$-sphere of each of the $\epsilon$-necks +is isotopic in ${\mathcal T}$ to the $S^2$-factors of the product +structure. + +An *open $\epsilon$-tube* is one without boundary. It is a union +of $\epsilon$-necks with the central spheres that are isotopic to +the $2$-spheres of the product structure. + +A *$C$-capped $\epsilon$-tube* +in $M$ is a connected submanifold that is the union of a +$(C,\epsilon)$-cap ${\mathcal C}$ + and an open $\epsilon$-tube where the intersection of ${\mathcal C}$ with the +$\epsilon$-tube is diffeomorphic to $S^2\times (0,1)$ and contains +an end of the $\epsilon$-tube and an end of the cap. + A *doubly $C$-capped +$\epsilon$-tube* in $M$ is a +closed, connected submanifold of $M$ that is the union of two +$(C,\epsilon)$-caps ${\mathcal C}_1$ and ${\mathcal C}_2$ and an +open $\epsilon$-tube. Furthermore, we require (i) that the cores +$Y_1$ and $Y_2$ of ${\mathcal C}_1$ and ${\mathcal C}_2$ have +disjoint closures, (ii) that the union of either ${\mathcal C}_i$ +with the $\epsilon$-tube is a capped $\epsilon$-tube and ${\mathcal +C}_1$ and ${\mathcal C}_2$ contain the opposite ends of the +$\epsilon$-tube. There is one further closely related notion, that +of an *$\epsilon$-fibration*. By +definition an $\epsilon$-fibration is a closed, connected manifold +that fibers over the circle with fibers $S^2$ that is also a union +of $\epsilon$-necks with the property that the central $2$-sphere of +each neck is isotopic to a fiber of the fibration structure. We +shall not see this notion again until the appendix, but because it +is clearly closely related to the notion of an $\epsilon$-tube, we +introduce it here. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/04518fa1ac2f.md b/projects/poincare-conjecture/.astrolabe/atoms/04518fa1ac2f.md new file mode 100644 index 00000000..69415a35 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/04518fa1ac2f.md @@ -0,0 +1,32 @@ +--- +chapter: 2 +generator: tools/poincare_tex_extract.py +label: lengthcompar +labels: +- lengthcompar +mtref: '2.4' +ref: +- 04518fa1ac2f +sort: theorem +source: tex +src: morgan-tian +tex_file: prelim +--- +**(Length comparison)** +Let $(M,g)$ be a manifold of non-negative curvature. Suppose that +$\triangle(a,b,c)$ is a triangle in $M$ and let +$\triangle(a',b',c')$ be a Euclidean triangle. + +1. Suppose that the corresponding +sides of $\triangle(a,b,c)$ and $\triangle(a',b',c')$ have the same +lengths. Then the angle at each vertex of the Euclidean triangle is +no larger than the corresponding angle of $\triangle(a,b,c)$. +Furthermore, for any $\alpha$ and $\beta$ less than $|s_{ab}|$ and +$|s_{ac}|$ respectively, let $x$, resp. $x'$, be the point on +$s_{ab}$, resp. $s_{a'b'}$, at distance $\alpha$ from $a$, resp. +$a'$, and let $y$, resp. $y'$, be the point on $s_{ac}$, resp. +$s_{a'c'}$, at distance $\beta$ from $a$, resp. $a'$. Then +$d(x,y)\ge d(x',y')$. +1. Suppose that $|s_{ab}|=|s_{a'b'}|$, that +$|s_{ac}|=|s_{a'c'}|$ and that $\angle_a=\angle_{a'}$. Then +$|s_{b'c'}|\ge |s_{bc}|$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/04dd3c3ea1c8.md b/projects/poincare-conjecture/.astrolabe/atoms/04dd3c3ea1c8.md new file mode 100644 index 00000000..c9e2a3e2 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/04dd3c3ea1c8.md @@ -0,0 +1,27 @@ +--- +chapter: 19 +generator: tools/poincare_tex_extract.py +label: topiso +labels: +- topiso +mtref: '19.10' +ref: +- 04dd3c3ea1c8 +sort: lemma +source: tex +src: morgan-tian +tex_file: canonnbhd +--- +Given any $\alpha>0$ there is $\epsilon(\alpha)>0$ such that the +following holds for any $0<\epsilon\le \epsilon(\alpha)$. Suppose +that $N$ is an $\epsilon$-neck centered at $x$ in a connected +manifold $M$ (here we are not assuming that $R(x)=1$) and that $z$ +is a point outside the middle two-thirds of $N$. We suppose that the +central two-sphere of $N$ separates $M$. Let $p$ be a point in the +middle sixth of $N$ at distance $d$ from $z$. Then the intersection +of the boundary of the metric ball $B(z,d)$ with $N$ is a +topological $2$-sphere contained in the middle quarter of $N$ that +maps homeomorphically onto $S^2$ under the projection mapping $N\to +S^2$ determined by the $\epsilon$-neck structure. Furthermore, if +$p'\in \partial B(z,d)$ then $|s(p)-s(p')|<\alpha +R(x)^{-1/2}\epsilon^{-1}$; see Fig. 19.2. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/0545613262dd.md b/projects/poincare-conjecture/.astrolabe/atoms/0545613262dd.md new file mode 100644 index 00000000..e5d9e75e --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/0545613262dd.md @@ -0,0 +1,13 @@ +--- +chapter: 18 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '18.21' +ref: +- 0545613262dd +sort: claim +source: tex +src: morgan-tian +tex_file: energy1 +--- +If $t$ is not a surgery time, then $W_\xi(t)$ is continuous at $t$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/06ac7beab852.md b/projects/poincare-conjecture/.astrolabe/atoms/06ac7beab852.md new file mode 100644 index 00000000..f45baa0a --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/06ac7beab852.md @@ -0,0 +1,20 @@ +--- +chapter: 7 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '7.21' +ref: +- 06ac7beab852 +sort: definition +source: tex +src: morgan-tian +tex_file: newcomp2 +--- +For any continuous function $\psi$ defined on $B_{(q,t)}'\setminus +\left(S_{(q,t)}\cap B_{(q,t)}'\right)$ we define + +$$ +\int_{(B_{(q,t)}')^*}\psi d\mathit{vol}(g(t))=\mathit{lim}_{\epsilon\rightarrow +0}\int_{B_{(q,t)}'\setminus \nu_\epsilon(S_{(q,t)})\cap +B_{(q,t)}'}\psi d\mathit{vol}(g(t)). +$$ diff --git a/projects/poincare-conjecture/.astrolabe/atoms/07232fc42f3a.md b/projects/poincare-conjecture/.astrolabe/atoms/07232fc42f3a.md new file mode 100644 index 00000000..38d64b3e --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/07232fc42f3a.md @@ -0,0 +1,22 @@ +--- +chapter: 9 +generator: tools/poincare_tex_extract.py +label: gradshrink +labels: +- gradshrink +mtref: '9.32' +ref: +- 07232fc42f3a +sort: proposition +source: tex +src: morgan-tian +tex_file: temp2kappa +--- +The following equation holds on $M_\infty\times (-\infty,0)$: + +$$ +\mathit{Ric}_{g_{\infty}(t)} + \mathit{Hess}^{g_\infty(t)}(l_\infty(\cdot,\tau)) +-\frac{1}{2\tau}g_{\infty}(t) = 0, +$$ + + where $\tau=-t$, diff --git a/projects/poincare-conjecture/.astrolabe/atoms/074e02b067b8.md b/projects/poincare-conjecture/.astrolabe/atoms/074e02b067b8.md new file mode 100644 index 00000000..1a61ca64 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/074e02b067b8.md @@ -0,0 +1,27 @@ +--- +chapter: 11 +generator: tools/poincare_tex_extract.py +label: hexist +labels: +- hexist +mtref: '11.31' +ref: +- 074e02b067b8 +sort: theorem +source: tex +src: morgan-tian +tex_file: singlimit2 +--- +Fix $\rho>0$. Then for any $\delta>0$ there is an +$0r_0^{-2}$ one of the +following holds: + +- **(1)** $(x,T)$ is the center of a strong $2\epsilon$-neck in +$(\widehat{\mathcal M},\widehat G)$. +- **(2)** There is a +$(2C,2\epsilon)$-cap in $(\Omega(T),\widehat G(T))$ whose core +contains $(x,T)$. +- **(3)** There is a $2C$-component of +$\Omega(T)$ that contains $(x,T)$. +- **(4)** There is a $2\epsilon$-round component of $\Omega(T)$ that +contains $(x,T)$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/07ac984fbe88.md b/projects/poincare-conjecture/.astrolabe/atoms/07ac984fbe88.md new file mode 100644 index 00000000..e642f5bf --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/07ac984fbe88.md @@ -0,0 +1,27 @@ +--- +chapter: 18 +generator: tools/poincare_tex_extract.py +label: limitcs +labels: +- limitcs +mtref: '18.56' +ref: +- 07ac984fbe88 +sort: claim +source: tex +src: morgan-tian +tex_file: energy1 +--- +After passing to a subsequence of $\{\lambda_n\}$, either: + +- **(1)** for each $n$ sufficiently large there is $t_n\in J_B(c)$ with +the length of $\widetilde\Gamma_c^{\lambda_n}(t_n)<\delta^2B^{-1}$, +or +- **(2)** for each component $J_i=[t_i^-,t_i^+]$ of $ J_B(c)$, after +composing $\widetilde\Gamma_c^{\lambda_n}(x,t)$ by a +reparameterization of the domain circle (fixed in $t$ but a +different reparameterization for each $n$) so that the $\widetilde +\Gamma_c^{\lambda_n}(t_i^-)$ have constant speed, there is a smooth +limiting curve-shrinking flow denoted $\widetilde\Gamma_c(t)$, for +$t\in J_i$ for the sequence $p_1\widetilde\Gamma_c^{\lambda_n}(t),\ +t_i^-\le t\le t_i^+$. The limiting flow consists of immersions. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/08412f0ffb82.md b/projects/poincare-conjecture/.astrolabe/atoms/08412f0ffb82.md new file mode 100644 index 00000000..d99ac3e0 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/08412f0ffb82.md @@ -0,0 +1,18 @@ +--- +chapter: 18 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '18.44' +ref: +- 08412f0ffb82 +sort: corollary +source: tex +src: morgan-tian +tex_file: energy1 +--- +Given curve-shrinking flows $c_1(\cdot,t)$ and $c_2(\cdot,t)$ for +ramps of degree one in $(\bar M,\bar g(t))\times (S^1_\lambda,ds^2)$ +the minimal area of an annulus connecting $c_1(\cdot,t)$ and +$c_2(\cdot,t)$ grows at most exponentially with time with an +exponent determined by an upper bound on the sectional curvature of +the ambient flow, which in particular is independent of $\lambda$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/08c3ff9d0a1e.md b/projects/poincare-conjecture/.astrolabe/atoms/08c3ff9d0a1e.md new file mode 100644 index 00000000..7eb73398 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/08c3ff9d0a1e.md @@ -0,0 +1,26 @@ +--- +chapter: 11 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '11.26' +ref: +- 08c3ff9d0a1e +sort: definition +source: tex +src: morgan-tian +tex_file: singlimit2 +--- +A *strong double $2\epsilon$-horn* in +$(\Omega,g(T))$ is a component of $\Omega$ diffeomorphic to +$S^2\times (0,1)$ with the property that every point of this +component is the center of a strong $2\epsilon$-neck in $\widehat +{\mathcal M}$. This means that a strong double $2\epsilon$-horn is a +$2\epsilon$-tube and hence is a component of $\Omega$ diffeomorphic +to $S^2\times(-1,1)$. Notice that each end of a strong double +$2\epsilon$-horn contains a strong $2\epsilon$-horn. + + For any $C'<\infty$, a $C'$-*capped +$2\epsilon$-horn* in $(\Omega,g(T))$ is a +component of $\Omega$ that is a the union of a the core of a +$(C',2\epsilon)$-cap and a strong $2\epsilon$-horn. Such a component +is diffeomorphic to an open $3$-ball or to a punctured $\Ar P^3$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/0936bbe2abb8.md b/projects/poincare-conjecture/.astrolabe/atoms/0936bbe2abb8.md new file mode 100644 index 00000000..4a8eef56 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/0936bbe2abb8.md @@ -0,0 +1,17 @@ +--- +chapter: 18 +generator: tools/poincare_tex_extract.py +label: ubd +labels: +- ubd +mtref: '18.41' +ref: +- 0936bbe2abb8 +sort: corollary +source: tex +src: morgan-tian +tex_file: energy1 +--- +If $c(x,t),\ t_0\le t< t'_1<\infty$, is a solution of the curve +shrinking flow in $( M, g(t))\times (S^1_\lambda,ds^2)$ and if +$c(t_0)$ a ramp, then $c(t)$ is a ramp for all $t\in [t_0,t'_1)$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/096bebd3ecff.md b/projects/poincare-conjecture/.astrolabe/atoms/096bebd3ecff.md new file mode 100644 index 00000000..22d43362 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/096bebd3ecff.md @@ -0,0 +1,31 @@ +--- +chapter: 12 +generator: tools/poincare_tex_extract.py +label: stdsolnlimit +labels: +- stdsolnlimit +mtref: '12.36' +ref: +- 096bebd3ecff +sort: corollary +source: tex +src: morgan-tian +tex_file: stdsoln +--- +For any $\epsilon>0$ let $C'=C'(\epsilon)$ be as in +Theorem 12.32. Suppose that we have a sequence of +generalized Ricci flows $({\mathcal M}_n,G_n)$, points $x_n\in +{\mathcal M}_n$ with $\mathbf{t}(x_n)=0$, neighborhoods $U_n$ of $x_n$ +in the zero time-slice of ${\mathcal M}_n$, and a constant +$0<\theta<1$. Suppose that there are embeddings $\rho_n\colon +U_n\times [0,\theta)\to {\mathcal M}_n$ compatible with time and the +vector field so that the Ricci flows $\rho_n^*G_n$ on $U_n$ based at +$x_n$ converge geometrically to the restriction of the standard +solution to $[0,\theta)$. Then for all $n$ sufficiently large, and +any point $y_n$ in the image of $\rho_n$ one of the following holds: + +- **(1)** $y_n$ is contained in the core of a +$(C'(\epsilon),\epsilon)$-cap +- **(2)** $y_n$ is the center of a strong $\epsilon$-neck +- **(3)** $y_n$ is the center of an evolving $\epsilon$-neck whose +initial time-slice is at time $0$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/09af70d51aac.md b/projects/poincare-conjecture/.astrolabe/atoms/09af70d51aac.md new file mode 100644 index 00000000..26ebbcec --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/09af70d51aac.md @@ -0,0 +1,30 @@ +--- +chapter: 6 +generator: tools/poincare_tex_extract.py +label: Llength +labels: +- Llength +mtref: '6.2' +ref: +- 09af70d51aac +sort: definition +source: tex +src: morgan-tian +tex_file: newcompar +--- +Let $\gamma\colon [\tau_{1},\tau_{2}]\rightarrow {\mathcal M},\ 0 +\le \tau_{1} < \tau_{2}$, be a $C^{1}$-path parameterized by +backward time. We define $X_\gamma(\tau)$ to be the horizontal +projection of the tangent vector $d\gamma(\tau)/d\tau$, so that +$d\gamma/d\tau=-\chi+X_\gamma(\tau)$ with $X_\gamma(\tau)\in +{\mathcal HT}{\mathcal M}$. We define the $\mathcal{L}$-*length* of $\gamma$ to be: + +$$ +\mathcal{L}(\gamma) = \int_{\tau_{1}}^{\tau_{2}} +\sqrt{\tau}\left(R(\gamma(\tau)) + +\abs{X_\gamma(\tau)}^{2}\right)d\tau, +$$ + +where the norm of $X_\gamma(\tau)$ is measured using the metric +$G_{T-\tau}$ on ${\mathcal HT}{\mathcal M}$. When $\gamma$ is clear +from the context, we write $X$ for $X_\gamma$; see Fig. 0.2 from the Introduction. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/0a1eac346e21.md b/projects/poincare-conjecture/.astrolabe/atoms/0a1eac346e21.md new file mode 100644 index 00000000..37c376d2 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/0a1eac346e21.md @@ -0,0 +1,19 @@ +--- +chapter: 8 +generator: tools/poincare_tex_extract.py +label: inside +labels: +- inside +mtref: '8.7' +ref: +- 0a1eac346e21 +sort: claim +source: tex +src: morgan-tian +tex_file: noncoll +--- +Suppose $\varepsilon_0\le 1/4n(n-1)$ is the constant from the last +claim. Set $\tau_1=\varepsilon r^2$, and suppose that +$\varepsilon\le \varepsilon_0$. Lastly, assume that $|Z|\le +\frac{1}{8\sqrt{\varepsilon}}$. Then $\gamma_Z(\tau) \in B(x,T,r/2)$ +for all $\tau\le \tau_1$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/0a41b5c61dd2.md b/projects/poincare-conjecture/.astrolabe/atoms/0a41b5c61dd2.md new file mode 100644 index 00000000..75f76fbd --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/0a41b5c61dd2.md @@ -0,0 +1,31 @@ +--- +chapter: 13 +generator: tools/poincare_tex_extract.py +label: Rhatmatrix +labels: +- Rhatmatrix +mtref: '13.9' +ref: +- 0a41b5c61dd2 +sort: lemma +source: tex +src: morgan-tian +tex_file: surgery +--- +Set $\delta'_4=\mathit{min}(\delta_2',\delta_3')$. Suppose +that $\delta\le \mathit{min}(\delta'_4,C_0^{-1})$. Then in the basis +$\{f_0,f_1,f_2\}$ for $T_y(S^2\times I)$ as in Corollary 13.8 we +have + +$$ +(\hat R_{ijkl}(y)) =e^{-2f}\left[\begin{pmatrix} \lambda & 0 & 0 +\\ 0 & \alpha & \beta \\ 0 & \beta & \gamma\end{pmatrix} ++\begin{pmatrix} -\frac{q^2}{s^4}f^2 & 0 \\ 0 & +\left(\frac{q^2}{s^4}-\frac{2q}{s^3}\right)f\begin{pmatrix} 1 & 0 +\\ 0 & 1\end{pmatrix} +\end{pmatrix}+\left(\frac{q^2}{s^4}fO(\delta)\right)\right] +$$ + +where $\lambda,\alpha,\beta,\gamma$ are the constants in +Lemma 13.8 and the first matrix is the expression for +$(R_{ijkl}(y))$ in this basis. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/0a4d6f17810d.md b/projects/poincare-conjecture/.astrolabe/atoms/0a4d6f17810d.md new file mode 100644 index 00000000..1262e9a7 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/0a4d6f17810d.md @@ -0,0 +1,48 @@ +--- +chapter: 16 +generator: tools/poincare_tex_extract.py +label: bigell +labels: +- bigell +mtref: '16.13' +ref: +- 0a4d6f17810d +sort: proposition +source: tex +src: morgan-tian +tex_file: surgery +--- +For any $\ell<\infty$ there is $A_0=A_0(\ell)<\infty$, +$0<\theta_0=\theta_0(\ell)<1$, and for any $A\ge A_0$ for the +constant +$\delta''=\delta''(A)=\delta''_0(A,\theta_0,\overline\delta_0)>0$ +from Proposition 16.5 the following holds. Suppose that +$({\mathcal M},G)$ is a Ricci flow with surgery defined for $0\le +t< T<\infty$. Suppose that it satisfies the strong +$(C,\epsilon)$-canonical neighborhood assumption at all points $x$ +with $R(x)\ge r_{i+1}^{-2}$. Suppose also that the solution has +curvature pinched toward positive. Suppose that there is a surgery +at some time $\bar t$ with $T_{i-1}\le \bar t\ell. +$$ + +See Fig. 16.2. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/0ac34e2feca6.md b/projects/poincare-conjecture/.astrolabe/atoms/0ac34e2feca6.md new file mode 100644 index 00000000..059c2962 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/0ac34e2feca6.md @@ -0,0 +1,14 @@ +--- +chapter: 4 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '4.28' +ref: +- 0ac34e2feca6 +sort: claim +source: tex +src: morgan-tian +tex_file: maxprin +--- +For each $x\in M$ and each $t\ge 0$, the fiber $Z(x,t)$ of +${\mathcal Z}(t)$ over $x$ is a convex subset of $\mathit{Sym}^2(\wedge^2T^*M)$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/0b07beec84c0.md b/projects/poincare-conjecture/.astrolabe/atoms/0b07beec84c0.md new file mode 100644 index 00000000..53cddc7f --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/0b07beec84c0.md @@ -0,0 +1,26 @@ +--- +chapter: 12 +generator: tools/poincare_tex_extract.py +label: A_0 +labels: +- A_0 +mtref: '12.3' +ref: +- 0b07beec84c0 +sort: lemma +source: tex +src: morgan-tian +tex_file: stdsoln +--- +There is $A_0<\infty$ such that + +$$ +(\Ar^3\setminus B(0,A_0),g(0)) +$$ + +is isometric to the product of a round metric on $S^2$ of scalar +curvature $1$ with the Euclidean metric on $[0,\infty)$. There is a +constant $K<\infty$ such that the volume of $B_{g(0)}(0,A_0)$ is at +most $K$. Furthermore, there is a constant $D<\infty$ so that the +scalar curvature of standard initial metric $(\Ar^3,g(0))$ is +bounded above by $D$ and below by $D^{-1}$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/0bfc692d99b1.md b/projects/poincare-conjecture/.astrolabe/atoms/0bfc692d99b1.md new file mode 100644 index 00000000..433a1d8c --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/0bfc692d99b1.md @@ -0,0 +1,26 @@ +--- +chapter: 7 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '7.4' +ref: +- 0bfc692d99b1 +sort: definition +source: tex +src: morgan-tian +tex_file: newcomp2 +--- +Suppose that $(M,g(t)),\ 0\le t\le T<\infty$, is a Ricci +flow, complete of bounded curvature. We define the function +$L_{x}\colon M\times [0,T)\to \Ar$ by assigning to each $(q,t)$ the +length of any ${\mathcal L}$-minimizing ${\mathcal L}$-geodesic from +$x$ to $y=(q,t)\in M\times [0,T)$. Clearly, the restriction of this +function to ${\mathcal U}_{x}$ agrees with the smooth function +$L_{x}$ given in Definition 6.26. We define + $L_{x}^\tau\colon M\to \Ar$ to be the restriction of $L_x$ to + $M\times \{T-\tau\}$. +Of course, the restriction of $L_{x}^\tau$ to ${\mathcal +U}_{x}(\tau)$ agrees with the smooth function $L_{x}^\tau$ defined +in the last chapter. We define $l_{x}\colon M\times [0,T)\to \Ar$ by +$l_{x}(y)=L_{x}(y)/2\sqrt{\tau}$, where, as always $\tau=T-t$, and +we define $l_{x}^\tau(q)=l_{x}(q,T-\tau)$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/0c451aa3e5cc.md b/projects/poincare-conjecture/.astrolabe/atoms/0c451aa3e5cc.md new file mode 100644 index 00000000..c1ddab54 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/0c451aa3e5cc.md @@ -0,0 +1,43 @@ +--- +chapter: 4 +generator: tools/poincare_tex_extract.py +label: quad +labels: +- quad +mtref: '4.13' +ref: +- 0c451aa3e5cc +sort: lemma +source: tex +src: morgan-tian +tex_file: maxprin +--- +Let $(M,g)$ be a Riemannian $3$-manifold. Let ${\mathcal T}\in \mathit{Sym}^2(\wedge^2T_x^*M)$ be the curvature operator written with +respect to the evolving frame as in Proposition 3.19. Then +the evolution equation given in Proposition 3.19 is: + +$$ +\frac{\partial {\mathcal T}}{\partial t}=\triangle {\mathcal T}+\psi({\mathcal T}) +$$ + +where + +$$ +\psi({\mathcal T})={\mathcal T}^2+{\mathcal T}^\sharp. +$$ + +In particular, in an orthonormal basis in which + +$$ +{\mathcal T}=\begin{pmatrix} \lambda & 0 & 0 \\ 0 & \mu & 0 \\ 0 & +0 & \nu\end{pmatrix} +$$ + + with $\lambda\ge \mu\ge \nu$, the vector +field is given by + +$$ +\psi({\mathcal T})={\mathcal T}^2+{\mathcal T}^\sharp=\begin{pmatrix} +\lambda^2+\mu\nu & 0 & 0 \\ 0 & \mu^2+\lambda\nu & 0 \\ 0 & 0 & +\nu^2+\lambda\mu\end{pmatrix}. +$$ diff --git a/projects/poincare-conjecture/.astrolabe/atoms/0c6b379e7dca.md b/projects/poincare-conjecture/.astrolabe/atoms/0c6b379e7dca.md new file mode 100644 index 00000000..89023212 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/0c6b379e7dca.md @@ -0,0 +1,27 @@ +--- +chapter: 1 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '1.1' +ref: +- 0c6b379e7dca +sort: definition +source: tex +src: morgan-tian +tex_file: prelim +--- +Let $M$ be an $n$-dimensional manifold. A Riemannian metric $g$ on $M$ is a smooth section of $T^*M\otimes T^*M$ defining a +positive definite symmetric bilinear form on $T_pM$ for each $p\in +M$. In local coordinates $(x^1,\cdots, x^n)$, one has a natural +local basis $\{\partial_1,\cdots,\partial_n\}$ for $TM$, where +$\partial_i=\frac{\partial}{\partial x^i}$. The metric tensor +$g=g_{ij} dx^i\otimes dx^j$ is represented by a smooth matrix-valued +function + +$$ +g_{ij}=g(\partial_i,\partial_j). +$$ + + The +pair $(M,g)$ is a *Riemannian manifold*. We denote by $(g^{ij})$ +the inverse of the matrix $(g_{ij})$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/0cf778ecc2ec.md b/projects/poincare-conjecture/.astrolabe/atoms/0cf778ecc2ec.md new file mode 100644 index 00000000..231e2a3d --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/0cf778ecc2ec.md @@ -0,0 +1,19 @@ +--- +chapter: 1 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '1.27' +ref: +- 0cf778ecc2ec +sort: lemma +source: tex +src: morgan-tian +tex_file: prelim +--- +The *Laplacian operator* acting on scalar +functions on $M$ is given in local coordinates by + +$$ +\triangle = \frac{1}{\sqrt{\mathit{det}(g)}}\partial_i +\left(g^{ij}\sqrt{\mathit{det}(g)}\partial_j\right). +$$ diff --git a/projects/poincare-conjecture/.astrolabe/atoms/0d0c8f66ca5d.md b/projects/poincare-conjecture/.astrolabe/atoms/0d0c8f66ca5d.md new file mode 100644 index 00000000..a4cef375 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/0d0c8f66ca5d.md @@ -0,0 +1,17 @@ +--- +chapter: 12 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '12.12' +ref: +- 0d0c8f66ca5d +sort: corollary +source: tex +src: morgan-tian +tex_file: stdsoln +--- +The standard solution $g(t),\ t\in [0,T)$, consists of a family of metrics +all of +which are +rotationally symmetric by the standard action of $SO(3)$ on +$\mathbb{R}^3$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/0d534e8417eb.md b/projects/poincare-conjecture/.astrolabe/atoms/0d534e8417eb.md new file mode 100644 index 00000000..47a2de59 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/0d534e8417eb.md @@ -0,0 +1,17 @@ +--- +chapter: 2 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '2.15' +ref: +- 0d534e8417eb +sort: theorem +source: tex +src: morgan-tian +tex_file: prelim +--- +**(The Maximum Principle)** +Let $f$ be a real-valued continuous function on a connected +Riemannian manifold with $\Delta f\ge 0$ in the weak sense. Then $f$ +is locally constant near any local maximum. In particular, if $f$ +achieves its maximum then it is a constant. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/0d6f74e37efb.md b/projects/poincare-conjecture/.astrolabe/atoms/0d6f74e37efb.md new file mode 100644 index 00000000..114b6e32 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/0d6f74e37efb.md @@ -0,0 +1,16 @@ +--- +chapter: 18 +generator: tools/poincare_tex_extract.py +label: Dxacross +labels: +- Dxacross +mtref: '18.74' +ref: +- 0d6f74e37efb +sort: lemma +source: tex +src: morgan-tian +tex_file: energy1 +--- +There is no $x$ for which $D_x$ is an embedded arc with both +endpoints on $c_0$ and otherwise disjoint from $\partial A$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/0db415314964.md b/projects/poincare-conjecture/.astrolabe/atoms/0db415314964.md new file mode 100644 index 00000000..e24053b9 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/0db415314964.md @@ -0,0 +1,18 @@ +--- +chapter: 4 +generator: tools/poincare_tex_extract.py +label: rm>0 +labels: +- rm>0 +mtref: '4.14' +ref: +- 0db415314964 +sort: corollary +source: tex +src: morgan-tian +tex_file: maxprin +--- +Let $(M,g(t)),\ 0\le t\le T$, be a Ricci flow with $M$ a compact, +connected $3$-manifold. Suppose that $\mathit{Rm}(x,0)\ge 0$ for all +$x\in M$. Then $\mathit{Rm}(x,t)\ge 0$ for all $x\in M$ and all $t\in +[0,T]$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/0ddfe578961e.md b/projects/poincare-conjecture/.astrolabe/atoms/0ddfe578961e.md new file mode 100644 index 00000000..582c6461 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/0ddfe578961e.md @@ -0,0 +1,35 @@ +--- +chapter: 11 +generator: tools/poincare_tex_extract.py +label: controlnbhd +labels: +- controlnbhd +mtref: '11.2' +ref: +- 0ddfe578961e +sort: lemma +source: tex +src: morgan-tian +tex_file: singlimit2 +--- +Let $({\mathcal M},G)$ be a generalized $3$-dimensional Ricci flow. +Suppose that $r_0>0$ and that any $z\in {\mathcal M}$ with $R(z)\ge +r_0^{-2}$ has a strong $(C,\epsilon)$-canonical +neighborhood. Suppose $z\in +{\mathcal M}$ and $\mathbf{t}(z)=t_0$. Set + +$$ +r=\frac{1}{2C\sqrt{\mathit{max}(R(z),r_0^{-2})}} +$$ + + and + +$$ +\Delta t=\frac{1}{16C\left(R(z)+r_0^{-2}\right)}. +$$ + +Suppose that $r'\le r$ and that $|t'-t_0|\le \Delta t$ and let $I$ +be the interval with endpoints $t_0$ and $t'$. Suppose that there is +an embedding of $j\colon B(z,t_0,r')\times I$ into ${\mathcal M}$ +compatible with time and with the vector field. Then $R(y)\le +2\left(R(z)+r_0^{-2}\right)$ for all $y$ in the image of $j$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/0e9703c15036.md b/projects/poincare-conjecture/.astrolabe/atoms/0e9703c15036.md new file mode 100644 index 00000000..6b081a21 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/0e9703c15036.md @@ -0,0 +1,17 @@ +--- +chapter: 9 +generator: tools/poincare_tex_extract.py +label: absconv +labels: +- absconv +mtref: '9.28' +ref: +- 0e9703c15036 +sort: corollary +source: tex +src: morgan-tian +tex_file: temp2kappa +--- +The distributions $|\nabla l_\infty|^2e^{-l_\infty}$, +$Re^{-l_\infty}$, $|(\partial l_\infty/\partial \tau)|e^{-l_\infty}$ +are absolutely convergent in the sense of the above claim. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/0e9a1a167caa.md b/projects/poincare-conjecture/.astrolabe/atoms/0e9a1a167caa.md new file mode 100644 index 00000000..359cce20 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/0e9a1a167caa.md @@ -0,0 +1,33 @@ +--- +chapter: 19 +generator: tools/poincare_tex_extract.py +label: Xcontainedin +labels: +- Xcontainedin +mtref: '19.21' +ref: +- 0e9a1a167caa +sort: proposition +source: tex +src: morgan-tian +tex_file: canonnbhd +--- +For any $C<\infty$ the following holds. Suppose that $X$ is a +connected subset of a Riemannian three-manifold $(M,g)$. Suppose +that every point of $X$ is either the center of an $\epsilon$-neck +or is contained in the core of a $(C,\epsilon)$-cap. Then one of the +following holds: + +- **(1)** $X$ is contained in a component of $M$ that is the union of +two $(C,\epsilon)$-caps. This component is diffeomorphic to $S^3$, +$\Ar P^3$ or $\Ar P^3\#\Ar P^3$. +- **(2)** $X$ is contained in a component of $M$ that is a double $C$-capped $\epsilon$-tube. + This component is diffeomorphic to $S^3$, $\Ar P^3$ or $\Ar +P^3\#\Ar P^3$. +- **(3)** $X$ is contained in a single $(C,\epsilon)$-cap. +- **(4)** $X$ is contained in a $C$-capped $\epsilon$-tube. +- **(5)** $X$ is contained in an $\epsilon$-tube. +- **(6)** $X$ is contained in a component of $M$ that is an +$\epsilon$-fibration, which itself is a union of $\epsilon$-necks. + +(See Fig. 19.4.) diff --git a/projects/poincare-conjecture/.astrolabe/atoms/0ea74aba0c14.md b/projects/poincare-conjecture/.astrolabe/atoms/0ea74aba0c14.md new file mode 100644 index 00000000..9c9a62db --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/0ea74aba0c14.md @@ -0,0 +1,41 @@ +--- +chapter: 16 +generator: tools/poincare_tex_extract.py +label: J +labels: +- J +mtref: '16.18' +ref: +- 0ea74aba0c14 +sort: definition +source: tex +src: morgan-tian +tex_file: surgery +--- +Suppose that $\bar t$ is a surgery +time, that $\tau_1>0$, and that there are no other surgery times in +the interval $(\bar t-\tau_1,\bar t]$. Let $\{\Sigma_i(\bar t)\}_i$ +be the $2$-spheres on which we do surgery at time $\bar t$. Each +$\Sigma_i$ is the central $2$-sphere of a strong $\delta$-neck +$N_i$. We can flow the cylinders $J_0(\bar +t)=\cup_is_{N_i}^{-1}(-25,0])$ backward to any time + $t\in (\bar t-\tau_1,\bar t]$. Let $J_0(t)$ be +the result. There is an induced function, denoted +$\coprod_is_{N_i}(t)$, on $J_0(t)$. It takes values in $ (-25,0]$. +We denote the boundary of $J_0(t)$ by $\coprod_i\Sigma_i(t)$. Of +course, this boundary is the result of flowing +$\coprod_i\Sigma_i(\bar t)$ backward to time $t$. (These backward +flows are possible since there are no surgery times in $(\bar +t-\tau_1,\bar t)$.) For each $t\in [\bar t-\tau_1,\bar t)$ we also +have the disappearing region $D_t$: -- the region that disappears at +time $\bar t$. It is an open submanifold whose boundary is +$\coprod_i\Sigma_i(t)$. Thus, for every $t\in (\bar t-\tau_1,\bar +t)$ the subset $J(t)=J_0(t)\cup D_t$ is an open subset of $M_t$. We +define + +$$ +J(\bar t-\tau_1,\bar t)=\cup_{t\in (\bar t-\tau_1,\bar t)}J(t). +$$ + +Then $J(\bar t-\tau_1,\bar t)$ is an open subset of ${\mathcal M}$. +See Fig. 16.4. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/0f76a0750b40.md b/projects/poincare-conjecture/.astrolabe/atoms/0f76a0750b40.md new file mode 100644 index 00000000..d115326f --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/0f76a0750b40.md @@ -0,0 +1,33 @@ +--- +chapter: 3 +generator: tools/poincare_tex_extract.py +label: geosuff +labels: +- geosuff +mtref: '3.23' +ref: +- 0f76a0750b40 +sort: claim +source: tex +src: morgan-tian +tex_file: flowbasics +--- +Suppose that for every minimal $g(t_0)$-geodesic $\gamma$ from $x_0$ +to $x_1$ the function $\ell_t(\gamma)$ which is the $g(t)$-length of +$\gamma$ satisfies + +$$ +\frac{d(\ell_t(\gamma))}{dt}\Bigl|_{t=t_0}\Bigr.\ge C. +$$ + +Then + +$$ +\frac{d(d_t(x_0,x_1))}{dt}\Bigl|_{t=t_0}\Bigr.\ge C, +$$ + +where, as in the statement of the proposition, if the distance +function is not differentiable at $t_0$ then the inequality in the +conclusion is interpreted by replacing the derivative on the +left-hand side with the $\mathit{liminf}$ of the forward difference +quotients of $d_t(x_0,x_1)$ at $t_0$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/0f8fc616f97c.md b/projects/poincare-conjecture/.astrolabe/atoms/0f8fc616f97c.md new file mode 100644 index 00000000..d9d216f5 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/0f8fc616f97c.md @@ -0,0 +1,25 @@ +--- +chapter: 3 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '3.41' +ref: +- 0f8fc616f97c +sort: definition +source: tex +src: morgan-tian +tex_file: flowbasics +--- +Let $({\mathcal M},G)$ be a generalized Ricci flow and let $x$ be a +point of space-time. Set $t=\mathbf{t}(x)$. For any $r>0$ we define +$B(x,t,r)\subset M_t$ to be the metric ball of radius $r$ centered +at $x$ in the Riemannian manifold $(M_t,g(t))$. For any $\Delta t>0$ +we say that $P(x,t,r,\Delta t)$, +respectively, $P(x,r,t,-\Delta +t)$, *exists* in ${\mathcal M}$ if there is an embedding +$B(x,t,r)\times [t,t+\Delta t]$, respectively, $B(x,t,r)\times +[t-\Delta t,t]$, into ${\mathcal M}$ compatible with time and the +vector field. When this embedding exists, its image is defined to be the *forward parabolic neighborhood* +$P(x,t,r,\Delta t)$, respectively the *backward parabolic +neighborhood* $P(x,t,r,-\Delta t)$. +See Fig. 3.3. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/0fc5cfacef3f.md b/projects/poincare-conjecture/.astrolabe/atoms/0fc5cfacef3f.md new file mode 100644 index 00000000..fe601134 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/0fc5cfacef3f.md @@ -0,0 +1,24 @@ +--- +chapter: 6 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '6.58' +ref: +- 0fc5cfacef3f +sort: definition +source: tex +src: morgan-tian +tex_file: newcompar +--- +Let $y\in {\mathcal M}$ with $\mathbf{t}(y)=t$ and suppose that for +some $\epsilon>0$ there is an embedding $\iota\colon B(y,t,r)\times +(t-\epsilon,t+\epsilon)\to {\mathcal M}$ that is compatible with +time and the vector field. Then we denote by $\widetilde +P(y,r,\epsilon)\subset {\mathcal M}$ the image of $\iota$. Whenever +we introduce $\widetilde P(y,r,\epsilon)\subset {\mathcal M}$ +implicitly we are asserting that such an embedding exists. + +For $A\subset {\mathcal M}$, if $\widetilde +P(a,\epsilon,\epsilon)\subset {\mathcal M}$ exists for every $a\in +A$, then we denote by $\nu_\epsilon(A)$ the union over all $a\in A$ +of $\widetilde P(a,\epsilon,\epsilon)$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/1068f89476b6.md b/projects/poincare-conjecture/.astrolabe/atoms/1068f89476b6.md new file mode 100644 index 00000000..fbbc9fcd --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/1068f89476b6.md @@ -0,0 +1,34 @@ +--- +chapter: 14 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '14.3' +ref: +- 1068f89476b6 +sort: definition +source: tex +src: morgan-tian +tex_file: surgery +--- +Let ${\mathcal M}$ be a surgery space-time. Given a space $K$ and an +interval $J\subset \Ar$ we say that an embedding $K\times J\to +{\mathcal M}$ is *compatible with time and the vector field* +if: (i) the restriction of $\mathbf{t}$ to the image agrees with the +projection onto the second factor and (ii) for each $x\in X$ the +image of $\{x\}\times J$ is the integral curve for the vector field +$\chi$. + If in addition $K$ is a subset of $M_{t}$ +we require that $t\in J$ and that the map $K\times\{t\}\to M_{t}$ be +the identity. Clearly, by the uniqueness of integral curves for vector fields, two such +embeddings agree on their common interval of definition, so that, +given $K\subset M_{t}$ there is a maximal interval $J_K$ containing +$t$ such that such an embedding is defined on $K\times J_K$. In the +special case when $K=\{x\}$ for a point $x\in M_{t}$ we say that +such an embedding is *the maximal flow line* through $x$. The +embedding of the maximal interval through $x$ compatible with time +and the vector field $\chi$ is called *the domain of definition* +of the flow line through $x$. For a more general subset $K\subset +M_{t}$ there is an embedding $K\times J$ compatible with time and +the vector field $\chi$ if and only if, for every $x\in K$, the +interval $J$ is contained in the domain of definition of the flow +line through $x$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/10dace7fb165.md b/projects/poincare-conjecture/.astrolabe/atoms/10dace7fb165.md new file mode 100644 index 00000000..477ec601 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/10dace7fb165.md @@ -0,0 +1,32 @@ +--- +chapter: 18 +generator: tools/poincare_tex_extract.py +label: muarea +labels: +- muarea +mtref: '18.58' +ref: +- 10dace7fb165 +sort: lemma +source: tex +src: morgan-tian +tex_file: energy1 +--- +There is a $\mu>0$ such that the following holds. Let $c,\hat c\in +S^2$. Suppose that there is an arc $\omega$ in $S^2$ connecting $c$ +to $\hat c$ with the area of the annulus +$\widetilde\Gamma_\omega^\lambda(t_0)$ in $M\times S^1_\lambda$ less +than $\mu$. Let $v_{\hat c}$, resp., $v_c$, be the solution to +Equation (18.4) with initial condition $v_{\hat +c}(t_0)=A(\widetilde\Gamma(\hat c))$, resp., +$v_c(t_0)=A(\widetilde\Gamma(c))$. If + +$$ +A(p_1\widetilde\Gamma^\lambda_{\hat c}(t_1))\le v_{\hat c}+\zeta/2, +$$ + +then + +$$ +A(p_1(\widetilde\Gamma^\lambda_c(t_1))\le v_c+\zeta. +$$ diff --git a/projects/poincare-conjecture/.astrolabe/atoms/111551d7831b.md b/projects/poincare-conjecture/.astrolabe/atoms/111551d7831b.md new file mode 100644 index 00000000..560a36c7 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/111551d7831b.md @@ -0,0 +1,24 @@ +--- +chapter: 17 +generator: tools/poincare_tex_extract.py +label: lemstdcan +labels: +- lemstdcan +mtref: '17.7' +ref: +- 111551d7831b +sort: lemma +source: tex +src: morgan-tian +tex_file: surgery +--- +Suppose that there are $A',D',t'<\infty$ such that the following +holds for all $a$ sufficiently large. There is a point $y_a\in +B_{\widetilde G_a}(\tilde x_a,0,A')$ and a flow line of $\chi$ + beginning at $y_a$, defined for backward time and ending at a + point $z_a$ in a +surgery cap ${\mathcal C}_a$ at time $-t_a$ for some $t_a\le t'$. We +denote this flow line by $y_a(t), -t_a\le t\le 0$. Furthermore, +suppose that the scalar curvature on the flow line from $y_a$ to +$z_a$ is bounded by $D'$. Then for all $a$ sufficiently large, $x_a$ +has a strong $(C,\epsilon)$-canonical neighborhood. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/11159b600ca2.md b/projects/poincare-conjecture/.astrolabe/atoms/11159b600ca2.md new file mode 100644 index 00000000..e7b2b847 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/11159b600ca2.md @@ -0,0 +1,18 @@ +--- +chapter: 5 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '5.24' +ref: +- 11159b600ca2 +sort: definition +source: tex +src: morgan-tian +tex_file: converge2 +--- +For based metric spaces $(X_k,x_k)$ (not necessarily of finite +diameter) to *converge in the Gromov-Hausdorff sense to a based +metric space* $(Y,y)$ means +that for each $r>0$ there is a sequence $\delta_k\rightarrow 0$ such +that the sequence of balls $B(x_k,r+\delta_k)$ in $(X_k,x_k)$ +converges in the Gromov-Hausdorff sense to the ball $B(y,r)$ in $Y$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/1138b028cc69.md b/projects/poincare-conjecture/.astrolabe/atoms/1138b028cc69.md new file mode 100644 index 00000000..4f970bd9 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/1138b028cc69.md @@ -0,0 +1,16 @@ +--- +chapter: 9 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '9.82' +ref: +- 1138b028cc69 +sort: definition +source: tex +src: morgan-tian +tex_file: temp2kappa +--- +A *strong $\epsilon$-tube* in a +generalized Ricci flow is an $\epsilon$-tube with the property that +each point of the tube is the center of a strong $\epsilon$-neck in +the generalized flow. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/1160f53cc783.md b/projects/poincare-conjecture/.astrolabe/atoms/1160f53cc783.md new file mode 100644 index 00000000..c5939c6a --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/1160f53cc783.md @@ -0,0 +1,16 @@ +--- +chapter: 4 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '4.27' +ref: +- 1160f53cc783 +sort: remark +source: tex +src: morgan-tian +tex_file: maxprin +--- +This theorem tells us, among other things, that as the scalar +curvature goes to infinity then absolute values of all the +negative eigenvalues (if any) of $\mathit{Rm}$ are arbitrarily small +with respect to the scalar curvature. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/11e3d00eb60d.md b/projects/poincare-conjecture/.astrolabe/atoms/11e3d00eb60d.md new file mode 100644 index 00000000..2815bf69 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/11e3d00eb60d.md @@ -0,0 +1,18 @@ +--- +chapter: 6 +generator: tools/poincare_tex_extract.py +label: Hclaim +labels: +- Hclaim +mtref: '6.44' +ref: +- 11e3d00eb60d +sort: claim +source: tex +src: morgan-tian +tex_file: newcompar +--- +$$ +\sum_\alpha +H(X,Y_\alpha)=\frac{(\sqrt{\tau}-\sqrt{\tau_1})^2}{(\sqrt{\bar\tau}-\sqrt{\tau_1})^2}H(X). +$$ diff --git a/projects/poincare-conjecture/.astrolabe/atoms/11ec367eae93.md b/projects/poincare-conjecture/.astrolabe/atoms/11ec367eae93.md new file mode 100644 index 00000000..63271eba --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/11ec367eae93.md @@ -0,0 +1,19 @@ +--- +chapter: 18 +generator: tools/poincare_tex_extract.py +label: csforramps +labels: +- csforramps +mtref: '18.42' +ref: +- 11ec367eae93 +sort: lemma +source: tex +src: morgan-tian +tex_file: energy1 +--- +Let $(M,g(t)),\ t_0\le t\le t_1$, be a Ricci flow. Suppose that +$c\colon S^1\to ( M\times S^1_\lambda, g(t)\times ds^2)$ is a ramp. +Then there is a curve-shrinking flow $c(x,t)$ defined for all $t\in +[t_0,t_1]$ with $c$ as the initial condition at time $t=t_0$. The +curves $c(\cdot,t)$ are all ramps. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/124fe3535807.md b/projects/poincare-conjecture/.astrolabe/atoms/124fe3535807.md new file mode 100644 index 00000000..ca9b525f --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/124fe3535807.md @@ -0,0 +1,35 @@ +--- +chapter: 14 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '14.7' +ref: +- 124fe3535807 +sort: definition +source: tex +src: morgan-tian +tex_file: surgery +--- +By a *horizontal metric $G$ on a surgery space-time ${\mathcal +M}$* we mean a $C^\infty$ metric on ${\mathcal HT}{\mathcal M}$. For +each $t$, the horizontal metric $G$ induces a Riemannian metric, +denoted $G(t)$, on the time-slice $M_t$. Associated to a horizontal +metric $G$ we have the *horizontal covariant derivative*, +denoted $\nabla$. This is a pairing between horizontal vector fields + +$$ +X\otimes Y\mapsto \nabla_XY. +$$ + +On each time slice $M_t$ it is the usual Levi-Civita connection associated to +the Riemannian metric $G(t)$. Given a function $F$ on space-time, by its +gradient $\nabla F$ we mean its horizontal gradient. The value of this gradient +at a point $q\in M_t$ is the usual $G(t)$-gradient of $F|_{M_t}$. In +particular, $\nabla F$ is a smooth horizontal vector field on space-time. The +horizontal metric $G$ on space-time has its (horizontal) curvatures $\mathit{Rm}_G$. +These are smooth symmetric endomorphisms of the second exterior power of +${\mathcal HT}{\mathcal M}$. The value of $\mathit{Rm}_G$ at a point $q\in M_t$ is +simply the usual Riemann curvature operator of $G(t)$ at the point $q$. +Similarly, we have the (horizontal) Ricci curvature $\mathit{Ric}=\mathit{Ric}_G$, a +section of the symmetric square of the horizontal cotangent bundle, and the +(horizontal) scalar curvature denoted $R=R_G$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/127db1dc9315.md b/projects/poincare-conjecture/.astrolabe/atoms/127db1dc9315.md new file mode 100644 index 00000000..21db6dbd --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/127db1dc9315.md @@ -0,0 +1,27 @@ +--- +chapter: 0 +generator: tools/poincare_tex_extract.py +label: finiteext +labels: +- finiteext +mtref: '0.4' +ref: +- 127db1dc9315 +sort: theorem +source: tex +src: morgan-tian +tex_file: intro +--- +Let $M$ be a closed $3$-manifold whose fundamental group is a free +product of finite groups and infinite cyclic groups (In +[Perelman3] Perelman states the result for $3$-manifolds +without prime factors that are acyclic. It is a standard exercise in +$3$-manifold topology to show that Perelman's condition is +equivalent to the group theory hypothesis stated here; see +Corollary 0.5.). Let $g_0$ be any Riemannian metric on +$M$. Then $M$ admits no locally separating $\Ar P^2$, so that there +is a Ricci flow with surgery defined for all positive time with +$(M,g_0)$ as initial metric as described in Theorem 0.3. +This Ricci flow with surgery becomes extinct after some time +$T<\infty$, in the sense that the manifolds $M_t$ are empty for all +$t\ge T$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/1280a1901e2e.md b/projects/poincare-conjecture/.astrolabe/atoms/1280a1901e2e.md new file mode 100644 index 00000000..f9a705fa --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/1280a1901e2e.md @@ -0,0 +1,22 @@ +--- +chapter: 9 +generator: tools/poincare_tex_extract.py +label: partialtR +labels: +- partialtR +mtref: '9.74' +ref: +- 1280a1901e2e +sort: claim +source: tex +src: morgan-tian +tex_file: temp2kappa +--- +Suppose that $(M,g(t))$ is a Ricci flow and that $\left({\mathcal +C},g(t)|_{\mathcal C}\right)$ is a subset of a $t$ time-slice. Then +Condition (8) above is equivalent to + +$$ +\mathit{sup}_{(x,t)\in {\mathcal C}}\frac{\left|\frac{\partial R(x,t)}{\partial +t}\right|}{R^2(x,t)}0$. Let $(M,g)$ be a compact, connected $3$-manifold. +Then $(M,g)$ is *within $\epsilon$ of round in the +$C^{[1/\epsilon]}$-topology* if there exist a constant $R>0$, a +compact manifold $(Z,g_0)$ of constant curvature $+1$, and a +diffeomorphism $\varphi\colon Z\to M$ with the property that the +pull back under $\varphi$ of $Rg$ is within $\epsilon$ in the +$C^{[1/\epsilon]}$-topology of $g_0$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/1515151790b2.md b/projects/poincare-conjecture/.astrolabe/atoms/1515151790b2.md new file mode 100644 index 00000000..500e9c4e --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/1515151790b2.md @@ -0,0 +1,31 @@ +--- +chapter: 3 +generator: tools/poincare_tex_extract.py +label: thmRmevol +labels: +- thmRmevol +mtref: '3.13' +ref: +- 1515151790b2 +sort: theorem +source: tex +src: morgan-tian +tex_file: flowbasics +--- +The curvature tensor $R_{ijkl}$, the Ricci curvature $\mathit{Ric}_{ij}$, the scalar +curvature $R$, and the volume form $d\mathit{vol}(x,t)$ satisfy the following evolution +equations under Ricci flow: + +$$ +\begin{aligned} +\frac{\partial R_{ijkl}}{\partial t} &= & \triangle R_{ijkl} +2(B_{ijkl} - +B_{ijlk}-B_{iljk} + B_{ikjl})\nonumber \\& & - g^{pq}(R_{pjkl}\mathit{Ric}_{qi} + +R_{ipkl}\mathit{Ric}_{qj} + R_{ijpl}\mathit{Ric}_{qk} + +R_{ijkp}\mathit{Ric}_{ql}) \\ + \frac{\partial}{\partial t}\mathit{Ric}_{jk} & = & \triangle +\mathit{Ric}_{jk}+2g^{pq}g^{rs}R_{pjkr}\mathit{Ric}_{qs}-2g^{pq}\mathit{Ric}_{jp}\mathit{Ric}_{qk} \\ +\frac{\partial}{\partial t}R & = & \Delta R+2|\mathit{Ric}|^2 \\ + \frac{\partial}{\partial t}d\mathit{vol}(x,t) & = & +-R(x,t)d\mathit{vol}(x,t). +\end{aligned} +$$ diff --git a/projects/poincare-conjecture/.astrolabe/atoms/151c2c13281d.md b/projects/poincare-conjecture/.astrolabe/atoms/151c2c13281d.md new file mode 100644 index 00000000..fdc3658c --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/151c2c13281d.md @@ -0,0 +1,35 @@ +--- +chapter: 4 +generator: tools/poincare_tex_extract.py +label: pinchd +labels: +- pinchd +mtref: '4.34' +ref: +- 151c2c13281d +sort: definition +source: tex +src: morgan-tian +tex_file: maxprin +--- +Let $({\mathcal M},G)$ be a generalized Ricci flow whose domain of definition +is contained in $[0,\infty)$. Then we say that $({\mathcal M},G)$ has *curvature pinched toward positive* if +for every $x\in {\mathcal M}$ the following two conditions hold: + +1. + +$$ +R(x)\ge \frac{-6}{4\mathbf{t}(x)+1} +$$ + +1. + +$$ +R(x)\geq 2X(x)\left(\mathit{log}X(x)+\mathit{log}(1+\mathbf{t}(x))-3\right), +$$ + + whenever +$00$ be given. We +say that $p\in U$ is a *$\delta$-regular* +point if for every $r'<\delta$ the +metric ball $B(p,r')$ has compact closure in $U$. Equivalently, $p$ +is $\delta$-regular if the exponential mapping at $p$ is defined on +the open ball of radius $\delta$ centered at the origin in $T_pU$, +i.e., if each geodesic ray emanating from $p$ extends to a geodesic +defined on $[0,\delta)$. We denote by $\mathit{Reg}_\delta(U,g)$ the +subset of $\delta$-regular points in $(U,g)$. For any $x\in \mathit{Reg}_\delta(U,g)$ we denote by $\mathit{Reg}_\delta(U,g,x)$ the +connected component of $\mathit{Reg}_\delta(U,g)$ containing $x$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/1596df6ade35.md b/projects/poincare-conjecture/.astrolabe/atoms/1596df6ade35.md new file mode 100644 index 00000000..32849021 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/1596df6ade35.md @@ -0,0 +1,17 @@ +--- +chapter: 17 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '17.5' +ref: +- 1596df6ade35 +sort: remark +source: tex +src: morgan-tian +tex_file: surgery +--- +Notice that even though $x_{(a,b)}$ is the center of a strong +$\epsilon$-neck, the canonical neighborhoods of the $x_n$ constructed +in the second case are not a strong $\epsilon$-necks but rather are +$(C,\epsilon)$-caps coming from applying the flow to a +neighborhood of the surgery cap ${\mathcal C}$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/16fbe263a7c8.md b/projects/poincare-conjecture/.astrolabe/atoms/16fbe263a7c8.md new file mode 100644 index 00000000..a6280ce1 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/16fbe263a7c8.md @@ -0,0 +1,31 @@ +--- +chapter: 7 +generator: tools/poincare_tex_extract.py +label: lipcomplete +labels: +- lipcomplete +mtref: '7.5' +ref: +- 16fbe263a7c8 +sort: proposition +source: tex +src: morgan-tian +tex_file: newcomp2 +--- +Let $(M,g(t)),\ 0\le t\le T<\infty$, be a Ricci flow that is +complete and of bounded curvature. Let $p\in M$, let $x=(p,T)\in +M\times [0,T]$, and let $\tau\in (0,T)$. + +- **(1)** The functions $L_{x}$ and $l_{x}$ are locally Lipschitz functions +on $M\times (0,T)$. +- **(2)** ${\mathcal L}\mathit{exp}_{x}^\tau$ + is a diffeomorphism from $\widetilde{\mathcal U}_{x}(\tau)$ onto +an open subset ${\mathcal U}_{x}(\tau)$ of $M$. +- **(3)** The complement of ${\mathcal U}_{x}(\tau)$ in $M$ is a closed +subset of $M$ of zero Lebesgue measure. +- **(4)** For every +$\tau<\tau'0$ and $R<\infty$ there is an $N<\infty$ such that +$N(\delta,R,X_k)\le N$ for all $k$. On the other hand, if the +sequence $(X_k,x_k)$ has a Gromov-Hausdorff limit, then for every +$\delta>0$ and $R<\infty$ the $N(\delta,R,X_k)$ are bounded +independent of $k$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/1766f37a13cf.md b/projects/poincare-conjecture/.astrolabe/atoms/1766f37a13cf.md new file mode 100644 index 00000000..1c59df07 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/1766f37a13cf.md @@ -0,0 +1,31 @@ +--- +chapter: 9 +generator: tools/poincare_tex_extract.py +label: absconv1 +labels: +- absconv1 +mtref: '9.27' +ref: +- 1766f37a13cf +sort: claim +source: tex +src: morgan-tian +tex_file: temp2kappa +--- +Fix a compact interval $[-\tau,-\tau_0]\subset +(-\infty,0)$. Let $f$ be a locally Lipschitz function that is +defined on $M_\infty\times [-\tau,-\tau_0]$ and such that there is a +constant $C$ with the property that $f(q,\tau')$ by $C$ times $\mathit{max}(l_\infty(q,\tau'),1)$ . Then the distribution ${\mathcal +D}_1=fe^{-l_\infty}$ is absolutely convergent in the following +sense. For any bounded smooth function $\varphi$ defined on all of +$M_\infty\times [-\tau,-\tau_0]$ and any sequence of compactly +supported, non-negative smooth functions $\psi_k$, bounded above by +$1$ everywhere that are eventually $1$ on every compact subset, the +following limit exists and is finite: + +$$ +\mathit{lim}_{k\rightarrow\infty}{\mathcal D}_1(\varphi\psi_k). +$$ + + Furthermore, the limit is independent of the choice of the +$\psi_k$ with the given properties. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/188acafae968.md b/projects/poincare-conjecture/.astrolabe/atoms/188acafae968.md new file mode 100644 index 00000000..4bb71dfc --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/188acafae968.md @@ -0,0 +1,19 @@ +--- +chapter: 18 +generator: tools/poincare_tex_extract.py +label: curvflowcompare +labels: +- curvflowcompare +mtref: '18.92' +ref: +- 188acafae968 +sort: claim +source: tex +src: morgan-tian +tex_file: energy1 +--- +$$ +Y=\frac{\nabla_ZZ-\langle\Gamma(Z,Z),\partial_z\rangle_{h'} Z}{|Z|^2} +=\nabla_SS+\left(\frac{\langle \nabla_ZZ,Z\rangle}{|Z|^4} +-\frac{\langle\Gamma(Z,Z),\partial_z\rangle_{h'} }{|Z|^2}\right)Z. +$$ diff --git a/projects/poincare-conjecture/.astrolabe/atoms/18ee66757f32.md b/projects/poincare-conjecture/.astrolabe/atoms/18ee66757f32.md new file mode 100644 index 00000000..f858d45f --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/18ee66757f32.md @@ -0,0 +1,16 @@ +--- +chapter: 9 +generator: tools/poincare_tex_extract.py +label: compact2D +labels: +- compact2D +mtref: '9.40' +ref: +- 18ee66757f32 +sort: corollary +source: tex +src: morgan-tian +tex_file: temp2kappa +--- +There is no two-dimensional flow satisfying the hypotheses of +Proposition 9.39. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/198e7475ad2f.md b/projects/poincare-conjecture/.astrolabe/atoms/198e7475ad2f.md new file mode 100644 index 00000000..909a9833 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/198e7475ad2f.md @@ -0,0 +1,20 @@ +--- +chapter: 8 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '8.10' +ref: +- 198e7475ad2f +sort: theorem +source: tex +src: morgan-tian +tex_file: noncoll +--- +Fix positive constants $\omega>0$ and $T_0<\infty$. Then there is +$\kappa>0$ depending only on these constants such that the following +holds. Let $(M,g(t)),\ 0\le t0$ with $r^2\le t_0$ and any $(p,t_0)\in M\times +\{t_0\}$, if $|\mathit{Rm}(q,t)|\le r^{-2}$ on $B(p,t_0,r)\times +[t_0-r^2,t_0]$ then $\mathit{Vol} B(p,t_0,r)\ge \kappa r^3$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/1998da8dab3d.md b/projects/poincare-conjecture/.astrolabe/atoms/1998da8dab3d.md new file mode 100644 index 00000000..220132f0 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/1998da8dab3d.md @@ -0,0 +1,22 @@ +--- +chapter: 9 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '9.55' +ref: +- 1998da8dab3d +sort: definition +source: tex +src: morgan-tian +tex_file: temp2kappa +--- +Let $(M,g)$ be a complete, connected, non-compact Riemannian manifold of +non-negative curvature. Fix a point $p\in M$. We define the *asymptotic +scalar curvature* + +$$ +{\mathcal R}(M,g)=\mathit{limsup}_{x\rightarrow\infty}R(x)d^2(x,p). +$$ + + Clearly, this limit is +independent of $p$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/19afab2e3115.md b/projects/poincare-conjecture/.astrolabe/atoms/19afab2e3115.md new file mode 100644 index 00000000..e5d466e7 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/19afab2e3115.md @@ -0,0 +1,19 @@ +--- +chapter: 1 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '1.24' +ref: +- 19afab2e3115 +sort: definition +source: tex +src: morgan-tian +tex_file: prelim +--- +There is an open neighborhood $U_p\subset T_pM$ of $0$ consisting +of all $v\in T_pM$ for which: (i) $\gamma_v$ is the unique minimal +geodesic from $p$ to $\gamma_v(1)$, and (ii) $ \mathit{exp}_p$ is a +local diffeomorphism at $v$. We set ${\mathcal C}_p\subset M$ equal +to $M\setminus \mathit{exp}_p(U_p)$. Then ${\mathcal C}_p$ is called +the *cut locus from $p$*. It is a closed subset +of measure $0$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/19cc95ff1cb9.md b/projects/poincare-conjecture/.astrolabe/atoms/19cc95ff1cb9.md new file mode 100644 index 00000000..b5ecff8e --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/19cc95ff1cb9.md @@ -0,0 +1,20 @@ +--- +chapter: 12 +generator: tools/poincare_tex_extract.py +label: cl12.20 +labels: +- cl12.20 +mtref: '12.17' +ref: +- 19cc95ff1cb9 +sort: claim +source: tex +src: morgan-tian +tex_file: stdsoln +--- +For any fixed $t$ the function $r\colon \Ar^n \to[0,\infty)$ is a +function only of $\hat r$. Considered as a function of two +variables, + $r(\hat r,t)$ is a smooth function defined for $\hat r\ge 0$. It is + an odd function of $\hat r$. For fixed $t$ it is an increasing + function of $\hat r$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/19d01820dad0.md b/projects/poincare-conjecture/.astrolabe/atoms/19d01820dad0.md new file mode 100644 index 00000000..e1d3c113 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/19d01820dad0.md @@ -0,0 +1,22 @@ +--- +chapter: 18 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '18.23' +ref: +- 19d01820dad0 +sort: definition +source: tex +src: morgan-tian +tex_file: energy1 +--- +Let $(M,g(t)),\ t_0\le t\le t_1$, be a Ricci flow on a compact +$3$-manifold. For any $a$ and any $t'\in [t_0,t_1]$ let +$w_{a,t'}(t)$ be the solution to the differential equation + +$$ +\frac{dw_{a,t'}}{dt}=-2\pi-\frac{1}{2}R_\mathit{min}(t)w_{a,t'}(t) +$$ + +with initial condition $w_{a,t'}(t')=a$. We also denote $w_{a,t_0}$ +by $w_a$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/1a1b2c4fdc9b.md b/projects/poincare-conjecture/.astrolabe/atoms/1a1b2c4fdc9b.md new file mode 100644 index 00000000..d1b766b1 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/1a1b2c4fdc9b.md @@ -0,0 +1,28 @@ +--- +chapter: 1 +generator: tools/poincare_tex_extract.py +label: BishopGromov +labels: +- BishopGromov +mtref: '1.34' +ref: +- 1a1b2c4fdc9b +sort: theorem +source: tex +src: morgan-tian +tex_file: prelim +--- +(Relative Volume Comparison, Bishop-Gromov 1964-1980) Suppose $(M, +g)$ is a Riemannian manifold. Fix a point $p\in M$, and suppose that +$B(p,R)$ has compact closure in $M$. Suppose that for some $k\ge 0$ +we have $\mathit{Ric} \geq -(n-1)k$ on $B(p,R)$. Recall that $H^n_k$ is +the simply connected, complete manifold of constant curvature $-k$ +and $q_k\in H^n_k$ is a point. Then + +$$ +\frac{\mathit{Vol} B(p,r)}{\mathit{Vol} B_{H^n_k}B(q_k,r)} +$$ + +is a non-increasing function of $r$ for $r0$ is a positive constant. Then we can define a new generalized +Ricci flow by setting $G'=QG$, $\mathbf{t'}=Q\mathbf{t}$ and +$\chi'=Q^{-1}\chi$. It is easy to see that the result still +satisfies the generalized Ricci flow equation. We denote this new +generalized Ricci flow by $(Q{\mathcal M},QG)$ where the changes in +$\mathbf{t}$ and $\chi$ are denoted by the factor of $Q$ in front of +${\mathcal M}$. + +It is also possible to translate a generalized solution $({\mathcal M},G)$ by +replacing the time function $\mathbf{t}$ by $\mathbf{t'}=\mathbf{t}+a$ for any constant +$a$, leaving $G$ and $\chi$ unchanged. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/1ac0cdbfd850.md b/projects/poincare-conjecture/.astrolabe/atoms/1ac0cdbfd850.md new file mode 100644 index 00000000..8dab57a4 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/1ac0cdbfd850.md @@ -0,0 +1,22 @@ +--- +chapter: 6 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '6.35' +ref: +- 1ac0cdbfd850 +sort: definition +source: tex +src: morgan-tian +tex_file: newcompar +--- +Let $q\in {\mathcal U}_x(\bar\tau)$ and let $\gamma\colon +[\tau_1,\bar\tau]\to {\mathcal M}$ be the unique minimizing +${\mathcal L}$-geodesic from $x$ to $q$. We say that a horizontal +vector field $Y(\tau)$ along $\gamma$ is *adapted* if it solves +the following ODE on $[\tau_1,\bar \tau]$: + +$$ +\nabla_{X}Y(\tau) = -\mathit{Ric}(Y(\tau),\cdot)^* + +\frac{1}{2\sqrt{\tau}(\sqrt{\tau}-\sqrt{\tau_1})}Y(\tau). +$$ diff --git a/projects/poincare-conjecture/.astrolabe/atoms/1b517dfd4bd7.md b/projects/poincare-conjecture/.astrolabe/atoms/1b517dfd4bd7.md new file mode 100644 index 00000000..dc4bb612 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/1b517dfd4bd7.md @@ -0,0 +1,26 @@ +--- +chapter: 1 +generator: tools/poincare_tex_extract.py +label: conecurv +labels: +- conecurv +mtref: '1.15' +ref: +- 1b517dfd4bd7 +sort: proposition +source: tex +src: morgan-tian +tex_file: prelim +--- +Let $N$ be a Riemannian manifold of dimension $n-1$. Fix $(x,s)\in +c(N)=N\times (0,\infty)$. With respect to the coordinates +$(x^0,\ldots,x^n)$ the curvature operator ${\mathit{Rm}}_{\tilde +g}(p,s)$ of the cone decomposes as + +$$ +\begin{pmatrix} 0 & 0 \\ s^2(\mathit{Rm}_g(p)-\wedge^2g(p)) & 0 +\end{pmatrix}, +$$ + +where $\wedge^2g(p)$ is the symmetric form on $\wedge^2T_pN$ induced +by $g$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/1b6c3ac94c37.md b/projects/poincare-conjecture/.astrolabe/atoms/1b6c3ac94c37.md new file mode 100644 index 00000000..bad9dcd2 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/1b6c3ac94c37.md @@ -0,0 +1,19 @@ +--- +chapter: 19 +generator: tools/poincare_tex_extract.py +label: sinside +labels: +- sinside +mtref: '19.24' +ref: +- 1b6c3ac94c37 +sort: claim +source: tex +src: morgan-tian +tex_file: canonnbhd +--- +Suppose that there is a two-sphere $\Sigma\subset N'$ contained in +the closure of the positive half of $N'$ and also contained in +$\widetilde C$. Suppose that $\Sigma$ is isotopic in $N'$ to the +central two-sphere $S'$ of $N'$. Then $\widetilde C\cup C'$ is a +component of $M$, a component containing $X$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/1b78c6bffb78.md b/projects/poincare-conjecture/.astrolabe/atoms/1b78c6bffb78.md new file mode 100644 index 00000000..fe6a82cf --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/1b78c6bffb78.md @@ -0,0 +1,15 @@ +--- +chapter: 9 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '9.4' +ref: +- 1b78c6bffb78 +sort: theorem +source: tex +src: morgan-tian +tex_file: temp2kappa +--- +Every orientable, $2$-dimensional $\kappa$-solution is a rescaling +of the previous example, i.e., is a family of shrinking round +$2$-spheres. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/1b8d34cae3a8.md b/projects/poincare-conjecture/.astrolabe/atoms/1b8d34cae3a8.md new file mode 100644 index 00000000..6f46a675 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/1b8d34cae3a8.md @@ -0,0 +1,19 @@ +--- +chapter: 18 +generator: tools/poincare_tex_extract.py +label: C0claim +labels: +- C0claim +mtref: '18.87' +ref: +- 1b8d34cae3a8 +sort: claim +source: tex +src: morgan-tian +tex_file: energy1 +--- +There is a constant $D_0<\infty$, depending only on $\Theta$ and the +curvature bound of the ambient Ricci flow such that for every $t\in +[t',t'+\delta r^2]$ and every sub-arc $\gamma_{t'}$ whose length is +at most $r$, we have $\int_{\gamma_t} kds0$ there is a neighborhood $U$ of $(q,t)$ in + $M\times [0,T]$ + and an upper barrier $\varphi$ for $l_x$ at this point defined on $U$ +satisfying + +$$ +\frac{\partial \varphi}{\partial \tau}(q,\tau)+\triangle \varphi(q,\tau) +\le \frac{(n/2)-l_x(q,\tau)}{\tau}+\epsilon. +$$ diff --git a/projects/poincare-conjecture/.astrolabe/atoms/1f2e2e449b34.md b/projects/poincare-conjecture/.astrolabe/atoms/1f2e2e449b34.md new file mode 100644 index 00000000..0bf7f334 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/1f2e2e449b34.md @@ -0,0 +1,13 @@ +--- +chapter: 15 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '15.6' +ref: +- 1f2e2e449b34 +sort: definition +source: tex +src: morgan-tian +tex_file: surgery +--- +We set $t_0=2^{-5}$, and for any $i\ge 0$ we define $T_i=2^it_0$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/1f7d2e438fef.md b/projects/poincare-conjecture/.astrolabe/atoms/1f7d2e438fef.md new file mode 100644 index 00000000..64d8ef4b --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/1f7d2e438fef.md @@ -0,0 +1,26 @@ +--- +chapter: 18 +generator: tools/poincare_tex_extract.py +label: claim1885 +labels: +- claim1885 +mtref: '18.89' +ref: +- 1f7d2e438fef +sort: claim +source: tex +src: morgan-tian +tex_file: energy1 +--- +There is a constant $D_2$ depending only on the curvature bound of +the ambient Ricci flow and $\Theta$ and a constant $D_3$ depending +only on the curvature bound of the ambient Ricci flow, such that for +any $t\in [t',t_2]$ and any sub-arc $\gamma_{t'}$ of length $r$, we +have + +$$ +\left|\frac{d}{dt}\int_{\gamma_t}\varphi +kds\right|\le +D_2\left(1+\frac{1}{r\sqrt{t-t'}}\right)+\frac{D_2}{r^2}+D_3\int_{\gamma_t}\varphi +kds. +$$ diff --git a/projects/poincare-conjecture/.astrolabe/atoms/1ff58113d89b.md b/projects/poincare-conjecture/.astrolabe/atoms/1ff58113d89b.md new file mode 100644 index 00000000..5a75123e --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/1ff58113d89b.md @@ -0,0 +1,51 @@ +--- +chapter: 16 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '16.20' +ref: +- 1ff58113d89b +sort: definition +source: tex +src: morgan-tian +tex_file: surgery +--- +Fix $\ell<\infty$. Let $\theta_0=\theta_0(\ell)$ be as in +Proposition 16.13. For each surgery time $\bar t\in +[T_{i-1},T]$, let $h(\bar t)$ be the scale of the surgery. + Let $p_1,\ldots,p_k$ be +the tips of the surgery caps at time $\bar t$. For each $1\le j\le k$, +we consider $B_j(\bar +t)=B(p_j,\bar t,(A_0+10)h(\bar t))$, and we let $\Delta t_j\le \mathit{min}(\theta_0,(T-\bar t)/h^2(\bar t))$ be maximal subject to the condition that +there is an embedding $\rho_j\colon B_j(\bar t)\times [\bar t,\bar t+h^2(\bar +t)\Delta t_j)$ into ${\mathcal M}$ compatible with time and the vector field. +Clearly, $B'_j=B(p_j,\bar t,(10+A_0)h)\cap C_{\bar t}$ is contained in $J(\bar +t)$. Let $\bar t'$ be the previous surgery time if there is one, otherwise set +$\bar t'=0$. Also for each $\bar t$ we set $\tau_1(\ell,\bar t)=\mathit{min}\left(h(\bar t)^2/\ell,\bar t-\bar t'\right)$. For each $t\in (\bar +t-\tau_1(\ell,\bar t),\bar t)$ let $\widetilde J(t)\subset J(t)$ be the union +of $D_t$, the disappearing region at time $t$, and $\coprod_iB_i'(t)$, the +result of flowing $\coprod_iB'_i$ backward to time $t$. Then we set +$\widetilde J(\bar t-\tau_1(\ell,\bar t),\bar t)\subset J(\bar +t-\tau_1(\ell,\bar t),\bar t)$ equal to the union over $t\in (\bar +t-\tau_1(\ell,\bar t),\bar t)$ of $\widetilde J(t)$. + + By construction, +for each surgery time $\bar t$, the union + +$$ +\nu_\mathit{sing}(\ell,\bar +t)=\widetilde J(\bar t-\tau_1(\ell,\bar t),\bar t)\cup\bigcup +_iB_i\times [\bar t,\bar t+h^2(\bar t)\Delta t_i) +$$ + + is an open +subset of ${\mathcal M}$ containing all the exposed regions and +singular points at time $\bar t$. + +We define $Y(\ell )\subset \mathbf{t}^{-1}([T_{i-1},T])$ to be the +complement of the $\cup_{\bar t}\nu_\mathit{sing}(\ell,\bar t)$ where +the union is over all surgery times $\bar t\in [T_{i-1},T]$. +Clearly, $Y(\ell )$ is a closed subset of $\mathbf{t}^{-1}([T_{i-1},T])$ and hence $Y(\ell )$ is a compact subset +contained in the open subset of smooth points of ${\mathcal M}$. +(Notice that $Y(\ell )$ depends on $\ell $ because $\tau_1(\ell,\bar +t)$ and $\theta_0$ depend on $\ell$.) diff --git a/projects/poincare-conjecture/.astrolabe/atoms/200f280a8e3b.md b/projects/poincare-conjecture/.astrolabe/atoms/200f280a8e3b.md new file mode 100644 index 00000000..819ff1a0 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/200f280a8e3b.md @@ -0,0 +1,18 @@ +--- +chapter: 6 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '6.11' +ref: +- 200f280a8e3b +sort: definition +source: tex +src: morgan-tian +tex_file: newcompar +--- +A field $Y(\tau)$ along an ${\mathcal L}$-geodesic is +called an *${\mathcal L}$-Jacobi field* if it satisfies the ${\mathcal L}$-Jacobi +equation, Equation (6.6), and if it vanishes at $\tau_1$. +For any horizontal vector field $Y$ along $\gamma$ we denote by +$\mathit{Jac}(Y)$ the expression on the left-hand +side of Equation (6.6). diff --git a/projects/poincare-conjecture/.astrolabe/atoms/20ae3cd37b93.md b/projects/poincare-conjecture/.astrolabe/atoms/20ae3cd37b93.md new file mode 100644 index 00000000..dd0d43a2 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/20ae3cd37b93.md @@ -0,0 +1,18 @@ +--- +chapter: 12 +generator: tools/poincare_tex_extract.py +label: stdsolncap +labels: +- stdsolncap +mtref: '12.34' +ref: +- 20ae3cd37b93 +sort: corollary +source: tex +src: morgan-tian +tex_file: stdsoln +--- +Fix $\epsilon>0$. Suppose that $(q,t)$ is a point in the standard +solution with $t\le R(q,t)^{-1}(1+\epsilon))$ and with $(q,0)\in +B(p_0,0,(\epsilon^{-1}/2)+A_0+5)$. Then $(q,t)$ is contained in an +$(C'(\epsilon),\epsilon)$-cap. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/2104e6f56e28.md b/projects/poincare-conjecture/.astrolabe/atoms/2104e6f56e28.md new file mode 100644 index 00000000..35221d36 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/2104e6f56e28.md @@ -0,0 +1,16 @@ +--- +chapter: 9 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '9.41' +ref: +- 2104e6f56e28 +sort: definition +source: tex +src: morgan-tian +tex_file: temp2kappa +--- +Let $(M,g)$ be an $n$-dimensional complete Riemannian manifold and +fix $\kappa>0$. We say that $(M,g)$ is *$\kappa$-non-collapsed* +if for every $p\in M$ and any $r>0$, if $|\mathit{Rm}_g|\le r^{-2}$ on +$B(p,r)$ then $\mathit{Vol} B(p,r)\ge \kappa r^n$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/21f9e3c12b4c.md b/projects/poincare-conjecture/.astrolabe/atoms/21f9e3c12b4c.md new file mode 100644 index 00000000..9840b8c7 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/21f9e3c12b4c.md @@ -0,0 +1,14 @@ +--- +chapter: 14 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '14.9' +ref: +- 21f9e3c12b4c +sort: remark +source: tex +src: morgan-tian +tex_file: surgery +--- +Notice that at an exposed point and at points at the initial and the final time + the Lie derivative is a one-sided derivative. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/22c40a5664ac.md b/projects/poincare-conjecture/.astrolabe/atoms/22c40a5664ac.md new file mode 100644 index 00000000..97043622 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/22c40a5664ac.md @@ -0,0 +1,31 @@ +--- +chapter: 15 +generator: tools/poincare_tex_extract.py +label: surgeryparams +labels: +- surgeryparams +mtref: '15.7' +ref: +- 22c40a5664ac +sort: definition +source: tex +src: morgan-tian +tex_file: surgery +--- +*Surgery parameter sequences* are sequences + +- **(i)** $\mathbf{r}=r_0\ge r_1\ge r_2\ge \cdots >0$, with $r_0=\epsilon$, +- **(ii)** $\mathbf{K}=\kappa_0\ge \kappa_1\ge \kappa_2\ge \cdots >0$ +with $\kappa_0$ as in Claim 15.1, and +- **(iii)** $ +\Delta=\delta_0\ge \delta_1\ge \delta_2\ge \cdots +>0$ with $\delta_0=\mathit{min}(\beta\epsilon/3,\delta_0',K^{-1},D^{-1})$ + where $\delta_0'$ is the constant +from Theorem 13.2 and $\beta<1/2$ is the constant from +Proposition 15.2, $\epsilon$ is the constant that we have +already fixed, and $K$ and $D$ are the constants from +Lemma 12.3. + +We shall also refer to partial sequences defined for indices $0,\ldots, i$ for some $i>0$ +as surgery parameter sequences if they are positive, non-increasing and if their initial +terms satisfy the conditions given above. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/2336b554a054.md b/projects/poincare-conjecture/.astrolabe/atoms/2336b554a054.md new file mode 100644 index 00000000..2cad854f --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/2336b554a054.md @@ -0,0 +1,21 @@ +--- +chapter: 7 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '7.1' +ref: +- 2336b554a054 +sort: definition +source: tex +src: morgan-tian +tex_file: newcomp2 +--- +Let $(M,g(t)),\ a\le t\le b$, be a Ricci flow. We say that the flow +is *complete of bounded curvature* if for each $t\in [a,b]$ the +Riemannian manifold $(M,g(t))$ is complete and if there is +$C<\infty$ such that $|\mathit{Rm}|(p,t)\le C$ for all $p\in M$ and all +$t\in [a,b]$. Let $I$ be an interval and let $(M,g(t)),\ t\in I$, be +a Ricci flow. Then we say that the flow is *complete with +curvature locally bounded in time* if for each compact subinterval +$J\subset I$ the restriction of the flow to $(M,g(t)),\ t\in J$, is +complete of bounded curvature. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/243579230a8c.md b/projects/poincare-conjecture/.astrolabe/atoms/243579230a8c.md new file mode 100644 index 00000000..0b75b9ae --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/243579230a8c.md @@ -0,0 +1,13 @@ +--- +chapter: 5 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '5.18' +ref: +- 243579230a8c +sort: lemma +source: tex +src: morgan-tian +tex_file: converge2 +--- +The Gromov-Hausdorff distance satisfies the triangle inequality. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/246771506ee5.md b/projects/poincare-conjecture/.astrolabe/atoms/246771506ee5.md new file mode 100644 index 00000000..fc8f491a --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/246771506ee5.md @@ -0,0 +1,21 @@ +--- +chapter: 12 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '12.33' +ref: +- 246771506ee5 +sort: remark +source: tex +src: morgan-tian +tex_file: stdsoln +--- +At first glance it may seem impossible for a point $(x,t)$ in the +standard solution to be the center of an evolving $\epsilon$-neck +defined for rescaled time $1+\epsilon$ since the standard solution +itself is only defined for time $1$. But this is indeed possible. +The reason is because the scale referred to for an evolving neck +centered at $(x,t)$ is $R(x,t)^{-1/2}$. As $t$ approaches one, +$R(x,t)$ goes to infinity, so that rescaled time $1$ at $(x,t)$ is +an arbitrarily small time interval measured in the scale of the +standard solution. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/254e8dae97e6.md b/projects/poincare-conjecture/.astrolabe/atoms/254e8dae97e6.md new file mode 100644 index 00000000..d5393a17 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/254e8dae97e6.md @@ -0,0 +1,31 @@ +--- +chapter: 17 +generator: tools/poincare_tex_extract.py +label: extend +labels: +- extend +mtref: '17.1' +ref: +- 254e8dae97e6 +sort: proposition +source: tex +src: morgan-tian +tex_file: surgery +--- +Suppose that for some $i\ge 0$ we have surgery parameter sequences +$\delta_0\ge \delta_1\ge\cdots\ge \delta_i>0$, $\epsilon=r_0\ge +r_1\ge \cdots\ge r_i>0$ and $\kappa_0\ge \kappa_1\ge \cdots\ge +\kappa_i>0$. For any $r_{i+1}\le r_i$, let $\delta(r_{i+1})>0$ be +the constant in Proposition 16.1 associated to these +three sequences and to $r_{i+1}$. + Then there are positive +constants $r_{i+1}\le r_i$ and $\delta_{i+1}\le\delta(r_{i+1})$ such that the +following holds. Suppose that $({\mathcal M},G)$ is a Ricci flow with surgery +defined for $0\le t0$ depending on $\bar\tau_0$ such that, if +$\varepsilon\le \varepsilon_0$, we have + +$$ +\int_{\widetilde +W_\mathit{sm}}\tau_1^{-n/2}e^{-\tilde l(Z,\tau_1)}{\mathcal +J}(Z,\tau_1)dZ\le 2\varepsilon^{\frac{n}{2}}. +$$ diff --git a/projects/poincare-conjecture/.astrolabe/atoms/274cda9be677.md b/projects/poincare-conjecture/.astrolabe/atoms/274cda9be677.md new file mode 100644 index 00000000..96c19e33 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/274cda9be677.md @@ -0,0 +1,21 @@ +--- +chapter: 9 +generator: tools/poincare_tex_extract.py +label: noncompkappa +labels: +- noncompkappa +mtref: '9.88' +ref: +- 274cda9be677 +sort: corollary +source: tex +src: morgan-tian +tex_file: temp2kappa +--- +There is $\bar\epsilon_2>0$ such that for any $0<\epsilon\le \bar +\epsilon_2$ the following holds. + There +is $C_0=C_0(\epsilon)$ such that for any $\kappa>0$ and any non-compact +$3$-dimensional $\kappa$-solution not containing an embedded $\Ar P^2$ with +trivial normal bundle, the zero time-slice is either a strong $\epsilon$-tube +or a $C_0$-capped strong $\epsilon$-tube. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/27af18c808d2.md b/projects/poincare-conjecture/.astrolabe/atoms/27af18c808d2.md new file mode 100644 index 00000000..66a582e0 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/27af18c808d2.md @@ -0,0 +1,16 @@ +--- +chapter: 14 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '14.6' +ref: +- 27af18c808d2 +sort: definition +source: tex +src: morgan-tian +tex_file: surgery +--- +We call the codimension-one subbundle of the tangent bundle of +${\mathcal M}$ described in the previous lemma the *horizontal +subbundle*, and we denote it +${\mathcal HT}({\mathcal M})$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/27b452d6cb48.md b/projects/poincare-conjecture/.astrolabe/atoms/27b452d6cb48.md new file mode 100644 index 00000000..b1731bf5 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/27b452d6cb48.md @@ -0,0 +1,24 @@ +--- +chapter: 18 +generator: tools/poincare_tex_extract.py +label: nonembeasycase +labels: +- nonembeasycase +mtref: '18.32' +ref: +- 27b452d6cb48 +sort: lemma +source: tex +src: morgan-tian +tex_file: energy1 +--- +Suppose that the dimension of $M$ is at least $3$ and that +$c(\cdot,t)$ is a $C^2$-family of homotopically trivial, immersed +curves satisfying the curve-shrinking equation defined for $t^-\le +t\le t^+$. For each $t$, let $A(t)$ be the infimum of the areas of +spanning disks for $c(\cdot,t)$. Then $A(t)$ is a continuous +function and, with $\psi$ as above, we have + +$$ +A(t^+)\le \psi(t^+). +$$ diff --git a/projects/poincare-conjecture/.astrolabe/atoms/27bef66b1ee8.md b/projects/poincare-conjecture/.astrolabe/atoms/27bef66b1ee8.md new file mode 100644 index 00000000..bd9201d1 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/27bef66b1ee8.md @@ -0,0 +1,14 @@ +--- +chapter: 3 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '3.20' +ref: +- 27bef66b1ee8 +sort: remark +source: tex +src: morgan-tian +tex_file: flowbasics +--- +Notice that neither ${\mathcal T}^{2}$ nor ${\mathcal T}^{\sharp}$ +satisfies the Bianchi identity, but their sum does. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/27c072f30ccf.md b/projects/poincare-conjecture/.astrolabe/atoms/27c072f30ccf.md new file mode 100644 index 00000000..b410b80c --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/27c072f30ccf.md @@ -0,0 +1,18 @@ +--- +chapter: 6 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '6.55' +ref: +- 27c072f30ccf +sort: definition +source: tex +src: morgan-tian +tex_file: newcompar +--- +Let $({\mathcal M},G)$ be a generalized Ricci flow and let $x\in +M_{T-\tau_1}\subset {\mathcal M}$. The reduced length function $l_x$ is defined +on the subset of ${\mathcal M}$ consisting of all points $y\in {\mathcal M}$ +for which there is a minimizing ${\mathcal L}$-geodesic from $x$ to $y$. The +value $l_x(y)$ is the quotient of ${\mathcal L}$-length of any such minimizing +${\mathcal L}$-geodesic divided by $2\sqrt{\tau}$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/289ab251dd83.md b/projects/poincare-conjecture/.astrolabe/atoms/289ab251dd83.md new file mode 100644 index 00000000..03798bbb --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/289ab251dd83.md @@ -0,0 +1,16 @@ +--- +chapter: 15 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '15.13' +ref: +- 289ab251dd83 +sort: remark +source: tex +src: morgan-tian +tex_file: surgery +--- +If we have a non-separating surgery $2$-sphere then there will a +component $X(T)$ with surgery caps on both sides of the surgery +$2$-sphere and hence we cannot extend the map even continuously +over all of $X(t')$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/29b84db22679.md b/projects/poincare-conjecture/.astrolabe/atoms/29b84db22679.md new file mode 100644 index 00000000..bd5eac94 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/29b84db22679.md @@ -0,0 +1,17 @@ +--- +chapter: 3 +generator: tools/poincare_tex_extract.py +label: ortho +labels: +- ortho +mtref: '3.16' +ref: +- 29b84db22679 +sort: claim +source: tex +src: morgan-tian +tex_file: flowbasics +--- +Suppose that ${\mathcal F}(0)=\{F_a\}_a$ is a local $g(0)$-orthonormal frame, + and suppose that ${\mathcal F}(t)$ evolves according to Equation (3.6). Then + for all $t\in [0,T)$ the frame ${\mathcal F}(t)$ is a local $g(t)$-orthonormal frame. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/29f2350a588a.md b/projects/poincare-conjecture/.astrolabe/atoms/29f2350a588a.md new file mode 100644 index 00000000..c4a7e037 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/29f2350a588a.md @@ -0,0 +1,19 @@ +--- +chapter: 11 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '11.35' +ref: +- 29f2350a588a +sort: claim +source: tex +src: morgan-tian +tex_file: singlimit2 +--- +For any $z\in M_\infty$ for all $n$ sufficiently large, $z\in V_n$, +so that $\varphi_n(z)$ is defined. Furthermore, for all $n$ +sufficiently large, there is a backward flow line through +$\varphi_n(z)$ in the generalized Ricci flow $(Q_n{\mathcal +M}_n,Q_nG_n)$ defined on the interval +$(-T-(R_{Q_nG_n}^{-1}(\varphi_n(z),0)/2),0]$. The scalar curvature +is bounded above on this entire flow line by $R(\varphi_n(z),0)$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/2a16041cf547.md b/projects/poincare-conjecture/.astrolabe/atoms/2a16041cf547.md new file mode 100644 index 00000000..fc7b873d --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/2a16041cf547.md @@ -0,0 +1,30 @@ +--- +chapter: 11 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '11.28' +ref: +- 2a16041cf547 +sort: lemma +source: tex +src: morgan-tian +tex_file: singlimit2 +--- +Suppose that $0<\rho0$. Let $(M,g(t)),\ a0$. We say that $(M,g(t))$ is *$\kappa$-non-collapsed on scales at +most $r_0$* if the following holds for any $(p,t)\in M\times (a,b]$ +and any $00$ and $C<\infty$, such that any point +$x\in {\mathcal M}$ with $R(x)\ge r_0^{-2}$ has a strong +$(C,\epsilon)$-canonical neighborhood. In particular, for every +$x\in {\mathcal M}$ with $R(x)\ge r_0^{-2}$ the following two +inequalities hold: + +$$ +\left|\frac{\partial R(x)}{\partial t}\right|< CR^2(x), +$$ + +$$ +|\nabla R(x)|< CR^{3/2}(x). +$$ diff --git a/projects/poincare-conjecture/.astrolabe/atoms/30414a5a14a9.md b/projects/poincare-conjecture/.astrolabe/atoms/30414a5a14a9.md new file mode 100644 index 00000000..c74d4a00 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/30414a5a14a9.md @@ -0,0 +1,18 @@ +--- +chapter: 6 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '6.64' +ref: +- 30414a5a14a9 +sort: claim +source: tex +src: morgan-tian +tex_file: newcompar +--- +There is a constant $C'_1$ depending only on +$C_0$, $C'$, and $\bar\tau_0$ such that + +$$ +l(\gamma_1)\le l(\gamma|_{[\tau_1,\bar\tau-2\alpha]})+C'_1\alpha +$$ diff --git a/projects/poincare-conjecture/.astrolabe/atoms/31216ebde119.md b/projects/poincare-conjecture/.astrolabe/atoms/31216ebde119.md new file mode 100644 index 00000000..3964c63a --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/31216ebde119.md @@ -0,0 +1,15 @@ +--- +chapter: 10 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '10.3' +ref: +- 31216ebde119 +sort: claim +source: tex +src: morgan-tian +tex_file: bddcurvbdddist +--- +There is a geodesic $\hat\gamma_n$ in $X_n$ with endpoints $z_n$ and +$y_n$. This geodesic is minimizing among all paths in $X_n$ from +$z_n$ to $y_n$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/317c62593611.md b/projects/poincare-conjecture/.astrolabe/atoms/317c62593611.md new file mode 100644 index 00000000..8cbc57cf --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/317c62593611.md @@ -0,0 +1,25 @@ +--- +chapter: 10 +generator: tools/poincare_tex_extract.py +label: distcomp +labels: +- distcomp +mtref: '10.25' +ref: +- 317c62593611 +sort: lemma +source: tex +src: morgan-tian +tex_file: bddcurvbdddist +--- +Suppose that $\gamma$ and $\mu$ are unique shortest geodesic rays +from points $x$ and $y$ to the end ${\mathcal E}$. Let $[\gamma]$ +and $[\mu]$ be the points of $X({\mathcal E})$ represented by these +two geodesics rays. Let $a$, resp. $b$, be the distance from $x$, +resp. $y$, to ${\mathcal E}$. Denote by $x'$, resp. $y'$, the image +in $C_{\mathcal E}$ of the point $([\gamma],a)$, resp. $([\mu],b)$, +of $X({\mathcal E})\times [0,\infty)$. Then + +$$ +d_{g_\infty}(x,y)\le d_{g_{\mathcal E}}(x',y'). +$$ diff --git a/projects/poincare-conjecture/.astrolabe/atoms/3227c39e1892.md b/projects/poincare-conjecture/.astrolabe/atoms/3227c39e1892.md new file mode 100644 index 00000000..c71f636a --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/3227c39e1892.md @@ -0,0 +1,20 @@ +--- +chapter: 9 +generator: tools/poincare_tex_extract.py +label: bdvolbdcurv +labels: +- bdvolbdcurv +mtref: '9.62' +ref: +- 3227c39e1892 +sort: corollary +source: tex +src: morgan-tian +tex_file: temp2kappa +--- +For any $\nu>0$ there is a $C=C(\nu)<\infty$ depending only on the +dimension $n$ such that the following holds. Suppose that +$(M,g(t)),\ -\infty0$ are such that $\mathit{Vol} B(p,0,r)\ge \nu r^n$. Then $r^2R(q,0)\le C$ for all $q\in +B(p,0,r)$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/327067f8535b.md b/projects/poincare-conjecture/.astrolabe/atoms/327067f8535b.md new file mode 100644 index 00000000..6197e5b3 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/327067f8535b.md @@ -0,0 +1,17 @@ +--- +chapter: 19 +generator: tools/poincare_tex_extract.py +label: cccomponent +labels: +- cccomponent +mtref: '19.23' +ref: +- 327067f8535b +sort: claim +source: tex +src: morgan-tian +tex_file: canonnbhd +--- +If $C'$ is a $(C,\epsilon)$-cap whose core contains a point of the +frontier of $\widetilde C$, then $\widetilde C\cup C'$ is a +component of $M$ containing $X$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/329d0175aed8.md b/projects/poincare-conjecture/.astrolabe/atoms/329d0175aed8.md new file mode 100644 index 00000000..9f65465d --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/329d0175aed8.md @@ -0,0 +1,24 @@ +--- +chapter: 9 +generator: tools/poincare_tex_extract.py +label: global +labels: +- global +mtref: '9.63' +ref: +- 329d0175aed8 +sort: corollary +source: tex +src: morgan-tian +tex_file: temp2kappa +--- +Fix the dimension $n$. Given $\nu>0$, there is a function +$K(A)<\infty$, defined for $A\in (0,\infty)$, such that if +$(M,g(t)),\ -\infty0$ sufficiently small, +an $\epsilon$-round component satisfies the first condition in the above definition + for some $\kappa>0$ depending only on +the order of the fundamental group of the underlying manifold, but there is no universal +$\kappa>0$ that works for all $\epsilon$-round manifolds. Fixing an integer $N$ let ${\mathcal C}_N$ +be the class of closed $3$-manifolds + with the property that any finite free factor of $\pi_1(M)$ +has order at most $N$. Then any $\epsilon$-round component of any +time-slice of any Ricci flow $({\mathcal M},G)$ whose initial +conditions consist of a manifold in ${\mathcal C}_N$ will have +fundamental group of order at most $N$ and hence will satisfy the first condition +in the above definition for some $\kappa>0$ depending only on $N$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/33f705095319.md b/projects/poincare-conjecture/.astrolabe/atoms/33f705095319.md new file mode 100644 index 00000000..70c26484 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/33f705095319.md @@ -0,0 +1,25 @@ +--- +chapter: 13 +generator: tools/poincare_tex_extract.py +label: delta0 +labels: +- delta0 +mtref: '13.5' +ref: +- 33f705095319 +sort: corollary +source: tex +src: morgan-tian +tex_file: surgery +--- +There is $\delta'_2>0$, depending on $C_0$ and $q$, +such that if $\delta\le \delta'_2$ then we have + +$$ +\left(\hat +R_{ijkl}\right)=e^{-2f}\left[\left(R_{ijkl}\right)+\begin{pmatrix} +-\frac{q^2}{s^4}f^2 & 0 \\ 0 & +\left(\frac{q^2}{s^4}-\frac{2q}{s^3}\right)f\begin{pmatrix} 1 & 0 +\\ 0 & 1\end{pmatrix} +\end{pmatrix}+\left(\frac{q}{s^2}fO(\delta)\right)\right]. +$$ diff --git a/projects/poincare-conjecture/.astrolabe/atoms/347896f0385b.md b/projects/poincare-conjecture/.astrolabe/atoms/347896f0385b.md new file mode 100644 index 00000000..ff18fe58 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/347896f0385b.md @@ -0,0 +1,33 @@ +--- +chapter: 13 +generator: tools/poincare_tex_extract.py +label: '13.10' +labels: +- '13.10' +mtref: '13.10' +ref: +- 347896f0385b +sort: corollary +source: tex +src: morgan-tian +tex_file: surgery +--- +Assuming that $\delta\le \mathit{min}(\delta'_4,C_0^{-1})$, there is an +$h$-orthonormal basis $\{f_0,f_1,f_2\}$ so that in the associated basis for +$\wedge^2T_y(S^2\times I)$ the matrix $(R_{ijkl}(y))$ is diagonal and given by + +$$ +\begin{pmatrix} \lambda & 0 & 0 \\ 0 & \mu & 0 \\ 0 & 0 & +\nu\end{pmatrix} +$$ + + with $|\lambda-1/2|\le O(\delta)$ and +$|\mu|,|\nu|\le O(\delta)$. Furthermore, in this same basis the +matrix $(\hat R_{ijkl}(y))$ is + +$$ +e^{-2f}\left[\begin{pmatrix}\lambda & 0 & 0 \\ 0 & \mu & 0 \\ 0 & 0 & +\nu\end{pmatrix}+\begin{pmatrix} -\frac{q^2}{s^4}f^2 & 0 \\ 0 & +\left(\frac{q^2}{s^4}-\frac{2q}{s^3}\right)f\begin{pmatrix} 1 & 0 \\ +0 & 1\end{pmatrix}\end{pmatrix}+\frac{q^2}{s^4}fO(\delta)\right]. +$$ diff --git a/projects/poincare-conjecture/.astrolabe/atoms/35010eaa585e.md b/projects/poincare-conjecture/.astrolabe/atoms/35010eaa585e.md new file mode 100644 index 00000000..efa74696 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/35010eaa585e.md @@ -0,0 +1,21 @@ +--- +chapter: 8 +generator: tools/poincare_tex_extract.py +label: volvomp +labels: +- volvomp +mtref: '8.8' +ref: +- 35010eaa585e +sort: claim +source: tex +src: morgan-tian +tex_file: noncoll +--- +There is a universal constant $\varepsilon_0>0$ such that if +$\varepsilon\le \varepsilon_0$, for any open subset $U$ of +$B(x,T,r)$, and for any $0\le \tau_1\le \tau_0$, we have + +$$ +0.9\le \mathit{Vol}_{g(T)}U/\mathit{Vol}_{g(T-\tau_1)}U\le 1.1. +$$ diff --git a/projects/poincare-conjecture/.astrolabe/atoms/35089ff7c79e.md b/projects/poincare-conjecture/.astrolabe/atoms/35089ff7c79e.md new file mode 100644 index 00000000..f9caf090 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/35089ff7c79e.md @@ -0,0 +1,16 @@ +--- +chapter: 1 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '1.13' +ref: +- 35089ff7c79e +sort: example +source: tex +src: morgan-tian +tex_file: prelim +--- +Let $M$ be an $n$-dimensional manifold with $n$ being either $2$ or $3$. +If $(M,g)$ is Einstein with Einstein constant $\lambda$, one can +easily show that $M$ has constant sectional curvature +$\frac{\lambda}{n-1}$, so that in fact $M$ is a space-form. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/351728d82639.md b/projects/poincare-conjecture/.astrolabe/atoms/351728d82639.md new file mode 100644 index 00000000..1c2e66be --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/351728d82639.md @@ -0,0 +1,24 @@ +--- +chapter: 9 +generator: tools/poincare_tex_extract.py +label: 4times +labels: +- 4times +mtref: '9.37' +ref: +- 351728d82639 +sort: lemma +source: tex +src: morgan-tian +tex_file: temp2kappa +--- +Let $(M,g)$ be a Riemannian manifold and let $p\in M$ and $r>0$ be +given. Suppose that $B(p,2r)$ has compact closure in $M$ and suppose +that $f\colon B(p,2r)\times (-2r,0]\to \Ar$ is a continuous, bounded +function with $f(p,0)>0$. Then there is a point $(q,t)\in +B(p,2r)\times (-2r,0]$ with the following properties: + +- **(1)** $f(q,t)\ge f(p,0)$. +- **(2)** Setting $\alpha=f(p,0)/f(q,t)$ we have $d(p,q)\le 2r(1-\alpha)$ +and $t\ge -2r(1-\alpha)$. +- **(3)** $f(q',t')<2f(q,t)$ for all $(q',t')\in B(q,\alpha r)\times (t-\alpha r,t]$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/351e4ac14e4e.md b/projects/poincare-conjecture/.astrolabe/atoms/351e4ac14e4e.md new file mode 100644 index 00000000..4146024d --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/351e4ac14e4e.md @@ -0,0 +1,19 @@ +--- +chapter: 9 +generator: tools/poincare_tex_extract.py +label: 3DGSSkappa +labels: +- 3DGSSkappa +mtref: '9.53' +ref: +- 351e4ac14e4e +sort: corollary +source: tex +src: morgan-tian +tex_file: temp2kappa +--- +Let $(M,g(t)),\ -\infty0$ the +flow $(M,g(t)),\ -\infty0$ there is a +$0<\zeta<\eta/2$ such that any $C^1$-loop $c\colon S^1\to X$ of length less +than $\eta$ bounds a disk in $X$ of area less than $\eta$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/37d7d059a526.md b/projects/poincare-conjecture/.astrolabe/atoms/37d7d059a526.md new file mode 100644 index 00000000..d743687a --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/37d7d059a526.md @@ -0,0 +1,42 @@ +--- +chapter: 2 +generator: tools/poincare_tex_extract.py +label: fordiffmax +labels: +- fordiffmax +mtref: '2.23' +ref: +- 37d7d059a526 +sort: proposition +source: tex +src: morgan-tian +tex_file: prelim +--- +Let $M$ be a smooth manifold with a smooth vector field $\chi$ and a +smooth function $\mathbf{t}\colon M\to [a,b]$ with $\chi(\mathbf{t})=1$. +Suppose also that $F\colon M\to \Ar$ is a smooth function with the +properties: + +1. for each $t_0\in [a,b]$ the restriction of $F$ to the level set $\mathbf{t}^{-1}(t_0)$ achieves its maximum, and +1. the subset ${\mathcal Z}$ of $M$ +consisting of all $x$ for which $F(x)\ge F(y)$ for all $y\in \mathbf{t}^{-1}(\mathbf{t}(x))$ is a compact set. + + Suppose also that at each $x\in {\mathcal Z}$ we have $\chi(F(x))\le \psi(\mathbf{t}(x),F(x))$. Set $F_\mathit{max}(t)=\mathit{max}_{x\in \mathbf{t}^{-1}(t)}F(x)$. Then $F_\mathit{max}(t)$ is a continuous function and + +$$ +\frac{dF_\mathit{max}}{dt}(t)\le \psi(t,F_\mathit{max}(t)) +$$ + +in the sense of forward difference quotients. + Suppose that $G(t)$ satisfies the +differential equation + +$$ +G'(t)=\psi(t,G(t)) +$$ + + and has initial condition $F_\mathit{max}(a)\le G(a)$. Then for all $t\in [a,b]$ we have + +$$ +F_\mathit{max}(t)\le G(t). +$$ diff --git a/projects/poincare-conjecture/.astrolabe/atoms/37efbeaa9eec.md b/projects/poincare-conjecture/.astrolabe/atoms/37efbeaa9eec.md new file mode 100644 index 00000000..f7b9dcb4 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/37efbeaa9eec.md @@ -0,0 +1,16 @@ +--- +chapter: 18 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '18.8' +ref: +- 37efbeaa9eec +sort: claim +source: tex +src: morgan-tian +tex_file: energy1 +--- +Let $({\mathcal M},G)$ be a Ricci flow with surgery and for each $t$ set +$s(t)=s(M_t)$. If $t'0$, such that for any $y\in M$ with $d(x_0,y)=d\ge D$, +there is $x\in M$ with $d(y,x)=d$ and with $d(x_0,x)>3d/2$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/38712e8cea3d.md b/projects/poincare-conjecture/.astrolabe/atoms/38712e8cea3d.md new file mode 100644 index 00000000..47cb0420 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/38712e8cea3d.md @@ -0,0 +1,36 @@ +--- +chapter: 18 +generator: tools/poincare_tex_extract.py +label: CSSHI +labels: +- CSSHI +mtref: '18.52' +ref: +- 38712e8cea3d +sort: lemma +source: tex +src: morgan-tian +tex_file: energy1 +--- +Let $(M,g(t)),\ t_0\le t\le t_1$, be a Ricci flow. Then there exist +constants $\delta>0$ and $\widetilde C_i<\infty$ for +$i=0,1,2,\ldots$, depending only on $t_1-t_0$ and a bound for the +norm of the curvature of the Ricci flow, such that the following +holds. Let $c(x,t)$ be a curve-shrinking flow that is an immersion +for each $t$. Suppose that at a time $t'$ for some $00$ +depending on $t_0(A)$ and $Q(A)$, and a constant $Q'(A)<\infty$ +depending only on $Q(A)$, and, for all $n$ sufficiently large, an +embedding + +$$ +B_{Q_nG_n}(x_n,0,A)\times (-t'_0(A),0]\to{\mathcal +M}_n +$$ + + compatible with time and with the vector field with the +property that the scalar curvature of the restriction of $Q_nG_n$ +to the image of this subset is bounded by $Q'(A)$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/399dd63f056b.md b/projects/poincare-conjecture/.astrolabe/atoms/399dd63f056b.md new file mode 100644 index 00000000..7e801331 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/399dd63f056b.md @@ -0,0 +1,34 @@ +--- +chapter: 14 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '14.18' +ref: +- 399dd63f056b +sort: definition +source: tex +src: morgan-tian +tex_file: surgery +--- +Fix constants $(C,\epsilon)$ and a constant $r$. We say that a Ricci +flow with surgery $({\mathcal M},G)$ *satisfies the strong +$(C,\epsilon)$-canonical neighborhood assumption with parameter $r$* +if every point $x\in {\mathcal M}$ with $R(x)\ge r^{-2}$ has a strong +$(C,\epsilon)$-canonical neighborhood in ${\mathcal M}$. In all +cases except that of the strong $\epsilon$-neck, the strong +canonical neighborhood of $x$ is a subset of the time-slice +containing $x$, and the notion of a $(C,\epsilon)$-canonical +neighborhood has exactly the same meaning as in the case of an +ordinary Ricci flow. In the case of a strong $\epsilon$-neck +centered at $x$ this means that there is an embedding +$\left(S^2\times (-\epsilon^{-1},\epsilon^{-1})\right)\times (\mathbf{t}(x)-R(x)^{-1},\mathbf{t}(x)]\to {\mathcal M}$, mapping $(q_0,0)$ to $x$, + where $q_0$ is the basepoint of $S^2$, an embedding +compatible with time and the vector field, +such that the pullback +of $G$ is a Ricci flow on $S^2\times (-\epsilon^{-1},\epsilon^{-1})$ +which, when the time is shifted by $-\mathbf{t}(x)$ and then the flow +is rescaled by $R(x)$, is within $\epsilon$ in the +$C^{[1/\epsilon]}$-topology of the standard evolving round cylinder +$\left(S^2\times(-\epsilon^{-1},\epsilon^{-1}),h_0(t)\times +ds^2\right),\ -10$. Let $({\mathcal M},G)$ be +a generalized Ricci flow. An *evolving $\epsilon$-neck defined +for an interval of normalized time of length +$t'>0$* centered at a point $x\in +{\mathcal M}$ with $\mathbf{t}(x)=t$ is an embedding $\psi\colon +S^2\times (-\epsilon^{-1},\epsilon^{-1})\stackrel{\cong}{\to} +N\subset M_t$ with $x\in \psi(S^2\times\{0\})$ satisfying the +following properties: + +- **(1)** There is an embedding $N\times (t-R(x)^{-1}t',t]\to {\mathcal +M}$ compatible with time and the vector field. +- **(2)** The pullback under $\psi$ of the one-parameter family of +metrics on $N$ determined by restricting $R(x)G$ to the image of +this embedding is within $\epsilon$ in the +$C^{[1/\epsilon]}$-topology of the standard family $(h(t),ds^2), +-t'0$ such that for all $a$ +sufficiently large $P_{\widetilde G_a}(\tilde x_a,0,A,-t(A))$ exists in +$\widetilde {\mathcal M}_a$ and $|\mathit{Rm}|$ is bounded by $D(A)$ on this +backward parabolic neighborhood, or +- **(2)** each $x_a$ has a strong +$(C,\epsilon)$-canonical neighborhood. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/3defa9b2209c.md b/projects/poincare-conjecture/.astrolabe/atoms/3defa9b2209c.md new file mode 100644 index 00000000..4d975c78 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/3defa9b2209c.md @@ -0,0 +1,43 @@ +--- +chapter: 8 +generator: tools/poincare_tex_extract.py +label: THM +labels: +- THM +mtref: '8.1' +ref: +- 3defa9b2209c +sort: theorem +source: tex +src: morgan-tian +tex_file: noncoll +--- +Fix positive constants $\bar\tau_0<\infty$, $l_0<\infty$, and $V>0$. + Then there is $\kappa>0$ depending on $\bar\tau_0$, $V$, and $l_0$ + and the dimension $n$ such +that the following holds. Let $({\mathcal M},G)$ be a generalized +$n$-dimensional Ricci flow, and let $0<\tau_0\le \bar\tau_0$. Let +$x\in {\mathcal M}$ be fixed. Set $T=\mathbf{t}(x)$. Suppose that +$00$ +such that for any $\tau$ with $\tau_1<\tau<\tau_1+\delta$ the map +${\mathcal L}\mathit{exp}_x^{\tau}$ is a local diffeomorphism from a +neighborhood of $Z$ in $T_xM_{T-\tau_1}$ to $M_{T-\tau}$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/41a5f172b2dc.md b/projects/poincare-conjecture/.astrolabe/atoms/41a5f172b2dc.md new file mode 100644 index 00000000..bcde355c --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/41a5f172b2dc.md @@ -0,0 +1,45 @@ +--- +chapter: 13 +generator: tools/poincare_tex_extract.py +label: improve +labels: +- improve +mtref: '13.11' +ref: +- 41a5f172b2dc +sort: corollary +source: tex +src: morgan-tian +tex_file: surgery +--- +There is a constant $A<\infty$ such that the following holds for the given value of $q$ +and any $C_0$ provided that $\delta$ is sufficiently small. +Suppose that the eigenvalues for the curvature matrix of $h$ at $y$ +are $\lambda\ge \mu\ge \nu$. +Then the eigenvalues for the curvature of $\hat h$ at the point $y$ +are given by $\lambda',\mu',\nu'$, where + +$$ +\left|\lambda'-e^{2f}\left(\lambda-\frac{q^2}{s^4}f^2\right)\right| +\le \frac{q^2}{s^4}fA\delta +$$ + +$$ +\left|\mu'-e^{2f}\left(\mu+\left(\frac{q^2}{s^4}-\frac{2q}{s^3}\right)f\right)\right|\le +\frac{q^2}{s^4}fA\delta +$$ + +$$ +\left|\nu'-e^{2f}\left(\nu+\left(\frac{q^2}{s^4}-\frac{2q}{s^3}\right)f\right)\right|\le +\frac{q^2}{s^4}fA\delta. +$$ + +In particular, we have + +$$ +\nu'\ge e^{2f}\left(\nu+\frac{q^2}{2s^4}f\right) +$$ + +$$ +\mu'\ge e^{2f}\left(\mu+\frac{q^2}{2s^4}f\right). +$$ diff --git a/projects/poincare-conjecture/.astrolabe/atoms/41b9e2369bf0.md b/projects/poincare-conjecture/.astrolabe/atoms/41b9e2369bf0.md new file mode 100644 index 00000000..90356c99 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/41b9e2369bf0.md @@ -0,0 +1,18 @@ +--- +chapter: 9 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '9.3' +ref: +- 41b9e2369bf0 +sort: example +source: tex +src: morgan-tian +tex_file: temp2kappa +--- +Let $(S^{2},g_{0})$ be the standard round $2$-sphere of scalar +curvature $1$ (and hence Ricci tensor $g_0/2$). Set +$g(t)=(1-t)g_{0}$. Then $\partial g(t)/\partial t=-2\mathit{Ric}(g(t))$, $-\inftyt_0$ and a solution +to Equation (12.18) with initial condition +$\tilde{\rho}(r,t_0)=0$ defined on the time-interval $[t_0,T]$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/44032fc43e5d.md b/projects/poincare-conjecture/.astrolabe/atoms/44032fc43e5d.md new file mode 100644 index 00000000..bf443d69 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/44032fc43e5d.md @@ -0,0 +1,16 @@ +--- +chapter: 7 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '7.3' +ref: +- 44032fc43e5d +sort: lemma +source: tex +src: morgan-tian +tex_file: newcomp2 +--- +For any $0\le \tau_1<\tau_2\tau_1$ we set ${\mathcal U}_x(\tau)={\mathcal U}_x\cap +M_{T-\tau}$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/4415092c2079.md b/projects/poincare-conjecture/.astrolabe/atoms/4415092c2079.md new file mode 100644 index 00000000..e72cff63 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/4415092c2079.md @@ -0,0 +1,20 @@ +--- +chapter: 18 +generator: tools/poincare_tex_extract.py +label: pi2 +labels: +- pi2 +mtref: '18.9' +ref: +- 4415092c2079 +sort: proposition +source: tex +src: morgan-tian +tex_file: energy1 +--- +Let $({\mathcal M},G)$ be a Ricci flow with surgery satisfying the conclusion +of Theorem 15.9. Then there is some time $T_1<\infty$ such that every +component of $M_T$ for any $T\ge T_1$ has trivial $\pi_2$. For every $T\ge +T_1$, each component of $M_T$ either has finite fundamental group, and hence +has a homotopy $3$-sphere as universal covering, or has contractible universal +covering. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/443843fff58d.md b/projects/poincare-conjecture/.astrolabe/atoms/443843fff58d.md new file mode 100644 index 00000000..742aef47 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/443843fff58d.md @@ -0,0 +1,15 @@ +--- +chapter: 13 +generator: tools/poincare_tex_extract.py +label: surgerycap +labels: +- surgerycap +mtref: '13.3' +ref: +- 443843fff58d +sort: definition +source: tex +src: morgan-tian +tex_file: surgery +--- +The image in ${\mathcal S}$ of $B_{g_0}(p_0,0,A_0+4)$ is called the *surgery cap*. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/44abf07e60ec.md b/projects/poincare-conjecture/.astrolabe/atoms/44abf07e60ec.md new file mode 100644 index 00000000..ff8a2310 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/44abf07e60ec.md @@ -0,0 +1,21 @@ +--- +chapter: 6 +generator: tools/poincare_tex_extract.py +label: Jacobi +labels: +- Jacobi +mtref: '6.40' +ref: +- 44abf07e60ec +sort: lemma +source: tex +src: morgan-tian +tex_file: newcompar +--- +Let $\gamma$ be a minimizing ${\mathcal L}$-geodesic from $x$, and +let $Y(\tau)$ be an ${\mathcal L}$-Jacobi field along $\gamma$. Then + +$$ +2\sqrt{\bar\tau}\langle\nabla_XY(\bar\tau),Y(\bar\tau)\rangle = \mathit{Hess}(L_x^{\bar\tau})(Y(\bar\tau), +Y(\bar\tau)). +$$ diff --git a/projects/poincare-conjecture/.astrolabe/atoms/451b4cc590c1.md b/projects/poincare-conjecture/.astrolabe/atoms/451b4cc590c1.md new file mode 100644 index 00000000..1593aadc --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/451b4cc590c1.md @@ -0,0 +1,21 @@ +--- +chapter: 6 +generator: tools/poincare_tex_extract.py +label: tildelK +labels: +- tildelK +mtref: '6.54' +ref: +- 451b4cc590c1 +sort: corollary +source: tex +src: morgan-tian +tex_file: newcompar +--- +Suppose that $x\in M_T$ so that +$\tau_1=0$. Then + +$$ +\frac{\partial}{\partial\tau}\widetilde +l(Z,\tau)=-\frac{K^\tau(\gamma_Z)}{2\tau^{\frac{3}{2}}}. +$$ diff --git a/projects/poincare-conjecture/.astrolabe/atoms/45c783d384c1.md b/projects/poincare-conjecture/.astrolabe/atoms/45c783d384c1.md new file mode 100644 index 00000000..9afc2976 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/45c783d384c1.md @@ -0,0 +1,22 @@ +--- +chapter: 18 +generator: tools/poincare_tex_extract.py +label: laest +labels: +- laest +mtref: '18.66' +ref: +- 45c783d384c1 +sort: lemma +source: tex +src: morgan-tian +tex_file: energy1 +--- +Suppose that we have a focusing triangle ${\mathcal T}$ with base $\xi$ bounding a disk $B$ + in $A$. + Suppose that the length of $\xi$ is at most one. + Then + +$$ +l(\xi)\le \left(\int_\xi k_\mathit{geod}ds+\mathit{Area}(B)\right). +$$ diff --git a/projects/poincare-conjecture/.astrolabe/atoms/45d33631de13.md b/projects/poincare-conjecture/.astrolabe/atoms/45d33631de13.md new file mode 100644 index 00000000..ba73a302 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/45d33631de13.md @@ -0,0 +1,23 @@ +--- +chapter: 10 +generator: tools/poincare_tex_extract.py +label: refdistcomp +labels: +- refdistcomp +mtref: '10.26' +ref: +- 45d33631de13 +sort: corollary +source: tex +src: morgan-tian +tex_file: bddcurvbdddist +--- +Let $\gamma,\mu,x,y$ be as in the previous lemma. Fix $\lambda>0$. +Let $a=d_{\lambda g_\infty}(x,{\mathcal E})$ and $b=d_{\lambda +g_\infty}(y,{\mathcal E})$. Set $x'_\lambda$ and $y'_\lambda$ equal +to the points in the cone $([\gamma],a)$ and $([\mu],b)$. Then we +have + +$$ +d_{\lambda g_\infty}(x,y)\le d_{g_{\mathcal E}}(x'_\lambda,y'_\lambda). +$$ diff --git a/projects/poincare-conjecture/.astrolabe/atoms/4673eb9f4424.md b/projects/poincare-conjecture/.astrolabe/atoms/4673eb9f4424.md new file mode 100644 index 00000000..56d10e48 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/4673eb9f4424.md @@ -0,0 +1,30 @@ +--- +chapter: 5 +generator: tools/poincare_tex_extract.py +label: partialflowlimit +labels: +- partialflowlimit +mtref: '5.14' +ref: +- 4673eb9f4424 +sort: proposition +source: tex +src: morgan-tian +tex_file: converge2 +--- +Fix constants $-\infty\le T'\le 0\le T\le \infty$ and suppose that $T'0$ and any $R<\infty$ for all $k$ sufficiently large, +$x_k\in \mathit{Reg}_\delta(U_k,g_k)$ and for any $\ell\ge k$ the image +$\varphi_\ell(V_k)$ contains $B(x_\ell,R)\cap\mathit{Reg}_\delta(U_\ell,g_\ell,x_\ell)$. + +We also say that the sequence *converges geometrically +to* $(U_\infty,g_\infty,x_\infty)$ if +there exist $(V_k,\varphi_k)$ as required in the above definition. +We also say that $(U_\infty,g_\infty,x_\infty)$ is the *geometric limit* of the sequence. + +More generally, given $(U_\infty,g_\infty,x_\infty)$, a sequence of open +subsets and $\{V_k\}_{k=1}^\infty$ satisfying (1) above, and smooth maps +$\varphi_k\colon V_k\to U_k$ satisfying (2a) above, we say that +$(U_\infty,g_\infty,x_\infty)$ is a *partial geometric +limit* of the sequence. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/46f9d27654be.md b/projects/poincare-conjecture/.astrolabe/atoms/46f9d27654be.md new file mode 100644 index 00000000..14087a7a --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/46f9d27654be.md @@ -0,0 +1,18 @@ +--- +chapter: 18 +generator: tools/poincare_tex_extract.py +label: homess +labels: +- homess +mtref: '18.7' +ref: +- 46f9d27654be +sort: proposition +source: tex +src: morgan-tian +tex_file: energy1 +--- +Let $({\mathcal M},G)$ be a Ricci flow with surgery satisfying +Assumptions (1) -- (7) in Chapter 14. Then there can +be only finitely many homotopically essential surgeries along +$2$-spheres in $({\mathcal M},G)$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/47e16059fb12.md b/projects/poincare-conjecture/.astrolabe/atoms/47e16059fb12.md new file mode 100644 index 00000000..1aa7a500 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/47e16059fb12.md @@ -0,0 +1,20 @@ +--- +chapter: 18 +generator: tools/poincare_tex_extract.py +label: expcomp +labels: +- expcomp +mtref: '18.65' +ref: +- 47e16059fb12 +sort: claim +source: tex +src: morgan-tian +tex_file: energy1 +--- +There is $\delta'>0$ such that the following holds. + The +exponential mapping defines a map $\mathit{exp}\colon S_X(\delta')\to +A$ which is a local diffeomorphism and the pullback of the metric on +$A$ defines a metric on $S_X(\delta')$ which is at least +$(1-\delta)^2$ times the given product metric. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/48f3d2b5dea3.md b/projects/poincare-conjecture/.astrolabe/atoms/48f3d2b5dea3.md new file mode 100644 index 00000000..322d3f62 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/48f3d2b5dea3.md @@ -0,0 +1,58 @@ +--- +chapter: 0 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '0.6' +ref: +- 48f3d2b5dea3 +sort: remark +source: tex +src: morgan-tian +tex_file: intro +--- +(i) The use of the sphere theorem is unnecessary in the above +argument for what we actually prove is that if every prime factor of +$M$ has non-trivial $\pi_2$ or non-trivial $\pi_3$, then the Ricci +flow with surgery with $(M,g_0)$ as initial metric becomes extinct +after a finite time. In fact, the sphere theorem for closed +three-manifolds follows from the results here. + + (ii) If the initial manifold is simpler then all the +time-slices are simpler: If $({\mathcal M},G)$ is a Ricci flow with +surgery whose initial manifold is prime, then every time-slice is a +disjoint union of connected components, all but at most one being +diffeomorphic to a three-sphere and if there is one not +diffeomorphic to a three-sphere, then it is diffeomorphic to the +initial manifold. If the initial manifold is a simply connected +manifold $M_0$, then every component of every time-slice $M_T$ must +be simply connected, and thus *a posteriori* every time-slice is +a disjoint union of manifolds diffeomorphic to the three-sphere. +Similarly, if the initial manifold has finite fundamental group, +then every connected component of every time-slice is either simply +connected or has the same fundamental group as the initial manifold. + + (iii) The conclusion of this result is a natural +generalization of Hamilton's conclusion in analyzing the Ricci flow +on manifolds of positive Ricci curvature in [Hamilton3MPRC]. +Namely, under appropriate hypotheses, during the evolution process +of Ricci flow with surgery the manifold breaks into components each +of which disappears in finite time. As a component disappears at +some finite time, the metric on that component is well enough +controlled to show that the disappearing component admits a +non-flat, homogeneous Riemannian metric of non-negative sectional +curvature, i.e., a metric locally isometric to either a round $S^3$ +or to a product of a round $S^2$ with the usual metric on $ \Ar$. +The existence of such a metric on a component immediately gives the +topological conclusion of Theorem 0.1 for that component, +i.e., that it is diffeomorphic to a $3$-dimensional spherical +space-form, to $S^2\times S^1$ to a non-orientable $2$-sphere bundle +over $S^1$, or to $\Ar P^3\#\Ar P^3$. The biggest difference between +these two results is that Hamilton's hypothesis is geometric +(positive Ricci curvature) whereas Perelman's is homotopy theoretic +(information about the fundamental group). + + (iv) It is also worth pointing out that it follows from +Corollary 0.5 that the manifolds that satisfy the four equivalent +conditions in that corollary are exactly the closed, connected, three-manifolds +that admit a Riemannian metric of positive scalar curvature, cf, +[SchoenYau2] and [GromovLawson]. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/490290f4c322.md b/projects/poincare-conjecture/.astrolabe/atoms/490290f4c322.md new file mode 100644 index 00000000..ca64344d --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/490290f4c322.md @@ -0,0 +1,21 @@ +--- +chapter: 7 +generator: tools/poincare_tex_extract.py +label: ltauineq +labels: +- ltauineq +mtref: '7.24' +ref: +- 490290f4c322 +sort: corollary +source: tex +src: morgan-tian +tex_file: newcomp2 +--- +Let $\phi\colon B_{(q,t)}'\to \Ar$ be a non-negative, smooth +function with compact support. Then + +$$ +\int_{B_{(q,t)}'}l^\tau\triangle \phi d\mathit{vol}(g(t))\le \int_{(B_{(q,t)}')^*}\phi +\triangle l^\tau d\mathit{vol}(g(t)). +$$ diff --git a/projects/poincare-conjecture/.astrolabe/atoms/4917e81d7684.md b/projects/poincare-conjecture/.astrolabe/atoms/4917e81d7684.md new file mode 100644 index 00000000..3166683d --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/4917e81d7684.md @@ -0,0 +1,33 @@ +--- +chapter: 6 +generator: tools/poincare_tex_extract.py +label: Ueqn +labels: +- Ueqn +mtref: '6.50' +ref: +- 4917e81d7684 +sort: theorem +source: tex +src: morgan-tian +tex_file: newcompar +--- +Suppose that $x\in M_T$ so that $\tau_1=0$. For any $q\in{\mathcal +U}_x(\bar\tau)$, let $Z\in \widetilde{\mathcal U}_x(\bar\tau)$ be +the pre-image of $q$ and let $\gamma_Z$ be the ${\mathcal +L}$-geodesic determined by $Z$. As usual, let $X(\tau)$ be the +horizontal projection of $\gamma_Z'(\tau)$. Then we have + +$$ +\begin{aligned} +\frac{\partial l_x}{\partial \tau}(q) & = & +R(q)-\frac{l_x(q)}{\bar\tau}+\frac{K^{\bar\tau}(\gamma_Z)}{2\bar\tau^{3/2}} +\\ +|\nabla l_x^{\bar\tau}(q)|^2 & = & |X(\bar\tau)|^2 = +\frac{l_x^{\bar\tau}(q)}{\bar\tau}-\frac{K^{\bar\tau} +(\gamma_Z)}{\bar\tau^{3/2}}-R(q) \\ +\triangle l_x^{\bar\tau}(q) & = & +\frac{1}{2\sqrt{\bar\tau}}\triangle L_x^{\bar\tau}(q)\le \frac{n}{2 +\bar\tau}-R(q)-\frac{K^{\bar\tau}(\gamma_Z)}{2\bar\tau^{3/2}}. +\end{aligned} +$$ diff --git a/projects/poincare-conjecture/.astrolabe/atoms/494bc110f8ab.md b/projects/poincare-conjecture/.astrolabe/atoms/494bc110f8ab.md new file mode 100644 index 00000000..eff5159e --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/494bc110f8ab.md @@ -0,0 +1,18 @@ +--- +chapter: 9 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '9.90' +ref: +- 494bc110f8ab +sort: claim +source: tex +src: morgan-tian +tex_file: temp2kappa +--- +Assuming that $(M,g(0))$ is compact but not of constant positive +sectional curvature, for each $\epsilon>0$ there is $C_1$ such that +if the diameter of $(M,g(0))$ is greater than $C_1(\mathit{max}_{x\in +M}R(x,0))^{-1/2}$ then every point of $(M,g(0))$ is either contained +in the core of $(C_0(\epsilon),\epsilon)$-cap or is the center of a +strong $\epsilon$-neck in $(M,g(t))$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/49c665aedacc.md b/projects/poincare-conjecture/.astrolabe/atoms/49c665aedacc.md new file mode 100644 index 00000000..89878e38 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/49c665aedacc.md @@ -0,0 +1,19 @@ +--- +chapter: 7 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '7.18' +ref: +- 49c665aedacc +sort: claim +source: tex +src: morgan-tian +tex_file: newcomp2 +--- +Suppose that $\beta\colon [-a,a]\to \Ar$ is a continuous function +and that at each $s\in (-a,a)$ there is an upper barrier $\hat b_s$ +for $\beta$ at $s$ with $\hat b_s''\le -3/2$. Then + +$$ +\frac{\beta(a)+\beta(-a)}{2}\le \beta(0)-\frac{3}{4}a^2. +$$ diff --git a/projects/poincare-conjecture/.astrolabe/atoms/49e865228b19.md b/projects/poincare-conjecture/.astrolabe/atoms/49e865228b19.md new file mode 100644 index 00000000..2c1fe592 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/49e865228b19.md @@ -0,0 +1,17 @@ +--- +chapter: 9 +generator: tools/poincare_tex_extract.py +label: compactkappa +labels: +- compactkappa +mtref: '9.64' +ref: +- 49e865228b19 +sort: theorem +source: tex +src: morgan-tian +tex_file: temp2kappa +--- +Let $(M_k,g_k(t),(p_k,0))$ be a sequence of based $3$-dimensional +$\kappa$-solutions satisfying $R(p_k,0)=1$. Then there is a +subsequence converging smoothly to a based $\kappa$-solution. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/4a0173b99805.md b/projects/poincare-conjecture/.astrolabe/atoms/4a0173b99805.md new file mode 100644 index 00000000..da56439e --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/4a0173b99805.md @@ -0,0 +1,19 @@ +--- +chapter: 1 +generator: tools/poincare_tex_extract.py +label: localdiffeo +labels: +- localdiffeo +mtref: '1.32' +ref: +- 4a0173b99805 +sort: lemma +source: tex +src: morgan-tian +tex_file: prelim +--- +Fix $K\ge 0$. +If $|\mathit{Rm}(x)|\le K$ for all $x\in B(p,\pi/\sqrt{K})$, then +$\mathit{exp}_p$ is a local diffeomorphism from the ball +$B(0,\pi/\sqrt{K})$ in $T_pM$ to the ball $B(p,\pi/\sqrt{K})$ in +$M$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/4a4c8aae3a8d.md b/projects/poincare-conjecture/.astrolabe/atoms/4a4c8aae3a8d.md new file mode 100644 index 00000000..17e73f93 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/4a4c8aae3a8d.md @@ -0,0 +1,22 @@ +--- +chapter: 7 +generator: tools/poincare_tex_extract.py +label: flat +labels: +- flat +mtref: '7.25' +ref: +- 4a4c8aae3a8d +sort: lemma +source: tex +src: morgan-tian +tex_file: newcomp2 +--- +If $(M,g(t))$ is flat Euclidean $n$-space (independent of $t$), then +for any $x\in \Ar^n\times (-\infty,\infty)$ we have + +$$ +\widetilde V_x(M,\tau)=(4\pi)^{n/2} +$$ + + for all $\tau>0$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/4b41e7ff8317.md b/projects/poincare-conjecture/.astrolabe/atoms/4b41e7ff8317.md new file mode 100644 index 00000000..26b86faf --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/4b41e7ff8317.md @@ -0,0 +1,24 @@ +--- +chapter: 8 +generator: tools/poincare_tex_extract.py +label: large +labels: +- large +mtref: '8.9' +ref: +- 4b41e7ff8317 +sort: lemma +source: tex +src: morgan-tian +tex_file: noncoll +--- +There is a universal positive constant $\varepsilon_0>0$ such that +if $\varepsilon\le \varepsilon_0$, we have + +$$ +\widetilde V_x(W_\mathit{lg}(\tau_1)) \le \int_{{\widetilde +U}(\tau_1)\cap \{Z\bigl|\bigr. |Z|\geq +\frac{1}{8}\varepsilon^{-\frac{1}{2}}\}}(\tau_1)^{-\frac{n}{2}} +e^{-\tilde l(q,\tau_1)}\mathcal{J}(Z,\tau_1)dZ\le +\varepsilon^{\frac{n}{2}}. +$$ diff --git a/projects/poincare-conjecture/.astrolabe/atoms/4b6ee3b7a329.md b/projects/poincare-conjecture/.astrolabe/atoms/4b6ee3b7a329.md new file mode 100644 index 00000000..d37d3cf4 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/4b6ee3b7a329.md @@ -0,0 +1,32 @@ +--- +chapter: 16 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '16.8' +ref: +- 4b6ee3b7a329 +sort: lemma +source: tex +src: morgan-tian +tex_file: surgery +--- +Suppose that we have a sequence of $3$-dimensional Ricci flows +with surgeries $({\mathcal M}_n,G_n)$ that satisfy the strong +$(C,\epsilon)$-canonical neighborhood assumption with parameter +$r_{i+1}$, and have curvature pinched toward positive. Suppose that +there are surgeries in ${\mathcal M}_n$ at times $t_n$ with surgery +control parameters $\delta'_n$ and scales $h_n$. Let $p_n$ be the +tip of a surgery cap for the surgery at time $ t_n$. Also suppose +that there is $0\le\theta_n<1$ such that for every $A<\infty$, +for all $n$ sufficiently large +there are embeddings +$B(p_a, t_n,A h_n)\times [t_n, t_n+ h^2_n\theta_n)\to {\mathcal +M}_n$ compatible with time and the vector field. Suppose that +$\delta'_n\rightarrow 0$ and $\theta_n\rightarrow \theta<1$ as $n\rightarrow +\infty$. Let $({\mathcal M}'_n,G'_n,p_n)$ be the Ricci flow with +surgery obtained by shifting time by $-t_n$ so that surgery occurs +at $\mathbf{t}=0$ and rescaling by $h_n^{-2}$ so that the scale of the +surgery becomes one. Then, after passing to a subsequence, the +sequence converges smoothly to a limiting flow +$(M_\infty,g_\infty(t),(p_\infty,0)),\ 0\le t<\theta$. This limiting flow is isomorphic to +the restriction of the standard flow to time $0\le t<\theta$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/4b8a44aa316f.md b/projects/poincare-conjecture/.astrolabe/atoms/4b8a44aa316f.md new file mode 100644 index 00000000..6118fcc5 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/4b8a44aa316f.md @@ -0,0 +1,22 @@ +--- +chapter: 18 +generator: tools/poincare_tex_extract.py +label: extinct +labels: +- extinct +mtref: '18.1' +ref: +- 4b8a44aa316f +sort: theorem +source: tex +src: morgan-tian +tex_file: energy1 +--- +Let $(M,g(0))$ be a compact, connected normalized Riemannian $3$-manifold. +Suppose that the fundamental group of $M$ is a free product of finite groups +and infinite cyclic groups. Then $M$ contains no $\Ar P^2$ with trivial normal +bundle. Let $({\mathcal M},G)$ be the Ricci flow with surgery defined for all +$t\in [0,\infty)$ with $(M,g(0))$ as initial conditions given by +Theorem 15.9. This Ricci flow with surgery becomes extinct after a finite +time in the sense that the time-slices $M_T$ of ${\mathcal M}$ are empty for +all $T$ sufficiently large. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/4b9af3660539.md b/projects/poincare-conjecture/.astrolabe/atoms/4b9af3660539.md new file mode 100644 index 00000000..7d0b8eac --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/4b9af3660539.md @@ -0,0 +1,21 @@ +--- +chapter: 6 +generator: tools/poincare_tex_extract.py +label: tausmooth +labels: +- tausmooth +mtref: '6.31' +ref: +- 4b9af3660539 +sort: proposition +source: tex +src: morgan-tian +tex_file: newcompar +--- +Let $\gamma$ be a minimizing ${\mathcal L}$-geodesic defined for +$[\tau_1,\bar\tau]$. Fix $0\le\tau_1<\tau_2<\bar\tau$, and set +$q_2=\gamma(\tau_2)$, and $Z_2=\sqrt{\tau_2}X_{\gamma}(\tau_2)$. +Then, the map ${\mathcal L}\mathit{exp}_{q_2}$ is diffeomorphism from +a neighborhood of $\{Z_2\}\times (\tau_2,\bar\tau]$ in +$T_{q}M_{T-\tau_2}\times (\tau_2,\infty)$ onto a neighborhood of the +image of $\gamma|_{(\tau_2,\bar\tau]}$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/4bff7f1e7bc4.md b/projects/poincare-conjecture/.astrolabe/atoms/4bff7f1e7bc4.md new file mode 100644 index 00000000..3f6d978c --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/4bff7f1e7bc4.md @@ -0,0 +1,17 @@ +--- +chapter: 9 +generator: tools/poincare_tex_extract.py +label: rho +labels: +- rho +mtref: '9.70' +ref: +- 4bff7f1e7bc4 +sort: corollary +source: tex +src: morgan-tian +tex_file: temp2kappa +--- +Fix $\delta>0$ the universal constant of the last +claim. Then $R(q',t)\le \delta^{-2}$ for all $q'\in B(p,0,\delta)$ +and all $t\le 0$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/4ce8b5da9c58.md b/projects/poincare-conjecture/.astrolabe/atoms/4ce8b5da9c58.md new file mode 100644 index 00000000..06deab11 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/4ce8b5da9c58.md @@ -0,0 +1,31 @@ +--- +chapter: 16 +generator: tools/poincare_tex_extract.py +label: KAPPALIMIT +labels: +- KAPPALIMIT +mtref: '16.1' +ref: +- 4ce8b5da9c58 +sort: proposition +source: tex +src: morgan-tian +tex_file: surgery +--- +Suppose that for some $i\ge 0$ we have surgery parameter sequences $\delta_0\ge +\delta_1\ge\cdots\ge \delta_i>0$, $\epsilon=r_0\ge r_1\ge \cdots\ge +r_i>0$ and $\kappa_0\ge \kappa_1\ge \cdots\ge \kappa_i>0$. + Then there is $0<\kappa\le \kappa_i$ and for any $00$ such that the map +${\mathcal L}\mathit{exp}_x$ is defined on $B(0,A)\times (0,\delta)$, where +$B(0,A)$ is the ball of radius $A$ centered at the origin in $T_xM_T$. +Moveover, ${\mathcal L}\mathit{exp}_x$ defines a diffeomorphism of $B(0,A)\times +(0,\delta)$ onto an open subset of ${\mathcal M}$. Furthermore, + +$$ +\mathit{lim}_{\tau\rightarrow 0}\tau^{-n/2}e^{-\tilde l(Z,\tau)}{\mathcal J}(Z,\tau)=2^ne^{-|Z|^2}, +$$ + +where the convergence is uniform on each compact subset of $T_xM_T$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/4ee1bc9acc48.md b/projects/poincare-conjecture/.astrolabe/atoms/4ee1bc9acc48.md new file mode 100644 index 00000000..5955fd41 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/4ee1bc9acc48.md @@ -0,0 +1,15 @@ +--- +chapter: 16 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '16.16' +ref: +- 4ee1bc9acc48 +sort: remark +source: tex +src: morgan-tian +tex_file: surgery +--- +Recall that $(A_0+4)\bar h$ is the radius of the surgery cap (measured in the +rescaled version of the standard initial metric) that is glued in when +performing surgery with scale $\bar h$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/4f253de31e0c.md b/projects/poincare-conjecture/.astrolabe/atoms/4f253de31e0c.md new file mode 100644 index 00000000..b761e28e --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/4f253de31e0c.md @@ -0,0 +1,19 @@ +--- +chapter: 4 +generator: tools/poincare_tex_extract.py +label: sphsf +labels: +- sphsf +mtref: '4.24' +ref: +- 4f253de31e0c +sort: theorem +source: tex +src: morgan-tian +tex_file: maxprin +--- +Let $(M,g)$ be a compact $3$-dimensional soliton of positive Ricci +curvature. Then $(M,g)$ is round. In particular, $(M,g)$ is the +quotient of $S^3$ with a round metric by a finite subgroup of $O(4)$ +acting freely; that is to say, $M$ is a $3$-dimensional spherical +space-form. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/4f373966523b.md b/projects/poincare-conjecture/.astrolabe/atoms/4f373966523b.md new file mode 100644 index 00000000..1ede7e53 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/4f373966523b.md @@ -0,0 +1,16 @@ +--- +chapter: 5 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '5.26' +ref: +- 4f373966523b +sort: example +source: tex +src: morgan-tian +tex_file: converge2 +--- +Let $(M_n,g_n,x_n)$ be a sequence of based Riemannian manifolds +converging geometrically to $(M_\infty,g_\infty,x_\infty)$. Then the +sequence also converges in the Gromov-Hausdorff sense to the same +limit. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/4fc24fa1d31e.md b/projects/poincare-conjecture/.astrolabe/atoms/4fc24fa1d31e.md new file mode 100644 index 00000000..76be4338 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/4fc24fa1d31e.md @@ -0,0 +1,31 @@ +--- +chapter: 18 +generator: tools/poincare_tex_extract.py +label: 1stcurvshr +labels: +- 1stcurvshr +mtref: '18.34' +ref: +- 4fc24fa1d31e +sort: lemma +source: tex +src: morgan-tian +tex_file: energy1 +--- +Assume that $(M,g(t)),\ t_0\le t\le t_1$, is a Ricci flow and that +$c=c(x,t)$ is a solution to the curve-shrinking flow. We have vector +fields $X=\partial /\partial x$ and $H=\partial /\partial t$ defined +on the domain surface. We denote by $|X|_{c^*g}^2$ the function on +the domain surface whose value at $(x,t)$ is $|(X(x,t))|_{g(t)}^2$. +We define $S=|X|_{c^*g}^{-1}X$, the unit vector in the $x$-direction +measured in the evolving metric. Then, + +$$ +\frac{\partial}{\partial t}(|X|_{c^*g}^2)(x,t)=-2\mathit{Ric}_{g(t)}(X(x,t),X(x,t))-2k^2|X(x,t)|_{g(t)}^2, +$$ + +and + +$$ +[H,S](x,t)=\left(k^2+\mathit{Ric}_{g(t)}(S(x,t),S(x,t))\right)S(x,t). +$$ diff --git a/projects/poincare-conjecture/.astrolabe/atoms/50312568fd70.md b/projects/poincare-conjecture/.astrolabe/atoms/50312568fd70.md new file mode 100644 index 00000000..9b2f6093 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/50312568fd70.md @@ -0,0 +1,32 @@ +--- +chapter: 9 +generator: tools/poincare_tex_extract.py +label: Ccomponent +labels: +- Ccomponent +mtref: '9.75' +ref: +- 50312568fd70 +sort: definition +source: tex +src: morgan-tian +tex_file: temp2kappa +--- +Fix a positive constant $C$. A compact connected Riemannian manifold +$(M,g)$ is called *a $C$-component* if + +- **(1)** $M$ is diffeomorphic to either $S^3$ or $\Ar P^3$. +- **(2)** $(M,g)$ has positive sectional curvature. +- **(3)** + +$$ +C^{-1}<\frac{\mathit{inf}_{P}K(P)}{\mathit{sup}_{y\in M}R(y)} +$$ + +where $P$ varies over all $2$-planes in $TX$ (and $K(P)$ denotes the +sectional curvature in the $P$-direction). +- **(4)** + +$$ +C^{-1}\mathit{sup}_{y\in M}\left(R(y)^{-1/2}\right)<\mathit{diam}(M)0$ such that the following hold if $\delta\le +\delta_3'$. Let $\{e_0,e_1,e_2\}$ be an orthonormal basis for the +tangent space at a point $y\in S^2\times I$ for the metric +$h_0\times ds^2$ with the property that $e_0$ points in the +$I$-direction. Then there is a basis $\{f_0,f_1,f_2\}$ for this +tangent space so that the following hold: + +- **(1)** The basis is orthonormal in the metric $h$. +- **(2)** The change of basis matrix expressing the $\{f_0,f_1,f_2\}$ +in terms of $\{e_0,e_1,e_2\}$ is of the form $\mathit{Id}+O(\delta)$. +- **(3)** The Riemann curvature of $h$ in the basis $\{f_0\wedge f_1,f_1\wedge f_2,f_2\wedge f_0\}$ +of $\wedge^2T_y(S^2\times I)$ is + +$$ +\begin{pmatrix} 1/2 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 +\end{pmatrix}+O(\delta). +$$ diff --git a/projects/poincare-conjecture/.astrolabe/atoms/52f36e456355.md b/projects/poincare-conjecture/.astrolabe/atoms/52f36e456355.md new file mode 100644 index 00000000..0e5a8e08 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/52f36e456355.md @@ -0,0 +1,22 @@ +--- +chapter: 18 +generator: tools/poincare_tex_extract.py +label: mulength +labels: +- mulength +mtref: '18.60' +ref: +- 52f36e456355 +sort: claim +source: tex +src: morgan-tian +tex_file: energy1 +--- +There is $\mu>0$ such that the following holds. Suppose that $c,\hat +c\in S^2$ are such that there is an arc $\omega$ in $S^2$ connecting +$c$ and $\hat c$ such that the area of the annulus +$\widetilde\Gamma^\lambda(\omega)$ is at most $\mu$. Set +$t_2=t_1-B^{-1}$. If the length of $\widetilde\Gamma_{\hat +c}^\lambda(t)$ is less than $\zeta/2$ for all $t\in [t_2,t_1]$, then +the length of $p_1\widetilde\Gamma_c^\lambda(t_1)$ is less than +$\zeta$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/53758e75bb17.md b/projects/poincare-conjecture/.astrolabe/atoms/53758e75bb17.md new file mode 100644 index 00000000..8629e1e0 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/53758e75bb17.md @@ -0,0 +1,21 @@ +--- +chapter: 1 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '1.28' +ref: +- 53758e75bb17 +sort: exercise +source: tex +src: morgan-tian +tex_file: prelim +--- +(E.Calabi, 1958) +Let $f(x)=d(p,x)$ be the distance function from $p$. If $(M, g)$ has +$\mathit{Ric} \geq 0$, then + +$$ +\triangle f \leq \frac{n-1}{f} +$$ + + in the sense of distributions. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/541ad02dd5f8.md b/projects/poincare-conjecture/.astrolabe/atoms/541ad02dd5f8.md new file mode 100644 index 00000000..1ee07db8 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/541ad02dd5f8.md @@ -0,0 +1,14 @@ +--- +chapter: 18 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '18.62' +ref: +- 541ad02dd5f8 +sort: claim +source: tex +src: morgan-tian +tex_file: energy1 +--- +For all $\epsilon>0$ the Gaussian curvature of $(f+\epsilon)|dz|^2$ +is at most $2C''$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/541f71387f84.md b/projects/poincare-conjecture/.astrolabe/atoms/541f71387f84.md new file mode 100644 index 00000000..419a7f73 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/541f71387f84.md @@ -0,0 +1,29 @@ +--- +chapter: 6 +generator: tools/poincare_tex_extract.py +label: COR +labels: +- COR +mtref: '6.74' +ref: +- 541f71387f84 +sort: corollary +source: tex +src: morgan-tian +tex_file: newcompar +--- +With notation as above, and with the substitution $\tau'=Q\tau$, for +any $Z\in {\mathcal H}T_x{\mathcal M}$ and any $\tau>0$ we have + +$$ +{\mathcal +L}'\mathit{exp}_x(\sqrt{Q^{-1}}Z,\tau')=\iota({\mathcal L}\mathit{exp}_x(Z,\tau)) +$$ + + and + +$$ +\tilde l'(\sqrt{Q^{-1}}Z,\tau')=\tilde l(Z,\tau), +$$ + +whenever these are defined. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/545ca0a10566.md b/projects/poincare-conjecture/.astrolabe/atoms/545ca0a10566.md new file mode 100644 index 00000000..13760a68 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/545ca0a10566.md @@ -0,0 +1,21 @@ +--- +chapter: 12 +generator: tools/poincare_tex_extract.py +label: decrease +labels: +- decrease +mtref: '12.37' +ref: +- 545ca0a10566 +sort: lemma +source: tex +src: morgan-tian +tex_file: stdsoln +--- +Let $S^2$ be the unit sphere in $T_0\Ar^3$. Equip it with the metric +$h_0$ that is twice the usual metric (so that the scalar curvature +of $h_0$ is $1$). We define a map $\rho\colon S^2\times +[0,\infty)\to \Ar^3$ by sending the point $(x,s)$ to the point at +distance $s$ from the origin in the radial direction from $0$ given +by $x$ (all this measured in the metric $g_0$). Then $\rho^*g_0\le +h_0\times ds^2$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/554068bff2cf.md b/projects/poincare-conjecture/.astrolabe/atoms/554068bff2cf.md new file mode 100644 index 00000000..b026d1fb --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/554068bff2cf.md @@ -0,0 +1,20 @@ +--- +chapter: 2 +generator: tools/poincare_tex_extract.py +label: strnarrows +labels: +- strnarrows +mtref: '2.21' +ref: +- 554068bff2cf +sort: corollary +source: tex +src: morgan-tian +tex_file: prelim +--- +Fix $\epsilon>0$ sufficiently small so that Lemma 19.10 holds. Then +there is a constant $C<\infty$ depending on $\epsilon$ such that the following +holds. Suppose that $M$ is a non-compact $3$-manifold of positive sectional +curvature. Suppose that $N$ is an $\epsilon$-neck in $M$ centered at a point +$x$ and disjoint from a soul $p$ of $M$. Then for any $\epsilon$-neck $N'$ that +is separated from $p$ by $N$ with center $x'$ we have $R(x')\le CR(x)$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/5578a00f3308.md b/projects/poincare-conjecture/.astrolabe/atoms/5578a00f3308.md new file mode 100644 index 00000000..63b734c6 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/5578a00f3308.md @@ -0,0 +1,18 @@ +--- +chapter: 6 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '6.27' +ref: +- 5578a00f3308 +sort: definition +source: tex +src: morgan-tian +tex_file: newcompar +--- +The function $L_x\colon {\mathcal U}_x\to \Ar$ +assigns to each $q$ in ${\mathcal U}_x$ the length of any minimizing +${\mathcal L}$-geodesic from $x$ to $q$. For any $\tau>\tau_1$, we +denote by $L_x^\tau$ the restriction of $L_x$ +to the $T-\tau$ time-slice of ${\mathcal U}_x$, i.e., the +restriction of $L_x$ to ${\mathcal U}_x(\tau)$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/55b7b71de6e4.md b/projects/poincare-conjecture/.astrolabe/atoms/55b7b71de6e4.md new file mode 100644 index 00000000..72211825 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/55b7b71de6e4.md @@ -0,0 +1,29 @@ +--- +chapter: 18 +generator: tools/poincare_tex_extract.py +label: D_4cor +labels: +- D_4cor +mtref: '18.91' +ref: +- 55b7b71de6e4 +sort: corollary +source: tex +src: morgan-tian +tex_file: energy1 +--- +For $\gamma_{t'}\subset c(\cdot,t')$ a sub-arc of length at most $r$ + and for any $t\in [t',t_2]$, we have + +$$ +\int_{\gamma_t}kds\le +2D_4\sqrt{\delta}. +$$ + + For any $t\in [t',t_2]$ and any sub-arc +$J\subset c(\cdot,t)$ of length at most $r/2$ with respect to the +metric $h(t)$, we have + +$$ +\int_Jk(x,t)ds(x,t)\le 2D_4\sqrt{\delta}. +$$ diff --git a/projects/poincare-conjecture/.astrolabe/atoms/569be40ea88e.md b/projects/poincare-conjecture/.astrolabe/atoms/569be40ea88e.md new file mode 100644 index 00000000..085eb334 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/569be40ea88e.md @@ -0,0 +1,28 @@ +--- +chapter: 1 +generator: tools/poincare_tex_extract.py +label: riccurvcomp +labels: +- riccurvcomp +mtref: '1.33' +ref: +- 569be40ea88e +sort: theorem +source: tex +src: morgan-tian +tex_file: prelim +--- +**(Ricci curvature comparison)** Fix $k\ge 0$. Assume that $(M, +g)$ satisfies $\mathit{Ric} \geq -(n-1)k$. Let $\gamma\colon [0,r_0)\to +M$ be a minimal geodesic of unit speed. Then for any $r0$ and $\kappa>0$ such that for every $n$ the metric ball $B(x_n,0,r)\subset +(M_n,g_n(0))$ is $\kappa$-non-collapsed. +- **(2)** For each $A<\infty$ there is $C=C(A)<\infty$ such that the Riemannian +curvature on $B(x_n,0,A)\times (-T,0]$ is bounded by $C$. + +Then after passing to a subsequence there is a geometric limit which +is a based Ricci flow $(M_\infty,g_\infty(t),(x_\infty,0))$ defined +for $t\in (-T,0]$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/5727e4c714af.md b/projects/poincare-conjecture/.astrolabe/atoms/5727e4c714af.md new file mode 100644 index 00000000..6d0399f2 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/5727e4c714af.md @@ -0,0 +1,22 @@ +--- +chapter: 18 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '18.4' +ref: +- 5727e4c714af +sort: definition +source: tex +src: morgan-tian +tex_file: energy1 +--- +Let $X$ be a compact $3$-manifold (possibly disconnected). An +embedded $2$-sphere in $X$ is said to be *homotopically +essential* if the inclusion of the $2$-sphere into $X$ is not +homotopic to a point map of the $2$-sphere to $X$. More generally, +let $F=\{\Sigma_1,\ldots,\Sigma_n\}$ be a family of disjointly +embedded $2$-spheres in $X$. We say that the family is *homotopically essential* if + +- **(i)** each $2$-sphere in the family is homotopically essential, and +- **(ii)** for any $1\le i0$ such that for all $0<\epsilon\le +\epsilon_0$ the following is true. Suppose that $N$ and $N'$ are +$\epsilon$-necks centered at $x$ and $x'$, respectively, in a +Riemannian manifold $M$. Suppose that $x'$ is not contained in $N$ +but is contained in the closure of $N$ in $M$. Suppose also that the +two-spheres of the neck structure on $N$ and $N'$ separate $M$. +Then, possibly after reversing the $\epsilon$-neck structures on $N$ +and/or $N'$, the pair $\{N,N'\}$ forms a balanced chain. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/59189adac6be.md b/projects/poincare-conjecture/.astrolabe/atoms/59189adac6be.md new file mode 100644 index 00000000..4043c391 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/59189adac6be.md @@ -0,0 +1,24 @@ +--- +chapter: 8 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '8.6' +ref: +- 59189adac6be +sort: claim +source: tex +src: morgan-tian +tex_file: noncoll +--- +There is a +positive constant $\varepsilon_0\le 1/4n(n-1)$ depending on +$\bar\tau_0$, such that the following holds. Suppose that + $\varepsilon\le \varepsilon_0$ and $\tau_1' \leq +\tau_1=\varepsilon r^2$. Let $Z\in T_xM_T$ and let $\gamma_Z$ be the +associated ${\mathcal L}$-geodesic from $x$. Suppose that +$\gamma_Z(\tau) \in B(x,T,r/2)$ for all $\tau < \tau_1'$. Then for +all $\tau<\tau_1'$ we have + +$$ +\abs{|\sqrt{\tau}X(\tau)|_{g(T)} - |Z|} \leq 2\varepsilon (1+|Z|). +$$ diff --git a/projects/poincare-conjecture/.astrolabe/atoms/5982e4029637.md b/projects/poincare-conjecture/.astrolabe/atoms/5982e4029637.md new file mode 100644 index 00000000..7e1e527f --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/5982e4029637.md @@ -0,0 +1,22 @@ +--- +chapter: 1 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '1.11' +ref: +- 5982e4029637 +sort: theorem +source: tex +src: morgan-tian +tex_file: prelim +--- +**(Uniformization Theorem)** If +$(M^n,g)$ is a complete, simply-connected Riemannian manifold of +constant sectional curvature $\lambda$, then: + +1. If $\lambda=0$, then $M^n$ is isometric to Euclidean $n$-space. +1. If $\lambda>0$ there is a diffeomorphism $\phi \colon M \rightarrow S^{n}$ + such that g = $\lambda^{-1}\phi^{*}(g_\mathit{st})$ where $g_{st}$ is the usual metric on the unit sphere in $\Ar^{n+1}$. +1. If $\lambda<0$ there is a diffeomorphism $\phi \colon M \rightarrow {\mathbb H}^{n}$ + such that g = $\abs{\lambda}^{-1}\phi^{*}(g_\mathit{st})$ where $g_{st}$ is the Poincaré metric of constant curvature +$-1$ on ${\mathbb H}^n$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/598f6059280c.md b/projects/poincare-conjecture/.astrolabe/atoms/598f6059280c.md new file mode 100644 index 00000000..bf6c9fd0 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/598f6059280c.md @@ -0,0 +1,16 @@ +--- +chapter: 9 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '9.22' +ref: +- 598f6059280c +sort: claim +source: tex +src: morgan-tian +tex_file: temp2kappa +--- +$$ +\mathit{lim}_{k\rightarrow\infty}\int_{M\times\{t\}}\langle\nabla^{h_k}(l_k-l_\infty-\epsilon_k), +\varphi\nabla^{h_k} l_k\rangle_{h_k} d\mathit{vol}(h_k)\le 0. +$$ diff --git a/projects/poincare-conjecture/.astrolabe/atoms/59dc7d470f3e.md b/projects/poincare-conjecture/.astrolabe/atoms/59dc7d470f3e.md new file mode 100644 index 00000000..f6736fed --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/59dc7d470f3e.md @@ -0,0 +1,18 @@ +--- +chapter: 5 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '5.13' +ref: +- 59dc7d470f3e +sort: definition +source: tex +src: morgan-tian +tex_file: converge2 +--- +For $00$, then $h(x,t)>0$ for all +$(x,t)\in \mathit{int}(\overline U)\times(0,T]$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/5b34d0c84cbb.md b/projects/poincare-conjecture/.astrolabe/atoms/5b34d0c84cbb.md new file mode 100644 index 00000000..b42ff49b --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/5b34d0c84cbb.md @@ -0,0 +1,33 @@ +--- +chapter: 10 +generator: tools/poincare_tex_extract.py +label: thetaconv +labels: +- thetaconv +mtref: '10.21' +ref: +- 5b34d0c84cbb +sort: lemma +source: tex +src: morgan-tian +tex_file: bddcurvbdddist +--- +For all $\gamma,s,\mu,s'$ as in the previous definition we have + +$$ +0\le \theta(\gamma,s,\mu,s')\le \pi. +$$ + + Furthermore, $\theta(\gamma,s,\mu,s')$ +is a non-increasing function of $s$ when $\gamma,\mu,s'$ are held +fixed, and symmetrically it is a non-increasing function of $s'$ +when $\gamma,s,\mu$ are held fixed. In particular, fixing $\gamma$ +and $\mu$, the function $\theta(\gamma,s,\mu,s')$ is non-decreasing +as $s$ and $s'$ tend to zero. Thus, there is a well-defined limit as +$s$ and $s'$ go to zero, denoted $ \theta(\gamma,\mu)$. This limit +is greater than or equal to $\theta(\gamma,s,\mu,s')$ for all $s$ +and $s'$ for which the latter is defined. We have +$0\le\theta(\gamma,\mu)\le \pi$. The angle $\theta(\gamma,\mu)=0$ if +and only if $\gamma$ and $\mu$ are equivalent. Furthermore, if +$\gamma$ is equivalent to $\gamma'$ and $\mu$ is equivalent to +$\mu'$, then $\theta(\gamma,\mu)=\theta(\gamma',\mu')$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/5b4c94102f41.md b/projects/poincare-conjecture/.astrolabe/atoms/5b4c94102f41.md new file mode 100644 index 00000000..1498b482 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/5b4c94102f41.md @@ -0,0 +1,24 @@ +--- +chapter: 9 +generator: tools/poincare_tex_extract.py +label: absconv2 +labels: +- absconv2 +mtref: '9.29' +ref: +- 5b4c94102f41 +sort: claim +source: tex +src: morgan-tian +tex_file: temp2kappa +--- +Suppose that $\varphi$ and $\psi_k$ are as in Claim 9.27, +but in addition $\varphi$ and all the $\psi_k$ are uniformly +Lipschitz. Then + +$$ +\mathit{lim}_{k\rightarrow\infty}\int_{M_\infty}\varphi\psi_k\triangle +e^{-l_\infty}d\mathit{vol}_{g_\infty} +$$ + + converges absolutely. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/5b649fe97568.md b/projects/poincare-conjecture/.astrolabe/atoms/5b649fe97568.md new file mode 100644 index 00000000..62370ec3 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/5b649fe97568.md @@ -0,0 +1,19 @@ +--- +chapter: 18 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '18.2' +ref: +- 5b649fe97568 +sort: definition +source: tex +src: morgan-tian +tex_file: energy1 +--- +Let $I$ be an interval (which is allowed to be open or closed at +each end and finite or infinite at each end). By a *path of +components* of a Ricci flow with surgery $({\mathcal M},G)$ defined +for all $t\in I$ we mean a connected, open subset ${\mathcal +X}\subset \mathbf{t}^{-1}(I)$ with the property that for every $t\in I$ +the intersection ${\mathcal X}(t)$ of ${\mathcal X}$ with each +time-slice $M_t$ is a connected component of $M_t$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/5b79e248c5cb.md b/projects/poincare-conjecture/.astrolabe/atoms/5b79e248c5cb.md new file mode 100644 index 00000000..19a93efb --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/5b79e248c5cb.md @@ -0,0 +1,35 @@ +--- +chapter: 12 +generator: tools/poincare_tex_extract.py +label: dtuniqtouniq +labels: +- dtuniqtouniq +mtref: '12.16' +ref: +- 5b79e248c5cb +sort: corollary +source: tex +src: morgan-tian +tex_file: stdsoln +--- +Suppose that $(M,g_1(t)),\ t_0\le t\le T$, and $(M,g_2(t)),\ t_0\le +t\le T$, are solutions to the Ricci flow equation for which there +are solutions + +$$ +\psi_{1,t}\colon (M,g_1(t))\to (M,g_1(0)) +$$ + + and + +$$ +\psi_{2,t}\colon +(M,g_2(t))\to (M,g_2(0)) +$$ + + to the harmonic map equation with +$\psi_{1,t_0}=\psi_{2,t_0}=\mathit{Id}$. Let $\hat +g_1(t)=(\psi_{1,t}^{-1})^*g_1(t)$ and $\hat +g_2(t)=(\psi_{2,t}^{-1})^*g_2(t)$ be the corresponding solutions to +the Ricci-DeTurck flow. Suppose that $\hat g_1(t)=\hat g_2(t)$ for +all $t\in [t_0,T]$. Then $g_1(t)=g_2(t)$ for all $t\in [t_0,T]$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/5ba0ec2c68db.md b/projects/poincare-conjecture/.astrolabe/atoms/5ba0ec2c68db.md new file mode 100644 index 00000000..1e08c07b --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/5ba0ec2c68db.md @@ -0,0 +1,15 @@ +--- +chapter: 16 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '16.23' +ref: +- 5ba0ec2c68db +sort: claim +source: tex +src: morgan-tian +tex_file: surgery +--- +For every surgery time $\bar t\in [T_{i-1},T]$, the path $\gamma$ +starting at $x$ with ${\mathcal L}(\gamma)\le L$ is disjoint from +$J(\bar t,\bar t-\tau_1(\bar t))$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/5be0c15e901a.md b/projects/poincare-conjecture/.astrolabe/atoms/5be0c15e901a.md new file mode 100644 index 00000000..7db51615 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/5be0c15e901a.md @@ -0,0 +1,21 @@ +--- +chapter: 5 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '5.22' +ref: +- 5be0c15e901a +sort: lemma +source: tex +src: morgan-tian +tex_file: converge2 +--- +Let $(X_k,x_k)$ be a sequence of metric spaces whose diameters are uniformly +bounded. Then the $(X_k,x_k)$ converge in the Gromov-Hausdorff +sense to $(X,x)$ if and only if the +following holds for every $\delta>0$. For every $\delta$-net $L\subset X$, for +every $\eta>0$, and for every $k$ sufficiently large, there is a +$(\delta+\eta)$-net $L_k\subset X_k$ and a bijection $L_k\to L$ sending $x_k$ +to $x$ so that the push forward of the metric on $L_k$ induced from that of +$X_k$ is $(1+\eta)$-bi-Lipschitz equivalent to the metric on $L$ induced from +$X$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/5cfd7c3aee29.md b/projects/poincare-conjecture/.astrolabe/atoms/5cfd7c3aee29.md new file mode 100644 index 00000000..5fef9284 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/5cfd7c3aee29.md @@ -0,0 +1,25 @@ +--- +chapter: 9 +generator: tools/poincare_tex_extract.py +label: volcomp +labels: +- volcomp +mtref: '9.60' +ref: +- 5cfd7c3aee29 +sort: proposition +source: tex +src: morgan-tian +tex_file: temp2kappa +--- +Fix the dimension $n$. +For every $\nu>0$ there is $A<\infty$ such that the following holds. +Suppose that $(M_k,g_k(t)),\ -t_k\le t\le 0$, is a sequence of (not +necessarily complete) $n$-dimensional Ricci flows of non-negative +curvature operator. Suppose in addition we have points $p_k\in M_k$ +and radii $r_k>0$ with the property that for each $k$ the ball +$B(p_k,0,r_k)$ has compact closure in $M_k$. Let $Q_k=R(p_k,0)$ and +suppose that $R(q,t)\le 4Q_k$ for all $q\in B(p_k,0,r_k)$ and for +all $t\in [-t_k,0]$, and suppose that $t_kQ_k\rightarrow\infty$ and +$r_k^2Q_k\rightarrow\infty$ as $k\rightarrow\infty$. Then $\mathit{Vol} B(p_k,0,A/\sqrt{Q_k})< \nu(A/\sqrt{Q_k})^n$ for all $k$ +sufficiently large. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/5d895f9c3918.md b/projects/poincare-conjecture/.astrolabe/atoms/5d895f9c3918.md new file mode 100644 index 00000000..17cefcd1 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/5d895f9c3918.md @@ -0,0 +1,27 @@ +--- +chapter: 3 +generator: tools/poincare_tex_extract.py +label: I.8.3 +labels: +- I.8.3 +mtref: '3.21' +ref: +- 5d895f9c3918 +sort: proposition +source: tex +src: morgan-tian +tex_file: flowbasics +--- +Let $t_0\in \Ar$ and let $(M,g(t))$ be a Ricci flow defined for $t$ in an interval containing $t_0$ + with $(M,g(t))$ complete for every $t$ in this interval. Fix a constant $K<\infty$. Let $x_0,x_1$ be two points +of $M$ and let $r_0>0$ such that $d_{t_0}(x_0,x_1)\ge 2r_0$. Suppose +that $\mathit{Ric}(x,t_0)\le (n-1)K$ for all $x\in B(x_0,r_0,t_0)\cup +B(x_1,r_0,t_0)$. Then + +$$ +\frac{d(d_t(x_0,x_1))}{dt}\Bigl|_{t=t_0}\Bigr.\ge -2(n-1)\left(\frac{2}{3}Kr_0+r_0^{-1}\right). +$$ + + If the distance function $d_{t}(x_0,x_1)$ is not a differentiable +function of $t$ at $t=t_0$ then this inequality is understood as an +inequality for the forward difference quotient. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/5d9f1b75170d.md b/projects/poincare-conjecture/.astrolabe/atoms/5d9f1b75170d.md new file mode 100644 index 00000000..9166a6c9 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/5d9f1b75170d.md @@ -0,0 +1,18 @@ +--- +chapter: 9 +generator: tools/poincare_tex_extract.py +label: asympscal +labels: +- asympscal +mtref: '9.56' +ref: +- 5d9f1b75170d +sort: proposition +source: tex +src: morgan-tian +tex_file: temp2kappa +--- +Suppose that $(M,g(t)),\ -\infty0$ as in the statement of +Proposition 16.4. For any $1<\ell<\infty$ the following +holds. Suppose that $\bar t\in [T_{i-1},T)$ is a surgery time and +that $\gamma(\tau)$ is a path with $\gamma(\tau)\in M_{\bar +t-\tau}$. Let $\{p_1,\ldots,p_k\}$ be the tips of all the surgery +caps at time $\bar t$ and let $\bar h$ be the scale of surgery at +time $\bar t$. Suppose that for some $0<\tau_1\le \ell^{-1} \bar +h^2$ there are no surgery times in the interval $(\bar t-\tau_1,\bar +t)$. We identify all $M_t$ for $t\in [\bar t-\tau_0,\bar t)$ with +$M_{\bar t-\tau_1}$ using the flow. Suppose that $\gamma(0)\in +M_{\bar t}\setminus \cup_{i=1}^kB(p_i,\bar t,(50+A_0)\bar h)$, and +lastly, suppose that + +$$ +\int_{0}^{\tau_1}|X_\gamma(\tau)|^2d\tau\le \ell. +$$ + +Then $\gamma$ is disjoint from the open subset $J(\bar t-\tau_1,\bar +t))$ of ${\mathcal M}$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/5ef5906bb370.md b/projects/poincare-conjecture/.astrolabe/atoms/5ef5906bb370.md new file mode 100644 index 00000000..5c5181ba --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/5ef5906bb370.md @@ -0,0 +1,25 @@ +--- +chapter: 19 +generator: tools/poincare_tex_extract.py +label: coreint +labels: +- coreint +mtref: '19.8' +ref: +- 5ef5906bb370 +sort: corollary +source: tex +src: morgan-tian +tex_file: canonnbhd +--- +The following holds for all $\epsilon>0$ sufficiently small and any +$C<\infty$. Let $X$ be an $(C,\epsilon)$-cap in a complete +Riemannian manifold $(M,g)$, and let $Y$ be its core and let $S$ be +the central two-sphere of the $\epsilon$-neck $N=X-\overline Y$. We +orient the $s$-direction in $N$ so that $Y$ lies off the negative +end of $N$. Let $\widehat Y$ be the union of $\overline Y$ and the +closed negative half of $N$ and let $S$ be the boundary of $\widehat +Y$. Suppose that $\gamma$ is a minimal geodesic in $(M,g)$ that +contains a point of the core $Y$. Then the intersection of $\gamma$ +with $\widehat Y$ is an interval containing an endpoint of $\gamma$; +see Fig. 19.1. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/5f77e346e12a.md b/projects/poincare-conjecture/.astrolabe/atoms/5f77e346e12a.md new file mode 100644 index 00000000..d20e20bd --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/5f77e346e12a.md @@ -0,0 +1,14 @@ +--- +chapter: 18 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '18.80' +ref: +- 5f77e346e12a +sort: claim +source: tex +src: morgan-tian +tex_file: energy1 +--- +The restriction of the exponential mapping to $S_{X'_J}(B)$ is an +embedding. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/5fbeadbba61c.md b/projects/poincare-conjecture/.astrolabe/atoms/5fbeadbba61c.md new file mode 100644 index 00000000..3f46f640 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/5fbeadbba61c.md @@ -0,0 +1,26 @@ +--- +chapter: 12 +generator: tools/poincare_tex_extract.py +label: harmapflow +labels: +- harmapflow +mtref: '12.25' +ref: +- 5fbeadbba61c +sort: proposition +source: tex +src: morgan-tian +tex_file: stdsoln +--- +For a standard solution $(\mathbb{R}^n,g(t)),\ 0\le t0$ such that for all $n$ sufficiently large + +$$ +\mathit{Vol}(B_{Q_nG_n}(x_n,0,\eta))\ge \kappa\eta^3. +$$ diff --git a/projects/poincare-conjecture/.astrolabe/atoms/611a73900deb.md b/projects/poincare-conjecture/.astrolabe/atoms/611a73900deb.md new file mode 100644 index 00000000..9f394eb9 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/611a73900deb.md @@ -0,0 +1,26 @@ +--- +chapter: 9 +generator: tools/poincare_tex_extract.py +label: lxkest +labels: +- lxkest +mtref: '9.16' +ref: +- 611a73900deb +sort: corollary +source: tex +src: morgan-tian +tex_file: temp2kappa +--- +There is a positive continuous function $C_1(\tau)$ defined for +$\tau>0$ such that for any $q\in M_k$ we have: + +$$ +l_{x_k}(q,\tau)\le +\left(\sqrt{\frac{3}{\tau}}d_{g_k(-\tau)}(q_k,q)+C_1(\tau)\right)^2, +$$ + +$$ +|\nabla l_{x_k}(q,\tau)|\le +\frac{3}{\tau}d_{g_k(-\tau)}(q_k,q)+\sqrt{\frac{3}{\tau}}C_1(\tau). +$$ diff --git a/projects/poincare-conjecture/.astrolabe/atoms/613fb3994956.md b/projects/poincare-conjecture/.astrolabe/atoms/613fb3994956.md new file mode 100644 index 00000000..d1eb78bf --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/613fb3994956.md @@ -0,0 +1,20 @@ +--- +chapter: 7 +generator: tools/poincare_tex_extract.py +label: 4picase +labels: +- 4picase +mtref: '7.27' +ref: +- 613fb3994956 +sort: proposition +source: tex +src: morgan-tian +tex_file: newcomp2 +--- +Suppose that $(M,g(\tau)),\ 0\le \tau\le T$, + is a solution to the backward Ricci flow +equation, complete of bounded curvature. Let $x=(p,T)\in M\times +\{T\}$, and suppose that $0<\bar\tau0$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/6262ca70063e.md b/projects/poincare-conjecture/.astrolabe/atoms/6262ca70063e.md new file mode 100644 index 00000000..1a241ef7 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/6262ca70063e.md @@ -0,0 +1,16 @@ +--- +chapter: 7 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '7.9' +ref: +- 6262ca70063e +sort: remark +source: tex +src: morgan-tian +tex_file: newcomp2 +--- +The operator $\triangle$ in the above statement is the horizontal +Laplacian, i.e., the Laplacian of the restriction of the indicated +function to the slice $M\times\{t=T-\tau\}$ as defined using the +metric $g(T-\tau)$ on this slice. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/6345ee49174f.md b/projects/poincare-conjecture/.astrolabe/atoms/6345ee49174f.md new file mode 100644 index 00000000..6209c163 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/6345ee49174f.md @@ -0,0 +1,16 @@ +--- +chapter: 9 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '9.6' +ref: +- 6345ee49174f +sort: example +source: tex +src: morgan-tian +tex_file: temp2kappa +--- +Consider the product $S^{2}\times \mathbb{R} $, with the metric $ +g(t) =(1-t)g_{0}+ds^{2}.$ This is a $\kappa$-solution for any +$\kappa$ at most the volume of a ball of radius one in the product +of the unit $2$-sphere with $\Ar$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/64100c8689ff.md b/projects/poincare-conjecture/.astrolabe/atoms/64100c8689ff.md new file mode 100644 index 00000000..261eb21d --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/64100c8689ff.md @@ -0,0 +1,18 @@ +--- +chapter: 9 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '9.36' +ref: +- 64100c8689ff +sort: claim +source: tex +src: morgan-tian +tex_file: temp2kappa +--- +If, for some $t\in (-\infty,0)$, the Riemannian manifold +$(M_\infty,g_\infty(t))$ is flat, then there is an isometry from +$\Ar^n$ to $(M_\infty,g_\infty(t))$ and the pullback under this +isometry of the function $l_\infty(x,\tau)$ is the function +$|x|^2/4\tau+\langle x,a\rangle+b\cdot \tau$ for some $a\in \Ar^n$ +and $b\in \Ar$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/6448a516850a.md b/projects/poincare-conjecture/.astrolabe/atoms/6448a516850a.md new file mode 100644 index 00000000..5484e6e1 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/6448a516850a.md @@ -0,0 +1,13 @@ +--- +chapter: 9 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '9.57' +ref: +- 6448a516850a +sort: claim +source: tex +src: morgan-tian +tex_file: temp2kappa +--- +$S_\infty(M,p)$ is isometric to a round $2$-sphere. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/64a0f82323d3.md b/projects/poincare-conjecture/.astrolabe/atoms/64a0f82323d3.md new file mode 100644 index 00000000..c7e554ab --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/64a0f82323d3.md @@ -0,0 +1,24 @@ +--- +chapter: 2 +generator: tools/poincare_tex_extract.py +label: homeo +labels: +- homeo +mtref: '2.10' +ref: +- 64a0f82323d3 +sort: corollary +source: tex +src: morgan-tian +tex_file: prelim +--- +Let $(M,g)$ be a complete, non-compact manifold of non-negative +sectional curvature. Let $p\in M$ and define a function $f\colon +M\to \Ar$ by $f(q)=d(p,q)$. Then there is $R<\infty$ such that for +$R\le st_0$ with the property that each $c(\cdot,t)$ is an immersion. +Either the curve-shrinking flow extends to a curve-shrinking flow +that is a family of immersions defined at $t_1'$ and beyond, or +$\mathit{max}_{x\in S^1}k(x,t)$ blows up as $t$ approaches $t_1'$ from +below. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/6590af0738c7.md b/projects/poincare-conjecture/.astrolabe/atoms/6590af0738c7.md new file mode 100644 index 00000000..e16f763c --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/6590af0738c7.md @@ -0,0 +1,27 @@ +--- +chapter: 18 +generator: tools/poincare_tex_extract.py +label: mugrowth +labels: +- mugrowth +mtref: '18.43' +ref: +- 6590af0738c7 +sort: lemma +source: tex +src: morgan-tian +tex_file: energy1 +--- +Suppose that the dimension $n$ of $ M$ +is at least three. Let $c_1(x,t)$ and $c_2(x,t)$ be ramp solutions +in $( M, g)\times (S^1_\lambda,ds^2)$ with the image under the +projection to $S^1_\lambda$ of each $c_i$ being of degree one. Let +$\mu(t)$ be the infimum of the areas of annuli in $( M\times +S^1_\lambda, g(t)\times ds^2)$ with boundary $c_1(x,t)\cup +c_2(x,t)$. Then $\mu(t)$ is a continuous function of $t$ and + +$$ +\frac{d}{dt}\mu(t)\le (2n-1)\mathit{max}_{x\in M}|\mathit{Rm}(x,t)|\mu(t), +$$ + +in the sense of forward difference quotients. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/666538bf577e.md b/projects/poincare-conjecture/.astrolabe/atoms/666538bf577e.md new file mode 100644 index 00000000..f8a7e2d7 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/666538bf577e.md @@ -0,0 +1,26 @@ +--- +chapter: 5 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '5.32' +ref: +- 666538bf577e +sort: definition +source: tex +src: morgan-tian +tex_file: converge2 +--- +Let $({\mathcal M}_k,G_k,x_k)$ be a sequence of based +generalized Ricci flows. We suppose that $\mathbf{t}(x_k)=0$ for all +$n$. We set $Q_k$ equal to $R(x_k)$. We denote by $(Q_k{\mathcal +M}_k,Q_kG_k,x_k)$ the family of generalized flows that have been +rescaled so that $R_{Q_kG_k}(x_k)=1$. + Suppose that $\mathit{lim}_{k\rightarrow\infty}Q_k=\infty$ and that after passing to a +subsequence there is a geometric limit of the sequence +$(Q_k{\mathcal M}_k,Q_kG_k,x_k)$ which is a Ricci flow defined for +$-T0$, an ancient solution is *$\kappa$-non-collapsed* if it is $\kappa$-non-collapsed on all +scales. We also use the terminology $\kappa$-*solution* for a +$\kappa$-non-collapsed, ancient solution. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/67090b75df54.md b/projects/poincare-conjecture/.astrolabe/atoms/67090b75df54.md new file mode 100644 index 00000000..aa40d35f --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/67090b75df54.md @@ -0,0 +1,15 @@ +--- +chapter: 10 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '10.5' +ref: +- 67090b75df54 +sort: claim +source: tex +src: morgan-tian +tex_file: bddcurvbdddist +--- +For all $n$ sufficiently large, $3R(z'_n)^{-1/2}\epsilon^{-1}/2$ is +less than $A_1$, and hence $U_n$ contains the strong $\epsilon$-neck +$N(z_n')$ centered at $z'_n$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/671402eb454b.md b/projects/poincare-conjecture/.astrolabe/atoms/671402eb454b.md new file mode 100644 index 00000000..657fa994 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/671402eb454b.md @@ -0,0 +1,39 @@ +--- +chapter: 9 +generator: tools/poincare_tex_extract.py +label: asympGSS +labels: +- asympGSS +mtref: '9.11' +ref: +- 671402eb454b +sort: theorem +source: tex +src: morgan-tian +tex_file: temp2kappa +--- +Let $(M,g(t)),\ -\infty0$ sufficiently small, let $\nu_\epsilon(S_{(q,t)})$ be the +$\epsilon$-neighborhood (with respect to the Euclidean metric) in +$B_{(q,t)}$ of $S_{(q,t)}\cap B_{(q,t)}$. Then, as $k\rightarrow +\infty$ the restrictions of $f_k$ to $B_{(q,t)}'\setminus +\left(B_{(q,t)}'\cap \nu_\epsilon(S_{(q,t)})\right)$ converge +uniformly in the $C^\infty$-topology to the restriction of +$\beta_{(q,t)}$ to this subset. +1. For each $k$, and for any $z\in B_{(q,t)}'$ and any $v\in T_zM$ we have + +$$ +\mathit{Hess}(f_k)(v,v)\le -|v|_{g(t)}^2/2. +$$ + +That is to say, $f_k$ is strictly convex with respect to the metric +$g(t)$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/67b855d0d6b5.md b/projects/poincare-conjecture/.astrolabe/atoms/67b855d0d6b5.md new file mode 100644 index 00000000..635743ae --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/67b855d0d6b5.md @@ -0,0 +1,21 @@ +--- +chapter: 18 +generator: tools/poincare_tex_extract.py +label: 3rdcurvshr +labels: +- 3rdcurvshr +mtref: '18.36' +ref: +- 67b855d0d6b5 +sort: claim +source: tex +src: morgan-tian +tex_file: energy1 +--- +There is a constant $C_1<\infty$ depending only on an +upper bound for the norm of the sectional curvatures of the ambient +manifolds in the Ricci flow $(M,g(t)),\ t_0\le t\le t_1$, such that + +$$ +\frac{\partial}{\partial t} k\le k''+k^3+C_1k. +$$ diff --git a/projects/poincare-conjecture/.astrolabe/atoms/67c378f87457.md b/projects/poincare-conjecture/.astrolabe/atoms/67c378f87457.md new file mode 100644 index 00000000..4deb75a3 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/67c378f87457.md @@ -0,0 +1,21 @@ +--- +chapter: 2 +generator: tools/poincare_tex_extract.py +label: fordiffquot +labels: +- fordiffquot +mtref: '2.22' +ref: +- 67c378f87457 +sort: lemma +source: tex +src: morgan-tian +tex_file: prelim +--- +Suppose that $f\colon [a,b]\to \Ar$ is a continuous function. Suppose that +$\psi$ is a $C^1$-function on $[a,b]\times \Ar$ and suppose that +$\frac{df}{dt}(t)\le \psi(t,f(t))$ for every $t\in [a,b)$ in the sense of +forward difference quotients. Suppose also that there is a function $G(t)$ +defined on $[a,b]$ that satisfies the differential equation +$G'(t)=\psi(t,G(t))$ and has $f(a)\le G(a)$. Then $f(t)\le G(t)$ for all $t\in +[a,b]$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/683bb06c1059.md b/projects/poincare-conjecture/.astrolabe/atoms/683bb06c1059.md new file mode 100644 index 00000000..e74ee229 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/683bb06c1059.md @@ -0,0 +1,27 @@ +--- +chapter: 16 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '16.10' +ref: +- 683bb06c1059 +sort: claim +source: tex +src: morgan-tian +tex_file: surgery +--- +Suppose that $\theta_n<\mathit{min}(\theta,(T'_n- t_n)/ h_n^2)$. Let +$\Sigma_{1},\ldots,\Sigma_{k}$ be the $2$-spheres along which +we do surgery at time $ t_n+h_n^2\theta_n$. Then for any $t< t_n+ +h_n^2\theta_n$ sufficiently close to $ t_n+ h_n^2\theta_n$ the +following holds provided that $\delta_n'$ is sufficiently small. The image + +$$ +\rho_n\left(B_{g_n}(x,t_n,A h_n)\times +\{t\}\right) +$$ + + is disjoint from the images $\{\Sigma_{i}(t)\}$ of the $\{\Sigma_{i}\}$ +under the +backward flow to time $t$ of the spheres $\Sigma_{i}$ along which +we do surgery at time $t_n+h_n^2\theta_n$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/68624c3d26bd.md b/projects/poincare-conjecture/.astrolabe/atoms/68624c3d26bd.md new file mode 100644 index 00000000..72bf3b04 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/68624c3d26bd.md @@ -0,0 +1,18 @@ +--- +chapter: 2 +generator: tools/poincare_tex_extract.py +label: htpytype +labels: +- htpytype +mtref: '2.11' +ref: +- 68624c3d26bd +sort: corollary +source: tex +src: morgan-tian +tex_file: prelim +--- +Let $(M,g)$ be a complete, non-compact Riemannian $n$-manifold of positive +curvature. Then for any point $p\in M$ there is a constant $R=R(p)$ such that +for any $s0$ there is a uniform bound, independent of $n$, for +the curvature on $\mathit{Reg}_\delta(U_n,g'_n)$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/6a1625220073.md b/projects/poincare-conjecture/.astrolabe/atoms/6a1625220073.md new file mode 100644 index 00000000..c42fbf8e --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/6a1625220073.md @@ -0,0 +1,13 @@ +--- +chapter: 18 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '18.19' +ref: +- 6a1625220073 +sort: claim +source: tex +src: morgan-tian +tex_file: energy1 +--- +${\mathcal X}(T_1)$ has non-trivial $\pi_3$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/6a52537bef2f.md b/projects/poincare-conjecture/.astrolabe/atoms/6a52537bef2f.md new file mode 100644 index 00000000..e06a566b --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/6a52537bef2f.md @@ -0,0 +1,20 @@ +--- +chapter: 6 +generator: tools/poincare_tex_extract.py +label: Amono +labels: +- Amono +mtref: '6.80' +ref: +- 6a52537bef2f +sort: theorem +source: tex +src: morgan-tian +tex_file: newcompar +--- +Fix $x\in M_T\subset {\mathcal M}$. Let $A\subset{\mathcal +U}_x\subset {\mathcal M}$ be an open subset. We suppose that for +any $0<\tau\le \bar\tau$ and any $y\in A_\tau=A\cap M_{T-\tau}$ the +minimizing ${\mathcal L}$-geodesic from $x$ to $y$ contained in +$A\cup \{x\}$. Then $\widetilde V_x(A_\tau)$ is a non-increasing +function of $\tau$ for all $0<\tau\le \bar\tau$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/6a961a764f44.md b/projects/poincare-conjecture/.astrolabe/atoms/6a961a764f44.md new file mode 100644 index 00000000..5fb0c740 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/6a961a764f44.md @@ -0,0 +1,16 @@ +--- +chapter: 6 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '6.1' +ref: +- 6a961a764f44 +sort: definition +source: tex +src: morgan-tian +tex_file: newcompar +--- +Let $0\le \tau_1<\tau_2$ be given and let $\gamma\colon +[\tau_1,\tau_2]\to {\mathcal M}$ be a continuous map. We say that +$\gamma$ is *parameterized by backward time* provided that $\gamma(\tau)\in M_{T-\tau}$ for all +$\tau\in [\tau_1,\tau_2]$ diff --git a/projects/poincare-conjecture/.astrolabe/atoms/6aa2c1fdf94e.md b/projects/poincare-conjecture/.astrolabe/atoms/6aa2c1fdf94e.md new file mode 100644 index 00000000..234d6a5a --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/6aa2c1fdf94e.md @@ -0,0 +1,28 @@ +--- +chapter: 3 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '3.34' +ref: +- 6aa2c1fdf94e +sort: definition +source: tex +src: morgan-tian +tex_file: flowbasics +--- +By *space-time* we mean a smooth $(n+1)$-dimensional manifold +${\mathcal M}$ (possibly with boundary), equipped with a smooth +function $\mathbf{t}\colon {\mathcal M}\to \Ar$, called *time* and +a smooth vector field $\chi$ subject to the following axioms: + +1. The image of $\mathbf{t}$ is an interval $I$ (possibly infinite) and +the boundary of ${\mathcal M}$ is the preimage under $\mathbf{t}$ of +$\partial I$. +1. For each $x\in {\mathcal M}$ there is an open neighborhood $U\subset +{\mathcal M}$ of $x$ and a diffeomorphism $f\colon V\times J\to U$, +where $V$ is an open subset in $\Ar ^n$ and $J$ is an interval with +the property that (i) $\mathbf{t}$ is the composition of $f^{-1}$ +followed by the projection onto the interval $J$ and (ii) $\chi$ is +the image under $f$ of the unit vector field in the positive +direction tangent to the foliation by the lines $\{v\}\times J$ of +$V\times J$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/6ace86bb0b70.md b/projects/poincare-conjecture/.astrolabe/atoms/6ace86bb0b70.md new file mode 100644 index 00000000..101c9817 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/6ace86bb0b70.md @@ -0,0 +1,15 @@ +--- +chapter: 3 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '3.8' +ref: +- 6ace86bb0b70 +sort: definition +source: tex +src: morgan-tian +tex_file: flowbasics +--- +A shrinking soliton $(M,g(t)),\ 0\le t0$ such +that the following holds for any $0<\epsilon\le \epsilon_4$. Suppose +that $N$ is an $\epsilon$-neck centered at $x$ in a connected +manifold $M$ (here we are not assuming that $R(x)=1$). We suppose +that the central $2$-sphere of $N$ separates $M$. Let $z$ be a point +outside of the middle two-thirds of $N$ and lying on the negative +side of the central $2$-sphere of $N$. (We allow both the case when +$z\in N$ and when $z\not\in N$.) Let $p$ be a point in the middle +half of $N$. Let $\mu\colon [0,a]\to N$ be a straight line segment +(with respect to the standard product metric) in the positive +$s$-direction in $N$ beginning at $p$ and ending at a point $q$ of +$N$. Then + +$$ +(1-\alpha)(s(q)-s(p))\le d(z,q)-d(z,p)\le (1+\alpha)(s(q)-s(p)). +$$ diff --git a/projects/poincare-conjecture/.astrolabe/atoms/6d6227668950.md b/projects/poincare-conjecture/.astrolabe/atoms/6d6227668950.md new file mode 100644 index 00000000..cb04aefd --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/6d6227668950.md @@ -0,0 +1,34 @@ +--- +chapter: 3 +generator: tools/poincare_tex_extract.py +label: strongepsneck +labels: +- strongepsneck +mtref: '3.6' +ref: +- 6d6227668950 +sort: definition +source: tex +src: morgan-tian +tex_file: flowbasics +--- +Let $(M,g(t))$ be a Ricci flow. An *evolving $\epsilon$-neck +centered at $(x,t_0)$ and defined for rescaled time +$t_1$* is an $\epsilon$-neck + +$$ +\varphi\colon S^2\times +(-\epsilon^{-1},\epsilon^{-1})\buildrel\cong\over\longrightarrow +N\subset (M,g(t)) +$$ + + centered at $(x,t_0)$ with the property that +pull-back via $\varphi$ of the family of metrics $R(x,t_0)g(t')|_N,\ +-t_1< t'\le 0$, where $t_1=R(x,t_0)^{-1}(t-t_0)$, is within +$\epsilon$ in the $C^{[1/\epsilon]}$-topology of the product of the +standard metric on the interval with evolving round metric on $S^2$ +with scalar curvature $1/(1-t')$ at time $t'$. A *strong +$\epsilon$-neck centered at +$(x,t_0)$* in a Ricci flow is an +evolving $\epsilon$-neck centered at $(x,t_0)$ and defined for +rescaled time $1$, see Fig. 3.1. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/6d6b9e050054.md b/projects/poincare-conjecture/.astrolabe/atoms/6d6b9e050054.md new file mode 100644 index 00000000..7345e706 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/6d6b9e050054.md @@ -0,0 +1,14 @@ +--- +chapter: 10 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '10.31' +ref: +- 6d6b9e050054 +sort: claim +source: tex +src: morgan-tian +tex_file: bddcurvbdddist +--- +The ratio $\rho_n=\lambda_n'/\lambda_n$ is bounded above and below +by positive constants. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/6dc87ad116fc.md b/projects/poincare-conjecture/.astrolabe/atoms/6dc87ad116fc.md new file mode 100644 index 00000000..13f046d5 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/6dc87ad116fc.md @@ -0,0 +1,27 @@ +--- +chapter: 19 +generator: tools/poincare_tex_extract.py +label: addaneck +labels: +- addaneck +mtref: '19.18' +ref: +- 6dc87ad116fc +sort: lemma +source: tex +src: morgan-tian +tex_file: canonnbhd +--- +There exists $\epsilon_0>0$ such that for all $0<\epsilon\le +\epsilon_0$ the following is true. Suppose that +$\{N_a,\ldots,N_b\}$ is a balanced chain in a Riemannian manifold +$M$ with $U=\cup_{i=a}^bN_i$. Suppose that the two-spheres of the +neck structure of $N_a$ separate $M$. + Suppose that $x$ +is a point of the frontier of $U$ contained in the closure of the +plus end of $N_b$ that is also the center of an $\epsilon$-neck $N$. +Then possibly after reversing the direction of $N$, we have that +$\{N_a,\ldots,N_b,N\}$ is a balanced chain. Similarly, if $x$ is in +the closure of the minus end of $N_a$, then (again after possibly +reversing the direction of $N$) we have that $\{N,N_a,\ldots,N_b\}$ +is a balanced $\epsilon$-chain. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/6e9739b0c59b.md b/projects/poincare-conjecture/.astrolabe/atoms/6e9739b0c59b.md new file mode 100644 index 00000000..a2e946f1 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/6e9739b0c59b.md @@ -0,0 +1,27 @@ +--- +chapter: 18 +generator: tools/poincare_tex_extract.py +label: annlengthest +labels: +- annlengthest +mtref: '18.63' +ref: +- 6e9739b0c59b +sort: proposition +source: tex +src: morgan-tian +tex_file: energy1 +--- +Fix $0<\delta<1/100$. For each $00$ such that the following holds. Suppose that $A$ is an +annulus with boundary components $c_0$ and $c_1$. Denote by $l(c_0)$ +and $l(c_1)$ the lengths of $c_0$ and $c_1$, respectively. Suppose +that the Gaussian curvature of $A$ is bounded above by $C''$. +Suppose that $l(c_0)>r$ and that for each sub-interval $I$ of $c_0$ +of length $r$, the integral of the absolute value of the geodesic +curvature along $I$ is less than $\delta$. Suppose that the area of +$A$ is less than $\mu$. Then + +$$ +l(c_1)\ge \frac{3}{4}l(c_0). +$$ diff --git a/projects/poincare-conjecture/.astrolabe/atoms/6ed4f379cb9f.md b/projects/poincare-conjecture/.astrolabe/atoms/6ed4f379cb9f.md new file mode 100644 index 00000000..829fb260 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/6ed4f379cb9f.md @@ -0,0 +1,16 @@ +--- +chapter: 5 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '5.8' +ref: +- 6ed4f379cb9f +sort: lemma +source: tex +src: morgan-tian +tex_file: converge2 +--- +Suppose that $(U_k,g_k,x_k)$ is a sequence of based Riemannian +manifolds and that there is a partial geometric limit +$(U_\infty,g_\infty,x_\infty)$ that is a complete Riemannian +manifold. Then this partial geometric limit is a geometric limit. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/6efe562bcc06.md b/projects/poincare-conjecture/.astrolabe/atoms/6efe562bcc06.md new file mode 100644 index 00000000..c27a453f --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/6efe562bcc06.md @@ -0,0 +1,21 @@ +--- +chapter: 17 +generator: tools/poincare_tex_extract.py +label: bddsurgery +labels: +- bddsurgery +mtref: '17.12' +ref: +- 6efe562bcc06 +sort: lemma +source: tex +src: morgan-tian +tex_file: surgery +--- +Given a Ricci flow with surgery $({\mathcal M},G)$ defined on $[0,T)$ with +$T\le T_{i+1}$ with surgery control parameter $\overline\delta$ a +non-increasing positive function defined on $[0,T_{i+1}]$ satisfying the +hypotheses of Theorem 15.9 on its entire time-domain of definition, there +is a constant $N$ depending only on the volume of $(M_0,g(0))$, on $T_{i+1}$, +on $r_{i+1}$, and on $\overline\delta(T_{i+1})$ such that this Ricci flow with +surgery defined on the interval $[0,T)$ has at most $N$ surgery times. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/6f08d8fab637.md b/projects/poincare-conjecture/.astrolabe/atoms/6f08d8fab637.md new file mode 100644 index 00000000..cc2893b5 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/6f08d8fab637.md @@ -0,0 +1,25 @@ +--- +chapter: 6 +generator: tools/poincare_tex_extract.py +label: redvol +labels: +- redvol +mtref: '6.70' +ref: +- 6f08d8fab637 +sort: definition +source: tex +src: morgan-tian +tex_file: newcompar +--- +Let $A\subset {\mathcal U}_x(\tau)$ be a measurable subset of +$M_{T-\tau}$. The $\mathcal{L}$-*reduced volume of $A$ from $x$* +(or the *reduced volume* for short) is +defined to be + +$$ +\widetilde V_x(A) = \int_{A} \tau^{-\frac{n}{2}}\mathit{exp}(-l_x(q))dq +$$ + + where $dq$ is the volume +element of the metric $G(T-\tau)$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/6ff28dba4e56.md b/projects/poincare-conjecture/.astrolabe/atoms/6ff28dba4e56.md new file mode 100644 index 00000000..15ae2afc --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/6ff28dba4e56.md @@ -0,0 +1,23 @@ +--- +chapter: 12 +generator: tools/poincare_tex_extract.py +label: stdsolncannbhd +labels: +- stdsolncannbhd +mtref: '12.32' +ref: +- 6ff28dba4e56 +sort: theorem +source: tex +src: morgan-tian +tex_file: stdsoln +--- +For any $\epsilon>0$ there is $C'(\epsilon)<\infty$ such that for +any point $x$ in the standard solution one of the following holds +(see Fig. 12.2). + +- **(1)** $(x,t)$ is contained in the core of a $(C'(\epsilon),\epsilon)$-cap. +- **(2)** $(x,t)$ is the center of an evolving $\epsilon$-neck $N$ whose initial time-slice is +$t=0$ and whose initial time-slice is disjoint from the surgery cap. +- **(3)** $(x,t)$ is the center of an evolving $\epsilon$-neck defined for rescaled time +$1+\epsilon$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/7028ebd40f18.md b/projects/poincare-conjecture/.astrolabe/atoms/7028ebd40f18.md new file mode 100644 index 00000000..0bb1bd52 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/7028ebd40f18.md @@ -0,0 +1,15 @@ +--- +chapter: 16 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '16.3' +ref: +- 7028ebd40f18 +sort: proposition +source: tex +src: morgan-tian +tex_file: surgery +--- +Let $x\in {\mathcal M}$ with $\mathbf{t}(x)=t$ and with $R(x)=r^{-2}\ge r_{i+1}^{-2}$. Then there is $\kappa>0$ depending only +on $C$ such that ${\mathcal M}$ is $\kappa$-non-collapsed at $x$; that is to say, +if $R(x)=r^{-2}$ with $r\le r_{i+1}$, then $\mathit{Vol} B(x,t,r)\ge \kappa r^3$, or $x$ is contained in a component of $M_t$ with positive sectional curvature. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/703d5ca9da05.md b/projects/poincare-conjecture/.astrolabe/atoms/703d5ca9da05.md new file mode 100644 index 00000000..53cab86e --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/703d5ca9da05.md @@ -0,0 +1,36 @@ +--- +chapter: 9 +generator: tools/poincare_tex_extract.py +label: prodatinf +labels: +- prodatinf +mtref: '9.39' +ref: +- 703d5ca9da05 +sort: proposition +source: tex +src: morgan-tian +tex_file: temp2kappa +--- +Suppose that $(M,g(t)),\ -\infty 0$, then for all $n$ sufficiently +large, the restriction of $G_n'$ to the image +$B_{G_n'}(p_n,0,A)\times [0,\theta_n)$ is within $\delta''$ in the +$C^{[1/\delta'']}$-topology of the restriction of the standard +solution to the ball of radius $A$ about the tip for time $0\le +t<\theta_n$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/71669e7fd5ec.md b/projects/poincare-conjecture/.astrolabe/atoms/71669e7fd5ec.md new file mode 100644 index 00000000..f933dd58 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/71669e7fd5ec.md @@ -0,0 +1,31 @@ +--- +chapter: 7 +generator: tools/poincare_tex_extract.py +label: compactcoro +labels: +- compactcoro +mtref: '7.12' +ref: +- 71669e7fd5ec +sort: corollary +source: tex +src: morgan-tian +tex_file: newcomp2 +--- +Suppose that $({\mathcal M},G)$ is a generalized $n$-dimensional Ricci flow and +that $x\in {\mathcal M}$ is given and set $t_0=\mathbf{t}(x)$. We suppose that +there is an open subset $U\subset \mathbf{t}^{-1}(-\infty,t_0)$ with the following +properties: + +1. For every $y\in U$ there is a minimizing ${\mathcal L}$-geodesic from $x$ +to $y$. +1. There are $r>0$ and $\Delta t>0$ such that the backward parabolic neighborhood +$P(x,t_0,r,-\Delta t)$ of $x$ exists in ${\mathcal M}$ and has the property +that $P\cap \mathbf{t}^{-1}(-\infty,t_0)$ is contained in $U$. +1. For each compact interval (including the case of degenerate intervals +consisting of a single point) $I\subset (-\infty,t_0)$ the subset of points +$y\in \mathbf{t}^{-1}(I)\cap U$ for which ${\mathcal L}(y)=\mathit{inf}_{z\in \mathbf{t}^{-1}(\mathbf{t}(y))\cap U}{\mathcal L}(z)$ is compact and non-empty. + + Then +for every $t0$ (resp., $K(P)<0$) for every +2-plane $P$. There are analogous notions of non-negative and +non-positive sectional curvature. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/7237f6ec73b4.md b/projects/poincare-conjecture/.astrolabe/atoms/7237f6ec73b4.md new file mode 100644 index 00000000..be28ec37 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/7237f6ec73b4.md @@ -0,0 +1,13 @@ +--- +chapter: 6 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '6.73' +ref: +- 7237f6ec73b4 +sort: remark +source: tex +src: morgan-tian +tex_file: newcompar +--- +Notice that $|Z|_G^2=|\sqrt{Q^{-1}}Z|^2_{G'}$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/72949e6350d4.md b/projects/poincare-conjecture/.astrolabe/atoms/72949e6350d4.md new file mode 100644 index 00000000..1a430984 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/72949e6350d4.md @@ -0,0 +1,25 @@ +--- +chapter: 6 +generator: tools/poincare_tex_extract.py +label: adapjab +labels: +- adapjab +mtref: '6.41' +ref: +- 72949e6350d4 +sort: lemma +source: tex +src: morgan-tian +tex_file: newcompar +--- +Suppose that $q\in {\mathcal U}_x(\bar\tau)$, that $Z\in \widetilde +{\mathcal U}_x(\bar\tau)$ is the pre-image of $q$, and that +$\gamma_Z$ is the ${\mathcal L}$-geodesic to $q$ determined by $Z$. +Suppose further that $Y(\tau)$ is a horizontal vector field along +$\gamma$ that is both adapted and an ${\mathcal L}$-Jacobi field. +Then, we have + +$$ +\frac{1}{2\sqrt{\bar\tau}(\sqrt{\bar\tau}-\sqrt{\tau_1})}|Y(\bar\tau)|^2 +=\frac{1}{2\sqrt{\bar\tau}}\mathit{Hess}(L_x^{\bar\tau})(Y(\bar\tau),Y(\bar\tau))+\mathit{Ric}(Y(\bar\tau),Y(\bar\tau)). +$$ diff --git a/projects/poincare-conjecture/.astrolabe/atoms/729d9a0428c4.md b/projects/poincare-conjecture/.astrolabe/atoms/729d9a0428c4.md new file mode 100644 index 00000000..79e32c5f --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/729d9a0428c4.md @@ -0,0 +1,13 @@ +--- +chapter: 12 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '12.29' +ref: +- 729d9a0428c4 +sort: theorem +source: tex +src: morgan-tian +tex_file: stdsoln +--- +For the standard flow $T=1$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/72be1154f65a.md b/projects/poincare-conjecture/.astrolabe/atoms/72be1154f65a.md new file mode 100644 index 00000000..a339cc68 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/72be1154f65a.md @@ -0,0 +1,16 @@ +--- +chapter: 10 +generator: tools/poincare_tex_extract.py +label: notray +labels: +- notray +mtref: '10.24' +ref: +- 72be1154f65a +sort: proposition +source: tex +src: morgan-tian +tex_file: bddcurvbdddist +--- +$(C_{\mathcal E},g_{\mathcal E})$ is a metric cone that is not +homeomorphic to a ray. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/744167bbcef7.md b/projects/poincare-conjecture/.astrolabe/atoms/744167bbcef7.md new file mode 100644 index 00000000..28b73318 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/744167bbcef7.md @@ -0,0 +1,18 @@ +--- +chapter: 9 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '9.92' +ref: +- 744167bbcef7 +sort: remark +source: tex +src: morgan-tian +tex_file: temp2kappa +--- +For a round $\kappa$-solution $(M,g(t))$ we have $R(x,0)=R(y,0)$ for +all $x,y\in M$, and the volume of $(M,g(0))$ is bounded above by a +constant times $R(x,0)^{-3/2}$. There is no universal lower bound to +the volume in terms of the curvature. The lower bound takes the form +$C_2|\pi_1(M)|^{-1}R(x,0)^{-3/2}$, where $|\pi_1(M)|$ is the order +of the fundamental group $\pi_1(M)$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/7464980c4085.md b/projects/poincare-conjecture/.astrolabe/atoms/7464980c4085.md new file mode 100644 index 00000000..0d1cb5ce --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/7464980c4085.md @@ -0,0 +1,37 @@ +--- +chapter: 10 +generator: tools/poincare_tex_extract.py +label: pinchdef +labels: +- pinchdef +mtref: '10.1' +ref: +- 7464980c4085 +sort: definition +source: tex +src: morgan-tian +tex_file: bddcurvbdddist +--- +Let $({\mathcal M},G)$ be a generalized three-dimensional Ricci flow +whose domain of definition is contained in $[0,\infty)$. For each +$x\in {\mathcal M}$, let $\nu(x)$ be the smallest eigenvalue of +$\mathit{Rm}(x)$ on $\wedge^2T_xM_{\mathbf{t}(x)}$, as measured with +respect to a $G(x)$-orthonormal basis for the horizontal space at +$x$, and set $X(x)=\mathit{max}(0,-\nu(x))$. We say that $({\mathcal +M},G)$ has curvature *pinched toward positive* if, for all $x\in +{\mathcal M}$, if the following two inequalities hold: + +- **(1)** + +$$ +R(x)\ge \frac{-6}{1+4\mathbf{t}(x)}, +$$ + +- **(2)** + +$$ +R(x)\geq 2X(x)\left(\mathit{log}X(x)+\mathit{log}(1+\mathbf{t}(x))-3\right), +$$ + + whenever +$00$ and any $\kappa$-solution $(M,g(t))$ the asymptotic volume +${\mathcal V}_\infty(M,g(t))$ is identically zero. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/74c7682a68ed.md b/projects/poincare-conjecture/.astrolabe/atoms/74c7682a68ed.md new file mode 100644 index 00000000..f800d1da --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/74c7682a68ed.md @@ -0,0 +1,41 @@ +--- +chapter: 5 +generator: tools/poincare_tex_extract.py +label: basicconv +labels: +- basicconv +mtref: '5.6' +ref: +- 74c7682a68ed +sort: theorem +source: tex +src: morgan-tian +tex_file: converge2 +--- +Suppose that $\{(U_k,g_k,x_k)\}_{k=1}^\infty$ is a sequence of +based, connected, $n$-dimensional Riemannian manifolds. In addition, +suppose the following: + +- **(1)** There is $\delta>0$ such that $x_k\in \mathit{Reg}_\delta(U_k,g_k)$ for all $k$. +- **(2)** For each $R<\infty$ and $\delta>0$ there is a constant $V(R,\delta)<\infty$ such +that $\mathit{Vol}(B(x_k,R)\cap \mathit{Reg}_\delta(U_k,x_k))\le +V(R,\delta)$ for all $k$ sufficiently large. +- **(3)** For each non-negative integer $\ell$, each $\delta>0$, and each $R<\infty$, + there is a constant +$C(\ell,\delta,R)$ such that for every $k$ sufficiently large we +have + +$$ +|\nabla^\ell \mathit{Rm}(g_k)|\le C(\ell,\delta,R) +$$ + +on all of $B(x_k,R)\cap\mathit{Reg}_\delta(U_k,g_k)$. +- **(4)** For every $R<\infty$ there are $r_0>0$ and $\kappa>0$ such that +for every $k$ sufficiently large, for every $\delta\le r_0$ and +every $x\in B(x_k,R)\cap \mathit{Reg}_\delta(U_k,g_k,x_k)$ the volume +of the metric ball $B(x,\delta)\subset U_k$ is at least +$\kappa\delta^n$. + +Then, after passing to a subsequence, there exists a based +Riemannian manifold $(U_\infty,g_\infty,x_\infty)$ that is a +geometric limit of the sequence $\{(U_k,g_k,x_k)\}_{k=1}^\infty$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/752133843e1e.md b/projects/poincare-conjecture/.astrolabe/atoms/752133843e1e.md new file mode 100644 index 00000000..2284d995 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/752133843e1e.md @@ -0,0 +1,23 @@ +--- +chapter: 13 +generator: tools/poincare_tex_extract.py +label: capball +labels: +- capball +mtref: '13.4' +ref: +- 752133843e1e +sort: lemma +source: tex +src: morgan-tian +tex_file: surgery +--- +The surgery cap in $({\mathcal S},\tilde g)$ has a metric that +differs from the one coming from a rescaled version of the standard +solution. Thus, the image of this cap is not necessarily a metric +ball. Nevertheless for $\epsilon<1/200$ the image of this cap will +be contained in the metric ball in ${\mathcal S}$ centered at $p_0$ +of radius $R(x_0)^{-1/2}(A_0+5)$ and will contain the metric ball +centered at $p_0$ of radius $R(x_0)^{-1/2}(A_0+3)$. Notice also that +the complement of the closure of the surgery cap in ${\mathcal S}$ +is isometrically identified with $N^-$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/7549bace9302.md b/projects/poincare-conjecture/.astrolabe/atoms/7549bace9302.md new file mode 100644 index 00000000..2f8c6b32 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/7549bace9302.md @@ -0,0 +1,18 @@ +--- +chapter: 17 +generator: tools/poincare_tex_extract.py +label: D(A)delta(A) +labels: +- D(A)delta(A) +mtref: '17.9' +ref: +- 7549bace9302 +sort: claim +source: tex +src: morgan-tian +tex_file: surgery +--- +Given any $A<\infty$ there is $D(A)<\infty$ and $\delta(A)>0$ such that for all $a$ +sufficiently large, $|\mathit{Rm}|$ is bounded by $D(A)$ along all backward flow lines +beginning at a point of $B_{\widetilde +G_a}(\tilde x_a,0,A)$ and defined for backward time at most $\delta(A)$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/755d47f17b43.md b/projects/poincare-conjecture/.astrolabe/atoms/755d47f17b43.md new file mode 100644 index 00000000..90f9b063 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/755d47f17b43.md @@ -0,0 +1,44 @@ +--- +chapter: 2 +generator: tools/poincare_tex_extract.py +label: epsclose +labels: +- epsclose +mtref: '2.16' +ref: +- 755d47f17b43 +sort: definition +source: tex +src: morgan-tian +tex_file: prelim +--- +Suppose that we have a fixed metric $g_0$ on a manifold $M$ and an +open submanifold $X\subset M$. We say that another metric $g$ on $X$ +is *within $\epsilon$ of $g_0|_X$ in the +$C^{[1/\epsilon]}$-topology* if, setting $k=[1/\epsilon]$ we have + +$$ +\mathit{sup}_{x\in X}\left(|g(x)-g_0(x)|_{g_0}^2+ +\sum_{\ell=1}^k|\nabla_{g_0}^\ell g(x)|_{g_0}^2\right)< +\epsilon^2, +$$ + + where the covariant derivative +$\nabla^\ell_{g_0}$ is the Levi-Civita connection of $g_0$ and +norms are the pointwise $g_0$-norms on + +$$ +\mathit{Sym}^2T^*M\otimes \underbrace{T^*M\otimes\cdots\otimes +T^*M}_{\ell-\mathit{times}}. +$$ + +More generally, given two smooth families of metrics $g(t)$ and +$g_0(t)$ on $M$ defined for $t$ in some interval $I$ we say that +the family $g(t)|_X$ is within $\epsilon$ of the family $g_0(t)|_X$ +in the $C^{[1/\epsilon]}$-topology if we have + +$$ +\mathit{sup}_{(x,t)\in X\times I}\left(\left| g(x,t) +-g_0(x,t)\right|_{g_0(t)}^2 +\sum_{\ell=1}^k\left|\nabla_{g_0}^\ell +g(x,t)\right|_{g_0}^2\right)<\epsilon^2. +$$ diff --git a/projects/poincare-conjecture/.astrolabe/atoms/75695d1b1b54.md b/projects/poincare-conjecture/.astrolabe/atoms/75695d1b1b54.md new file mode 100644 index 00000000..9784bc0f --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/75695d1b1b54.md @@ -0,0 +1,21 @@ +--- +chapter: 6 +generator: tools/poincare_tex_extract.py +label: EL +labels: +- EL +mtref: '6.4' +ref: +- 75695d1b1b54 +sort: lemma +source: tex +src: morgan-tian +tex_file: newcompar +--- +The +Euler-Lagrange equation for critical paths for the $\mathcal{L}$-length is + +$$ +\nabla_{X}X - \frac{1}{2}\nabla R ++ \frac{1}{2\tau}X + 2\mathit{Ric}(X,\cdot)^* = 0. +$$ diff --git a/projects/poincare-conjecture/.astrolabe/atoms/757e83881e62.md b/projects/poincare-conjecture/.astrolabe/atoms/757e83881e62.md new file mode 100644 index 00000000..63f45f32 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/757e83881e62.md @@ -0,0 +1,58 @@ +--- +chapter: 9 +generator: tools/poincare_tex_extract.py +label: epscap +labels: +- epscap +mtref: '9.72' +ref: +- 757e83881e62 +sort: definition +source: tex +src: morgan-tian +tex_file: temp2kappa +--- +Fix constants $0<\epsilon<1/2$ and $C<\infty$. Let $(M,g)$ be a +Riemannian $3$-manifold. A $(C,\epsilon)$-cap in $(M,g)$ is an open +submanifold $({\mathcal C},g|_{\mathcal C})$ together with an open +submanifold $N\subset {\mathcal C}$ with the following properties: + +- **(1)** ${\mathcal C}$ is diffeomorphic either to an open $3$-ball or to a +punctured $\Ar P^3$. +- **(2)** $N$ is an $\epsilon$-neck with compact complement in +${\mathcal C}$. +- **(3)** $\overline Y={\mathcal C}\setminus N$ is a compact submanifold with boundary. +Its interior, $Y$, is called + the *core* of ${\mathcal C}$. The frontier of $Y$, which is $\partial \overline Y$, + is a central + $2$-sphere of an $\epsilon$-neck contained in ${\mathcal C}$. +- **(4)** The scalar curvature $R(y)>0$ for every $y\in {\mathcal C}$ and + +$$ +\mathit{diam}({\mathcal C},g|_{\mathcal C})< C \left(\mathit{sup}_{y\in +{\mathcal C}}R(y)\right)^{-1/2}. +$$ + +- **(5)** $\mathit{sup}_{x,y\in {\mathcal C}}\left[ R(y)/R(x)\right]< C$. +- **(6)** $\mathit{Vol} {\mathcal C} < C(\mathit{sup}_{y\in {\mathcal C}}R(y))^{-3/2}$. +- **(7)** For any $y\in Y$ let $r_y$ be defined so +that $\mathit{sup}_{y'\in B(y,r_y)}R(y')=r_y^{-2}$. Then for each $y\in +Y$, the ball $B(y,r_y)$ lies in ${\mathcal C}$ and indeed has +compact closure in ${\mathcal C}$. Furthermore, + +$$ +C^{-1}<\mathit{inf}_{y\in +Y}\frac{\mathit{Vol} B(y,r_y)}{r_y^3}. +$$ + +- **(8)** Lastly, + +$$ +\mathit{sup}_{y\in {\mathcal C}}\frac{|\nabla R(y)|}{R(y)^{3/2}}0$ and $r_0>0$. We say that a Ricci flow with surgery +$({\mathcal M},G)$ is *$\kappa$-noncollapsed on scales $\le +r_0$* if the following holds for every point $x\in {\mathcal M}$ and +for every $r\le r_0$. Denote $\mathbf{t}(x)$ by $t$. If the parabolic +neighborhood $P(x,t,r,-r^2)$ exists in ${\mathcal M}$ and if +$|\mathit{Rm}_G|\le r^{-2}$ on $P(x,t,r,-r^2)$, then $\mathit{Vol} B(x,t,r)\ge \kappa r^3$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/76958d4968e1.md b/projects/poincare-conjecture/.astrolabe/atoms/76958d4968e1.md new file mode 100644 index 00000000..188a2e28 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/76958d4968e1.md @@ -0,0 +1,15 @@ +--- +chapter: 1 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '1.12' +ref: +- 76958d4968e1 +sort: definition +source: tex +src: morgan-tian +tex_file: prelim +--- +The Riemannian manifold +$(M,g)$ is said to be an *Einstein manifold with Einstein +constant* $\lambda$ if $\mathit{Ric}(g)=\lambda g$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/76c5f88ad2b4.md b/projects/poincare-conjecture/.astrolabe/atoms/76c5f88ad2b4.md new file mode 100644 index 00000000..318929e7 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/76c5f88ad2b4.md @@ -0,0 +1,22 @@ +--- +chapter: 6 +generator: tools/poincare_tex_extract.py +label: Qformula +labels: +- Qformula +mtref: '6.75' +ref: +- 76c5f88ad2b4 +sort: lemma +source: tex +src: morgan-tian +tex_file: newcompar +--- +With the notation as above let ${\mathcal J}'(Z,\tau')$ denote the +Jacobian determinant of ${\mathcal L}'\mathit{exp}_x$. Then, with the +substitution $\tau'=Q\tau$, we have + +$$ +(\tau')^{-n/2}e^{-\tilde l'(\sqrt{Q^{-1}}Z,\tau')}{\mathcal J}'(\sqrt{Q^{-1}}Z,\tau')= +\tau^{-n/2}e^{-\tilde l(Z,\tau)}{\mathcal J}(Z,\tau). +$$ diff --git a/projects/poincare-conjecture/.astrolabe/atoms/7744a698d8da.md b/projects/poincare-conjecture/.astrolabe/atoms/7744a698d8da.md new file mode 100644 index 00000000..2ea757d2 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/7744a698d8da.md @@ -0,0 +1,34 @@ +--- +chapter: 9 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '9.91' +ref: +- 7744a698d8da +sort: proposition +source: tex +src: morgan-tian +tex_file: temp2kappa +--- +Let $\bar\epsilon_2$ and $\bar \epsilon_3$ be as in +Corollary 9.88 and Theorem 9.89, respectively. +For each $0<\epsilon\le \mathit{min}(\bar \epsilon_2,\bar\epsilon_3)$ +let $C_1=C_1(\epsilon)$ be as in Theorem 9.89. There is +$C_2=C_2(\epsilon)<\infty$ such that for any $\kappa>0$ and any +compact $\kappa$-solution $(M,g(t))$ the following holds. If +$(M,g(0))$ is not of constant positive curvature and if $(M,g(0))$ +is of diameter less than $C_1(\mathit{max}_{x\in M}R(x,0))^{-1/2}$ then +for any $x\in M$ we have + +$$ +C_2^{-1}R(x,0)^{-3/2}< \mathit{Vol}(M,g(0))< C_2R(x,0)^{-3/2}. +$$ + +In addition, for any $y\in M$ and any $2$-plane $P_y$ in $T_yM$ we +have + +$$ +C_2^{-1}< \frac{K(P_y)}{R(x,0)}< C_2, +$$ + +where $K(P_y)$ is the sectional curvature in the $P_y$-direction. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/7744e53809c5.md b/projects/poincare-conjecture/.astrolabe/atoms/7744e53809c5.md new file mode 100644 index 00000000..e2ca0b0d --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/7744e53809c5.md @@ -0,0 +1,30 @@ +--- +chapter: 5 +generator: tools/poincare_tex_extract.py +label: seqconv +labels: +- seqconv +mtref: '5.21' +ref: +- 7744e53809c5 +sort: definition +source: tex +src: morgan-tian +tex_file: converge2 +--- +Suppose that $\{(X_k,x_k)\}_k$ converges in the Gromov-Hausdorff sense to +$(Y,y)$. Then a *realization sequence* for this convergence is a sequence +of triples $((Z_k,z_k),f_k,g_k)$ where, for each $k$, the pair $(Z_k,z_k)$ is a +based metric space, + +$$ +f_k\colon (X_k,x_k)\to +(Z_k,z_k) \ \ \ \ \mathit{and} \ \ \ g_k\colon (Y,y)\to (Z_k,z_k) +$$ + + are +isometric embeddings and $D_{GH}(f_k(X_k),g_k(Y))\rightarrow 0$ as +$k\rightarrow\infty$. Given a realization sequence for the +convergence, we say that a sequence $\ell_k\in X_k$ converges to +$\ell\in Y$ (relative to the given realization sequence) if +$d(f_k(\ell_k),g_k(\ell))\rightarrow 0$ as $i\rightarrow\infty$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/778d690c4e7c.md b/projects/poincare-conjecture/.astrolabe/atoms/778d690c4e7c.md new file mode 100644 index 00000000..800eef29 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/778d690c4e7c.md @@ -0,0 +1,22 @@ +--- +chapter: 17 +generator: tools/poincare_tex_extract.py +label: '16.2' +labels: +- '16.2' +mtref: '17.2' +ref: +- 778d690c4e7c +sort: lemma +source: tex +src: morgan-tian +tex_file: surgery +--- +For each $a$, and given $a$, for all $b$ sufficiently large there is $t_{(a,b)}\in +[T_i,T_{(a,b)})$ such that the restriction of $({\mathcal +M}_{(a,b)},G_{(a,b)})$ to $\mathbf{t}^{-1}\left([0,t_{(a,b)})\right)$ +satisfies the strong $(C,\epsilon)$-canonical neighborhood +assumption with parameter $r_a$ and furthermore, there is +$x\in{\mathcal M}_{(a,b)}$ with $\mathbf{t}(x_{(a,b)})=t_{(a,b)}$ at +which the strong $(C,\epsilon)$-canonical neighborhood assumption +with parameter $r_a$ fails. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/77ca4669ffb3.md b/projects/poincare-conjecture/.astrolabe/atoms/77ca4669ffb3.md new file mode 100644 index 00000000..775aa3db --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/77ca4669ffb3.md @@ -0,0 +1,19 @@ +--- +chapter: 12 +generator: tools/poincare_tex_extract.py +label: item5 +labels: +- item5 +mtref: '12.7' +ref: +- 77ca4669ffb3 +sort: proposition +source: tex +src: morgan-tian +tex_file: stdsoln +--- +Given $T'0$ there is a compact ball $B$ centered +at the origin of $\Ar^3$ such that the restriction of the flow +$(\Ar^3\setminus B,g(t)),\ 0\le t\le T'$, is within $\epsilon$ in +the $C^{[1/\epsilon]}$-topology of the standard evolving cylinder +$(S^2,h(t))\times (\Ar^+,ds^2)$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/77ed5048ae37.md b/projects/poincare-conjecture/.astrolabe/atoms/77ed5048ae37.md new file mode 100644 index 00000000..78c8eab6 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/77ed5048ae37.md @@ -0,0 +1,18 @@ +--- +chapter: 6 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '6.52' +ref: +- 77ed5048ae37 +sort: definition +source: tex +src: morgan-tian +tex_file: newcompar +--- +We define $\widetilde l\colon \widetilde {\mathcal U}_x\to +\Ar$ by + +$$ +\widetilde l(Z,\tau)=\frac{\widetilde L(Z,\tau)}{2\sqrt{\tau}}=l(\gamma_Z|_{[\tau_1,\bar\tau]}). +$$ diff --git a/projects/poincare-conjecture/.astrolabe/atoms/7822d06083e0.md b/projects/poincare-conjecture/.astrolabe/atoms/7822d06083e0.md new file mode 100644 index 00000000..9775ef73 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/7822d06083e0.md @@ -0,0 +1,29 @@ +--- +chapter: 15 +generator: tools/poincare_tex_extract.py +label: Tgood0good +labels: +- Tgood0good +mtref: '15.4' +ref: +- 7822d06083e0 +sort: corollary +source: tex +src: morgan-tian +tex_file: surgery +--- +Let $({\mathcal M},G)$ be a generalized Ricci flow satisfying Assumptions +(1) -- (7) with initial conditions $(M,g(0))$. Suppose +that for some $T$ the time-slice $M_T$ of this generalized flow +satisfies Thurston's Geometrization Conjecture. Then the same is true for the manifold +$M_t$ for any $t\le T$, and in particular $M$ satisfies Thurston's +Geometrization Conjecture. In addition, + +- **(1)** If for some $T>0$ the manifold $M_T$ is empty, then $M$ is a +connected sum of manifolds diffeomorphic to $2$-sphere bundles over +$S^1$ and $3$-dimensional space-forms, i.e., compact +$3$-manifolds that admit a metric of constant positive curvature. +- **(2)** If for some $T>0$ the manifold $M_T$ is empty and if $M$ is connected and simply connected, +then $M$ is diffeomorphic to $S^3$. +- **(3)** If for some $T>0$ the manifold $M_T$ is empty and if $M$ has finite fundamental group, then $M$ +is a $3$-dimensional space-form. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/786d89b0be9f.md b/projects/poincare-conjecture/.astrolabe/atoms/786d89b0be9f.md new file mode 100644 index 00000000..827b863d --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/786d89b0be9f.md @@ -0,0 +1,13 @@ +--- +chapter: 5 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '5.2' +ref: +- 786d89b0be9f +sort: lemma +source: tex +src: morgan-tian +tex_file: converge2 +--- +$\mathit{Reg}_\delta(U,g)$ is a closed subset of $U$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/78cab9a6159b.md b/projects/poincare-conjecture/.astrolabe/atoms/78cab9a6159b.md new file mode 100644 index 00000000..56bae30c --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/78cab9a6159b.md @@ -0,0 +1,28 @@ +--- +chapter: 12 +generator: tools/poincare_tex_extract.py +label: '12.9' +labels: +- '12.9' +mtref: '12.9' +ref: +- 78cab9a6159b +sort: lemma +source: tex +src: morgan-tian +tex_file: stdsoln +--- +With $X$ and its dual $X^*$ evolving by the above equations, set +$V=\nabla X^*$, so that $V$ is a contravariant two-tensor. In local +coordinates we have $V=V_{ij}dx^i\otimes dx^j$ with + +$$ +V_{ij}=(\nabla_iX)_j=g_{jk}(\nabla_iX)^k. +$$ + +This symmetric two-tensor satisfies + +$$ +\frac{\partial}{\partial t}V=\triangle +V-\left(2{{R_k}^{rl}}_jV_{rl}+\mathit{Ric}_k^lV_{lj} +\mathit{Ric}_j^lV_{kl}\right)dx^k\otimes dx^j. +$$ diff --git a/projects/poincare-conjecture/.astrolabe/atoms/78d9cbe9be13.md b/projects/poincare-conjecture/.astrolabe/atoms/78d9cbe9be13.md new file mode 100644 index 00000000..a7d41965 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/78d9cbe9be13.md @@ -0,0 +1,24 @@ +--- +chapter: 3 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '3.5' +ref: +- 78d9cbe9be13 +sort: example +source: tex +src: morgan-tian +tex_file: flowbasics +--- +Let $h_0$ be the round metric on $S^2$ with constant Gausssian +curvature $1/2$. Set $h(t)=(1-t)h_0$. Then the flow + +$$ +(S^2,h(t)),\ +-\infty0$. For any $n$ sufficiently large denote by +$\Gamma_n$ the family of loops defined by setting $\Gamma_n(c)$ +equal to the $n$-polygonal approximation to $\Gamma(c)$. There is +$N$ such that for all $n\ge N$ we have + +- **(1)** $\Gamma_n$ is a continuous family of $n$-polygonal loops in +$M$. +- **(2)** For each $c\in S^2$, the loop +$\Gamma_n(c)$ is a homotopically trivial loop in $M$ and its length +is within $\zeta$ of the length of $\Gamma(c)$. +- **(3)** For each $c\in S^2$, we have $|A(\Gamma_n(c))-A(\Gamma(c))|<\zeta$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/7aad50e49af2.md b/projects/poincare-conjecture/.astrolabe/atoms/7aad50e49af2.md new file mode 100644 index 00000000..8448d086 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/7aad50e49af2.md @@ -0,0 +1,24 @@ +--- +chapter: 6 +generator: tools/poincare_tex_extract.py +label: fullmeasure +labels: +- fullmeasure +mtref: '6.67' +ref: +- 7aad50e49af2 +sort: corollary +source: tex +src: morgan-tian +tex_file: newcompar +--- +Let $({\mathcal M},G)$ be a generalized Ricci flow and let $x\in {\mathcal M}$ +with $\mathbf{t}(x)=T-\tau_1$. Let + $A\subset {\mathcal +M}\cap \mathbf{t}^{-1}(-\infty,T-\tau_1)$ be a subset whose intersection with each +time-slice $M_t$ is measurable. Suppose that there is a subset $F\subset +{\mathcal M}$ such that $|\nabla R|$ and $|\mathit{Ric}|$ are bounded on $F$ and +such that every minimizing ${\mathcal L}$ geodesic from $x$ to any point in a +neighborhood, $\nu(A)$, of $A$ is contained in $F$. Then for each $\tau\in +(\tau_1,\bar\tau]$ the intersection of $A$ with ${\mathcal U}_x(\tau)$ is an +open subset of full measure in $A\cap M_{T-\tau}$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/7b914755dcf6.md b/projects/poincare-conjecture/.astrolabe/atoms/7b914755dcf6.md new file mode 100644 index 00000000..b6cac918 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/7b914755dcf6.md @@ -0,0 +1,16 @@ +--- +chapter: 4 +generator: tools/poincare_tex_extract.py +label: ivey +labels: +- ivey +mtref: '4.36' +ref: +- 7b914755dcf6 +sort: theorem +source: tex +src: morgan-tian +tex_file: maxprin +--- +Any 3-dimensional compact Ricci soliton +$g_0$ is Einstein. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/7bb0fa0c9412.md b/projects/poincare-conjecture/.astrolabe/atoms/7bb0fa0c9412.md new file mode 100644 index 00000000..4b2e0756 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/7bb0fa0c9412.md @@ -0,0 +1,26 @@ +--- +chapter: 18 +generator: tools/poincare_tex_extract.py +label: W_2 +labels: +- W_2 +mtref: '18.11' +ref: +- 7bb0fa0c9412 +sort: lemma +source: tex +src: morgan-tian +tex_file: energy1 +--- +$$ +\frac{d}{dt}W_2(t)\le -4\pi-\frac{1}{2}R_\mathit{min}(t)W_2(t) +$$ + +in the sense of forward difference quotients. If $t$ is not a +surgery time, then $W_2(t)$ is continuous at $t$, and if $t$ is a +surgery time, then + +$$ +W_2(t)\le \mathit{liminf}_{t'\rightarrow +t^-}W_2(t'). +$$ diff --git a/projects/poincare-conjecture/.astrolabe/atoms/7bb59bcd7682.md b/projects/poincare-conjecture/.astrolabe/atoms/7bb59bcd7682.md new file mode 100644 index 00000000..ca7ddc67 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/7bb59bcd7682.md @@ -0,0 +1,19 @@ +--- +chapter: 10 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '10.23' +ref: +- 7bb59bcd7682 +sort: definition +source: tex +src: morgan-tian +tex_file: bddcurvbdddist +--- +Let $X({\mathcal E})$ denote the set of directions at ${\mathcal +E}$. We define the metric on $X({\mathcal E})$ by setting +$d([\gamma],[\mu])=\theta(\gamma,\mu)$. We call this the *(metric) space of realized directions at* ${\mathcal E}$. The *metric space of directions at* ${\mathcal E}$ is the completion +$\bar X({\mathcal E})$ of $X({\mathcal E})$ with respect to the +given metric. We denote by $(C_{\mathcal E},g_{\mathcal E})$ the +cone on $\bar X({\mathcal E})$ with the cone metric as given in +Equation (10.2). (See Fig. 10.2.) diff --git a/projects/poincare-conjecture/.astrolabe/atoms/7c06c4b22b84.md b/projects/poincare-conjecture/.astrolabe/atoms/7c06c4b22b84.md new file mode 100644 index 00000000..78118446 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/7c06c4b22b84.md @@ -0,0 +1,18 @@ +--- +chapter: 2 +generator: tools/poincare_tex_extract.py +label: Blambda +labels: +- Blambda +mtref: '2.3' +ref: +- 7c06c4b22b84 +sort: proposition +source: tex +src: morgan-tian +tex_file: prelim +--- +Suppose that $M$ is complete and of non-negative Ricci curvature. +Then, for any minimizing geodesic ray $\lambda$, the Busemann +function $B_\lambda$ satisfies $\Delta B_\lambda\le 0$ in the weak +sense. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/7c2758f0e2b3.md b/projects/poincare-conjecture/.astrolabe/atoms/7c2758f0e2b3.md new file mode 100644 index 00000000..2b88158e --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/7c2758f0e2b3.md @@ -0,0 +1,43 @@ +--- +chapter: 9 +generator: tools/poincare_tex_extract.py +label: canonvary +labels: +- canonvary +mtref: '9.79' +ref: +- 7c2758f0e2b3 +sort: proposition +source: tex +src: morgan-tian +tex_file: temp2kappa +--- +The following holds for any $\epsilon<1/4$ and any $C<\infty$. Let +$({\mathcal M},G)$ be a generalized Ricci flow and let $x\in +{\mathcal M}$ be a point with $\mathbf{t}(x)=t$. + +(1) Suppose that $U\subset M_t$ is a $(C,\epsilon)$-canonical +neighborhood for $x$. Then for any horizontal metric $G'$ +sufficiently close to $G|_U$ in the $C^{[1/\epsilon]}$-topology, +$(U,G'|_U)$ is a $(C,\epsilon)$ neighborhood for any $x'\in U$ +sufficiently close to $x$. + +(2) Suppose that in $({\mathcal M},G)$ there is an evolving +$\epsilon$-neck $U$ centered at $(x,t)$ defined for an interval of +normalized time of length $a>1$. Then any Ricci flow on $U\times +(t-aR(x,t)^{-1},t]$ sufficiently close in the +$C^{[1/\epsilon]}$-topology to the pullback of $G$ contains a strong +$\epsilon$-neck centered at $(x,t)$. + +(3) Given $(C,\epsilon)$ and $(C',\epsilon')$ with $C'>C$ and +$\epsilon'>\epsilon$ there is $\delta>0$ such that the following +holds. Suppose that $R(x)\le 2$. If $(U,g)$ is a +$(C,\epsilon)$-canonical neighborhood of $x$ then for any metric +$g'$ within $\delta$ of $g$ in the $C^{[1/\epsilon]}$-topology +$(U,g')$ contains a $(C',\epsilon')$-neighborhood of $x$. + +(4) Suppose that $g(t),\ -10$ and $C_1<\infty$ +(independent of the based $3$-dimensional $\kappa$-solution +$(M,g(t),(p,0))$ with $R(p,0)=1$) such that $d(p,q)\ge \delta$. In +addition, $R(q',t)\le C_1$ for all $q'\in B(p,0,d)$ and all $t\le +0$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/7d130fba9693.md b/projects/poincare-conjecture/.astrolabe/atoms/7d130fba9693.md new file mode 100644 index 00000000..a33daa68 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/7d130fba9693.md @@ -0,0 +1,40 @@ +--- +chapter: 6 +generator: tools/poincare_tex_extract.py +label: upperbd +labels: +- upperbd +mtref: '6.62' +ref: +- 7d130fba9693 +sort: lemma +source: tex +src: morgan-tian +tex_file: newcompar +--- +Given constants $\epsilon>0$, $\bar\tau_0<\infty$, $C_0<\infty$, +and $l_0<\infty$, there are $C_3<\infty$ and $0<\delta_2\le +\epsilon/4$ depending only on the given constants such that the +following holds. Let $y\in {\mathcal M}$ be a point with $\mathbf{t}(y)=t_0=T-\bar\tau$ where $\tau_1+\epsilon\le \bar\tau\le +\bar\tau_0$. Suppose that there is a minimizing ${\mathcal +L}$-geodesic $\gamma$ from $x$ to $y$ with $l_x(\gamma)\le l_0$. +Suppose that $|\nabla R|$ and $|\mathit{Ric}|$ are bounded by $C_0$ +along $\gamma$. Suppose also that the ball $B(y,t_0,\epsilon)$ has +compact closure in $M_{t_0}$ and that there is an embedding + +$$ +\iota\colon B(y,t_0,\epsilon)\times (t_0-\epsilon,t_0+\epsilon)\stackrel +{\cong}{\longrightarrow} \tilde P(y,\epsilon,\epsilon)\subset +{\mathcal M} +$$ + + compatible with time and the vector field so that +the sectional curvatures of the restriction of $G$ to the image of +this embedding are bounded by $C_0$. Then for any point $b\in +B(y,t_0,\delta_2)$ and for any $t'\in (t_0-\delta_2,t_0+\delta_2)$ +there is a curve $\gamma_1$ from $x$ to the point $z=\iota(b,t')$, +parameterized by backward time, such that + +$$ +l(\gamma_1)\le l(\gamma)+C_3\sqrt{d_{t_0}(y,b)^2+|t_0-t'|^2}. +$$ diff --git a/projects/poincare-conjecture/.astrolabe/atoms/7d5d94efe4e9.md b/projects/poincare-conjecture/.astrolabe/atoms/7d5d94efe4e9.md new file mode 100644 index 00000000..66fecef3 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/7d5d94efe4e9.md @@ -0,0 +1,49 @@ +--- +chapter: 7 +generator: tools/poincare_tex_extract.py +label: weak +labels: +- weak +mtref: '7.13' +ref: +- 7d5d94efe4e9 +sort: theorem +source: tex +src: morgan-tian +tex_file: newcomp2 +--- +Let $(M,g(t)),\ 0\le t\le T<\infty$, be a Ricci flow, complete and +of bounded curvature, and let $x=(p,T)\in M\times[0,T]$. The last +two inequalities in Corollary 6.51, namely + +$$ +\frac{\partial l_x}{\partial \tau}+|\nabla l^\tau_x|^2-R+\frac{n}{2\tau} +- \triangle l^\tau_x\ge 0 +$$ + +$$ +-|\nabla l^\tau_x|^2+R+\frac{l^\tau_x-n}{\tau}+2\triangle l^\tau_x\le 0 +$$ + +hold in the weak or distributional sense on all of $M\times\{\tau\}$ +for all $\tau>0$. This means that for any $\tau>0$ and for any +non-negative, compactly supported, smooth function $\phi(q)$ on $M$ +we have the following two inequalities: + +$$ +\int_{M\times \{\tau\}}\Bigl[ \phi\cdot\left(\frac{\partial l_x}{\partial \tau}+|\nabla +l_x^\tau|^2-R+\frac{n}{2\tau}\right) - l_x^\tau\triangle \phi\Bigr]\ +d\mathit{vol}(g(t))\ge 0 +$$ + +$$ +\int_{M\times \{\tau\}}\Big[ \phi\cdot\left(-|\nabla +l_x^\tau|^2+R+\frac{l_x^\tau-n}{\tau}\right)+2l^\tau_x\triangle +\phi\Bigr]\ d\mathit{vol}(g(t))\le 0. +$$ + + Furthermore, equality holds in +either of these weak inequalities for all functions $\phi$ as above +and all $\tau$ if and only if it holds in both. In that case $l_x$ +is a smooth function on space-time and the equalities hold in the +usual smooth sense. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/7dec33ee6a59.md b/projects/poincare-conjecture/.astrolabe/atoms/7dec33ee6a59.md new file mode 100644 index 00000000..eee340df --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/7dec33ee6a59.md @@ -0,0 +1,25 @@ +--- +chapter: 6 +generator: tools/poincare_tex_extract.py +label: Kequal +labels: +- Kequal +mtref: '6.47' +ref: +- 7dec33ee6a59 +sort: lemma +source: tex +src: morgan-tian +tex_file: newcompar +--- +For any ${\mathcal L}$-geodesic $\gamma$ defined on $[0,\bar\tau]$ both +$K^{\bar\tau}_{\tau_1}(\gamma)$ and ${\mathcal K}^{\bar\tau}_{\tau_1}(\gamma)$ +are continuous in $\tau_1$ and at $\tau_1=0$ they take the same value. Also, + +$$ +\left(\frac{\tau_1}{\bar\tau}\right)^{3/2}\left(R(\gamma(\tau_1))+|X(\tau_1)|^2\right) +$$ + +is continuous for all $\tau_1> 0$ and has limit $0$ as +$\tau_1\rightarrow 0$. Here, as always, $X(\tau_1)$ is the +horizontal component of $\gamma'$ at $\tau=\tau_1$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/7e0b4b1362c0.md b/projects/poincare-conjecture/.astrolabe/atoms/7e0b4b1362c0.md new file mode 100644 index 00000000..d8b33e8f --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/7e0b4b1362c0.md @@ -0,0 +1,39 @@ +--- +chapter: 18 +generator: tools/poincare_tex_extract.py +label: C1C4delta +labels: +- C1C4delta +mtref: '18.54' +ref: +- 7e0b4b1362c0 +sort: claim +source: tex +src: morgan-tian +tex_file: energy1 +--- +Fix $C_4$ as in Claim 18.51 and fix a constant +$\widetilde A>0$. Given $\zeta>0$ there is $\delta'>0$ depending on +$C_4$, $t_1-t_0$, and $\widetilde A$ as well as the curvature bound +of the ambient Ricci flow such that the following holds. Suppose +that $f\colon [t_0,t_1]\to \Ar$ is a function and suppose that +$J\subset [t_0,t_1]$ is a finite union of intervals. Suppose that on +each interval $[a,b]$ of $J$ the function $f$ satisfies + +$$ +f(b)\le w_{f(a),a}(b). +$$ + + Suppose further that for any $t'0$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/7fec5a940413.md b/projects/poincare-conjecture/.astrolabe/atoms/7fec5a940413.md new file mode 100644 index 00000000..10b4fa37 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/7fec5a940413.md @@ -0,0 +1,21 @@ +--- +chapter: 1 +generator: tools/poincare_tex_extract.py +label: injvol +labels: +- injvol +mtref: '1.35' +ref: +- 7fec5a940413 +sort: proposition +source: tex +src: morgan-tian +tex_file: prelim +--- +Fix an integer $n>0$. For every $\epsilon>0$ there is $\delta>0$ +depending on $n$ and $\epsilon$ such that the following holds. +Suppose that $(M^n,g)$ is a complete Riemannian manifold of +dimension $n$ and that $p\in M$. Suppose that $|\mathit{Rm}(x)|\le +r^{-2}$ for all $x\in B(p,r)$. If the injectivity radius of $M$ at +$p$ is at least $\epsilon r$, then $\mathit{Vol}(B(p,r))\ge \delta +r^n$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/809fbb8ba32a.md b/projects/poincare-conjecture/.astrolabe/atoms/809fbb8ba32a.md new file mode 100644 index 00000000..ed46c569 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/809fbb8ba32a.md @@ -0,0 +1,18 @@ +--- +chapter: 19 +generator: tools/poincare_tex_extract.py +label: chainfrontier +labels: +- chainfrontier +mtref: '19.14' +ref: +- 809fbb8ba32a +sort: corollary +source: tex +src: morgan-tian +tex_file: canonnbhd +--- +Let $\{N_a,\ldots,N_b\}$ be a chain of $\epsilon$-necks. If a +connected set $Y$ meets both $U=\cup_{a\le i\le b}N_i$ and its +complement, then $Y$ either contains points of the frontier of the +negative end $N_a$ or of the positive end of $N_b$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/80e5fc6759b2.md b/projects/poincare-conjecture/.astrolabe/atoms/80e5fc6759b2.md new file mode 100644 index 00000000..5839bcb2 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/80e5fc6759b2.md @@ -0,0 +1,23 @@ +--- +chapter: 18 +generator: tools/poincare_tex_extract.py +label: '1887' +labels: +- '1887' +mtref: '18.88' +ref: +- 80e5fc6759b2 +sort: claim +source: tex +src: morgan-tian +tex_file: energy1 +--- +There is a constant $D_1$ depending only on the curvature bound for + the ambient Ricci flow such that for any sub-arc $\gamma_{t'}$ of + length $r$, defining $\varphi(x,t)$ as above, for all $x\in \gamma_t$ + and all $t\in [t',t_2]$ we have + +$$ +\left|\frac{\partial \varphi(x,t)}{\partial t}\right|\le +\frac{D_1}{r\sqrt{t-t'}}+D_1. +$$ diff --git a/projects/poincare-conjecture/.astrolabe/atoms/8141585de23a.md b/projects/poincare-conjecture/.astrolabe/atoms/8141585de23a.md new file mode 100644 index 00000000..a81dcc4c --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/8141585de23a.md @@ -0,0 +1,19 @@ +--- +chapter: 1 +generator: tools/poincare_tex_extract.py +label: star +labels: +- star +mtref: '1.23' +ref: +- 8141585de23a +sort: corollary +source: tex +src: morgan-tian +tex_file: prelim +--- +Suppose that $\gamma$ is a minimal geodesic parameterized by $[0,1]$ +starting at $p$. Let $X(0)=\gamma'(0)\in T_pM$. Then for each +$t_0<1$ the restriction $\gamma|_{[0,t_0]}$ is a minimal geodesic +and $\mathit{exp}_p\colon T_pM\to M$ is a local diffeomorphism near +$t_0X(0)$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/818772530e5a.md b/projects/poincare-conjecture/.astrolabe/atoms/818772530e5a.md new file mode 100644 index 00000000..56311dfc --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/818772530e5a.md @@ -0,0 +1,22 @@ +--- +chapter: 2 +generator: tools/poincare_tex_extract.py +label: neckseparate +labels: +- neckseparate +mtref: '2.20' +ref: +- 818772530e5a +sort: lemma +source: tex +src: morgan-tian +tex_file: prelim +--- +Suppose that $\epsilon>0$ is sufficiently small that Lemma 19.10 from +the appendix holds. Let $(M,g)$ be a non-compact $3$-manifold of positive +curvature and let $p\in M$ be a soul for it. Then for any $\epsilon$-neck $N$ +disjoint from $p$ + the central $2$-sphere of $N$ separates the soul from the end of the manifold. In particular, if + two $\epsilon$-necks $N_1$ and $N_2$ in $M$ are disjoint from each other and from $p$, then +the central $2$-spheres of $N_1$ and $N_2$ are the boundary +components of a region in $M$ diffeomorphic to $S^2\times I$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/81f8de5102de.md b/projects/poincare-conjecture/.astrolabe/atoms/81f8de5102de.md new file mode 100644 index 00000000..adde7924 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/81f8de5102de.md @@ -0,0 +1,28 @@ +--- +chapter: 3 +generator: tools/poincare_tex_extract.py +label: etaprop +labels: +- etaprop +mtref: '3.33' +ref: +- 81f8de5102de +sort: proposition +source: tex +src: morgan-tian +tex_file: flowbasics +--- +Fix constants $0<\alpha$ and the dimension $n$. Then there is a constant +$C_2'=C_2'(\alpha,n)$ and for each $r>0$ and $K<\infty$ there is a constant +$C_2=C_2(K,\alpha,r,n)$ such that the following holds. Suppose that $(U,g(t)),\ +0\le t\le T$, is an $n$-dimensional Ricci flow with $T\le \alpha/K$. Suppose +that $p\in U$ and that $B(p,0,r)$ has compact closure in $U$ and that the +exponential mapping from the ball of radius $r$ in $T_pU$ to $B(p,0,r)$ is a +diffeomorphism. Suppose that $|\mathit{Rm}(x,0)|\le K$ for all $x\in B(p,0,r)$. There is +a smooth function $\eta\colon B(p,0,r)\to [0,1]$ satisfying the following for +all $t\in [0,T]$: + +1. $\eta$ has compact support in $B(p,0,r/2)$ +1. The restriction of $\eta$ to $B(p,0,r/4)$ is identically $1$. +1. $|\Delta_{g(t)} \eta|\le C_2(K,\alpha,r,n)$. +1. $\frac{|\nabla \eta|_{g(t)}^2}{\eta}\le \frac{C'_2(\alpha,n)}{r^2}$ diff --git a/projects/poincare-conjecture/.astrolabe/atoms/8283cc66c217.md b/projects/poincare-conjecture/.astrolabe/atoms/8283cc66c217.md new file mode 100644 index 00000000..9d1ee864 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/8283cc66c217.md @@ -0,0 +1,18 @@ +--- +chapter: 9 +generator: tools/poincare_tex_extract.py +label: '11.20' +labels: +- '11.20' +mtref: '9.34' +ref: +- 8283cc66c217 +sort: claim +source: tex +src: morgan-tian +tex_file: temp2kappa +--- +$$ +\frac{\partial }{\partial t}\triangle +f=\triangle(\frac{\partial f}{\partial t})+2\langle \mathit{Ric},\mathit{Hess}(f)\rangle. +$$ diff --git a/projects/poincare-conjecture/.astrolabe/atoms/82f43e137897.md b/projects/poincare-conjecture/.astrolabe/atoms/82f43e137897.md new file mode 100644 index 00000000..a3fec27e --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/82f43e137897.md @@ -0,0 +1,19 @@ +--- +chapter: 19 +generator: tools/poincare_tex_extract.py +label: compint +labels: +- compint +mtref: '19.6' +ref: +- 82f43e137897 +sort: corollary +source: tex +src: morgan-tian +tex_file: canonnbhd +--- +The following holds for all $\epsilon>0$ sufficiently small. Let $N$ +be an $\epsilon$-neck centered at $x$. If $\gamma$ is a shortest +geodesic in $N$ between its endpoints and if +$|\gamma|>R(x)^{-1/2}\epsilon^{-1}/100$, then $\gamma$ crosses each +two-sphere in the neck structure on $N$ at most once. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/842f5a7c856b.md b/projects/poincare-conjecture/.astrolabe/atoms/842f5a7c856b.md new file mode 100644 index 00000000..bb699513 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/842f5a7c856b.md @@ -0,0 +1,26 @@ +--- +chapter: 14 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '14.8' +ref: +- 842f5a7c856b +sort: definition +source: tex +src: morgan-tian +tex_file: surgery +--- +A *Ricci flow with surgery* is a pair $({\mathcal M},G)$ +consisting of a surgery space-time ${\mathcal M}$ and a horizontal +metric $G$ on ${\mathcal M}$ such that for every $x\in {\mathcal M}$ + we have + +$$ +{\mathcal L}_\chi(G)(x)=-2\mathit{Ric}_G(x)) +$$ + + as sections +of the symmetric square of the dual to ${\mathcal HT}({\mathcal +M})$. If space-time is $(n+1)$-dimensional, then we say that the +Ricci flow with surgery is $n$-dimensional (meaning of course that +each time-slice is an $n$-dimensional manifold). diff --git a/projects/poincare-conjecture/.astrolabe/atoms/847af3ecd4a8.md b/projects/poincare-conjecture/.astrolabe/atoms/847af3ecd4a8.md new file mode 100644 index 00000000..e52ff043 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/847af3ecd4a8.md @@ -0,0 +1,27 @@ +--- +chapter: 0 +generator: tools/poincare_tex_extract.py +label: surgery +labels: +- surgery +mtref: '0.3' +ref: +- 847af3ecd4a8 +sort: theorem +source: tex +src: morgan-tian +tex_file: intro +--- +Let $(M,g_0)$ be a closed Riemannian $3$-manifold. +Suppose that there is no embedded, locally separating $\Ar P^2$ +contained (I.e., no embedded $\Ar P^2$ in $M$ with trivial +normal bundle. Clearly, all orientable manifolds satisfy this +condition.) in $M$. Then there is a Ricci flow with surgery defined +for all $t\in [0,\infty)$ with initial metric $(M,g_0)$. The set of +discontinuity times for this Ricci flow with surgery is a discrete +subset of $[0,\infty)$. The topological change in the $3$-manifold +as one crosses a surgery time is a connected sum decomposition +together with removal of connected components, each of which is +diffeomorphic to one of $S^2\times S^1$, $\Ar P^3\#\Ar P^3$, the +non-orientable $2$-sphere bundle over $S^1$, or a manifold admitting +a metric of constant positive curvature. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/84e573c0f2b2.md b/projects/poincare-conjecture/.astrolabe/atoms/84e573c0f2b2.md new file mode 100644 index 00000000..e0bfd072 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/84e573c0f2b2.md @@ -0,0 +1,21 @@ +--- +chapter: 4 +generator: tools/poincare_tex_extract.py +label: Rminprime +labels: +- Rminprime +mtref: '4.3' +ref: +- 84e573c0f2b2 +sort: claim +source: tex +src: morgan-tian +tex_file: maxprin +--- +$$ +\frac{d}{d t}(R_\mathit{min}(t))\ge \frac{2}{n}R_\mathit{min}^2(t), +$$ + +where, at times $t$ where $R_\mathit{min}(t)$ is not smooth, this +inequality is interpreted as an inequality for the forward +difference quotients. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/858af0fef413.md b/projects/poincare-conjecture/.astrolabe/atoms/858af0fef413.md new file mode 100644 index 00000000..e30acb26 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/858af0fef413.md @@ -0,0 +1,18 @@ +--- +chapter: 18 +generator: tools/poincare_tex_extract.py +label: grouptheory +labels: +- grouptheory +mtref: '18.3' +ref: +- 858af0fef413 +sort: lemma +source: tex +src: morgan-tian +tex_file: energy1 +--- +Suppose that $G$ is a finitely generated group, +say generated by $k$ elements. Let $G=G_1*\cdots*G_\ell$ be a free +product decomposition of $G$ with non-trivial free factors, i.e., +with $G_i\not=\{1\}$ for each $i=1,\ldots,\ell$. Then $\ell\le k$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/85e3c925177e.md b/projects/poincare-conjecture/.astrolabe/atoms/85e3c925177e.md new file mode 100644 index 00000000..135a305f --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/85e3c925177e.md @@ -0,0 +1,24 @@ +--- +chapter: 12 +generator: tools/poincare_tex_extract.py +label: R_0cannbhd +labels: +- R_0cannbhd +mtref: '12.28' +ref: +- 85e3c925177e +sort: theorem +source: tex +src: morgan-tian +tex_file: stdsoln +--- +Fix $0<\epsilon<1$. Then there is $r>0$ such that for any point +$(x_0,t_0)$ in the standard flow with $R(x_0,t_0)\ge r^{-2}$ the +following hold. + +- **(1)** $t_0>r^2$. +- **(2)** $(x_0,t_0)$ has a strong canonical +$(C(\epsilon),\epsilon)$-neighborhood. If this canonical neighborhood is a strong +$\epsilon$-neck centered at $(x_0,t_0)$, then the strong neck +extends to an evolving neck defined for backward rescaled time +$(1+\epsilon)$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/86043191a3c2.md b/projects/poincare-conjecture/.astrolabe/atoms/86043191a3c2.md new file mode 100644 index 00000000..7a95fcba --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/86043191a3c2.md @@ -0,0 +1,47 @@ +--- +chapter: 16 +generator: tools/poincare_tex_extract.py +label: L_0 +labels: +- L_0 +mtref: '16.15' +ref: +- 86043191a3c2 +sort: lemma +source: tex +src: morgan-tian +tex_file: surgery +--- +Given $L_0<\infty$, there is +$\overline\delta_1=\overline\delta_1(L_0,r_{i+1})>0$, independent of +$({\mathcal M},G)$ and $x$, such that if $\overline\delta(t)\le +\overline\delta_1$ for all $t\in [T_{i-1},T)$, then for any curve +$\gamma(\tau),\ 0\le \tau\le \tau_0$, with $\tau_0\le T-T_{i-1}$, +parameterized by backward time with $\gamma(0)=x$ and with + +$$ +{\mathcal L}_+(\gamma)=\int_0^{\tau_0}\sqrt{\tau} +\left(R_+(\gamma(\tau))+|X_\gamma|^2\right)d\tau< L_0 +$$ + + the +following two statements hold: + +- **(2)** Set + +$$ +\tau'=\mathit{min}\left(\frac{r_{i+1}^4}{(256)L_0^2},\mathit{ln}(\sqrt[3]{2})r_{i+1}^2\right). +$$ + +Then for all $\tau\le \mathit{min}(\tau',\tau_0)$ we have $\gamma(\tau)\in P(x,T,r/2,-r^2)$. +- **(2)** Suppose that $\bar t\in [T-\tau_0,T)$ is a surgery time with $p$ +being the tip of the surgery cap at time $\bar t$ and with the scale +of the surgery being $\bar h$. Suppose $t'\in [\bar t,\bar t+\bar +h^2/2]$ is such that there is an embedding + +$$ +\rho\colon B(p,\bar t,(50+A_0)\bar h)\times [\bar t,t']\to {\mathcal M} +$$ + +compatible with time and the vector field. Then the image of $\rho$ +is disjoint from the image of $\gamma$. See Fig. 16.3. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/8651cf20bd2d.md b/projects/poincare-conjecture/.astrolabe/atoms/8651cf20bd2d.md new file mode 100644 index 00000000..ad828e55 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/8651cf20bd2d.md @@ -0,0 +1,37 @@ +--- +chapter: 11 +generator: tools/poincare_tex_extract.py +label: hatM +labels: +- hatM +mtref: '11.22' +ref: +- 8651cf20bd2d +sort: definition +source: tex +src: morgan-tian +tex_file: singlimit2 +--- +Let + +$$ +\widehat {\mathcal M}= {\mathcal +M}\cup_{\Omega\times[T^-,T)} \left(\Omega\times +[T^-,T]\right). +$$ + + Since both +${\mathcal M}$ and $\Omega\times [T^-,T]$ have the structure of +space-times and the time functions and vector fields agree on the +overlap, $\widehat {\mathcal M}$ inherits the structure of a +space-time. Let $G'(t),\ T^-\le t\le T$, be the smooth family of +metrics on $\Omega$. The horizontal metrics, $G$, on ${\mathcal M}$ +and this family of metrics on $\Omega$ agree on the overlap and +hence define a horizontal metric $\widehat G$ on $\widehat {\mathcal +M}$. Clearly, this metric satisfies the Ricci flow equation, so that +$(\widehat {\mathcal M},\widehat G)$ is a generalized Ricci flow +extending $({\mathcal M},G)$. We call this the *maximal +extension of* $({\mathcal M},G)$ to time $T$. Notice that even +though the time-slices $M_t$ of ${\mathcal M}$ are compact, it will +not necessarily be the case that the time-slice $\Omega$ is +complete. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/867793768688.md b/projects/poincare-conjecture/.astrolabe/atoms/867793768688.md new file mode 100644 index 00000000..5af59bc8 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/867793768688.md @@ -0,0 +1,34 @@ +--- +chapter: 6 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '6.3' +ref: +- '867793768688' +sort: example +source: tex +src: morgan-tian +tex_file: newcompar +--- +Suppose that our generalized Ricci flow is a constant family of +Euclidean metrics on $\Ar^n\times [0,T]$. That is to say, $g(t) = +g_{0}$ is the usual Euclidean metric. Then we have +$R(\gamma(\tau))\equiv 0$. Using the change of variables +$s=\sqrt{\tau}$, we have + +$$ +\mathcal{L}(\gamma) = \frac{1}{2}\int_{\sqrt{\tau_1}}^{\sqrt{\tau_2}} \abs{A(s)}^{2}ds, +$$ + + which is the standard energy functional in Riemannian geometry +for the path $\gamma(s)$. The minimizers for this functional are the +maps $s\mapsto (\alpha(s),T-s^2)$ where $\alpha(s)$ is a straight +line in $\Ar^n$ parameterized at constant speed. Written in the +$\tau$ variables the minimizers are + +$$ +\gamma(\tau)=(x+\sqrt{\tau}v,T-\tau), +$$ + +straight lines parameterized at speed varying linearly with +$\sqrt{\tau}$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/86e5a5766bda.md b/projects/poincare-conjecture/.astrolabe/atoms/86e5a5766bda.md new file mode 100644 index 00000000..c2558708 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/86e5a5766bda.md @@ -0,0 +1,38 @@ +--- +chapter: 4 +generator: tools/poincare_tex_extract.py +label: flowtoround +labels: +- flowtoround +mtref: '4.23' +ref: +- 86e5a5766bda +sort: theorem +source: tex +src: morgan-tian +tex_file: maxprin +--- +Suppose that $(M,g(t)),\ 0\le t0$ for all $(x,t)\in M\times +(0,T)$ or $\mathit{Ric}(x,t)=0$ for all $(x,t)\in M\times[0,T)$. +Suppose that $\mathit{Ric}(x,t)>0$ for some $(x,t)$ and that the flow +is maximal in the sense that there is no $T'>T$ and an extension of +the given flow to a flow defined on the time interval $[0,T')$. For +each $(x,t)$, let $\lambda(x,t)$, resp. $\nu(x,t)$, denote the +largest, resp. smallest, eigenvalue of $\mathit{Rm}(x,t)$ on +$\wedge^2T_xM$. Then as $t$ tends to $T$ the Riemannian manifolds +$(M,g(t))$ are becoming round in the sense that + +$$ +\mathit{lim}_{t\rightarrow T}\frac{\mathit{max}_{x\in M}\lambda(x,t)}{\mathit{min}_{x\in M}\nu(x,t)}=1. +$$ + + Furthermore, for any $x\in M$ the +largest eigenvalue $\lambda(x,t)$ tends to $\infty$ as $t$ tends to +$T$, and rescaling $(M,g(t))$ by $\lambda(x,t)$ produces a family of +Riemannian manifolds converging smoothly as $t$ goes to $T$ to a +compact round manifold. In particular, the underlying smooth +manifold supports a Riemannian metric of constant positive curvature +so that the manifold is diffeomorphic to a $3$-dimensional spherical +space-form. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/8734da813eff.md b/projects/poincare-conjecture/.astrolabe/atoms/8734da813eff.md new file mode 100644 index 00000000..beedac4b --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/8734da813eff.md @@ -0,0 +1,29 @@ +--- +chapter: 12 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '12.18' +ref: +- 8734da813eff +sort: lemma +source: tex +src: morgan-tian +tex_file: stdsoln +--- +(a) Suppose that $f(w,t)$ is a smooth function defined for $w\ge 0$. +Define $\phi(r,t)=f(r^2,t)$. Then $\phi(r,t)$ is a smooth function +defined for all $r\in \Ar$. Now fix $k$ and let $\hat r\colon +\Ar^k\to [0,\infty)$ be the usual radial coordinate. Then we have a +smooth family of smooth functions on $\Ar^k$ defined by + +$$ +\hat\phi(x^1,\ldots,x^k,t)=\phi(\hat r(x^1,\ldots,x^k),t)=f(\sum_{i=1}^k(x^i)^2,t). +$$ + + (b) If $\psi(r,t)$ is a smooth function defined for $r\ge +0$ and if it is even in the sense that its Taylor expansion to all +orders along the line $r=0$ involves only even powers of $r$, then +there is a smooth function $f(w,t)$ defined for $w\ge 0$ such that +$\psi(r,t)=f(r^2,t)$. In particular, for any $k\ge 2$ the function +$\hat\psi((x^1,\ldots,x^k),t)=\psi(r(x^1,\ldots,x^k),t)$ is a smooth +family of smooth functions on $\Ar^k$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/882677eecdc1.md b/projects/poincare-conjecture/.astrolabe/atoms/882677eecdc1.md new file mode 100644 index 00000000..1542b130 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/882677eecdc1.md @@ -0,0 +1,53 @@ +--- +chapter: 6 +generator: tools/poincare_tex_extract.py +label: triL +labels: +- triL +mtref: '6.43' +ref: +- 882677eecdc1 +sort: proposition +source: tex +src: morgan-tian +tex_file: newcompar +--- +Suppose that $q\in {\mathcal U}_x(\bar\tau)$, that $Z\in \widetilde +{\mathcal U}_x(\bar\tau)$ is the pre-image of $q$ and that +$\gamma_Z$ is the ${\mathcal L}$-geodesic determined by $Z$. Then + +$$ +\triangle L_x^{\bar\tau}(q) \leq +\frac{n}{\sqrt{\bar\tau}-\sqrt{\tau_1}} -2\sqrt{\bar\tau}R(q) - +\frac{1}{(\sqrt{\bar\tau}-\sqrt{\tau_1})^2}{\mathcal +K}_{\tau_1}^{\bar\tau}(\gamma_Z), +$$ + +where, for any path $\gamma$ parameterized by backward time on the +interval + $[\tau_1,\bar\tau]$ taking + value $x$ at $\tau=\tau_1$ we define + +$$ +{\mathcal K}_{\tau_1}^{\bar\tau}(\gamma)= \int_{\tau_1}^{\bar\tau} +\sqrt{\tau}(\sqrt{\tau}-\sqrt{\tau_1})^2H(X)d\tau, +$$ + + with + +$$ +H(X)= -\frac{\partial R}{\partial \tau} - \frac{1}{\tau}R - 2\langle +\nabla R,X\rangle + 2\mathit{Ric}(X,X), +$$ + + where $X$ is the +horizontal projection of $\gamma'(\tau)$. Furthermore, +Inequality (6.14) is an equality if and only if for every +$Y\in T_q(M_{T-\bar\tau})$ the unique adapted vector field $Y(\tau)$ +along $\gamma$ satisfying $Y(\bar\tau)=Y$ is an ${\mathcal +L}$-Jacobi field. In this case + +$$ +\mathit{Ric}+\frac{1}{2\sqrt{\bar\tau}}\mathit{Hess}(L_x^{\bar\tau}) +=\frac{1}{2\sqrt{\bar\tau}(\sqrt{\bar\tau}-\sqrt{\tau_1})}G(T-\bar\tau). +$$ diff --git a/projects/poincare-conjecture/.astrolabe/atoms/883dba87d14c.md b/projects/poincare-conjecture/.astrolabe/atoms/883dba87d14c.md new file mode 100644 index 00000000..247f9ea0 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/883dba87d14c.md @@ -0,0 +1,32 @@ +--- +chapter: 3 +generator: tools/poincare_tex_extract.py +label: Mevol +labels: +- Mevol +mtref: '3.19' +ref: +- 883dba87d14c +sort: proposition +source: tex +src: morgan-tian +tex_file: flowbasics +--- +The evolution of the curvature operator ${\mathcal +T}(t)=\Phi_t^*\mathit{Rm}(g(t))$ is given by + +$$ +\frac{\partial {\mathcal T}_{\alpha\beta}}{\partial t} = \triangle +{\mathcal T}_{\alpha\beta} + {\mathcal T}^{2}_{\alpha\beta} + +{\mathcal T}^{\sharp}_{\alpha\beta}, +$$ + + where ${\mathcal T}^{2}_{\alpha\beta}={\mathcal T}_{\alpha\gamma}{\mathcal +T}_{\gamma\beta}$ is the operator square; ${\mathcal T}^{\sharp}_{\alpha\beta} + = +c_{\alpha\gamma\zeta}c_{\beta\delta\eta}{\mathcal T}_{\gamma\delta}{\mathcal +T}_{\zeta\eta}$ is the Lie algebra square; and $c_{\alpha\beta\gamma} = +\langle[\varphi^{\alpha},\varphi^{\beta}],\varphi^{\gamma}\rangle $ are the +structure constants of the Lie algebra $\mathit{so}(n)$ relative to the basis +$\{\varphi^{\alpha}\}$. The structure constants $c_{\alpha\beta\gamma}$ are +fully antisymmetric in the three indices. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/88571a22a2bc.md b/projects/poincare-conjecture/.astrolabe/atoms/88571a22a2bc.md new file mode 100644 index 00000000..74bd85d3 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/88571a22a2bc.md @@ -0,0 +1,19 @@ +--- +chapter: 9 +generator: tools/poincare_tex_extract.py +label: tausqtau +labels: +- tausqtau +mtref: '9.15' +ref: +- 88571a22a2bc +sort: claim +source: tex +src: morgan-tian +tex_file: temp2kappa +--- +For any $\tau>0$ we have + +$$ +l_{x_k}(q_k,\tau)\le \frac{n}{2\tau^2}+\frac{n\tau}{2}. +$$ diff --git a/projects/poincare-conjecture/.astrolabe/atoms/88a5a2233418.md b/projects/poincare-conjecture/.astrolabe/atoms/88a5a2233418.md new file mode 100644 index 00000000..99b7b210 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/88a5a2233418.md @@ -0,0 +1,31 @@ +--- +chapter: 19 +generator: tools/poincare_tex_extract.py +label: directions +labels: +- directions +mtref: '19.4' +ref: +- 88a5a2233418 +sort: lemma +source: tex +src: morgan-tian +tex_file: canonnbhd +--- +For any $\alpha>0$ there is $\epsilon_2=\epsilon_2(\alpha)>0$ +such that the following hold for all $0<\epsilon\le \epsilon_2$. +Suppose that $(N,g)$ is an $\epsilon$-neck centered at $x$ and +$R(x)=1$. Suppose that $\gamma$ is a minimal geodesic in $N$ from +$p$ to $q$. We suppose that $\gamma$ is parameterized by arc length, +is of length $\ell>\epsilon^{-1}/100$, and that $s(p)0$ is sufficiently small, +the angle (measured in $h(t)$) between $Y$ and $Z=(1,f_z)$ is +greater than $\pi/4$. Also, + +- **(1)** + +$$ +k\le |Y|<\sqrt{2}k. +$$ + +- **(2)** + +$$ +\left|\langle \nabla_ZZ,Z\rangle\right|<(k+ 2\delta) |Z|^3. +$$ + +- **(3)** + +$$ +\left|\nabla_ZZ\right|<2(|Y|+\delta). +$$ + +- **(4)** + +$$ +|\langle Y,Z\rangle |\le |Y||f_z|(1+3\delta). +$$ diff --git a/projects/poincare-conjecture/.astrolabe/atoms/89a6041c42b9.md b/projects/poincare-conjecture/.astrolabe/atoms/89a6041c42b9.md new file mode 100644 index 00000000..c92ab340 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/89a6041c42b9.md @@ -0,0 +1,18 @@ +--- +chapter: 2 +generator: tools/poincare_tex_extract.py +label: ends +labels: +- ends +mtref: '2.1' +ref: +- 89a6041c42b9 +sort: lemma +source: tex +src: morgan-tian +tex_file: prelim +--- +Suppose that $M$ is a complete, connected, non-compact Riemannian +manifold and let $p$ be a point of $M$. Then $M$ has a minimizing +geodesic ray emanating from $p$. If $M$ has more than one end, then +it has a minimizing line. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/89a60eb69c3f.md b/projects/poincare-conjecture/.astrolabe/atoms/89a60eb69c3f.md new file mode 100644 index 00000000..643c9bca --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/89a60eb69c3f.md @@ -0,0 +1,37 @@ +--- +chapter: 16 +generator: tools/poincare_tex_extract.py +label: delta0ri+1 +labels: +- delta0ri+1 +mtref: '16.4' +ref: +- 89a60eb69c3f +sort: proposition +source: tex +src: morgan-tian +tex_file: surgery +--- +For each $r_{i+1}$ with $00$ (depending implicitly on $t_0$, $C$, +$\epsilon$, and $i$) such that if $\overline\delta(t)\le \delta$ for +all $t\in [T_{i-1},T_{i+1}]$ then the following holds. Let +$({\mathcal M},G)$ be a Ricci flow with surgery satisfying the +hypothesis of Proposition 16.1 with respect to the given +sequences and $r_{i+1}$, and let $x\in {\mathcal M}$ have $\mathbf{t}(x)=T$ with $T\in [T_i,T_{i+1})$. Suppose that for some $r\ge +r_{i+1}$ the parabolic neighborhood $P(x,r,T,-r^2)$ exists in +${\mathcal M}$ and $|\mathit{Rm}|\le r^{-2}$ on this neighborhood. Then +there is an open subset $U$ of $\mathbf{t}^{-1}[T_{i-1},T)$ contained +in the open subset of smooth manifold points of ${\mathcal M}$ with +the following properties: + +- **(1)** For every $y$ in $U$ there is a minimizing ${\mathcal L}$-geodesic +connecting $x$ to $y$. +- **(2)** $U_t=U\cap \mathbf{t}^{-1}(t)$ is non-empty for every $t\in [T_{i-1},T)$ +- **(3)** For each $t\in [T_{i-1},T)$ the restriction of ${\mathcal +L}$ to $U_t$ achieves its minimum and that minimum is at most +$3\sqrt{(T-t)}$. +- **(4)** The subset of $U$ consisting of all $y$ with the property that +${\mathcal L}(y)\le {\mathcal L}(y')$ for all $y'\in \mathbf{t}^{-1}(\mathbf{t}(y))$ has the property that its intersection with +$\mathbf{t}^{-1}(I)$ is compact for every compact interval $I\subset +[T_{i-1},T)$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/8a667675febd.md b/projects/poincare-conjecture/.astrolabe/atoms/8a667675febd.md new file mode 100644 index 00000000..0bb57016 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/8a667675febd.md @@ -0,0 +1,24 @@ +--- +chapter: 3 +generator: tools/poincare_tex_extract.py +label: GSSgeneration +labels: +- GSSgeneration +mtref: '3.9' +ref: +- 8a667675febd +sort: proposition +source: tex +src: morgan-tian +tex_file: flowbasics +--- +Suppose we have a complete Riemannian manifold $(M,g(0))$, a +smooth function $f\colon M\to \Ar$, and a constant $\lambda>0$ such +that + +$$ +-\mathit{Ric}(g(0))=\mathit{Hess}(f)-\lambda g(0). +$$ + + Then there is $T>0$ and a gradient shrinking soliton +$(M,g(t))$ defined for $0\le t0$ we have + +$$ +|\bigtriangledown l_x(q,\tau))|^{2} +R(q,\tau) \leq \frac{3l_x(q,\tau)}{\tau}, +$$ + +$$ +-\frac{2l_x(q,\tau)}{\tau}\le \frac{\partial l_x(q,\tau)}{\partial \tau} +\le \frac{l_x(q,\tau)}{\tau}. +$$ + + where these inequalities are valid +in the sense of smooth functions on the open subset of full measure +of $M\times \{\tau\}$ on which $l_x$ is a smooth function, and are +valid as inequalities of $L^\infty_\mathit{loc}$-functions on all of +$M\times \{\tau\}$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/8b9b3403dbc1.md b/projects/poincare-conjecture/.astrolabe/atoms/8b9b3403dbc1.md new file mode 100644 index 00000000..ea3c2c67 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/8b9b3403dbc1.md @@ -0,0 +1,17 @@ +--- +chapter: 6 +generator: tools/poincare_tex_extract.py +label: inclusion +labels: +- inclusion +mtref: '6.30' +ref: +- 8b9b3403dbc1 +sort: proposition +source: tex +src: morgan-tian +tex_file: newcompar +--- +Let $\bar\tau'>\bar\tau$. Then $\widetilde{\mathcal +U}_x(\bar\tau')\subset \widetilde{\mathcal U}_x(\bar\tau)\subset +T_xM_{T-\tau_1}$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/8c090fbd21a0.md b/projects/poincare-conjecture/.astrolabe/atoms/8c090fbd21a0.md new file mode 100644 index 00000000..2bfbaf9f --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/8c090fbd21a0.md @@ -0,0 +1,24 @@ +--- +chapter: 9 +generator: tools/poincare_tex_extract.py +label: derivcor +labels: +- derivcor +mtref: '9.71' +ref: +- 8c090fbd21a0 +sort: corollary +source: tex +src: morgan-tian +tex_file: temp2kappa +--- +Given $\kappa>0$, there is $C<\infty$ such that for any +$3$-dimensional $\kappa$-solution $(M,g(t)),\ -\infty0$ cannot be chosen independent of $A$, then +after passing to a subsequence, the $x_a$ have strong +$(C,\epsilon)$-canonical neighborhoods. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/8de850960905.md b/projects/poincare-conjecture/.astrolabe/atoms/8de850960905.md new file mode 100644 index 00000000..3cad4632 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/8de850960905.md @@ -0,0 +1,15 @@ +--- +chapter: 17 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '17.6' +ref: +- 8de850960905 +sort: claim +source: tex +src: morgan-tian +tex_file: surgery +--- +For all $a$ sufficiently large, +every point $\tilde x\in(\widetilde{\mathcal M}_a,\widetilde G_a)$ with $R(\tilde x)>1$ and +$\mathbf{t}(\tilde x)=0$ has a $(2C,2\epsilon)$-canonical neighborhood. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/8e408e4f9091.md b/projects/poincare-conjecture/.astrolabe/atoms/8e408e4f9091.md new file mode 100644 index 00000000..64bc60ce --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/8e408e4f9091.md @@ -0,0 +1,15 @@ +--- +chapter: 2 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '2.8' +ref: +- 8e408e4f9091 +sort: remark +source: tex +src: morgan-tian +tex_file: prelim +--- +We only use the soul theorem for manifolds with positive curvature +and the fact that any soul of such a manifold is a point. A proof of +this result first appears in [GromollMeyer]. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/8e814eab5ddf.md b/projects/poincare-conjecture/.astrolabe/atoms/8e814eab5ddf.md new file mode 100644 index 00000000..0bb9e3e2 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/8e814eab5ddf.md @@ -0,0 +1,28 @@ +--- +chapter: 18 +generator: tools/poincare_tex_extract.py +label: BARMU +labels: +- BARMU +mtref: '18.59' +ref: +- 8e814eab5ddf +sort: lemma +source: tex +src: morgan-tian +tex_file: energy1 +--- +There is $\delta>0$ such that for any $r>0$ there is $\bar\mu>0$, +depending on $r$ and on the curvature bound for the ambient Ricci +flow such that the following holds. Suppose that $\gamma$ and $\hat +\gamma$ are ramps in $(M,g(t))\times S^1_\lambda$. Suppose that the +length of $\gamma$ is at least $r$ and suppose that on any +sub-interval $I$ of $\gamma$ of length $r$ we have + +$$ +\int_Ikds<\delta. +$$ + + Suppose also that there is an annulus connecting $\gamma$ and +$\hat\gamma$ of area less than $\bar\mu$. Then the length of +$\hat\gamma$ is at least $3/4$ the length of $\gamma$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/8ebd93835797.md b/projects/poincare-conjecture/.astrolabe/atoms/8ebd93835797.md new file mode 100644 index 00000000..0dc6df04 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/8ebd93835797.md @@ -0,0 +1,17 @@ +--- +chapter: 1 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '1.26' +ref: +- 8ebd93835797 +sort: definition +source: tex +src: morgan-tian +tex_file: prelim +--- +The *injectivity radius $\mathit{inj}_M(p)$ of $M$ at $p$* is the +supremum of the $r> 0$ for which the restriction of $\mathit{exp}_p\colon T_pM\to M$ to the ball $B(0,r)$ of radius $r$ in $T_pM$ +is a diffeomorphism into $M$. Clearly, $\mathit{inj}_M(p)$ is the +distance in $T_pM$ from $0$ to the frontier of $U_p$. It is also the +distance in $M$ from $p$ to the cut locus ${\mathcal C}_p$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/8ec855a2aab5.md b/projects/poincare-conjecture/.astrolabe/atoms/8ec855a2aab5.md new file mode 100644 index 00000000..33c1f047 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/8ec855a2aab5.md @@ -0,0 +1,24 @@ +--- +chapter: 4 +generator: tools/poincare_tex_extract.py +label: localprod +labels: +- localprod +mtref: '4.19' +ref: +- 8ec855a2aab5 +sort: corollary +source: tex +src: morgan-tian +tex_file: maxprin +--- +Fix $T>0$. Suppose that $(U,g(t)),\ 0\le t\le T$, is a Ricci flow +such that for each $t$, the Riemannian manifold $(U,g(t))$ is a +(not necessarily complete) connected, $3$-manifold of non-negative +sectional curvature. Suppose that $(U,g(0))$ is not flat and that +for some $p\in M$ the Ricci curvature at $(p,T)$ has a zero +eigenvalue. Then for each $t\in (0,T]$ the Riemannian manifold +$(U,g(t))$ splits locally as a product of a surface of positive +curvature and a line, and under this local splitting the flow is +locally the product of a Ricci flow on the surface and the trivial +flow on the line. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/8f737f2b0f7d.md b/projects/poincare-conjecture/.astrolabe/atoms/8f737f2b0f7d.md new file mode 100644 index 00000000..e570908b --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/8f737f2b0f7d.md @@ -0,0 +1,35 @@ +--- +chapter: 4 +generator: tools/poincare_tex_extract.py +label: varyingZ +labels: +- varyingZ +mtref: '4.8' +ref: +- 8f737f2b0f7d +sort: theorem +source: tex +src: morgan-tian +tex_file: maxprin +--- +Let $(M,g)$ be a compact Riemannian manifold. Let ${\mathcal V}\to M$ be a +tensor bundle and let ${\mathcal Z}\subset {\mathcal V}\times [0,T]$ be a +closed subset with the property that for each $t\in [0,T]$ the time-slice +${\mathcal Z}(t)$ is a convex subset of ${\mathcal V}\times\{t\}$ invariant +under the parallel translation induced by the Levi-Civita connection. Suppose +that $\psi$ is a fiberwise vector field defined on an open neighborhood of +${\mathcal Z}$ in ${\mathcal V}\times [0,T]$ that preserves the family +${\mathcal Z}(t)$ in the sense that any integral curve $\gamma(t),\ t_0\le t\le +t_1$, for $\psi$ with the property that $\gamma(t_0)\in {\mathcal Z}(t_0)$ has +$\gamma(t)\in {\mathcal Z}(t)$ for every $t\in [t_0,t_1]$. Suppose that +${\mathcal T}(x,t),\ 0\le t\le T$, is a one-parameter family of sections of +${\mathcal V}$ that evolves according to the parabolic equation + +$$ +\frac{\partial {\mathcal T}}{\partial t}=\Delta {\mathcal T}+\psi({\mathcal T}). +$$ + +If ${\mathcal T}(x,0)$ is contained in +${\mathcal Z}(0)$ for all $x\in M$, then ${\mathcal T}(x,t)$ is +contained in ${\mathcal Z}(t)$ for all $x\in M$ and for all $0\le +t\le T$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/9021a4eca03f.md b/projects/poincare-conjecture/.astrolabe/atoms/9021a4eca03f.md new file mode 100644 index 00000000..9bba2bcf --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/9021a4eca03f.md @@ -0,0 +1,37 @@ +--- +chapter: 15 +generator: tools/poincare_tex_extract.py +label: surgerydist +labels: +- surgerydist +mtref: '15.12' +ref: +- 9021a4eca03f +sort: proposition +source: tex +src: morgan-tian +tex_file: surgery +--- +Suppose that $({\mathcal M},G)$ is a Ricci flow with surgery +satisfying Assumptions (1) -- (7) in Section 15.1.1. +Suppose that $T$ is a surgery time, suppose that the surgery +control parameter $\delta(T)$ is less than $\delta_0$ in +Definition 15.7, and suppose that the scale of the +surgery $h(T)$ is less than $R_0^{-1/2}$ where $R_0$ is the constant +from Theorem 13.2. Fix $t'\tau_1$, we set $\widetilde {\mathcal +U}_x(\tau)\subset T_xM_{T-\tau_1}$ equal to the slice of $\widetilde{\mathcal U}_x$ at +$\tau$, i.e., $\widetilde {\mathcal U}_x(\tau)$ is determined by the +equation + +$$ +\widetilde{\mathcal +U}_x(\tau)\times\{\tau\}=\widetilde{\mathcal +U}_x\cap\left(T_xM_{T-\tau_1}\times\{\tau\}\right). +$$ diff --git a/projects/poincare-conjecture/.astrolabe/atoms/91876608df01.md b/projects/poincare-conjecture/.astrolabe/atoms/91876608df01.md new file mode 100644 index 00000000..0e2c325d --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/91876608df01.md @@ -0,0 +1,29 @@ +--- +chapter: 15 +generator: tools/poincare_tex_extract.py +label: hrestricted +labels: +- hrestricted +mtref: '15.5' +ref: +- 91876608df01 +sort: definition +source: tex +src: morgan-tian +tex_file: surgery +--- +We have: (i) *a canonical neighborhood parameter*, + $r(t)>0$, and +(ii) *a surgery control parameter* $\overline\delta(t)>0$. We use these +to define the surgery scale function $h(t)$. Set +$\rho(t)=\overline\delta(t)r(t)$. Let $h(t)= +h(\rho(t),\overline\delta(t))\le +\rho(t)\cdot\overline\delta(t)=\overline\delta^2(t)r(t)$ be the +function given by Theorem 11.31. We require that $h(0)\le +R_0^{-1/2}$ where $R_0$ is the constant from +Theorem 13.2. + + In addition, there is a +function $\kappa(t)>0$ called the *non-collapsing parameter*. +All three functions $r(t)$, $\overline\delta(t)$ and $\kappa(t)$ are +required to be positive, non-increasing functions of $t$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/91d88796541d.md b/projects/poincare-conjecture/.astrolabe/atoms/91d88796541d.md new file mode 100644 index 00000000..8dde3719 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/91d88796541d.md @@ -0,0 +1,15 @@ +--- +chapter: 6 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '6.5' +ref: +- 91d88796541d +sort: remark +source: tex +src: morgan-tian +tex_file: newcompar +--- +$\mathit{Ric}(X,\cdot)$ is a horizontal one-form along $\gamma$ and its +dual $\mathit{Ric}(X,\cdot)^*$ is a horizontal tangent vector field +along $\gamma$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/92827a66b5dd.md b/projects/poincare-conjecture/.astrolabe/atoms/92827a66b5dd.md new file mode 100644 index 00000000..f2a57372 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/92827a66b5dd.md @@ -0,0 +1,26 @@ +--- +chapter: 4 +generator: tools/poincare_tex_extract.py +label: pinch +labels: +- pinch +mtref: '4.26' +ref: +- 92827a66b5dd +sort: theorem +source: tex +src: morgan-tian +tex_file: maxprin +--- +**(Pinching toward positive curvature)** +Let $(M,g(t)),\ 0\le t0$. Suppose that $(U,g(T))$ is +isometric to a non-empty open subset of a cone over a Riemannian +manifold. Then $(U,g(t))$ is flat for every $t\in [0,T]$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/92ee3ee82b58.md b/projects/poincare-conjecture/.astrolabe/atoms/92ee3ee82b58.md new file mode 100644 index 00000000..b65d853f --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/92ee3ee82b58.md @@ -0,0 +1,16 @@ +--- +chapter: 9 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '9.54' +ref: +- 92ee3ee82b58 +sort: corollary +source: tex +src: morgan-tian +tex_file: temp2kappa +--- +Let $(M,g(t)),\ -\infty0$ we denote by $B(p,t,r)$ the metric ball +of radius $r$ centered at $(p,t)$ in the $t$ time-slice. For any +$\Delta t>0$ for which $[t-\Delta t,t]\subset I$, we define the *backwards parabolic neighborhood* +$P(x,t,r,-\Delta t)$ to be the product $B(x,t,r)\times +[t-\Delta t,t]$ in space-time. Notice that the intersection of +$P(x,t,r,-\Delta t)$ with a time-slice other that the $t$ time-slice +need not be a metric ball in that time-slice. There is the +corresponding notion of a forward parabolic neighborhood +$P(x,t,r,\Delta t)$ provided that $[t,t+\Delta t]\subset I$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/95fbdd1a8a2d.md b/projects/poincare-conjecture/.astrolabe/atoms/95fbdd1a8a2d.md new file mode 100644 index 00000000..2dba5137 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/95fbdd1a8a2d.md @@ -0,0 +1,58 @@ +--- +chapter: 14 +generator: tools/poincare_tex_extract.py +label: spacetime +labels: +- spacetime +mtref: '14.2' +ref: +- 95fbdd1a8a2d +sort: definition +source: tex +src: morgan-tian +tex_file: surgery +--- +A *surgery space-time* is a +space-time ${\mathcal M}$ equipped with a maximal atlas of charts +covering ${\mathcal M}$, each chart being of one of the three types +listed above, with the overlap functions being diffeomorphisms +preserving the functions $\mathbf{t}$ and the vector fields $\chi$. The +points with neighborhoods of the first type are called *smooth +points*, those with neighborhoods of the second type but not the +first type are called *exposed points*, +and all the other points are called *singular +points*. Notice that the union of +the set of smooth points and the set of exposed points forms a +smooth manifold with boundary (possibly disconnected). Each +component of the boundary of this manifold is contained in a single +time-slice. The union of those components contained in a time +distinct from the initial time and the final time is called the *exposed region*. and the boundary points of +the closure of the exposed region form the set of the singular +points of ${\mathcal M}$. (Technically, the exposed points are +singular, but we reserve this word for the most singular points.) An +$(n+1)$-dimensional surgery space-time is by definition of +homogeneous dimension $n+1$. + +By construction, the local smooth functions $\mathbf{t}$ are compatible +on the overlaps and hence fit together to define a global smooth +function $\mathbf{t}\colon {\mathcal M}\to \Ar$, called the *time* +function. The level sets of this function are called the *time-slices* of the space-time, and $\mathbf{t}^{-1}(t)$ is denoted +$M_t$. Similarly, the tangent bundles of the various charts are +compatible under the overlap diffeomorphisms and hence glue together +to give a global smooth tangent bundle on space-time. The smooth +sections of this vector bundle, the smooth vector fields on space +time, act as derivations on the smooth functions on space-time. The +tangent bundle of an $(n+1)$-dimensional surgery space-time is a +vector bundle of dimension $(n+1)$. Also, by construction the local +vector fields $\chi$ are compatible and hence glue together to +define a global vector field, denoted $\chi$. The vector field and +time function satisfy + +$$ +\chi(\mathbf{t})=1. +$$ + +At the manifold points (including the exposed points) it is a usual +vector field. Along the exposed region and the initial time-slice +the vector field points into the manifold; along the final +time-slice it points out of the manifold. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/96519f84d54e.md b/projects/poincare-conjecture/.astrolabe/atoms/96519f84d54e.md new file mode 100644 index 00000000..663f1746 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/96519f84d54e.md @@ -0,0 +1,28 @@ +--- +chapter: 10 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '10.4' +ref: +- 96519f84d54e +sort: claim +source: tex +src: morgan-tian +tex_file: bddcurvbdddist +--- +For every $n$ sufficiently large, there is a sub-geodesic +$\gamma'_n$ of $\gamma_n$ with end points $z_n'$ and $y_n'$ such +that the following hold: + +- **(1)** The length of $\gamma'_n$ is bounded independent of $n$. +- **(2)** $R(z_n')$ is bounded independent of $n$. +- **(3)** $R(y_n')$ tends to infinity as $n$ tends to infinity. +- **(4)** $\gamma_n'$ is contained in a strong $\epsilon$-tube +$T_n$ that is the union of a balanced +chain of strong +$\epsilon$-necks centered at points of $\gamma'_n$. The first +element in this chain is a strong $\epsilon$-neck $N(z'_n)$ centered +at $z'_n$. The last element is a strong $\epsilon$-neck containing +$y'_n$. +- **(5)** For every $x\in T_n$, we have $R(x)>3$ and $x$ is the center of a strong +$\epsilon$-neck in the flow $({\mathcal M}'_n,G'_n)$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/967c0f8bb774.md b/projects/poincare-conjecture/.astrolabe/atoms/967c0f8bb774.md new file mode 100644 index 00000000..0bd0947b --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/967c0f8bb774.md @@ -0,0 +1,49 @@ +--- +chapter: 6 +generator: tools/poincare_tex_extract.py +label: lformula +labels: +- lformula +mtref: '6.51' +ref: +- 967c0f8bb774 +sort: corollary +source: tex +src: morgan-tian +tex_file: newcompar +--- +Suppose that $x\in M_T$ so that $\tau_1=0$. Then for $q\in{\mathcal +U}^{\bar\tau}_x$ we have + +$$ +\frac{\partial l_x}{\partial \tau}(q)+\triangle l_x^{\bar\tau}(q)\le \frac{(n/2)-l_x^{\bar\tau}(q)}{\bar\tau}. +$$ + +$$ +\frac{\partial l_x}{\partial \tau}(q)-\triangle l_x^{\bar\tau}(q)+|\nabla l_x^{\bar\tau}(q)|^2-R(q)+\frac{n}{2\bar\tau}\ge 0 . +$$ + +$$ +2\triangle l_x^{\bar\tau}(q)-|\nabla l_x^{\bar\tau}(q)|^2+R(q)+\frac{l_x^{\bar\tau}(q)-n}{\bar\tau} +\le 0. +$$ + +In fact, setting + +$$ +\delta=\frac{n}{2 +\bar\tau}-R(q)-\frac{K^{\bar\tau}(\gamma_Z)}{2\bar\tau^{3/2}}-\triangle +l_x^{\bar\tau}(q), +$$ + + then $\delta\ge 0$ and + +$$ +\frac{\partial l_x}{\partial \tau}(q)-\triangle l_x^{\bar\tau}(q) ++|\nabla l_x^{\bar\tau}(q)|^2-R(q)+\frac{n}{2\bar\tau}=\delta +$$ + +$$ +2\triangle l_x^{\bar\tau}(q)-|\nabla l_x^{\bar\tau}(q)|^2+R(q) ++\frac{l_x^{\bar\tau}(q)-n}{\bar\tau}=-2\delta. +$$ diff --git a/projects/poincare-conjecture/.astrolabe/atoms/96ced8c484df.md b/projects/poincare-conjecture/.astrolabe/atoms/96ced8c484df.md new file mode 100644 index 00000000..5426e4b1 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/96ced8c484df.md @@ -0,0 +1,34 @@ +--- +chapter: 6 +generator: tools/poincare_tex_extract.py +label: lipaty +labels: +- lipaty +mtref: '6.59' +ref: +- 96ced8c484df +sort: proposition +source: tex +src: morgan-tian +tex_file: newcompar +--- +Given constants $\epsilon>0$, $\bar\tau_0<\infty$, $l_0<\infty$, +and $C_0<\infty$, there are constants $ C<\infty$ and +$0<\delta<\epsilon$ depending only the given constants such that +the following holds. Let $({\mathcal M},G)$ be a generalized Ricci +flow and let $x\in{\mathcal M}$ be a point with $\mathbf{t}(x)=T-\tau_1$. Let $y\in {\mathcal M}$ be a point with $\mathbf{t}(y)=t=T-\bar\tau$ where $\tau_1+\epsilon\le \bar\tau\le +\bar\tau_0$. Suppose that there is a minimizing ${\mathcal +L}$-geodesic $\gamma$ from $x$ to $y$ with $l(\gamma)\le l_0$. +Suppose that the ball $B(y,t,\epsilon)$ has compact closure in +$M_{t}$ and that $\widetilde P(y,\epsilon,\epsilon)\subset {\mathcal +M}$ exists and that the sectional curvatures of the restriction of +$G$ to this submanifold are bounded by $C_0$. Lastly, suppose that +for every point of the form $z\in \widetilde P(y,\delta,\delta)$ +there is a minimizing ${\mathcal L}$-geodesic from $x$ to $z$ with +$|\mathit{Ric}|$ and $|\nabla R|$ bounded by $C_0$ along this geodesic. +Then for all $(b,t')\in B(y,t,\delta)\times(t-\delta,t+\delta)$ we +have + +$$ +|l_x(y)-l_x(\iota(b,t'))|\le C\sqrt{d_{t}(y,b)^2+|t-t'|^2}. +$$ diff --git a/projects/poincare-conjecture/.astrolabe/atoms/9737d037f97b.md b/projects/poincare-conjecture/.astrolabe/atoms/9737d037f97b.md new file mode 100644 index 00000000..e05a8c35 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/9737d037f97b.md @@ -0,0 +1,16 @@ +--- +chapter: 19 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '19.22' +ref: +- 9737d037f97b +sort: claim +source: tex +src: morgan-tian +tex_file: canonnbhd +--- +It is not possible to have an infinite chain of $(C,\epsilon)$-caps +$C_0\subset C_1\subset \cdots$ in $M$ with the property that for +each $i\ge 1$, the closure of the core of $C_i$ contains a point of +the frontier of $C_{i-1}$ diff --git a/projects/poincare-conjecture/.astrolabe/atoms/97c2fdddbf04.md b/projects/poincare-conjecture/.astrolabe/atoms/97c2fdddbf04.md new file mode 100644 index 00000000..f9eeea37 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/97c2fdddbf04.md @@ -0,0 +1,20 @@ +--- +chapter: 5 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '5.29' +ref: +- 97c2fdddbf04 +sort: definition +source: tex +src: morgan-tian +tex_file: converge2 +--- +Now we define a metric space whose underlying space is the quotient +space of the equivalence classes of minimal geodesic rays emanating +from $p$, with two rays equivalent if and only if the angle at +infinity between them is zero. The pseudo-distance function +$\theta_\infty$ descends to a metric on this space. This space is a + length space [BGP]. Notice that the distance between +any two points in this metric space is at most $\pi$. We denote this +space by $S_\infty(M,p)$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/981cc21e82e4.md b/projects/poincare-conjecture/.astrolabe/atoms/981cc21e82e4.md new file mode 100644 index 00000000..a894db29 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/981cc21e82e4.md @@ -0,0 +1,17 @@ +--- +chapter: 18 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '18.22' +ref: +- 981cc21e82e4 +sort: claim +source: tex +src: morgan-tian +tex_file: energy1 +--- +Suppose that $t$ is a surgery time. Then + +$$ +W_\xi(t)\le \mathit{liminf}_{t'\rightarrow t^-}W_\xi(t'). +$$ diff --git a/projects/poincare-conjecture/.astrolabe/atoms/981d63c19e50.md b/projects/poincare-conjecture/.astrolabe/atoms/981d63c19e50.md new file mode 100644 index 00000000..dabf30f3 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/981d63c19e50.md @@ -0,0 +1,22 @@ +--- +chapter: 18 +generator: tools/poincare_tex_extract.py +label: onec +labels: +- onec +mtref: '18.47' +ref: +- 981d63c19e50 +sort: claim +source: tex +src: morgan-tian +tex_file: energy1 +--- +Given $\zeta>0$ and a $C^1$-map $c\colon S^1\to M$ then for all $n$ +sufficiently large the following hold for the $n$-polygonal +approximation $c_n$ of $c$. + +- **(a)** the length of $c_n$ is within $\zeta$ of the length of $c$. +- **(b)** there is a map of the annulus $S^1\times I$ to $M$ connecting +$c_n$ to $c$ with the property that the image is piecewise smooth +and of area less than $\zeta$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/98af9c99c3c2.md b/projects/poincare-conjecture/.astrolabe/atoms/98af9c99c3c2.md new file mode 100644 index 00000000..bf1a1d5e --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/98af9c99c3c2.md @@ -0,0 +1,39 @@ +--- +chapter: 1 +generator: tools/poincare_tex_extract.py +label: Hessformula +labels: +- Hessformula +mtref: '1.3' +ref: +- 98af9c99c3c2 +sort: lemma +source: tex +src: morgan-tian +tex_file: prelim +--- +The Hessian is a contravariant, symmetric two-tensor, i.e., for + vector fields $X$ and $Y$ we have + +$$ +\mathit{Hess}(f)(X,Y)=\mathit{Hess}(f)(Y,X) +$$ + +and + +$$ +\mathit{Hess}(f)(\phi X,\psi Y)=\phi\psi \mathit{Hess}(f)(X,Y) +$$ + +for all smooth functions $\phi,\psi$. Other formulas for the Hessian +are + +$$ +\mathit{Hess}(f)(X,Y)=\langle \nabla_X(\nabla f),Y\rangle=\nabla_X(\nabla_Y(f))=\nabla^2f(X,Y). +$$ + +Also, in local coordinates we have + +$$ +\mathit{Hess}(f)_{ij}=\partial_i\partial_jf-(\partial_kf)\Gamma^k_{ij}. +$$ diff --git a/projects/poincare-conjecture/.astrolabe/atoms/98c6a95f8556.md b/projects/poincare-conjecture/.astrolabe/atoms/98c6a95f8556.md new file mode 100644 index 00000000..1c30298a --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/98c6a95f8556.md @@ -0,0 +1,23 @@ +--- +chapter: 19 +generator: tools/poincare_tex_extract.py +label: s2isotopic +labels: +- s2isotopic +mtref: '19.3' +ref: +- 98c6a95f8556 +sort: corollary +source: tex +src: morgan-tian +tex_file: canonnbhd +--- +The following holds for any $\epsilon>0$ sufficiently small. Suppose +that $(N,g)$ is an $\epsilon$-neck and we have and an embedding +$f\colon S^2 \to N$ with the property that the restriction of $g$ to +the image of this embedding is within $\epsilon$ in the +$C^{[1/\epsilon]}$-topology to the round metric $h_0$ of scalar +curvature one on $S^2$ and with the norm of the second fundamental +form less than $\epsilon$. Then the two-sphere $f(S^2)$ is isotopic +in $N$ to any member of the family of two-spheres coming from the +$\epsilon$-neck structure on $N$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/9999efa2be2c.md b/projects/poincare-conjecture/.astrolabe/atoms/9999efa2be2c.md new file mode 100644 index 00000000..01f7daef --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/9999efa2be2c.md @@ -0,0 +1,20 @@ +--- +chapter: 0 +generator: tools/poincare_tex_extract.py +label: Theorem1 +labels: +- Theorem1 +mtref: '0.1' +ref: +- 9999efa2be2c +sort: theorem +source: tex +src: morgan-tian +tex_file: intro +--- +Let $M$ be a closed, connected $3$-manifold and +suppose that the fundamental group of $M$ is a free product of +finite groups and infinite cyclic groups. Then $M$ is diffeomorphic +to a connected sum of spherical space-forms, copies of $S^2\times +S^1$, and copies of the unique (up to diffeomorphism) non-orientable +$2$-sphere bundle over $S^1$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/9a27ed21e742.md b/projects/poincare-conjecture/.astrolabe/atoms/9a27ed21e742.md new file mode 100644 index 00000000..c8899e26 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/9a27ed21e742.md @@ -0,0 +1,34 @@ +--- +chapter: 5 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '5.12' +ref: +- 9a27ed21e742 +sort: definition +source: tex +src: morgan-tian +tex_file: converge2 +--- +Let $\{({\mathcal M}_k,G_k,x_k)\}_{k=1}^\infty$ be a sequence of based +generalized Ricci flows. We suppose that $\mathbf{t}(x_k)=0$ for all $k$ and we +denote by $(M_k,g_k)$ the time-slice of $({\mathcal M}_k,G_k)$. For some +$00$ for all $w\ge 0$ and +all $t$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/9b7f421ab0b9.md b/projects/poincare-conjecture/.astrolabe/atoms/9b7f421ab0b9.md new file mode 100644 index 00000000..8efc7342 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/9b7f421ab0b9.md @@ -0,0 +1,13 @@ +--- +chapter: 16 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '16.17' +ref: +- 9b7f421ab0b9 +sort: claim +source: tex +src: morgan-tian +tex_file: surgery +--- +$\int_0^{\tau''}|X_\gamma(\tau)|d\tau> r/2\sqrt{2}$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/9b87a4e6674a.md b/projects/poincare-conjecture/.astrolabe/atoms/9b87a4e6674a.md new file mode 100644 index 00000000..16066523 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/9b87a4e6674a.md @@ -0,0 +1,13 @@ +--- +chapter: 17 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '17.4' +ref: +- 9b87a4e6674a +sort: claim +source: tex +src: morgan-tian +tex_file: surgery +--- +For all $b$ sufficiently large, $w_{(a,b)}\in \psi_{(a,b)}(B(p_0,0,A)\times\{0\})$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/9b970f13e747.md b/projects/poincare-conjecture/.astrolabe/atoms/9b970f13e747.md new file mode 100644 index 00000000..ea11a57b --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/9b970f13e747.md @@ -0,0 +1,16 @@ +--- +chapter: 5 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '5.27' +ref: +- 9b970f13e747 +sort: definition +source: tex +src: morgan-tian +tex_file: converge2 +--- +A *length space* is a connected metric space +$(X,d)$ such that for any two points $x,y$ there is a rectifiable +arc $\gamma$ with endpoints $x$ and $y$ and with the length of +$\gamma$ equal to $d(x,y)$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/9baa1c0fea07.md b/projects/poincare-conjecture/.astrolabe/atoms/9baa1c0fea07.md new file mode 100644 index 00000000..d16f56a8 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/9baa1c0fea07.md @@ -0,0 +1,18 @@ +--- +chapter: 5 +generator: tools/poincare_tex_extract.py +label: epsnet +labels: +- epsnet +mtref: '5.17' +ref: +- 9baa1c0fea07 +sort: definition +source: tex +src: morgan-tian +tex_file: converge2 +--- +A $\delta$-net in $(X,x)$ is a subset $L$ of $X$ containing $x$ +whose $\delta$-neighborhood covers $X$ and for which there is some +$\delta'>0$ with $d(\ell_1,\ell_2)\ge \delta'$ for all $\ell_1\not= +\ell_2$ in $L$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/9c54e0d0c7d8.md b/projects/poincare-conjecture/.astrolabe/atoms/9c54e0d0c7d8.md new file mode 100644 index 00000000..9c1fc675 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/9c54e0d0c7d8.md @@ -0,0 +1,16 @@ +--- +chapter: 9 +generator: tools/poincare_tex_extract.py +label: kappa0 +labels: +- kappa0 +mtref: '9.58' +ref: +- 9c54e0d0c7d8 +sort: proposition +source: tex +src: morgan-tian +tex_file: temp2kappa +--- +There is a $\kappa_0>0$ such that any non-round $3$-dimensional +$\kappa$-solution is a $\kappa_0$-solution. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/9c83686198e8.md b/projects/poincare-conjecture/.astrolabe/atoms/9c83686198e8.md new file mode 100644 index 00000000..fda1afcd --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/9c83686198e8.md @@ -0,0 +1,17 @@ +--- +chapter: 1 +generator: tools/poincare_tex_extract.py +label: divRic +labels: +- divRic +mtref: '1.9' +ref: +- 9c83686198e8 +sort: lemma +source: tex +src: morgan-tian +tex_file: prelim +--- +$$ +dR=2\mathit{div}(\mathit{Ric})=2\nabla^*\mathit{Ric}. +$$ diff --git a/projects/poincare-conjecture/.astrolabe/atoms/9c8d4163f700.md b/projects/poincare-conjecture/.astrolabe/atoms/9c8d4163f700.md new file mode 100644 index 00000000..1e2301cd --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/9c8d4163f700.md @@ -0,0 +1,22 @@ +--- +chapter: 18 +generator: tools/poincare_tex_extract.py +label: muepsbd +labels: +- muepsbd +mtref: '18.72' +ref: +- 9c8d4163f700 +sort: corollary +source: tex +src: morgan-tian +tex_file: energy1 +--- +Suppose that ${\mathcal T}$ is a focusing triangle with base $\xi$ +of length at most one. Then the length of $\xi$ is at most +$\delta+\mu$. More generally, suppose we have a collection of +focusing triangles ${\mathcal T}_1,\ldots,{\mathcal T}_n$ whose +bases all lie in a fixed interval of length one in $c_0$. Suppose +also that the interiors of disks bounded by these focusing triangles +are disjoint. Then the sum of the lengths of the bases is at most +$\delta+\mu$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/9cf6ad839dbb.md b/projects/poincare-conjecture/.astrolabe/atoms/9cf6ad839dbb.md new file mode 100644 index 00000000..6b23dc56 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/9cf6ad839dbb.md @@ -0,0 +1,26 @@ +--- +chapter: 10 +generator: tools/poincare_tex_extract.py +label: statement +labels: +- statement +mtref: '10.12' +ref: +- 9cf6ad839dbb +sort: proposition +source: tex +src: morgan-tian +tex_file: bddcurvbdddist +--- +Let $(U_\infty,g_\infty,z_\infty)$ be as in the conclusion of +Proposition 10.7. Let $Q_\infty=R_{g_\infty}(z_\infty)$ and +let ${\mathcal E}$ be the end of $U_\infty$ where the scalar +curvature is unbounded. Let $\lambda_n$ be any sequence of positive +numbers with $\mathit{lim}_{n\rightarrow\infty}\lambda_n=+\infty$. Then +there is a sequence $x_n$ in $U_\infty$ such that for each $n$ the +distance from $x_n$ to ${\mathcal E}$ is $\lambda_n^{-1/2}$, and +such that the pointed Riemannian manifolds +$(U_\infty,\lambda_ng_\infty,x_n)$ converge in the Gromov-Hausdorff +sense to an open cone, an open cone not homeomorphic to +an open ray (i.e., not homeomorphic to the open cone on a point). + (see Fig. 10.2). diff --git a/projects/poincare-conjecture/.astrolabe/atoms/9d4d83df4874.md b/projects/poincare-conjecture/.astrolabe/atoms/9d4d83df4874.md new file mode 100644 index 00000000..e92324b2 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/9d4d83df4874.md @@ -0,0 +1,19 @@ +--- +chapter: 6 +generator: tools/poincare_tex_extract.py +label: Arnquantity +labels: +- Arnquantity +mtref: '6.77' +ref: +- 9d4d83df4874 +sort: claim +source: tex +src: morgan-tian +tex_file: newcompar +--- +In the case of the constant flow on Euclidean space we have + +$$ +f(\tau)=\tau^{-n/2}e^{-\tilde l(Z,\tau)}{\mathcal J}(Z,\tau)=2^ne^{-|Z|^2}. +$$ diff --git a/projects/poincare-conjecture/.astrolabe/atoms/9df99ae6f4cb.md b/projects/poincare-conjecture/.astrolabe/atoms/9df99ae6f4cb.md new file mode 100644 index 00000000..471a0b7a --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/9df99ae6f4cb.md @@ -0,0 +1,25 @@ +--- +chapter: 18 +generator: tools/poincare_tex_extract.py +label: xistronger +labels: +- xistronger +mtref: '18.53' +ref: +- 9df99ae6f4cb +sort: lemma +source: tex +src: morgan-tian +tex_file: energy1 +--- +Given $\zeta>0$, there is $1(t_1-t_0)^{-1}$, +depending only on $\Gamma$ and the curvature bounds on the ambient +Ricci flow $(M,g(t)),\ t_0\le t\le t_1$, such that the following +holds. Let $t_2=t_1-B^{-1}$. Fix $ c\in S^2$. Let $v_c$ be the +solution to Equation (18.4) with initial condition +$v_{c}(t_0)=A(p_1(\widetilde\Gamma(c)))$, so that in our previous +notation $v_c=w_{A(p_1(\widetilde\Gamma(c)))}$. Then for all +$\lambda>0$ sufficiently small, either +$A(p_1\widetilde\Gamma_c^\lambda(t_1))0$ and $\kappa>0$ with $\mathit{Vol} B(x_k,0,r_0)\ge \kappa r_0^n$ +for all $k$ sufficiently large. + + Then after passing to a subsequence there is a +flow $(M_\infty,g_\infty(t),(x_\infty,0))$ which is the geometric +limit. It is a solution to the Ricci flow equation defined for $t\in +(T',T)$. For every $t\in (T',T)$ the Riemannian manifold +$(M_\infty,g_\infty(t))$ is complete. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/a04af7434ef0.md b/projects/poincare-conjecture/.astrolabe/atoms/a04af7434ef0.md new file mode 100644 index 00000000..5e8b99fd --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/a04af7434ef0.md @@ -0,0 +1,15 @@ +--- +chapter: 18 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '18.20' +ref: +- a04af7434ef0 +sort: claim +source: tex +src: morgan-tian +tex_file: energy1 +--- +Let $X$ be a compact $3$-manifold. If $\pi_1(X)$ is a non-trivial +free product or if $\pi_1(X)$ is isomorphic to $\Zee$, then +$\pi_2(X)\not=0$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/a17e64a8c11e.md b/projects/poincare-conjecture/.astrolabe/atoms/a17e64a8c11e.md new file mode 100644 index 00000000..5aa1e250 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/a17e64a8c11e.md @@ -0,0 +1,14 @@ +--- +chapter: 6 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '6.38' +ref: +- a17e64a8c11e +sort: remark +source: tex +src: morgan-tian +tex_file: newcompar +--- +In spite of the notation, $H(X,Y)$ is a purely quadratic function of +the vector field $Y$ along $\gamma_Z$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/a43d37f014e8.md b/projects/poincare-conjecture/.astrolabe/atoms/a43d37f014e8.md new file mode 100644 index 00000000..7685e49e --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/a43d37f014e8.md @@ -0,0 +1,21 @@ +--- +chapter: 3 +generator: tools/poincare_tex_extract.py +label: '3.23' +labels: +- '3.23' +mtref: '3.24' +ref: +- a43d37f014e8 +sort: claim +source: tex +src: morgan-tian +tex_file: flowbasics +--- +Assuming the hypothesis of the proposition, for any + minimal $g(t_0)$-geodesic $\gamma$ from $x_0$ to $x_1$, we have + +$$ +\frac{d(\ell_t(\gamma))}{dt}\Bigl|_{t=t_0}\Bigr.\ge -2(n-1) +\left(\frac{2}{3}Kr_0+r_0^{-1}\right). +$$ diff --git a/projects/poincare-conjecture/.astrolabe/atoms/a46d72b87e07.md b/projects/poincare-conjecture/.astrolabe/atoms/a46d72b87e07.md new file mode 100644 index 00000000..da2f86f1 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/a46d72b87e07.md @@ -0,0 +1,74 @@ +--- +chapter: 19 +generator: tools/poincare_tex_extract.py +label: S2intersection +labels: +- S2intersection +mtref: '19.11' +ref: +- a46d72b87e07 +sort: proposition +source: tex +src: morgan-tian +tex_file: canonnbhd +--- +Given $0<\alpha\le 10^{-2}$, there is +$\epsilon_5=\epsilon_5(\alpha)>0$ such that the following hold for +all $0<\epsilon\le \epsilon_5$. Let $N$ and $N'$ be $\epsilon$-necks +centered at $x$ and $x'$, respectively, in a Riemannian manifold +$X$: + +- **(1)** If $N\cap N'\not=\emptyset$ then +$1-\alpha0$ such that for every $0<\epsilon\le \bar +\epsilon_3$ there is $C_1=C_1(\epsilon)<\infty$ such that one of +the following holds for any $\kappa>0$ and any compact +$3$-dimensional $\kappa$-solution $(M,g(t))$. + +- **(1)** The manifold $M$ is compact and of constant positive sectional curvature. +- **(2)** The diameter of $(M,g(0))$ is less than $C_1\cdot (\mathit{max}_{x\in +M}R(x,0))^{-1/2}$, and $M$ is diffeomorphic to either $S^3$ or $\Ar +P^3$. +- **(3)** $(M,g(0))$ + is a double $C_1$-capped strong $\epsilon$-tube. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/a501760fb0e6.md b/projects/poincare-conjecture/.astrolabe/atoms/a501760fb0e6.md new file mode 100644 index 00000000..cbdc0293 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/a501760fb0e6.md @@ -0,0 +1,20 @@ +--- +chapter: 12 +generator: tools/poincare_tex_extract.py +label: duniq +labels: +- duniq +mtref: '12.27' +ref: +- a501760fb0e6 +sort: corollary +source: tex +src: morgan-tian +tex_file: stdsoln +--- +Let $g_1(t)$ and $g_2(t)$ be standard solutions. Suppose that +$g_1(t)=g_2(t)$ for all $t\in [0,t_0]$ for some $t_0\ge 0$. The +Ricci-DeTurck solutions $\hat{g}_1(t)$ and $\hat{g}_2(t)$ +constructed from standard solutions $g_1(t)$ and $g_2(t)$ with +$g_1(t_0)=g_2(t_0)$ exist and satisfy $\hat{g}_1(t)=\hat{g}_1(t)$ +for $t \in [t_0,T']$ for some $T'>t_0$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/a560479acf35.md b/projects/poincare-conjecture/.astrolabe/atoms/a560479acf35.md new file mode 100644 index 00000000..c4ca2bfc --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/a560479acf35.md @@ -0,0 +1,13 @@ +--- +chapter: 3 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '3.10' +ref: +- a560479acf35 +sort: definition +source: tex +src: morgan-tian +tex_file: flowbasics +--- +In this case we say that $(M,g(0))$ and $f\colon M\to \Ar$ *generate* a gradient shrinking soliton. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/a5e421ef8adf.md b/projects/poincare-conjecture/.astrolabe/atoms/a5e421ef8adf.md new file mode 100644 index 00000000..a1ec687d --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/a5e421ef8adf.md @@ -0,0 +1,19 @@ +--- +chapter: 12 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '12.11' +ref: +- a5e421ef8adf +sort: claim +source: tex +src: morgan-tian +tex_file: stdsoln +--- +For any tensor $\phi$ we have + +$$ +\nabla(\nabla_r\phi)=\nabla_r(\nabla\phi)+dx^k\otimes +\mathit{Rm}(\partial_k,\partial_r)(\phi)-\nabla_r(dx^l)\otimes +\nabla_l(\phi). +$$ diff --git a/projects/poincare-conjecture/.astrolabe/atoms/a66fa0ea8720.md b/projects/poincare-conjecture/.astrolabe/atoms/a66fa0ea8720.md new file mode 100644 index 00000000..159aa3ab --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/a66fa0ea8720.md @@ -0,0 +1,21 @@ +--- +chapter: 11 +generator: tools/poincare_tex_extract.py +label: ancientlimit +labels: +- ancientlimit +mtref: '11.16' +ref: +- a66fa0ea8720 +sort: corollary +source: tex +src: morgan-tian +tex_file: singlimit2 +--- +Suppose that there is a geometric limit flow +$(M_\infty,g_\infty(t))$ of a subsequence defined on $(-T,0]$ for +some $T<\infty$. Suppose that the limit flow is complete of +non-negative curvature with the curvature locally bounded in time. +Then for every $A<\infty$ the scalar curvature $R_{g_\infty}(y,t)$ +is uniformly bounded for all $(y,t)\in B(x_\infty,0,A)\times +(-T,0]$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/a673ebd63624.md b/projects/poincare-conjecture/.astrolabe/atoms/a673ebd63624.md new file mode 100644 index 00000000..66a67141 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/a673ebd63624.md @@ -0,0 +1,21 @@ +--- +chapter: 6 +generator: tools/poincare_tex_extract.py +label: twosets +labels: +- twosets +mtref: '6.68' +ref: +- a673ebd63624 +sort: claim +source: tex +src: morgan-tian +tex_file: newcompar +--- +The complement of $\nu(A)\cap {\mathcal U}_{x}(\tau)$ +in $\nu(A)$ is contained in the union of two sets: The first is the set of +points $z$ where there is more than one minimizing ${\mathcal L}$-geodesic from +$x$ ending at $z$ and if $Z$ is the initial condition for any minimizing +${\mathcal L}$-geodesic to $z$ then the differential of ${\mathcal L}\mathit{exp}_x^\tau$ at any $Z$ is an isomorphism. The second is the intersection of +the set of critical values of ${\mathcal L}\mathit{exp}^\tau$ with $\nu(A)\cap +M_{T-\tau}$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/a679e0086677.md b/projects/poincare-conjecture/.astrolabe/atoms/a679e0086677.md new file mode 100644 index 00000000..74b32cd7 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/a679e0086677.md @@ -0,0 +1,21 @@ +--- +chapter: 18 +generator: tools/poincare_tex_extract.py +label: spherethm +labels: +- spherethm +mtref: '18.14' +ref: +- a679e0086677 +sort: corollary +source: tex +src: morgan-tian +tex_file: energy1 +--- +Suppose that $M$ is a closed, connected $3$-manifold containing no +embedded $\Ar P^2$ with trivial normal bundle, and suppose that +$\pi_2(M)\not= 0$. Then either $M$ can be written as a connected sum +$M_1\# M_2$ where neither of the $M_i$ is homotopy equivalent to +$S^3$ or $M_1$ has a prime factor that is a $2$-sphere bundle over +$S^1$. In either case, $M$ contains an embedded $2$-sphere which is +homotopically non-trivial. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/a6ea3cf8fb92.md b/projects/poincare-conjecture/.astrolabe/atoms/a6ea3cf8fb92.md new file mode 100644 index 00000000..1eb252ea --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/a6ea3cf8fb92.md @@ -0,0 +1,23 @@ +--- +chapter: 6 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '6.76' +ref: +- a6ea3cf8fb92 +sort: example +source: tex +src: morgan-tian +tex_file: newcompar +--- +Let the Ricci flow be the constant family of standard metrics on +$\Ar^n$. Fix $x=(p,T)\in \Ar ^n\times (-\infty,\infty)$. Then + +$$ +{\mathcal L}\mathit{exp}_x(Z,\tau)=(p+2\sqrt{\tau}Z,T-\tau). +$$ + +In particular, the Jacobian determinant of ${\mathcal L}\mathit{exp}_{(x,T)}^\tau$ is constant and equal to $2^n\tau^{n/2}$. The +$\widetilde l$-length of the ${\mathcal L}$-geodesic +$\gamma_Z(\tau)=(p+2\sqrt{\tau}Z, T-\tau),\ 0\le \tau\le \bar\tau$, +is $|Z|^2$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/a6ee3b2ab615.md b/projects/poincare-conjecture/.astrolabe/atoms/a6ee3b2ab615.md new file mode 100644 index 00000000..1dfbbfe4 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/a6ee3b2ab615.md @@ -0,0 +1,20 @@ +--- +chapter: 9 +generator: tools/poincare_tex_extract.py +label: lowerbdd +labels: +- lowerbdd +mtref: '9.25' +ref: +- a6ee3b2ab615 +sort: lemma +source: tex +src: morgan-tian +tex_file: temp2kappa +--- +There is a constant $c_1>0$ depending only on the dimension $n$ such +that for any $p,q\in M_k$ we have + +$$ +l_{x_k}(p,\tau)\ge -l_{x_k}(q,\tau)-1+c_1\frac{d^2_{g(-\tau)}(p,q)}{\tau}. +$$ diff --git a/projects/poincare-conjecture/.astrolabe/atoms/a75b1ef40db8.md b/projects/poincare-conjecture/.astrolabe/atoms/a75b1ef40db8.md new file mode 100644 index 00000000..6f49e6ed --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/a75b1ef40db8.md @@ -0,0 +1,26 @@ +--- +chapter: 6 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '6.34' +ref: +- a75b1ef40db8 +sort: claim +source: tex +src: morgan-tian +tex_file: newcompar +--- +Let +$\gamma(\tau)$ be a curve parameterized by backward time. Let $Y$ be +a horizontal vector field along $\gamma$ and let $X$ be the +horizontal component of $\partial\widetilde\gamma/\partial \tau$. +Then + +$$ +\begin{aligned} +\frac{\partial}{\partial\tau}\langle \nabla_{ X}Y, Y\rangle + &= \langle \nabla_{ X} Y, \nabla_{ X} Y\rangle + \langle +\nabla_{ X}\nabla_{ X} Y, Y\rangle +\\&\hspace{0.5cm} +2\mathit{Ric}(\nabla_XY,Y)) +(\nabla_X\mathit{Ric})(Y,Y)) +\end{aligned} +$$ diff --git a/projects/poincare-conjecture/.astrolabe/atoms/a7c2ebee07a3.md b/projects/poincare-conjecture/.astrolabe/atoms/a7c2ebee07a3.md new file mode 100644 index 00000000..c013dc3f --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/a7c2ebee07a3.md @@ -0,0 +1,25 @@ +--- +chapter: 6 +generator: tools/poincare_tex_extract.py +label: minmax +labels: +- minmax +mtref: '6.61' +ref: +- a7c2ebee07a3 +sort: corollary +source: tex +src: morgan-tian +tex_file: newcompar +--- +Given $\bar\tau_0<\infty$, $C_0<\infty$, $\epsilon>0$, and $l_0<\infty$, there +is a constant $C_2$ depending only on $C_0$, $l_0$, $\epsilon$ and $\bar\tau_0$ +such that the following holds. Let $\gamma$ be an ${\mathcal L}$-geodesic +defined on $[\tau_1,\bar\tau]$ with $\tau_1+\epsilon\le\bar\tau\le \bar\tau_0$ +and with $|\nabla R(\gamma(\tau))|\le C_0$ and $|\mathit{Ric}(\gamma(\tau))|\le +C_0$ for all $\tau\in [\tau_1,\bar\tau]$. Suppose also that $ l(\gamma)\le +l_0$. Then, we have + +$$ +\mathit{max}_{\tau}\left(\sqrt{\tau}|X_\gamma(\tau)|\right)\le C_2. +$$ diff --git a/projects/poincare-conjecture/.astrolabe/atoms/a86000e8dd35.md b/projects/poincare-conjecture/.astrolabe/atoms/a86000e8dd35.md new file mode 100644 index 00000000..94e3409b --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/a86000e8dd35.md @@ -0,0 +1,15 @@ +--- +chapter: 18 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '18.5' +ref: +- a86000e8dd35 +sort: lemma +source: tex +src: morgan-tian +tex_file: energy1 +--- +Let $X$ be a compact $3$-manifold (possibly disconnected). Then +there is a finite upper bound to the number of spheres in any +homotopically essential family of disjointly embedded $2$-spheres. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/a8675ea40ba2.md b/projects/poincare-conjecture/.astrolabe/atoms/a8675ea40ba2.md new file mode 100644 index 00000000..6a1c09e6 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/a8675ea40ba2.md @@ -0,0 +1,23 @@ +--- +chapter: 12 +generator: tools/poincare_tex_extract.py +label: '12.7' +labels: +- '12.7' +mtref: '12.15' +ref: +- a8675ea40ba2 +sort: lemma +source: tex +src: morgan-tian +tex_file: stdsoln +--- +Suppose that $g(t)$ solves the Ricci flow equation and suppose that +$\psi_t$ solves the harmonic map flow equation, +Equation (12.7); then $\hat g(t)=(\psi_t^{-1})^*g(t)$ solves +the Ricci-DeTurck flow, Equation (12.9) and $\psi_t$ +satisfies the following ODE: + +$$ +\frac{\partial \psi_t}{\partial t}=-\hat g^{ij}(t)W(t). +$$ diff --git a/projects/poincare-conjecture/.astrolabe/atoms/a89a833cf7cb.md b/projects/poincare-conjecture/.astrolabe/atoms/a89a833cf7cb.md new file mode 100644 index 00000000..7c741abb --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/a89a833cf7cb.md @@ -0,0 +1,15 @@ +--- +chapter: 6 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '6.57' +ref: +- a89a833cf7cb +sort: corollary +source: tex +src: morgan-tian +tex_file: newcompar +--- +With $A$ and $\nu_0(A)$ as in Proposition 6.56, the +restriction of $l_x$ to $\nu_0(A)\cap M_{T-\bar\tau}$ is a locally +Lipschitz function with respect to the metric $G_{T-\bar\tau}$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/a89be7fdb431.md b/projects/poincare-conjecture/.astrolabe/atoms/a89be7fdb431.md new file mode 100644 index 00000000..871a25ef --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/a89be7fdb431.md @@ -0,0 +1,34 @@ +--- +chapter: 4 +generator: tools/poincare_tex_extract.py +label: localMP +labels: +- localMP +mtref: '4.9' +ref: +- a89be7fdb431 +sort: theorem +source: tex +src: morgan-tian +tex_file: maxprin +--- +Let $(M,g)$ be a Riemannian manifold. Let $\overline U\subset M$ be a compact, +smooth, connected, codimension-$0$ submanifold. Let ${\mathcal V}\to M$ be a +tensor bundle and let ${\mathcal Z}\subset {\mathcal V}$ be a closed, convex +subset. Suppose that $\psi$ is a fiberwise vector field defined on an open +neighborhood of ${\mathcal Z}$ in ${\mathcal V}$ preserving ${\mathcal Z}$. +Suppose that ${\mathcal Z}$ is invariant under the parallel translation induced +by the Levi-Civita connection. Suppose that ${\mathcal T}(x,t),\ 0\le t\le T$, +is a one-parameter family of sections of ${\mathcal V}$ that evolves according +to the parabolic equation + +$$ +\frac{\partial {\mathcal T}}{\partial t}=\Delta {\mathcal T} ++\psi({\mathcal T}). +$$ + + If ${\mathcal T}(x,0)$ is contained in +${\mathcal Z}$ for all $x\in \overline U$ and if ${\mathcal +T}(x,t)\in {\mathcal Z}$ for all $x\in \partial \overline U$ and all +$0\le t\le T$, then ${\mathcal T}(x,t)$ is contained in ${\mathcal +Z}$ for all $x\in \overline U$ and all $0\le t\le T$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/a8ab8a19cc61.md b/projects/poincare-conjecture/.astrolabe/atoms/a8ab8a19cc61.md new file mode 100644 index 00000000..655617a5 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/a8ab8a19cc61.md @@ -0,0 +1,21 @@ +--- +chapter: 9 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '9.31' +ref: +- a8ab8a19cc61 +sort: claim +source: tex +src: morgan-tian +tex_file: temp2kappa +--- +$$ +\begin{aligned} +\int_{\tau_0}^{\tau_1}\int_{M_\infty\times\{-\tau\}}\triangle +e^{-l_\infty}d\mathit{vol}_{g_\infty}d\tau & = +&\int_{\tau_0}^{\tau_1}\int_{M_\infty\times\{-\tau\}}\left(|\nabla +l_\infty|^2-\triangle l_\infty\right)e^{-l_\infty}d\mathit{vol}_{g_\infty}d\tau \\ +& = & 0. +\end{aligned} +$$ diff --git a/projects/poincare-conjecture/.astrolabe/atoms/a8e10db32f90.md b/projects/poincare-conjecture/.astrolabe/atoms/a8e10db32f90.md new file mode 100644 index 00000000..1f430489 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/a8e10db32f90.md @@ -0,0 +1,23 @@ +--- +chapter: 9 +generator: tools/poincare_tex_extract.py +label: etimesdis +labels: +- etimesdis +mtref: '9.24' +ref: +- a8e10db32f90 +sort: corollary +source: tex +src: morgan-tian +tex_file: temp2kappa +--- +The functional + +$$ +\varphi\mapsto{\mathcal D}(e^{-l_\infty}\varphi) +$$ + + is a distribution and +its value on any non-negative, compactly supported $C^\infty$-function +$\varphi$ is $\ge 0$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/a9617f70db9f.md b/projects/poincare-conjecture/.astrolabe/atoms/a9617f70db9f.md new file mode 100644 index 00000000..a5693c8e --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/a9617f70db9f.md @@ -0,0 +1,14 @@ +--- +chapter: 18 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '18.69' +ref: +- a9617f70db9f +sort: claim +source: tex +src: morgan-tian +tex_file: energy1 +--- +For any $a\in (\partial A\cap B)$ there is a unique minimal geodesic +in $B$ from $a$ to $v$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/a992d6a89479.md b/projects/poincare-conjecture/.astrolabe/atoms/a992d6a89479.md new file mode 100644 index 00000000..a3b4f762 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/a992d6a89479.md @@ -0,0 +1,29 @@ +--- +chapter: 15 +generator: tools/poincare_tex_extract.py +label: neckglue +labels: +- neckglue +mtref: '15.2' +ref: +- a992d6a89479 +sort: proposition +source: tex +src: morgan-tian +tex_file: surgery +--- +There is $0<\beta<1/2$ such that the following holds for any +$\epsilon<1$. Let $(N\times [-t_0,0],g_1(t))$ be an evolving +$\beta\epsilon$-neck centered at $x$ with $R(x,0)=1$. Let $ +(N'\times (-t_1,-t_0],g_2(t))$ be a strong $\beta\epsilon/2$-neck. +Suppose we have an isometric embedding of $N\times \{-t_0\}$ with +$N'\times \{-t_0\}$ and the strong $\beta\epsilon/2$-neck structure on $N'\times +(-t_1,-t_0]$ is centered at the image of $(x,-t_0]$. Then the union + +$$ +N\times [-t_0,0]\cup N'\times (-t_1,-t_0] +$$ + + with the induced +one-parameter family of metrics contains a strong $\epsilon$-neck +centered at $(x,0)$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/a99536ac8110.md b/projects/poincare-conjecture/.astrolabe/atoms/a99536ac8110.md new file mode 100644 index 00000000..87762e7f --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/a99536ac8110.md @@ -0,0 +1,26 @@ +--- +chapter: 11 +generator: tools/poincare_tex_extract.py +label: Rproper +labels: +- Rproper +mtref: '11.21' +ref: +- a99536ac8110 +sort: lemma +source: tex +src: morgan-tian +tex_file: singlimit2 +--- +Suppose that $({\mathcal M},G)$ is a generalized Ricci flow defined +for $0\le t0$ sufficiently small. Let +$(M,g)$ be a connected Riemannian manifold. Suppose that every point +of $M$ is the center of an $\epsilon$-neck. Then either $M$ is +diffeomorphic to $S^2\times (0,1)$ and is an $\epsilon$-tube, or $M$ +is diffeomorphic to an $S^2$-fibration over $S^1$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/aa9d836895d8.md b/projects/poincare-conjecture/.astrolabe/atoms/aa9d836895d8.md new file mode 100644 index 00000000..f6f88498 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/aa9d836895d8.md @@ -0,0 +1,29 @@ +--- +chapter: 4 +generator: tools/poincare_tex_extract.py +label: Harnack +labels: +- Harnack +mtref: '4.37' +ref: +- aa9d836895d8 +sort: theorem +source: tex +src: morgan-tian +tex_file: maxprin +--- +Suppose that $(M,g(t))$ is a Ricci flow defined for $(T_0,T_1)$ with +$(M,g(t))$ a complete manifold of non-negative curvature operator +with bounded curvature for each $t\in (T_0,T_1)$. Then for any +time-dependent vector field $\chi(x,t)$ on $M$ we have: + +$$ +\frac{\partial R(x,t)}{\partial t}+\frac{R(x,t)}{t-T_0}+2\langle +\chi(x,t),\nabla R(x,t)\rangle +2\mathit{Ric}(x,t)(\chi(x,t),\chi(x,t))\ge 0. +$$ + + In particular, we have + +$$ +\frac{\partial R(x,t)}{\partial t}+\frac{R(x,t)}{t-T_0}\ge 0. +$$ diff --git a/projects/poincare-conjecture/.astrolabe/atoms/abd40a465b53.md b/projects/poincare-conjecture/.astrolabe/atoms/abd40a465b53.md new file mode 100644 index 00000000..e401599c --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/abd40a465b53.md @@ -0,0 +1,26 @@ +--- +chapter: 6 +generator: tools/poincare_tex_extract.py +label: symmY +labels: +- symmY +mtref: '6.16' +ref: +- abd40a465b53 +sort: corollary +source: tex +src: morgan-tian +tex_file: newcompar +--- +Let $\gamma$ be an ${\mathcal L}$-geodesic and let $Y_1,Y_2$ be vector fields +along $\gamma$ vanishing at $\tau_1$. Suppose $Y_1(\tau_2)=Y_2(\tau_2)=0$. + Then the +bilinear pairing + +$$ +-\int_{\tau_1}^{\tau_2}2\sqrt{\tau}\langle +\mathit{Jac}(Y_1),Y_2\rangle d\tau +$$ + + is a symmetric function of $Y_1$ +and $Y_2$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/abf9cbb10c68.md b/projects/poincare-conjecture/.astrolabe/atoms/abf9cbb10c68.md new file mode 100644 index 00000000..4928816b --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/abf9cbb10c68.md @@ -0,0 +1,22 @@ +--- +chapter: 6 +generator: tools/poincare_tex_extract.py +label: DtildeL +labels: +- DtildeL +mtref: '6.22' +ref: +- abf9cbb10c68 +sort: lemma +source: tex +src: morgan-tian +tex_file: newcompar +--- +Suppose that $Z\in {\mathcal D}^{\tau}_x$. Then for any $\widetilde +Y\in T_xM_{T-\tau_1}=T_Z({\mathcal D}^{\tau}_x)$ we have + +$$ +\langle \nabla\widetilde L^{\tau},\widetilde Y\rangle =2\sqrt{\tau}\langle +X(\tau),d_Z\left({\mathcal L}\mathit{exp}_x^{\tau}\right)(\widetilde +Y)\rangle. +$$ diff --git a/projects/poincare-conjecture/.astrolabe/atoms/ac4ac4fb06f8.md b/projects/poincare-conjecture/.astrolabe/atoms/ac4ac4fb06f8.md new file mode 100644 index 00000000..94f8dc0f --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/ac4ac4fb06f8.md @@ -0,0 +1,27 @@ +--- +chapter: 12 +generator: tools/poincare_tex_extract.py +label: asympt +labels: +- asympt +mtref: '12.23' +ref: +- ac4ac4fb06f8 +sort: claim +source: tex +src: morgan-tian +tex_file: stdsoln +--- +For any fixed $t$ we have the following asymptotic expansions at +spatial infinity. + +- **(1)** $e^{\tilde{h}(r^2,t)}$ is asymptotic to $\frac{1}{(1-t)r}$. +- **(2)** $\tilde{h}(r^2,t)$ is asymptotic to $-\log r$. +- **(3)** $\frac{\partial \tilde{h}}{\partial w}(r^2,t)$ is asymptotic to + $-\frac{1}{2r^2}$. +- **(4)** $r^{-1} \frac{\partial r}{\partial t}$ is asymptotic to +$\frac{C}{r}$. +- **(5)** $\frac{\partial B(r^2,t)}{\partial r}$ is asymptotic to $C$. +- **(6)** $|G( \tilde{\rho},r^2,t)| \leq C_*< \infty$ where +$C_* =C_*\left(\sup\{|\tilde{\rho}|,\tilde{h}\}\right)$ is a +constant depending only on $\sup\{|\tilde{\rho}|,\tilde{h}\}$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/acada7b5d698.md b/projects/poincare-conjecture/.astrolabe/atoms/acada7b5d698.md new file mode 100644 index 00000000..ec6c29d1 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/acada7b5d698.md @@ -0,0 +1,20 @@ +--- +chapter: 12 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '12.4' +ref: +- acada7b5d698 +sort: definition +source: tex +src: morgan-tian +tex_file: stdsoln +--- +A *partial standard Ricci flow* is a Ricci flow $(\Ar^3,g(t)),\ 0\le tT$ with +the property that the extension has curvature locally bounded in +time. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/ad2042196c95.md b/projects/poincare-conjecture/.astrolabe/atoms/ad2042196c95.md new file mode 100644 index 00000000..a5829101 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/ad2042196c95.md @@ -0,0 +1,25 @@ +--- +chapter: 13 +generator: tools/poincare_tex_extract.py +label: 2ndnewbasis +labels: +- 2ndnewbasis +mtref: '13.8' +ref: +- ad2042196c95 +sort: corollary +source: tex +src: morgan-tian +tex_file: surgery +--- +The following holds provided that $\delta\le \delta_3'$. It is +possible to choose the basis $\{f_0,f_1,f_2\}$ satisfying the +conclusions of Lemma 13.7 so that in addition the +curvature matrix for $(R_{ijkl}(y))$ is of the form + +$$ +\begin{pmatrix} \lambda & 0 & 0 \\ 0 & \alpha & \beta \\ 0 & \beta & \gamma\end{pmatrix} +$$ + +with $|\lambda-\frac{1}{2}|\le O(\delta)$ and +$|\alpha|,|\beta|,|\gamma|\le O(\delta)$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/ad36aef968b6.md b/projects/poincare-conjecture/.astrolabe/atoms/ad36aef968b6.md new file mode 100644 index 00000000..c496728c --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/ad36aef968b6.md @@ -0,0 +1,20 @@ +--- +chapter: 18 +generator: tools/poincare_tex_extract.py +label: noholes +labels: +- noholes +mtref: '18.83' +ref: +- ad36aef968b6 +sort: corollary +source: tex +src: morgan-tian +tex_file: energy1 +--- +Suppose that we have an arc $\xi$ of length at most $1$ on $c_0$. +Denote the endpoints of $\xi$ by $x$ and $y$ and suppose that +$D_x\cap D_y\not=\emptyset$. Let $D'_x$ and $D'_y$ be sub-geodesics +containing $x$ and $y$ respectively ending at the same point, $v$, +and otherwise disjoint. Then the loop $\xi*D'_y*(D'_x)^{-1}$ bounds +a disk in $A$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/ad48b42bb0a0.md b/projects/poincare-conjecture/.astrolabe/atoms/ad48b42bb0a0.md new file mode 100644 index 00000000..5d72b3be --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/ad48b42bb0a0.md @@ -0,0 +1,33 @@ +--- +chapter: 6 +generator: tools/poincare_tex_extract.py +label: Lrescale +labels: +- Lrescale +mtref: '6.72' +ref: +- ad48b42bb0a0 +sort: lemma +source: tex +src: morgan-tian +tex_file: newcompar +--- +Let $({\mathcal M},G)$ be a generalized Ricci flow and let $x\in +M_T\subset {\mathcal M}$. Fix $Q>0$ and let $({\mathcal M}',G')$ be +the $Q$ scaling and shifting of $({\mathcal M},G)$ as described in +the previous paragraph. Let $\iota\colon{\mathcal M}\to {\mathcal +M}'$ be the identity map. Suppose that $\gamma\colon [0,\bar\tau]\to +{\mathcal M}$ is a path parameterized by backward time with +$\gamma(0)=x$. Let $\beta\colon [0,Q\bar\tau]\to Q{\mathcal M}$ be +defined by + +$$ +\beta(\tau')=\iota(\gamma(\tau'/Q)). +$$ + +Then $\beta(0)=x$ and $\beta$ is parameterized by backward time in +$({\mathcal M}',G')$, and ${\mathcal L}(\beta)=\sqrt{Q}{\mathcal +L}(\gamma)$. Furthermore, $\beta$ is an ${\mathcal L}$-geodesic if +and only if $\gamma$ is. In this case, if $Z=\mathit{lim}_{\tau\rightarrow 0}\sqrt{\tau}X_\gamma(\tau)$ then +$\sqrt{Q^{-1}}Z=\mathit{lim}_{\tau'\rightarrow +0}\sqrt{\tau'}X_\beta(\tau')$ diff --git a/projects/poincare-conjecture/.astrolabe/atoms/ad4f67ea75bc.md b/projects/poincare-conjecture/.astrolabe/atoms/ad4f67ea75bc.md new file mode 100644 index 00000000..3a4ee516 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/ad4f67ea75bc.md @@ -0,0 +1,47 @@ +--- +chapter: 16 +generator: tools/poincare_tex_extract.py +label: altern +labels: +- altern +mtref: '16.5' +ref: +- ad4f67ea75bc +sort: proposition +source: tex +src: morgan-tian +tex_file: surgery +--- +Given $A<\infty$, $\delta''>0$ and $0<\theta<1$, there is +$\delta''_0=\delta''_0(A,\theta,\delta'')$ ($\delta_0''$ also +depends on $r_{i+1}$, $C$, and $\epsilon$, which are all now fixed) +such that the following holds. Suppose that $({\mathcal M},G)$ is a +Ricci flow with surgery defined for $0\le t0$ there are +$(n-1)$ zero eigenvalues of $\mathit{Rm}_{\tilde g}(p,s)$. The other +$(n-1)(n-2)/2$ eigenvalues of $\mathit{Rm}_{\tilde g}(p,s)$ are +$s^{-2}(\lambda_i-1)$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/adb6cc2705f2.md b/projects/poincare-conjecture/.astrolabe/atoms/adb6cc2705f2.md new file mode 100644 index 00000000..7a01e38b --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/adb6cc2705f2.md @@ -0,0 +1,14 @@ +--- +chapter: 9 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '9.19' +ref: +- adb6cc2705f2 +sort: remark +source: tex +src: morgan-tian +tex_file: temp2kappa +--- +**N.B.** We are not claiming that $l_\infty$ is the reduced length +function from a point of $M_\infty\times (-\infty,0)$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/ae6f1c6d00d9.md b/projects/poincare-conjecture/.astrolabe/atoms/ae6f1c6d00d9.md new file mode 100644 index 00000000..1d2f5ccc --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/ae6f1c6d00d9.md @@ -0,0 +1,30 @@ +--- +chapter: 7 +generator: tools/poincare_tex_extract.py +label: 4pi +labels: +- 4pi +mtref: '7.26' +ref: +- ae6f1c6d00d9 +sort: theorem +source: tex +src: morgan-tian +tex_file: newcomp2 +--- +Let $(M,g(t)),\ 0\le t\le T$, be a Ricci flow +of bounded curvature with the property that for each $t\in [0,T]$ +the Riemannian manifold $(M,g(t))$ is complete. Fix a point +$x=(p,T)\in M\times [0,T]$. For every $0<\tau0$ there is $K=K(\delta)<\infty$ so that for any set +of realized directions at ${\mathcal E}$ of cardinality $K$, +$\ell_1,\ldots,\ell_{K}$, it must be the case that there are $j$ and +$j'$ with $j\not=j'$ such that $\theta(\ell_j,\ell_{j'})<\delta$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/af00774dbaa9.md b/projects/poincare-conjecture/.astrolabe/atoms/af00774dbaa9.md new file mode 100644 index 00000000..57a9acd4 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/af00774dbaa9.md @@ -0,0 +1,47 @@ +--- +chapter: 11 +generator: tools/poincare_tex_extract.py +label: omega +labels: +- omega +mtref: '11.19' +ref: +- af00774dbaa9 +sort: theorem +source: tex +src: morgan-tian +tex_file: singlimit2 +--- +Suppose that $({\mathcal M},G)$ is a generalized Ricci flow defined +for $0\le tr_0^{-2}$ +has a strong $(2C,2\epsilon)$-canonical neighborhood in $\widehat{\mathcal M}$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/af79df2b9075.md b/projects/poincare-conjecture/.astrolabe/atoms/af79df2b9075.md new file mode 100644 index 00000000..1a611941 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/af79df2b9075.md @@ -0,0 +1,14 @@ +--- +chapter: 5 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '5.20' +ref: +- af79df2b9075 +sort: example +source: tex +src: morgan-tian +tex_file: converge2 +--- +A sequence of compact $n$-manifolds of diameter tending to zero has +a point as Gromov-Hausdorff limit. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/afc86a351f0c.md b/projects/poincare-conjecture/.astrolabe/atoms/afc86a351f0c.md new file mode 100644 index 00000000..6e9c6ab8 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/afc86a351f0c.md @@ -0,0 +1,22 @@ +--- +chapter: 9 +generator: tools/poincare_tex_extract.py +label: flatssplit +labels: +- flatssplit +mtref: '9.45' +ref: +- afc86a351f0c +sort: claim +source: tex +src: morgan-tian +tex_file: temp2kappa +--- +Suppose that $(M,g)$ and $f\colon M\to \Ar$ are as in the statement +of Theorem 9.42 and that $(M,g)$ does not have strictly +positive curvature. Then $n=3$ and the Ricci flow $(M,G(t))$ with +$G(-1)=g$ given in Equation (9.19) has a one- or two-sheeted +covering that is a product of a two-dimensional +$\kappa$-non-collapsed Ricci flow of positive curvature and a +constant flat copy of $\Ar$. The curvature is bounded on each +time-slice. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/b00ad5447ff9.md b/projects/poincare-conjecture/.astrolabe/atoms/b00ad5447ff9.md new file mode 100644 index 00000000..9eb2e5e8 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/b00ad5447ff9.md @@ -0,0 +1,21 @@ +--- +chapter: 1 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '1.22' +ref: +- b00ad5447ff9 +sort: definition +source: tex +src: morgan-tian +tex_file: prelim +--- +For any $p\in M$, we can define the *exponential +map* at $p$, $\mathit{exp}_{p}$. It is +defined on an open neighborhood $O_{p}$ of the origin in $T_{p}M$ +and is defined by $\mathit{exp}_{p}(v)=\gamma_v(1)$, the endpoint of +the unique geodesic $\gamma_v\colon [0,1]\rightarrow M$ starting +from $p$ with initial velocity vector $v$. We always take +$O_{p}\subset T_{p}M$ to be the maximal domain on which $\mathit{exp}_p$ is defined, so that $O_{p}$ is a star-shaped open +neighborhood of $0\in T_pM$. By the Hopf-Rinow Theorem, if $M$ is +complete, then the exponential map is define on all of $T_pM$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/b089fab5fd55.md b/projects/poincare-conjecture/.astrolabe/atoms/b089fab5fd55.md new file mode 100644 index 00000000..d17e9aab --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/b089fab5fd55.md @@ -0,0 +1,26 @@ +--- +chapter: 14 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '14.17' +ref: +- b089fab5fd55 +sort: definition +source: tex +src: morgan-tian +tex_file: surgery +--- +Let $({\mathcal M},G)$ be a $3$-dimensional Ricci flow with surgery, +whose time domain of definition is contained in $[0,\infty)$. For +any $x\in {\mathcal M}$ we denote the eigenvalues of $\mathit{Rm}(x)$ by +$\lambda(x)\ge \mu(x)\ge \nu(x)$ and we set $X(x)=\mathit{max}(0,-\nu(x))$. + We say +that its *curvature is pinched toward positive* if the following +hold for every $x\in {\mathcal M}$: + +- **(1)** $R(x)\ge \frac{-6}{1+4\mathbf{t}(x)}$. +- **(2)** $R(x)\geq 2X(x)\left(\mathit{log}X(x)+\mathit{log}(1+\mathbf{t}(x))-3\right)$, whenever $00$ we +say that $q=\gamma(t)$ is a *conjugate point along +$\gamma$* if there is a non-zero Jacobi field +along $\gamma$ vanishing at $\gamma(t)$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/b20ded1b2964.md b/projects/poincare-conjecture/.astrolabe/atoms/b20ded1b2964.md new file mode 100644 index 00000000..41c77fc5 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/b20ded1b2964.md @@ -0,0 +1,17 @@ +--- +chapter: 6 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '6.42' +ref: +- b20ded1b2964 +sort: claim +source: tex +src: morgan-tian +tex_file: newcompar +--- +$$ +\frac{\partial}{\partial v}\frac{\partial}{\partial w}{\mathcal L}(\mu)|_{u=v=0}= +\frac{\partial}{\partial w}\frac{\partial}{\partial v}{\mathcal +L}(\mu)|_{v=w=0}=0. +$$ diff --git a/projects/poincare-conjecture/.astrolabe/atoms/b2c992468118.md b/projects/poincare-conjecture/.astrolabe/atoms/b2c992468118.md new file mode 100644 index 00000000..264e1cf4 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/b2c992468118.md @@ -0,0 +1,16 @@ +--- +chapter: 6 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '6.8' +ref: +- b2c992468118 +sort: lemma +source: tex +src: morgan-tian +tex_file: newcompar +--- +Let $\gamma\colon [0,\tau_2]\to {\mathcal M}$ be an ${\mathcal L}$-geodesic. +Then $\mathit{lim}_{\tau\rightarrow 0}\sqrt{\tau}X_\gamma(\tau)$ exists. The +${\mathcal L}$-geodesic $\gamma$ is completely determined by this limit (and by +$\tau_2$). diff --git a/projects/poincare-conjecture/.astrolabe/atoms/b34d555cd7b5.md b/projects/poincare-conjecture/.astrolabe/atoms/b34d555cd7b5.md new file mode 100644 index 00000000..9ee428af --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/b34d555cd7b5.md @@ -0,0 +1,27 @@ +--- +chapter: 10 +generator: tools/poincare_tex_extract.py +label: secondstatment +labels: +- secondstatment +mtref: '10.29' +ref: +- b34d555cd7b5 +sort: proposition +source: tex +src: morgan-tian +tex_file: bddcurvbdddist +--- +Let $(U_\infty,g_\infty)$ be an incomplete Riemannian $3$-manifold +of non-negative curvature with an end ${\mathcal E}$ as in the +hypothesis of Proposition 10.12. Fix a minimizing geodesic +ray $\gamma$ limiting to ${\mathcal E}$. Let $\lambda_n$ be any +sequence of positive numbers tending to infinity. For each $n$ +sufficiently large let $x_n\in \gamma$ be the point at distance +$\lambda_n^{-1/2}$ from the end ${\mathcal E}$. Then the based +metric spaces $(U_\infty,\lambda_ng_\infty,x_n)$ converge in the +Gromov-Hausdorff sense to +$\left( C'_{\mathcal E},g_{\mathcal E},([\gamma],1)\right)$. Under +this convergence the distance function from the end ${\mathcal E}$ +in $(U_\infty,\lambda_ng_\infty)$ converges to the distance function +from the cone point in the open cone. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/b3b96a664796.md b/projects/poincare-conjecture/.astrolabe/atoms/b3b96a664796.md new file mode 100644 index 00000000..3448918e --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/b3b96a664796.md @@ -0,0 +1,16 @@ +--- +chapter: 11 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '11.17' +ref: +- b3b96a664796 +sort: claim +source: tex +src: morgan-tian +tex_file: singlimit2 +--- +For each $A<\infty$ and for all $n$ sufficiently large, there are +$\delta>0$ with $\delta\le T_0-T$ and a bound, independent of $n$, +on the scalar curvature of the restriction of $Q_nG_n$ to +$B_{Q_nG_n}(x_n,0,A)\times [-(T+\delta),0]$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/b3bf2f526cfc.md b/projects/poincare-conjecture/.astrolabe/atoms/b3bf2f526cfc.md new file mode 100644 index 00000000..fcce36ca --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/b3bf2f526cfc.md @@ -0,0 +1,23 @@ +--- +chapter: 6 +generator: tools/poincare_tex_extract.py +label: Ltau +labels: +- Ltau +mtref: '6.32' +ref: +- b3bf2f526cfc +sort: lemma +source: tex +src: morgan-tian +tex_file: newcompar +--- +Suppose that +$q\in {\mathcal U}_x$ with $\mathbf{t}(q)=T-\bar\tau$ for some $\bar\tau>\tau_1$. +Let $\gamma\colon [\tau_1,\bar\tau]\to {\mathcal M}$ be the unique minimizing +${\mathcal L}$-geodesic from $x$ to $q$. Then we have + +$$ +\frac{\partial L_x}{\partial \tau}(q)=2\sqrt{\bar\tau}R( +q)-\sqrt{\bar\tau}\left(R( q)+|X(\bar\tau)|^2\right). +$$ diff --git a/projects/poincare-conjecture/.astrolabe/atoms/b3fae89d3264.md b/projects/poincare-conjecture/.astrolabe/atoms/b3fae89d3264.md new file mode 100644 index 00000000..8adc035c --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/b3fae89d3264.md @@ -0,0 +1,17 @@ +--- +chapter: 11 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '11.15' +ref: +- b3fae89d3264 +sort: claim +source: tex +src: morgan-tian +tex_file: singlimit2 +--- +$y$ cannot lie in the core of a $(2C,2\epsilon)$-cap in +$(M_\infty,g_\infty(t))$, and hence it is the center of a +$2\epsilon$-neck $N$ in +$(M_\infty,g_\infty(t))$. Furthermore, minimal $g(t)$-geodesics from +$y$ to $x_\infty$ and $z$ exit out of opposite ends of $N$ (see Fig. 11.1). diff --git a/projects/poincare-conjecture/.astrolabe/atoms/b45765a3070e.md b/projects/poincare-conjecture/.astrolabe/atoms/b45765a3070e.md new file mode 100644 index 00000000..5b5dd6c6 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/b45765a3070e.md @@ -0,0 +1,20 @@ +--- +chapter: 5 +generator: tools/poincare_tex_extract.py +label: 2ndmfdconv +labels: +- 2ndmfdconv +mtref: '5.10' +ref: +- b45765a3070e +sort: corollary +source: tex +src: morgan-tian +tex_file: converge2 +--- +Suppose that $\{(M_k,g_k,x_k)\}_{k=1}^\infty$ is a sequence of based, connected +Riemannian manifolds. Suppose that the first two conditions in +Theorem 5.9 hold and suppose also that there are constants $\kappa>0$ +and $\delta>0$ such that $\mathit{Vol}_{g_k}B(x_k,\delta)\ge \kappa\delta^n$ for +all $k$. Then after passing to a subsequence there is a geometric limit which +is a complete Riemannian manifold. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/b49cbb9f66c6.md b/projects/poincare-conjecture/.astrolabe/atoms/b49cbb9f66c6.md new file mode 100644 index 00000000..da0b5362 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/b49cbb9f66c6.md @@ -0,0 +1,18 @@ +--- +chapter: 8 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '8.4' +ref: +- b49cbb9f66c6 +sort: corollary +source: tex +src: morgan-tian +tex_file: noncoll +--- +There is a constant $\varepsilon_0>0$ depending on $\bar\tau_0$ +such that, if $\varepsilon\le \varepsilon_0$, we have + +$$ +\widetilde V_x(W_\mathit{sm}(\tau_1))\le 2\varepsilon^{\frac{n}{2}}. +$$ diff --git a/projects/poincare-conjecture/.astrolabe/atoms/b4ef6801b3e2.md b/projects/poincare-conjecture/.astrolabe/atoms/b4ef6801b3e2.md new file mode 100644 index 00000000..1bbcc9a9 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/b4ef6801b3e2.md @@ -0,0 +1,20 @@ +--- +chapter: 1 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '1.30' +ref: +- b4ef6801b3e2 +sort: definition +source: tex +src: morgan-tian +tex_file: prelim +--- +We define a function $\mathit{sn}_k$ as follows: + +$$ +\mathit{sn}_k(r)=\begin{cases} +r \ \ & \mathit{if\ \}k=0 \\ +\frac{1}{\sqrt{k}}\mathit{sinh}(\sqrt{k}r) \ \ & \mathit{if \ \} k>0. +\end{cases} +$$ diff --git a/projects/poincare-conjecture/.astrolabe/atoms/b51fee7a4b41.md b/projects/poincare-conjecture/.astrolabe/atoms/b51fee7a4b41.md new file mode 100644 index 00000000..1673477e --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/b51fee7a4b41.md @@ -0,0 +1,14 @@ +--- +chapter: 3 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '3.30' +ref: +- b51fee7a4b41 +sort: remark +source: tex +src: morgan-tian +tex_file: flowbasics +--- +Clearly, the case $l=0$ of Theorem 3.29 is Shi's theorem +(Theorem 3.27). diff --git a/projects/poincare-conjecture/.astrolabe/atoms/b58a798d5eea.md b/projects/poincare-conjecture/.astrolabe/atoms/b58a798d5eea.md new file mode 100644 index 00000000..cc4d9f64 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/b58a798d5eea.md @@ -0,0 +1,27 @@ +--- +chapter: 7 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '7.17' +ref: +- b58a798d5eea +sort: claim +source: tex +src: morgan-tian +tex_file: newcomp2 +--- +For each $(q,t)\in M\times(0,T)$ there is a smooth function + +$$ +\psi_{(q,t)}\colon B_{(q,t)}\to \Ar +$$ + + with the property that at each +$z\in B_{(q,t)}$ there is an upper barrier $b_{(z,t)}$ for +$L^\tau+\psi_{(q,t)}$ at $z$ with + +$$ +\mathit{Hess}^h(b_{(z,t)})(v,v)\le -3|v|_h^2/2 +$$ + +for all $v\in T_zM$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/b5e6e97be37d.md b/projects/poincare-conjecture/.astrolabe/atoms/b5e6e97be37d.md new file mode 100644 index 00000000..8f8c99b8 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/b5e6e97be37d.md @@ -0,0 +1,22 @@ +--- +chapter: 3 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '3.39' +ref: +- b5e6e97be37d +sort: definition +source: tex +src: morgan-tian +tex_file: flowbasics +--- +We say that $t$ is a *regular time* if there is $\epsilon>0$ and +a diffeomorphism $M_{t}\times (t-\epsilon,t+\epsilon)\to \mathbf{t}^{-1}((t-\epsilon,t+\epsilon))$ compatible with time +and the vector field. A time is *singular* if it is not regular. +Notice that if all times are regular, then space-time is a product +$M_{t}\times I$ with $\mathbf{t}$ and $\chi$ coming from the second +factor. If the image $\mathbf{t}({\mathcal M})$ is an interval $I$ +bounded below, then the *initial time* for the flow is the +greatest lower bound for $I$. If $I$ includes $(-\infty,A]$ for some +$A$, then the initial time for the generalized Ricci flow is +$-\infty$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/b6bc786f7f44.md b/projects/poincare-conjecture/.astrolabe/atoms/b6bc786f7f44.md new file mode 100644 index 00000000..8dc235ee --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/b6bc786f7f44.md @@ -0,0 +1,21 @@ +--- +chapter: 3 +generator: tools/poincare_tex_extract.py +label: patch +labels: +- patch +mtref: '3.12' +ref: +- b6bc786f7f44 +sort: proposition +source: tex +src: morgan-tian +tex_file: flowbasics +--- +Suppose that $(U,g(t)),\ a\le t0$ be given. +Suppose that $B(p,2r)$ has compact closure in $M$ and suppose that $f\colon +B(p,2r)\to \Ar$ is a continuous, bounded function with $f(p)>0$. Then there is +a point $q\in B(x,2r)$ with the following properties: + +- **(1)** $f(q)\ge f(p)$. +- **(2)** Setting $\alpha=f(p)/f(q)$ we have $d(p,q)\le 2r(1-\alpha)$ +and $f(q')<2f(q)$ for all $q'\in B(q,\alpha r)$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/b74cd360f72f.md b/projects/poincare-conjecture/.astrolabe/atoms/b74cd360f72f.md new file mode 100644 index 00000000..c0db2632 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/b74cd360f72f.md @@ -0,0 +1,35 @@ +--- +chapter: 19 +generator: tools/poincare_tex_extract.py +label: neckgeo +labels: +- neckgeo +mtref: '19.5' +ref: +- b74cd360f72f +sort: corollary +source: tex +src: morgan-tian +tex_file: canonnbhd +--- +For any $\alpha>0$ there is $\epsilon_3=\epsilon_3(\alpha)>0$ such +that the following hold for any $0<\epsilon\le \epsilon_3$ and any +$\epsilon$-neck $N$ of scale $1$ centered at $x$. + +- **(1)** Suppose that $p$ and $q$ are points of $N$ +with either $|s(q)-s(p)|\ge \epsilon^{-1}/100$ or $d(p,q)\ge +\epsilon^{-1}/100$. Then we have + +$$ +(1-\alpha)|s(q)-s(p)|\le d(p,q)\le (1+\alpha)|s(q)-s(p)|. +$$ + +- **(2)** + +$$ +B(x,(1-\alpha)\epsilon^{-1})\subset +N\subset B(x,(1+\alpha)\epsilon^{-1}). +$$ + +- **(3)** Any geodesic that exits from both ends of $N$ has length at least $2(1-\alpha) +\epsilon^{-1}$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/b87df0b67468.md b/projects/poincare-conjecture/.astrolabe/atoms/b87df0b67468.md new file mode 100644 index 00000000..efd4da3c --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/b87df0b67468.md @@ -0,0 +1,29 @@ +--- +chapter: 18 +generator: tools/poincare_tex_extract.py +label: deltaprime +labels: +- deltaprime +mtref: '18.55' +ref: +- b87df0b67468 +sort: claim +source: tex +src: morgan-tian +tex_file: energy1 +--- +Fix $\zeta>0$, $A$ and $C_6, C_4$ as in the last claim, and let +$\delta'>0$ be as in the last claim. Suppose that we have $J\subset +[t_0,t_1]$ which is a finite disjoint union of intervals with +$t_1-t_0-|J|\le \delta'$. + Then there +is $\delta''>0$ ($\delta''$ is allowed to depend on $J$) such that +the following holds. Suppose that we have a function $f\colon +[t_0,t_1]\to \Ar$ such that: + +- **(1)** For all $t'0$ there is a constant $\widetilde +C=\widetilde C(c)$, depending only on $c$ and not on the +$3$-dimensional $\kappa$-solution, so that + +$$ +d_{g(-cQ^{-1})}(p,q)\le \tilde CQ^{-1/2}. +$$ diff --git a/projects/poincare-conjecture/.astrolabe/atoms/b92e1a990021.md b/projects/poincare-conjecture/.astrolabe/atoms/b92e1a990021.md new file mode 100644 index 00000000..9ff23201 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/b92e1a990021.md @@ -0,0 +1,18 @@ +--- +chapter: 16 +generator: tools/poincare_tex_extract.py +label: defnoverlinedelta0 +labels: +- defnoverlinedelta0 +mtref: '16.12' +ref: +- b92e1a990021 +sort: definition +source: tex +src: morgan-tian +tex_file: surgery +--- +Let $c>0$ be the constant from Proposition 12.31. Fix +$0<\overline\delta_0<1/4$ such that if $g$ is within +$\overline\delta_0$ of $g_0$ in the $C^{[1/\overline +\delta]}$-topology then $|R_{g'}(x)-R_{g_0}(x)|0$ such that for all $0<\epsilon\le +\epsilon_0$ the following is true. Let $X$ be a connected subset of +a Riemannian manifold $M$ with the property that every point $x\in +X$ is the center of an $\epsilon$-neck $N(x)$ in $M$. Suppose that +the central two-spheres of these necks do not separate $M$. Then +there is a subset $\{x_i\}$ of $X$ such that the necks $N(x_i)$ +(possibly after reversing their $s$-directions) form a balanced +chain of $\epsilon$-necks $\{N(x_i)\}$ whose union $U$ contains $X$. +The union $U$ is diffeomorphic to $S^2\times (0,1)$. It is an +$\epsilon$-tube. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/bb69d9bc83dc.md b/projects/poincare-conjecture/.astrolabe/atoms/bb69d9bc83dc.md new file mode 100644 index 00000000..f4f12799 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/bb69d9bc83dc.md @@ -0,0 +1,21 @@ +--- +chapter: 19 +generator: tools/poincare_tex_extract.py +label: genint +labels: +- genint +mtref: '19.7' +ref: +- bb69d9bc83dc +sort: lemma +source: tex +src: morgan-tian +tex_file: canonnbhd +--- +The following holds for every $\epsilon>0$ sufficiently small. +Suppose that $(M,g)$ is a Riemannian manifold and that $N\subset M$ +is an $\epsilon$-neck centered at $x$ and suppose that $\gamma$ is a +shortest geodesic in $M$ between its endpoints and that the length +of every component of $N\cap|\gamma|$ has length at least $ +R(x)^{-1/2}\epsilon^{-1}/8$. Then $\gamma$ crosses each two-sphere +in the neck structure on $N$ at most once; see Fig. 19.1. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/bb839ebe8376.md b/projects/poincare-conjecture/.astrolabe/atoms/bb839ebe8376.md new file mode 100644 index 00000000..7820c5f3 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/bb839ebe8376.md @@ -0,0 +1,28 @@ +--- +chapter: 9 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '9.12' +ref: +- bb839ebe8376 +sort: remark +source: tex +src: morgan-tian +tex_file: temp2kappa +--- +We are not claiming that the gradient shrinking soliton is a +$\kappa$-solution (or more precisely an extension forward in time of +a time-shifted version of a $\kappa$-solution) because we are not +claiming that the time-slices have bounded curvature operator. +Indeed, we do not know if this is true in general. We shall +establish below (see Corollary 9.50 and +Corollary 9.53) that in the case $n=2,3$, the gradient +shrinking soliton does indeed have time-slices of bounded curvature, +and hence is an extension of a $\kappa$-solution. We are also not +claiming at this point that the limiting flow is a gradient +shrinking soliton in the sense that there is a one-parameter family +of diffeomorphisms $\varphi_t\colon M_\infty\to M_\infty,\ t<0$, +with the property that $|t|\varphi_t^*g_\infty(-1)=g_\infty(t)$ and +with the property that the $\varphi_t$ are generated by the gradient +vector field of a function. We shall also + establish this result in dimensions $2$ and $3$ later in this chapter. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/bb88301ceeda.md b/projects/poincare-conjecture/.astrolabe/atoms/bb88301ceeda.md new file mode 100644 index 00000000..cd815a70 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/bb88301ceeda.md @@ -0,0 +1,17 @@ +--- +chapter: 18 +generator: tools/poincare_tex_extract.py +label: Ylengthrem +labels: +- Ylengthrem +mtref: '18.64' +ref: +- bb88301ceeda +sort: remark +source: tex +src: morgan-tian +tex_file: energy1 +--- +The condition on $k_\mathit{geod}$ implies that for any arc $J$ in +$c_0$ of length $1$ the total + length of $J\cap Y$ is less than $\delta$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/bdc4fc3c0d3e.md b/projects/poincare-conjecture/.astrolabe/atoms/bdc4fc3c0d3e.md new file mode 100644 index 00000000..f1e76400 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/bdc4fc3c0d3e.md @@ -0,0 +1,19 @@ +--- +chapter: 14 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '14.4' +ref: +- bdc4fc3c0d3e +sort: definition +source: tex +src: morgan-tian +tex_file: surgery +--- +Let ${\mathcal M}$ be a surgery space-time with $I$ as its time +interval of definition. We say that $t\in I$ is a *regular* time +if there is an interval $J\subset I$ which is an open neighborhood +in $I$ of $t$, and a diffeomorphism $M_{t}\times J\to \mathbf{t}^{-1}(J)\subset{\mathcal M}$ compatible with time and the vector +field. A time is *singular* if it is not regular. Notice that if +all times are regular, then space-time is a product $M_t\times I$ +with $\mathbf{t}$ and $\chi$ coming from the second factor. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/bdfb2841e15e.md b/projects/poincare-conjecture/.astrolabe/atoms/bdfb2841e15e.md new file mode 100644 index 00000000..93027c32 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/bdfb2841e15e.md @@ -0,0 +1,23 @@ +--- +chapter: 2 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '2.17' +ref: +- bdfb2841e15e +sort: remark +source: tex +src: morgan-tian +tex_file: prelim +--- +Notice that if we view a one-parameter family of metrics $g(t)$ as a +curve in the space of metrics on $X$ with the +$C^{[1/\epsilon]}$-topology then this is the statement that the two +paths are pointwise within $\epsilon$ of each other. It says nothing +about the derivatives of the paths, or equivalently about the time +derivatives of the metrics and of their covariant derivatives. We +will always be considering paths of metrics satisfying the Ricci +flow equation. In this context two one-parameter families of metrics +that are close in the $C^{2k}$-topology exactly when the $r^{th}$ +time derivatives of the $s^{th}$-covariant derivatives are close for +all $r,s$ with $s+2r\le 2k$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/beb6a354ee6a.md b/projects/poincare-conjecture/.astrolabe/atoms/beb6a354ee6a.md new file mode 100644 index 00000000..378df0fd --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/beb6a354ee6a.md @@ -0,0 +1,16 @@ +--- +chapter: 5 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '5.25' +ref: +- beb6a354ee6a +sort: lemma +source: tex +src: morgan-tian +tex_file: converge2 +--- +Let $(X_k,x_k)$ be a sequence of locally compact metric spaces. Suppose that +$(Y,y)$ and $(Y',y')$ are complete, locally compact, based metric spaces that +are limits of the sequence in the Gromov-Hausdorff sense. Then there is an +isometry $(Y,y)\to (Y',y')$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/bf8f356cfd7a.md b/projects/poincare-conjecture/.astrolabe/atoms/bf8f356cfd7a.md new file mode 100644 index 00000000..8a0d1144 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/bf8f356cfd7a.md @@ -0,0 +1,30 @@ +--- +chapter: 8 +generator: tools/poincare_tex_extract.py +label: nabR +labels: +- nabR +mtref: '8.5' +ref: +- bf8f356cfd7a +sort: claim +source: tex +src: morgan-tian +tex_file: noncoll +--- +There is a universal positive constant $\varepsilon'_0$ such that, if +$\varepsilon\le \varepsilon'_0$, then there is a constant $C_1<\infty$ +depending only on the dimension $n$ such that the following hold for all $y\in +B(x,T,r/2)$, and for all $t\in[T-\tau_1,T]$: + +- **(1)** + +$$ +|\nabla R(y,t)|\le \frac{C_1}{r^3} +$$ + +- **(2)** + +$$ +(1-C_1\varepsilon)\le \frac{g(y,t)}{g(y,T)}\le (1+C_1\varepsilon). +$$ diff --git a/projects/poincare-conjecture/.astrolabe/atoms/c1330c5a8be3.md b/projects/poincare-conjecture/.astrolabe/atoms/c1330c5a8be3.md new file mode 100644 index 00000000..e87cb829 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/c1330c5a8be3.md @@ -0,0 +1,21 @@ +--- +chapter: 4 +generator: tools/poincare_tex_extract.py +label: kappa0r0t0 +labels: +- kappa0r0t0 +mtref: '4.11' +ref: +- c1330c5a8be3 +sort: proposition +source: tex +src: morgan-tian +tex_file: maxprin +--- +There is $\kappa_0>0$ depending only on the dimension $n$ + such that the following holds. +Let $(M^n,g(t)),\ 0\le t\le T$, be a Ricci flow with bounded curvature, with +each $(M,g(t))$ being complete, and with normalized initial conditions. Then +$|\mathit{Rm}(x,t)|\le 2$ for all $x\in M$ and all $t\in [0,\mathit{min}(T,2^{-4})]$. +Furthermore, for any $t\in [0,\mathit{min}(T,2^{-4})]$ and any $x\in M$ and any +$r\le 1$ we have $\mathit{Vol} B(x,t,r)\ge \kappa_0r^n$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/c25742cc5546.md b/projects/poincare-conjecture/.astrolabe/atoms/c25742cc5546.md new file mode 100644 index 00000000..06390f37 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/c25742cc5546.md @@ -0,0 +1,19 @@ +--- +chapter: 6 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '6.69' +ref: +- c25742cc5546 +sort: claim +source: tex +src: morgan-tian +tex_file: newcompar +--- +Let $z\in M_{T-\tau}$. Suppose that there is a neighborhood of $z$ in +$M_{T-\tau}$ with the property that every point of the neighborhood is the +endpoint of a minimizing ${\mathcal L}$-geodesic from $x$, so that $L^\tau_x$ +is defined on this neighborhood of $z$. Suppose that there are two distinct, +minimizing ${\mathcal L}$-geodesics $\gamma_{Z_1}$ and $\gamma_{Z_2}$ from $x$ +ending at $z$ with the property that the differential of ${\mathcal L}\mathit{exp}^{\tau}$ is an isomorphism at both $Z_1$ and $Z_2$. Then the function +$L^\tau_x$ is non-differentiable at $z$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/c2ac821436de.md b/projects/poincare-conjecture/.astrolabe/atoms/c2ac821436de.md new file mode 100644 index 00000000..5de037b3 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/c2ac821436de.md @@ -0,0 +1,18 @@ +--- +chapter: 11 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '11.27' +ref: +- c2ac821436de +sort: definition +source: tex +src: morgan-tian +tex_file: singlimit2 +--- +Fix any $\rho,\ 0<\rho0$ such that the following is true for any +$0<\epsilon<\bar \epsilon$. There is $C=C(\epsilon)$ such that for +any $\kappa>0$ and any $\kappa$-solution $(M,g(t))$ one of the +following holds. + +- **(1)** $(M,g(t))$ is round for all $t\le 0$. In this case $M$ is +diffeomorphic to the quotient of $S^3$ by a finite subgroup of +$SO(4)$ acting freely. +- **(2)** $(M,g(0))$ is compact and of positive curvature. For any $x,y\in M$ +and any $2$-plane $P_y$ in $T_yM$ + we have + +$$ +\begin{aligned} +C^{-1/2}R(x,0)^{-1} & <\mathit{diam}(M,g(0)) + & 0$. In +particular, given two disjoint central $2$-spheres of + $\epsilon$-necks in $(M,g(0))$ the region of $M$ bounded by these $2$-spheres is + diffeomorphic + to $S^2\times [0,1]$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/c353298169dd.md b/projects/poincare-conjecture/.astrolabe/atoms/c353298169dd.md new file mode 100644 index 00000000..5bc217c0 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/c353298169dd.md @@ -0,0 +1,16 @@ +--- +chapter: 14 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '14.13' +ref: +- c353298169dd +sort: definition +source: tex +src: morgan-tian +tex_file: surgery +--- +Let $({\mathcal M},G)$ be a Ricci flow with surgery, and let $x$ be +a point of space-time. Set $t=\mathbf{t}(x)$. For any $r>0$ we define +$B(x,t,r)\subset M_t$ to be the metric ball of radius $r$ centered +at $x$ in the Riemannian manifold $(M_t,G(t))$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/c397ced906ac.md b/projects/poincare-conjecture/.astrolabe/atoms/c397ced906ac.md new file mode 100644 index 00000000..ad00a0c4 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/c397ced906ac.md @@ -0,0 +1,17 @@ +--- +chapter: 10 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '10.30' +ref: +- c397ced906ac +sort: remark +source: tex +src: morgan-tian +tex_file: bddcurvbdddist +--- +Notice that since the manifolds $(U_\infty,\lambda_ng_\infty,x_n)$ +are not complete, there can be more than one Gromov-Hausdorff limit. +For example we could take the full cone as a limit. Indeed, the cone +is the only Gromov-Hausdorff limit that is complete as a metric +space. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/c3d4eaed00fd.md b/projects/poincare-conjecture/.astrolabe/atoms/c3d4eaed00fd.md new file mode 100644 index 00000000..5307a44c --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/c3d4eaed00fd.md @@ -0,0 +1,25 @@ +--- +chapter: 6 +generator: tools/poincare_tex_extract.py +label: redvoltilde +labels: +- redvoltilde +mtref: '6.71' +ref: +- c3d4eaed00fd +sort: lemma +source: tex +src: morgan-tian +tex_file: newcompar +--- +Let $A\subset {\mathcal U}_x(\tau)$ be a measurable subset. Define +$\widetilde A\subset \widetilde{\mathcal U}_x(\tau)$ to be the +pre-image under ${\mathcal L}\mathit{exp}_x^{\tau}$ of $A$. Then + +$$ +\widetilde V_x(A)=\int_{\widetilde A}\tau^{-\frac{n}{2}}\mathit{exp}(-\tilde l(Z, \tau)){\mathcal J}(Z,\tau)dZ, +$$ + + where $dZ$ is the usual +Euclidean volume element and ${\mathcal J}(Z,\tau)$ is the Jacobian determinant +of ${\mathcal L}\mathit{exp}_x^{\tau}$ at $Z\in T_xM_T$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/c4848a1b6f80.md b/projects/poincare-conjecture/.astrolabe/atoms/c4848a1b6f80.md new file mode 100644 index 00000000..01448643 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/c4848a1b6f80.md @@ -0,0 +1,31 @@ +--- +chapter: 1 +generator: tools/poincare_tex_extract.py +label: Bianchi +labels: +- Bianchi +mtref: '1.5' +ref: +- c4848a1b6f80 +sort: claim +source: tex +src: morgan-tian +tex_file: prelim +--- +The Riemann curvature tensor ${\mathcal R}$ satisfies the following +properties: + +$\bullet$ (Symmetry) $R_{ijkl}=-R_{jikl}$, $R_{ijkl}=-R_{ijlk}$, +$R_{ijkl}=R_{klij}$, + +$\bullet$ (1st Bianchi identity) The sum of +$R_{ijkl}$ over the cyclic permutation of + + any three indices vanishes, + +$\bullet$ (2nd Bianchi identity) +$R_{ijkl,h}+R_{ijlh,k}+R_{ijhk,l}=0$, where + +$$ +R_{ijkl,h}=(\nabla_{\partial_h}{\mathcal R})_{ijkl}. +$$ diff --git a/projects/poincare-conjecture/.astrolabe/atoms/c4ccdd484637.md b/projects/poincare-conjecture/.astrolabe/atoms/c4ccdd484637.md new file mode 100644 index 00000000..b83284c4 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/c4ccdd484637.md @@ -0,0 +1,16 @@ +--- +chapter: 1 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '1.17' +ref: +- c4ccdd484637 +sort: definition +source: tex +src: morgan-tian +tex_file: prelim +--- +Let $I$ be an open interval. +A smooth curve $\gamma\colon I \rightarrow M$ is called a +*geodesic* if $\nabla_{\dot \gamma}\dot +\gamma =0$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/c5b8a4d0d061.md b/projects/poincare-conjecture/.astrolabe/atoms/c5b8a4d0d061.md new file mode 100644 index 00000000..4084805b --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/c5b8a4d0d061.md @@ -0,0 +1,41 @@ +--- +chapter: 1 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '1.8' +ref: +- c5b8a4d0d061 +sort: definition +source: tex +src: morgan-tian +tex_file: prelim +--- +The *Ricci curvature tensor*, +denoted $\mathit{Ric}$ or $\mathit{Ric}_g$ when it is necessary to specify +the metric, is a symmetric contravariant two-tensor. In local +coordinates it is defined by + +$$ +\mathit{Ric}(X,Y)=g^{kl}R(X,\partial_k,Y,\partial_l). +$$ + +The value of this tensor at a point $p\in M$ is given by +$\sum_{i=1}^nR(X(p),e_i,Y(p),e_i)$ where $\{e_{1},\cdots,e_{n}\}$ is +an orthonormal basis of $T_pM$. Clearly $\mathit{Ric}$ is a symmetric +bilinear form on $TM$, given in local coordinates by + +$$ +\mathit{Ric}=\mathit{Ric}_{ij}dx^i\otimes dx^j, +$$ + + where +$\mathit{Ric}_{ij}=\mathit{Ric}(\partial_i,\partial _j)$. The +*scalar curvature* is defined by: + +$$ +R=R_g=\mathit{tr}_g \mathit{Ric}=g^{ij}\mathit{Ric}_{ij}. +$$ + + We will say that $\mathit{Ric} \geq +k$ (or $\leq k$) if all the eigenvalues of $\mathit{Ric}$ are $\geq k$ +(or $\leq k$). diff --git a/projects/poincare-conjecture/.astrolabe/atoms/c5e007547ff3.md b/projects/poincare-conjecture/.astrolabe/atoms/c5e007547ff3.md new file mode 100644 index 00000000..e2f3a701 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/c5e007547ff3.md @@ -0,0 +1,28 @@ +--- +chapter: 12 +generator: tools/poincare_tex_extract.py +label: defn:stinmetr +labels: +- defn:stinmetr +mtref: '12.1' +ref: +- c5e007547ff3 +sort: definition +source: tex +src: morgan-tian +tex_file: stdsoln +--- +A *standard initial metric* +is a metric $g_0$ on $\Ar^3$ with the following properties: + +- $g_0$ is a complete metric. +- $g_0$ has non-negative sectional curvature at every point. +- $g_0$ is invariant under the usual $SO(3)$-action on $\Ar^3$. +- there is a compact ball $B\subset \Ar^3$ so +that the restriction of the metric $g_0$ to the complement of this +ball is isometric to the product $(S^2,h)\times (\Ar^+,ds^2)$ where +$h$ is the round metric of scalar curvature $1$ on $S^2$. +- $g_0$ has constant sectional curvature $1/4$ near the origin. (This point will be +denoted $p$ and is called the *tip* of the initial metric.) + + See Fig. 12.1. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/c5e225b067f1.md b/projects/poincare-conjecture/.astrolabe/atoms/c5e225b067f1.md new file mode 100644 index 00000000..07c08262 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/c5e225b067f1.md @@ -0,0 +1,48 @@ +--- +chapter: 12 +generator: tools/poincare_tex_extract.py +label: RDuniq +labels: +- RDuniq +mtref: '12.26' +ref: +- c5e225b067f1 +sort: proposition +source: tex +src: morgan-tian +tex_file: stdsoln +--- +Let $\hat g_1( t) $ and $\hat g_2( t),\ 0 \leq t\leq T $, be two +bounded solutions of the Ricci-DeTurck flow on complete and +noncompact manifold $M^{n}$ with initial metric $g_1( t_0) =g_2(t_ +0) =g$. Suppose that for some $10,$ there +is a $k_0$ arbitrarily large with + +$$ +\left\vert \hat g_1( t) -\hat g_2( t) \right\vert +_{C^{1}\left(\partial\Omega_{k_0}\right) ,g}\leq\epsilon, +$$ + +Then $ \hat g_1( t) =\hat g_2( t)$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/c67b6547955b.md b/projects/poincare-conjecture/.astrolabe/atoms/c67b6547955b.md new file mode 100644 index 00000000..9b61821b --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/c67b6547955b.md @@ -0,0 +1,15 @@ +--- +chapter: 3 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '3.17' +ref: +- c67b6547955b +sort: corollary +source: tex +src: morgan-tian +tex_file: flowbasics +--- +$$ +\Phi_t^*(g(t))=g(0). +$$ diff --git a/projects/poincare-conjecture/.astrolabe/atoms/c7a0e64aa85c.md b/projects/poincare-conjecture/.astrolabe/atoms/c7a0e64aa85c.md new file mode 100644 index 00000000..71f6146c --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/c7a0e64aa85c.md @@ -0,0 +1,22 @@ +--- +chapter: 3 +generator: tools/poincare_tex_extract.py +label: Cinftybd +labels: +- Cinftybd +mtref: '3.31' +ref: +- c7a0e64aa85c +sort: corollary +source: tex +src: morgan-tian +tex_file: flowbasics +--- +Suppose that $(M,g(t)),\ 0\le t\le T$, is a Ricci flow +with $(M,g(t))$ being complete and with $T<\infty$. Suppose that +$\operatorname*{Rm}(x,0)$ is bounded in the $C^\infty$-topology +independent of $x\in M$ and suppose that +$\left\vert\operatorname*{Rm}(x,t)\right\vert$ is bounded +independent of $x\in M$ and $t\in[0,T]$. Then the operator +$\operatorname*{Rm}(x,t)$ is bounded in the $C^\infty$-topology +independent of $(x,t)\in M\times [0,T]$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/c7dbaa8c2ca5.md b/projects/poincare-conjecture/.astrolabe/atoms/c7dbaa8c2ca5.md new file mode 100644 index 00000000..b7a2563b --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/c7dbaa8c2ca5.md @@ -0,0 +1,17 @@ +--- +chapter: 4 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '4.35' +ref: +- c7dbaa8c2ca5 +sort: theorem +source: tex +src: morgan-tian +tex_file: maxprin +--- +Let $(M,g)$ be a compact $3$-dimensional shrinking soliton, i.e., +there is a Ricci flow $(M,g(t)),\ 0\le t0$ such that $\mathit{inj}_{(M_k,g_k)}(x_k)\ge \delta$ for all $k$ sufficiently large. + + Then after passing to a subsequence there is a geometric limit which + is a complete Riemannian manifold. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/c86aa999ce8a.md b/projects/poincare-conjecture/.astrolabe/atoms/c86aa999ce8a.md new file mode 100644 index 00000000..bbc33801 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/c86aa999ce8a.md @@ -0,0 +1,27 @@ +--- +chapter: 15 +generator: tools/poincare_tex_extract.py +label: RFSexists +labels: +- RFSexists +mtref: '15.10' +ref: +- c86aa999ce8a +sort: corollary +source: tex +src: morgan-tian +tex_file: surgery +--- +Let $\mathbf{K}$, $\mathbf{r}$ and $\Delta$ be surgery parameter sequences +provided by the previous theorem. Let $\overline\delta(t)$ be a +non-increasing positive function with $\overline\delta(t)\le +\Delta$. Let $M$ be a compact $3$-manifold containing no $\Ar P^2$ +with trivial normal bundle. Then there is a Riemannian metric $g(0)$ +on $M$ and a Ricci flow with surgery defined for $0\le t<\infty$ +with initial metric $(M,g(0))$. This Ricci flow with surgery +satisfies the seven assumptions and is $\mathbf{K}$-non-collapsed on +scales $\le \epsilon$. It also satisfies the strong +$(C,\epsilon)$-canonical neighborhood assumption with parameter +$\mathbf{r}$ and has curvature pinched toward positive. Furthermore, +any surgery at a time $t\in [T_i,T_{i+1})$ is done using +$\overline\delta(t)$ and $r_{i+1}$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/c8e7931dbf65.md b/projects/poincare-conjecture/.astrolabe/atoms/c8e7931dbf65.md new file mode 100644 index 00000000..766b23cc --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/c8e7931dbf65.md @@ -0,0 +1,24 @@ +--- +chapter: 15 +generator: tools/poincare_tex_extract.py +label: '14.26' +labels: +- '14.26' +mtref: '15.11' +ref: +- c8e7931dbf65 +sort: lemma +source: tex +src: morgan-tian +tex_file: surgery +--- +Suppose that $({\mathcal M},G)$ is a Ricci flow with surgery going +singular at time $T\in [T_{i-1},T_i)$. We suppose that $({\mathcal +M},G)$ satisfies Assumptions (1) - (7), has curvature pinched toward +positive, satisfies the strong $(C,\epsilon)$-canonical neighborhood +assumption with parameter $\mathbf{r}$ and is $\mathbf{K}$ non-collapsed. +Then the result of the surgery operation at time $T$ on $({\mathcal M},G)$ is a + Ricci flow with surgery defined on $[0,T')$ for some $T'>T$. + The resulting Ricci flow with +surgery satisfies Assumptions (1) -- (7). It also has curvature pinched +toward positive. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/c954e95b5d89.md b/projects/poincare-conjecture/.astrolabe/atoms/c954e95b5d89.md new file mode 100644 index 00000000..d679586f --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/c954e95b5d89.md @@ -0,0 +1,32 @@ +--- +chapter: 4 +generator: tools/poincare_tex_extract.py +label: MP +labels: +- MP +mtref: '4.7' +ref: +- c954e95b5d89 +sort: theorem +source: tex +src: morgan-tian +tex_file: maxprin +--- +**(The maximum principle for tensors)** +Let $(M,g)$ be a compact Riemannian manifold. Let ${\mathcal V}\to M$ be a +tensor bundle and let ${\mathcal Z}\subset {\mathcal V}$ be a closed, convex +subset invariant under the parallel translation induced by the Levi-Civita +connection. Suppose that $\psi$ is a fiberwise vector field defined on an open +neighborhood of ${\mathcal Z}$ in ${\mathcal V}$ that preserves ${\mathcal Z}$. +Suppose that ${\mathcal T}(x,t),\ 0\le t\le T$, is a one-parameter family of +sections of ${\mathcal V}$ that evolves according to the parabolic equation + +$$ +\frac{\partial {\mathcal T}}{\partial t}=\Delta {\mathcal T} ++\psi({\mathcal T}). +$$ + + If ${\mathcal T}(x,0)$ is contained in +${\mathcal Z}$ for all $x\in M$, then ${\mathcal T}(x,t)$ is +contained in ${\mathcal Z}$ for all $x\in M$ and for all $0\le t\le +T$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/c958eb1d5427.md b/projects/poincare-conjecture/.astrolabe/atoms/c958eb1d5427.md new file mode 100644 index 00000000..bd14e967 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/c958eb1d5427.md @@ -0,0 +1,22 @@ +--- +chapter: 6 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '6.46' +ref: +- c958eb1d5427 +sort: definition +source: tex +src: morgan-tian +tex_file: newcompar +--- +For any ${\mathcal L}$-geodesic $\gamma$ parameterized by $[\tau_1,\bar\tau]$ +we define + +$$ +K_{\tau_1}^{\bar\tau}(\gamma)= +\int_{\tau_1}^{\bar\tau}\tau^{3/2}H(X)d\tau. +$$ + +In the special case when $\tau_1=0$ we denote this integral by +$K^{\bar\tau}(\gamma)$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/c9d8cb9d34c3.md b/projects/poincare-conjecture/.astrolabe/atoms/c9d8cb9d34c3.md new file mode 100644 index 00000000..260a911c --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/c9d8cb9d34c3.md @@ -0,0 +1,22 @@ +--- +chapter: 13 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '13.14' +ref: +- c9d8cb9d34c3 +sort: proposition +source: tex +src: morgan-tian +tex_file: surgery +--- +Fix $\delta>0$ sufficiently small. Suppose that for some $t\ge 0$ +and every point $p\in N$ the curvature of $h$ satisfies: + +- **(1)** $R(p)\ge \frac{-6}{1+4t}$, and +- **(2)** $R(p)\geq 2X(p)\left(\mathit{log}X(p)+\mathit{log}(1+t)-3\right)$ whenever $00$ sufficiently small. + Let $(M,g)$ be a complete, positively curved Riemannian +$3$-manifold. Then $(M,g)$ does not contain $\epsilon$-necks of +arbitrarily small scale. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/cbf568c0a596.md b/projects/poincare-conjecture/.astrolabe/atoms/cbf568c0a596.md new file mode 100644 index 00000000..5ffcf9ea --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/cbf568c0a596.md @@ -0,0 +1,17 @@ +--- +chapter: 18 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '18.70' +ref: +- cbf568c0a596 +sort: remark +source: tex +src: morgan-tian +tex_file: energy1 +--- +The same argument shows that from any $a\in (\partial A\cap B)$ +there is a unique embedded geodesic in $B$ from $v$ to $a$ with +length at most $(1/2)+\delta'$. (Such geodesics may cross more than +once, but the argument given in the lemma applies to sub-geodesics +from $v$ to the first point of intersection along $\gamma$.) diff --git a/projects/poincare-conjecture/.astrolabe/atoms/cc3a26ca3809.md b/projects/poincare-conjecture/.astrolabe/atoms/cc3a26ca3809.md new file mode 100644 index 00000000..ae2604ae --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/cc3a26ca3809.md @@ -0,0 +1,22 @@ +--- +chapter: 11 +generator: tools/poincare_tex_extract.py +label: compactbd +labels: +- compactbd +mtref: '11.11' +ref: +- cc3a26ca3809 +sort: lemma +source: tex +src: morgan-tian +tex_file: singlimit2 +--- +Suppose that there is a geometric limit flow defined on $(-T,0]$ for +some $00 +labels: +- nuprime>0 +mtref: '13.13' +ref: +- cc9110b4d2fe +sort: lemma +source: tex +src: morgan-tian +tex_file: surgery +--- +With these choices of $q$ and $C_0$ for any $\delta>0$ sufficiently small we have +$\nu'>0$ and $\mu'>0$ for $s\in [1,4+A_0]$ and $\lambda'>1/4$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/cd062526b91a.md b/projects/poincare-conjecture/.astrolabe/atoms/cd062526b91a.md new file mode 100644 index 00000000..72edd6a1 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/cd062526b91a.md @@ -0,0 +1,19 @@ +--- +chapter: 7 +generator: tools/poincare_tex_extract.py +label: geoexist +labels: +- geoexist +mtref: '7.2' +ref: +- cd062526b91a +sort: lemma +source: tex +src: morgan-tian +tex_file: newcomp2 +--- +Assume that $M$ is connected. +Given $p_1,p_2\in M$ and $0\le\tau_1<\tau_2\le T$, there is + a minimizing ${\mathcal L}$-geodesic: +$\gamma\colon[\tau_{1},\tau_{2}]\to M\times [0,T]$ connecting +$(p_1,\tau_1)$ to $(p_2,\tau_2)$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/cd2cfdd79bea.md b/projects/poincare-conjecture/.astrolabe/atoms/cd2cfdd79bea.md new file mode 100644 index 00000000..752728aa --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/cd2cfdd79bea.md @@ -0,0 +1,16 @@ +--- +chapter: 5 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '5.23' +ref: +- cd2cfdd79bea +sort: lemma +source: tex +src: morgan-tian +tex_file: converge2 +--- +Let $(X_k,x_k)$ be a sequence of based metric spaces whose diameters are +uniformly bounded. Suppose that $(Y,y)$ and $(Y',y')$ are limits in the +Gromov-Hausdorff sense of this sequence and each of $Y$ and $Y'$ are compact. +Then $(Y,y)$ is isometric to $(Y',y')$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/cdb7d427d07e.md b/projects/poincare-conjecture/.astrolabe/atoms/cdb7d427d07e.md new file mode 100644 index 00000000..b7e0a50f --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/cdb7d427d07e.md @@ -0,0 +1,19 @@ +--- +chapter: 9 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '9.77' +ref: +- cdb7d427d07e +sort: definition +source: tex +src: morgan-tian +tex_file: temp2kappa +--- +Fix $C<\infty$ and $\epsilon>0$. For any Riemannian manifold +$(M,g)$, an open neighborhood $U$ of a point $x\in M$ is a *$(C,\epsilon)$-canonical neighborhood* if one of the following holds: + +- **(1)** $U$ is an $\epsilon$-neck in $(M,g)$ centered at $x$. +- **(2)** $U$ a $(C,\epsilon)$-cap in $(M,g)$ whose core contains $x$. +- **(3)** $U$ is a $C$-component of $(M,g)$. +- **(4)** $U$ is an $\epsilon$-round component of $(M,g)$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/ce620437e5d6.md b/projects/poincare-conjecture/.astrolabe/atoms/ce620437e5d6.md new file mode 100644 index 00000000..e773cce0 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/ce620437e5d6.md @@ -0,0 +1,15 @@ +--- +chapter: 11 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '11.33' +ref: +- ce620437e5d6 +sort: claim +source: tex +src: morgan-tian +tex_file: singlimit2 +--- +For any $A<\infty$ for all $n$ sufficiently large, +$B(x_n,0,AQ_n^{-1/2})$ is contained in the $2\epsilon$-horn +${\mathcal H}_n$ and has compact closure in ${\mathcal M}_n$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/ce85fb7318bf.md b/projects/poincare-conjecture/.astrolabe/atoms/ce85fb7318bf.md new file mode 100644 index 00000000..a159f6a6 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/ce85fb7318bf.md @@ -0,0 +1,19 @@ +--- +chapter: 4 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '4.21' +ref: +- ce85fb7318bf +sort: remark +source: tex +src: morgan-tian +tex_file: maxprin +--- +Notice that there are only four possibilities for the cover required +by the corollary. It can be trivial, or a normal $\Zee$-cover or it +can be a two-sheeted cover or a normal infinite dihedral group +cover. In the first two cases, there is a unit vector field on $M$ +parallel under $g(t)$ for all $t$ spanning the null direction of +$\mathit{Ric}$. In the last two cases, there is no such vector field, +only a non-orientable line field. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/cf4b9ce8f8e3.md b/projects/poincare-conjecture/.astrolabe/atoms/cf4b9ce8f8e3.md new file mode 100644 index 00000000..ef282322 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/cf4b9ce8f8e3.md @@ -0,0 +1,14 @@ +--- +chapter: 18 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '18.77' +ref: +- cf4b9ce8f8e3 +sort: claim +source: tex +src: morgan-tian +tex_file: energy1 +--- +The interior of ${\mathcal T}_2$ is disjoint from the interior of +${\mathcal T}_1$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/cf61b731c6f2.md b/projects/poincare-conjecture/.astrolabe/atoms/cf61b731c6f2.md new file mode 100644 index 00000000..1f27ec1c --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/cf61b731c6f2.md @@ -0,0 +1,43 @@ +--- +chapter: 13 +generator: tools/poincare_tex_extract.py +label: LOCALSURGERY +labels: +- LOCALSURGERY +mtref: '13.2' +ref: +- cf61b731c6f2 +sort: theorem +source: tex +src: morgan-tian +tex_file: surgery +--- +There are constants $C_0,q,R_0<\infty$ and $\delta'_0>0$ such that the +following hold for the result $({\mathcal S},\widetilde g)$ of surgery on +$(N,g)$ provided that $R(x_0)\ge R_0$, $0<\delta\le \delta'_0$. Define $f(s)$ +as above with the constants $C_0,\delta$ and then use $f$ to define surgery on +a $\delta$-neck $N$ to produce $({\mathcal S},\tilde g)$. Then the following +hold. + +- Fix $t\ge 0$. For any $p\in N$, +let $X(p)=\mathit{max}(0,-\nu_{g}(p))$, where $\nu_ g(p)$ is the +smallest eigenvalue of $Rm_{ g}(p)$. Suppose that for all $p\in N$ +we have: + +- **(1)** $R(p)\ge \frac{-6}{1+4t}$, and +- **(2)** $R(p)\geq 2X(p)\left(\mathit{log}X(p)+\mathit{log}(1+t)-3\right)$, whenever $00$ there is $\delta'_1=\delta'_1(\delta'')>0$ such that if +$\delta\le \mathit{min}(\delta'_1,\delta'_0)$, then the restriction of +$\hat g$ to $B_{\hat g}(p_0,(\delta'')^{-1})$ in $({\mathcal S},\hat +g)$ is $\delta''$-close in the $C^{[1/\delta'']}$-topology to the +restriction of the standard initial metric $g_0$ to +$B_{g_0}(p_0,(\delta'')^{-1})$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/cf6eed7b1da4.md b/projects/poincare-conjecture/.astrolabe/atoms/cf6eed7b1da4.md new file mode 100644 index 00000000..19af7bc9 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/cf6eed7b1da4.md @@ -0,0 +1,18 @@ +--- +chapter: 12 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '12.30' +ref: +- cf6eed7b1da4 +sort: claim +source: tex +src: morgan-tian +tex_file: stdsoln +--- +For any $\epsilon>0$, there is $A<\infty$ such that for all $n$ +sufficiently large we have + +$$ +\mathit{Vol}(B_{Q_ng}(x,t_n,A))<\epsilon A^3. +$$ diff --git a/projects/poincare-conjecture/.astrolabe/atoms/cfb464e3cf35.md b/projects/poincare-conjecture/.astrolabe/atoms/cfb464e3cf35.md new file mode 100644 index 00000000..3432ec9a --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/cfb464e3cf35.md @@ -0,0 +1,14 @@ +--- +chapter: 16 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '16.22' +ref: +- cfb464e3cf35 +sort: claim +source: tex +src: morgan-tian +tex_file: surgery +--- +Any path $\gamma$ beginning at $x$ and parameterized by backward +time misses $P_0({\mathcal C})$ if ${\mathcal L}(\gamma)0$. Suppose that $(M,g(0))$ is not flat and that for some $x\in M$ the +endomorphism $\mathit{Rm}(x,T)$ has a zero eigenvalue. Then $M$ has a cover +$\widetilde M$ such that, denoting the induced family of metrics on this cover +by $\tilde g(t)$, we have that $(\widetilde M,\tilde g(t))$ splits as a product + +$$ +(N,h(t))\times (\Ar,ds^2) +$$ + +where $(N,h(t))$ is a surface of positive curvature for all $00$ then every metric ball +$B(x,t,r)$ has volume at least $Vr^n$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/d22d3cf47fbf.md b/projects/poincare-conjecture/.astrolabe/atoms/d22d3cf47fbf.md new file mode 100644 index 00000000..9743a7c0 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/d22d3cf47fbf.md @@ -0,0 +1,26 @@ +--- +chapter: 3 +generator: tools/poincare_tex_extract.py +label: 2ndI.8.3 +labels: +- 2ndI.8.3 +mtref: '3.25' +ref: +- d22d3cf47fbf +sort: corollary +source: tex +src: morgan-tian +tex_file: flowbasics +--- +Let $t_0\in \Ar$ and let $(M,g(t))$ be a Ricci flow defined for $t$ in an +interval +containing $t_0$ and with $(M,g(t))$ complete for every $t$ in this +interval. Fix a constant $K<\infty$. Suppose that $\mathit{Ric}(x,t_0)\le (n-1)K$ +for all $x\in M$. + Then for any points $x_0,x_1\in M$ we have + +$$ +\frac{d(d_t(x_0,x_1))}{dt}\Bigl|_{t=t_0}\Bigr.\ge -4(n-1)\sqrt{\frac{2K}{3}} +$$ + +in the sense of forward difference quotients. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/d23dc29f2aae.md b/projects/poincare-conjecture/.astrolabe/atoms/d23dc29f2aae.md new file mode 100644 index 00000000..472221c4 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/d23dc29f2aae.md @@ -0,0 +1,16 @@ +--- +chapter: 6 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '6.7' +ref: +- d23dc29f2aae +sort: definition +source: tex +src: morgan-tian +tex_file: newcompar +--- +A curve $\gamma$, parameterized by backward time, that is a critical +point of the ${\mathcal L}$-length is called an $\mathcal{L}$-*geodesic*. +Equation (6.3) is the *${\mathcal L}$-geodesic +equation*. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/d2e61ac52277.md b/projects/poincare-conjecture/.astrolabe/atoms/d2e61ac52277.md new file mode 100644 index 00000000..6d2c2a58 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/d2e61ac52277.md @@ -0,0 +1,28 @@ +--- +chapter: 6 +generator: tools/poincare_tex_extract.py +label: DLJacobi +labels: +- DLJacobi +mtref: '6.19' +ref: +- d2e61ac52277 +sort: lemma +source: tex +src: morgan-tian +tex_file: newcompar +--- +Let $Z\in {\mathcal D}_x^{\bar\tau}\subset T_xM_{T-\tau_1}$. The +differential of ${\mathcal L}\mathit{exp}^{\bar\tau}_x$ +at the point $Z$ is given as follows: For each $W\in +T_x(M_{T-\tau_1})$ there is a unique ${\mathcal L}$-Jacobi +field $Y_W(\tau)$ along +$\gamma_Z$ with the property that $\sqrt{\tau_1}Y_W(\tau_1)=0$ and +$\sqrt{\tau_1}\nabla_X(Y_W)(\tau_1)=W$. We have + +$$ +d_Z{\mathcal L}\mathit{exp}^{\bar\tau}_x(W)=Y_W(\bar\tau). +$$ + +Again, in case $\tau_1=0$, both of the conditions on $Y_W$ are interpreted as +the limits as $\tau\rightarrow 0$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/d3d4ce2e821e.md b/projects/poincare-conjecture/.astrolabe/atoms/d3d4ce2e821e.md new file mode 100644 index 00000000..2de9e545 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/d3d4ce2e821e.md @@ -0,0 +1,19 @@ +--- +chapter: 13 +generator: tools/poincare_tex_extract.py +label: stdballs +labels: +- stdballs +mtref: '13.15' +ref: +- d3d4ce2e821e +sort: lemma +source: tex +src: morgan-tian +tex_file: surgery +--- +Provided that $\delta>0$ is sufficiently small the following holds. Let $(N,g)$ +be a $\delta$-neck and let $({\mathcal S},\tilde g)$ be the result of surgery +along the cental $2$-sphere of this neck. Then for any $00$ is sufficiently small: + +- **(1)** $|Z(\psi(Z))|0,\kappa>0$. Let $({\mathcal +M}_n,G_n,x_n)$ be a sequence of based generalized $3$-dimensional +Ricci flows. We set $t_n=\mathbf{t}(x_n)$ and $Q_n=R(x_n)$. We denote +by $M_n$ the $t_n$ time-slice of ${\mathcal M}_n$. We suppose that: + +- **(1)** Each $({\mathcal M}_n,G_n)$ either has a time interval of definition contained in $[0,\infty)$ +and has curvature pinched toward positive, or has non-negative curvature. +- **(2)** Every point $y_n\in ({\mathcal M}_n,G_n)$ with $\mathbf{t}(y_n)\le +t_n$ and with $R(y_n)\ge 4R(x_n)$ has a strong +$(C,\epsilon)$-canonical neighborhood. +- **(3)** $\mathit{lim}_{n\rightarrow\infty}Q_n=\infty$. +- **(4)** For each $A<\infty$ the following holds for all $n$ sufficiently large. The ball +$B(x_n,t_n,AQ_n^{-1/2})$ has compact closure in $M_n$ and the flow +is $\kappa$-non-collapsed on scales $\le r$ at each point of +$B(x_n,t_n,AQ_n^{-1/2})$. +- **(5)** There is $\mu>0$ such that for every $A<\infty$ the following holds + for all $n$ sufficiently large. +For every $y_n\in B(x_n,t_n,AQ_n^{-1/2})$ the maximal flow line +through $y_n$ extends backwards for a time at least $\mu\left(\mathit{max}(Q_n,R(y_n))\right)^{-1}$. + +Then, after passing to a subsequence and shifting the times of each +of the generalized flows so that $t_n=0$ for every $n$, there is a +geometric limit $(M_\infty,g_\infty,x_\infty)$ of the sequence of +based Riemannian manifolds $(M_n,Q_nG_n(0),x_n)$. This limit is a +complete $3$-dimensional Riemannian manifold of bounded, +non-negative curvature. Furthermore, for some $t_0>0$ which depends +on the curvature bound for $(M_\infty,g_\infty)$ and on $\mu$, there +is a geometric limit Ricci flow defined on $(M_\infty,g_\infty(t)), +-t_0\le t\le 0$, with $g_\infty(0)=g_\infty$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/d5a1d88909aa.md b/projects/poincare-conjecture/.astrolabe/atoms/d5a1d88909aa.md new file mode 100644 index 00000000..fe6786f2 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/d5a1d88909aa.md @@ -0,0 +1,20 @@ +--- +chapter: 19 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '19.16' +ref: +- d5a1d88909aa +sort: lemma +source: tex +src: morgan-tian +tex_file: canonnbhd +--- +There is $\epsilon_0>0$ such that the following holds for all +$0<\epsilon\le \epsilon_0$. Suppose that $\{N_j\}_{j\in J}$ is a +chain of $\epsilon$-necks in a Riemannian manifold $M$. Let +$U=\cup_{j\in J}N_j$. Then there exist an interval $I$ and a smooth +map $p\colon U\to I$ such that every fiber of $p$ is a two-sphere, +and if $y$ is in the middle $7/8$'s of $N_j$ then the fiber +$p^{-1}(p(y))$ makes a small angle at every point with the family of +two-spheres in the $\epsilon$-neck $N_j$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/d6585ba392cf.md b/projects/poincare-conjecture/.astrolabe/atoms/d6585ba392cf.md new file mode 100644 index 00000000..4eb41705 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/d6585ba392cf.md @@ -0,0 +1,14 @@ +--- +chapter: 11 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '11.20' +ref: +- d6585ba392cf +sort: remark +source: tex +src: morgan-tian +tex_file: singlimit2 +--- +Recall that by definition a function $f$ is proper if the pre-image +under $f$ of every compact set is compact. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/d6919616448d.md b/projects/poincare-conjecture/.astrolabe/atoms/d6919616448d.md new file mode 100644 index 00000000..4fd6e7a4 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/d6919616448d.md @@ -0,0 +1,21 @@ +--- +chapter: 11 +generator: tools/poincare_tex_extract.py +label: maxT +labels: +- maxT +mtref: '11.10' +ref: +- d6919616448d +sort: proposition +source: tex +src: morgan-tian +tex_file: singlimit2 +--- +With the notation of, and under the hypotheses of +Theorem 11.8, suppose that there is a geometric limit flow +$(M_\infty,g_\infty(t))$ defined for $-T0$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/d74749898b25.md b/projects/poincare-conjecture/.astrolabe/atoms/d74749898b25.md new file mode 100644 index 00000000..19416228 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/d74749898b25.md @@ -0,0 +1,19 @@ +--- +chapter: 11 +generator: tools/poincare_tex_extract.py +label: epslimit +labels: +- epslimit +mtref: '11.24' +ref: +- d74749898b25 +sort: claim +source: tex +src: morgan-tian +tex_file: singlimit2 +--- +Fix a point $x\in\Omega$. Suppose that there is a sequence +$t_n\rightarrow T$ such that for every $n$, the point $(x,t_n)$ is +the center of a strong $\epsilon$-neck in $\widehat{\mathcal M}$. +Then $(x,T)$ is the center of a strong $2\epsilon$-neck in $\widehat +{\mathcal M}$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/d88ab31ec11f.md b/projects/poincare-conjecture/.astrolabe/atoms/d88ab31ec11f.md new file mode 100644 index 00000000..46c01968 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/d88ab31ec11f.md @@ -0,0 +1,28 @@ +--- +chapter: 18 +generator: tools/poincare_tex_extract.py +label: Wdefn +labels: +- Wdefn +mtref: '18.17' +ref: +- d88ab31ec11f +sort: definition +source: tex +src: morgan-tian +tex_file: energy1 +--- +Fix a homotopically trivial loop $\gamma\in \Lambda M$. We set +$A(\gamma)$ equal to the infimum of the areas of any spanning disks +for $\gamma$, where by definition a spanning disk is a Lipschitz map +$D^2\to M$ whose boundary is, up to reparameterization, $\gamma$. +Notice that $A(\gamma)$ is a continuous function of $\gamma$ in +$\Lambda M$. Also, notice that $A(\gamma)$ is invariant under +reparameterization of the curve $\gamma$. Now suppose that +$\Gamma\colon S^2\to \Lambda M$ is given with the image consisting +of homotopically trivial loops. + We define $W(\Gamma)$ to be equal to the maximum over all $c\in S^2$ of +$A(\Gamma(c))$. More generally, given a homotopy class $\xi\in \pi_2(\Lambda +M,*)$ we define $W(\xi)$ to be equal to the infimum over all (not necessarily +based) maps $\Gamma\colon S^2\to\Lambda M$ into the component of $\Lambda M$ +consisting of homotopically trivial loops representing $\xi$ of $W(\Gamma)$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/d9035559d0a4.md b/projects/poincare-conjecture/.astrolabe/atoms/d9035559d0a4.md new file mode 100644 index 00000000..4de9713e --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/d9035559d0a4.md @@ -0,0 +1,31 @@ +--- +chapter: 9 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '9.8' +ref: +- d9035559d0a4 +sort: example +source: tex +src: morgan-tian +tex_file: temp2kappa +--- +Consider the metric product $(S^{2},g_0) \times (S^{1}_{R},ds^2)$ +where $(S^1_R,ds^2)$ is the circle of radius $R$. We define +$g(t)=(1-t)g_0+ds^2$. This is an ancient solution to the Ricci flow. +But it is not $\kappa$-non-collapsed for any $\kappa>0$. The reason +is that + +$$ +|\mathit{Rm}(p,t)|=\frac{1}{1-t}, +$$ + + and + +$$ +\frac{\mathit{Vol}_{g(t)} B(p,\sqrt{1-t})}{(1-t)^{3/2}} +\le \frac{\mathit{Vol}_{g(t)}(S^{2}\times S^{1}_{R})}{(1-t)^{3/2}} +=\frac{2\pi R(1-t)4\pi}{(1-t)^{3/2}}=\frac{8\pi^2R}{\sqrt{1-t}}. +$$ + +Thus, as $t\rightarrow-\infty$ this ratio goes to zero. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/d91a6d9473a7.md b/projects/poincare-conjecture/.astrolabe/atoms/d91a6d9473a7.md new file mode 100644 index 00000000..5b86e2b6 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/d91a6d9473a7.md @@ -0,0 +1,22 @@ +--- +chapter: 18 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '18.61' +ref: +- d91a6d9473a7 +sort: claim +source: tex +src: morgan-tian +tex_file: energy1 +--- +Let $\psi\colon A\subset M\times S^1_\lambda$ be an area-minimizing +annulus of area at most $\mu$ with smoothly embedded boundary as +constructed above. Let $h$ be the induced (possibly singular) metric +on $A$ induced by pulling back $g\times ds^2$ by $\psi$, and let +$C''>0$ be an upper bound on the Gaussian curvature of $h$ (away +from the branch points). Then there is a deformation $\tilde h$ of +$h$, supported near the interior branch points, to a smooth metric +with the property that the area of the deformed smooth metric is at +most $2\mu$ and where the upper bound for the curvature of $\tilde +h$ is $2C''$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/d91d54ddbd1c.md b/projects/poincare-conjecture/.astrolabe/atoms/d91d54ddbd1c.md new file mode 100644 index 00000000..f1410780 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/d91d54ddbd1c.md @@ -0,0 +1,57 @@ +--- +chapter: 2 +generator: tools/poincare_tex_extract.py +label: epsneck +labels: +- epsneck +mtref: '2.18' +ref: +- d91d54ddbd1c +sort: definition +source: tex +src: morgan-tian +tex_file: prelim +--- +Let $(N,g)$ be a Riemannian manifold and $x\in N$ a point. Then *an $\epsilon$-neck structure on $(N,g)$ +centered at $x$* consists of a diffeomorphism + +$$ +\varphi\colon S^2\times (-\epsilon^{-1},\epsilon^{-1})\to N, +$$ + +with $x\in \varphi(S^2\times\{0\})$, such that the metric +$R(x)\varphi^*g$ is within $\epsilon$ in the +$C^{[1/\epsilon]}$-topology of the product of the usual Euclidean +metric on the open interval with the metric of constant Gaussian +curvature $1/2$ on $S^2$. We also use the terminology *$N$ is an +$\epsilon$-neck centered at $x$*. The image under $\varphi$ of the +family of submanifolds $S^2\times \{t\}$ is called the *family +of $2$-spheres of the $\epsilon$-neck*. The submanifold +$\varphi(S^2\times \{0\})$ is called *the central $2$-sphere* of +the $\epsilon$-neck structure. We denote by $s_N\colon N\to \Ar$ +the composition $p_2\circ \varphi^{-1}$, where $p_2$ is the +projection of $S^2\times (-\epsilon^{-1},\epsilon^{-1})$ to the +second factor. There is also the vector field $\partial/\partial +s_N$ on $N$ which is $\varphi_*$ of the standard vector field in the +interval-direction of the product. We also use the terminology of +the *plus* and *minus* end of the $\epsilon$-neck in the +obvious sense. The opposite (or reversed) $\epsilon$-neck structure +is the one obtained by composing the structure map with $\mathit{Id}_{S^2}\times -1$. We define the + *positive half of the neck* to be the region +$s_N^{-1}(0,\epsilon^{-1})$ and the *negative half* to be the +region $s_N^{-1} (-\epsilon^{-1},0)$. For any other fraction, e.g., +the left-hand three-quarters, the right-hand one-quarter, there are +analogous notions, all measured with respect to $s_N\colon N\to +(-\epsilon^{-1},\epsilon^{-1})$. We also use the terminology the +middle one-half, or middle one-third of the $\epsilon$-neck; again +these regions have their obvious meaning when measured via $s_N$. + +*An $\epsilon$-neck* in a Riemannian manifold $X$ is a +codimension-zero submanifold $N$ and an $\epsilon$-structure on $N$ +centered at some point $x\in N$. + +The *scale* of an $\epsilon$-neck $N$ centered +at $x$ is $R(x)^{-1/2}$. The scale of $N$ is denoted $r_N$. Intuitively, this +is a measure of the radius of the cross-sectional $S^2$ in the neck. In fact, +the extrinsic diameter of any $S^2$ factor in the neck is close to $\sqrt{2}\pi +r_N$. See Fig. 0.1 in the introduction. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/d93b1988d45d.md b/projects/poincare-conjecture/.astrolabe/atoms/d93b1988d45d.md new file mode 100644 index 00000000..134dce38 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/d93b1988d45d.md @@ -0,0 +1,19 @@ +--- +chapter: 16 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '16.2' +ref: +- d93b1988d45d +sort: remark +source: tex +src: morgan-tian +tex_file: surgery +--- +Implicitly, +$\kappa$ and $\delta(r_{i+1})$ are also allowed to depend on $t_0, \epsilon$, +and $C$, which are fixed, and also $i+1$. +Also recall that the non-collapsing condition allows for two outcomes: if $x$ is a point at which the +hypothesis of the non-collapsing hold, then +there is a lower bound on the volume of a ball centered at $x$, or $x$ +is contained in a component of its time-slice that has positive sectional curvature. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/d98ad25d9bc3.md b/projects/poincare-conjecture/.astrolabe/atoms/d98ad25d9bc3.md new file mode 100644 index 00000000..b0933226 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/d98ad25d9bc3.md @@ -0,0 +1,57 @@ +--- +chapter: 16 +generator: tools/poincare_tex_extract.py +label: Lgeoexist +labels: +- Lgeoexist +mtref: '16.21' +ref: +- d98ad25d9bc3 +sort: proposition +source: tex +src: morgan-tian +tex_file: surgery +--- +Fix $00$ such that for all $k$ sufficiently large the following +hold: + +- **(1)** the ball $B(x_k,0,A)$ has compact closure in $M_k$, +- **(2)** there is an embedding $B(x_k,0,A)\times (-\delta(A),0]\to {\mathcal M}_k$ compatible with +the time function and with the vector field, +- **(3)** $|\mathit{Rm}|\le C(A)$ +on the image of the embedding in the Item (2), and +- **(4)** there is $r_0>0$ +and $\kappa>0$ such that $\mathit{Vol} B(x_k,0,r_0)\ge \kappa r_0^n$ for all $k$ +sufficiently large. + + Then, after passing to a +subsequence, there is a geometric limit +$(M_\infty,g_\infty,x_\infty)$ of the $0$ time-slices +$(M_k,g_k,x_k)$. This limit is a complete Riemannian manifold. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/da7306fc733d.md b/projects/poincare-conjecture/.astrolabe/atoms/da7306fc733d.md new file mode 100644 index 00000000..250e875e --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/da7306fc733d.md @@ -0,0 +1,18 @@ +--- +chapter: 6 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '6.20' +ref: +- da7306fc733d +sort: claim +source: tex +src: morgan-tian +tex_file: newcompar +--- +If $\tau_1=0$, then + +$$ +\frac{\partial}{\partial u}\left(\mathit{lim}_{\tau\rightarrow 0}\sqrt{\tau}X(\tau,u)\right)|_{u=0}= +\mathit{lim}_{\tau\rightarrow 0}\sqrt{\tau}\frac{d}{d\tau}Y(\tau). +$$ diff --git a/projects/poincare-conjecture/.astrolabe/atoms/da767c71996b.md b/projects/poincare-conjecture/.astrolabe/atoms/da767c71996b.md new file mode 100644 index 00000000..d1ffc7cf --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/da767c71996b.md @@ -0,0 +1,26 @@ +--- +chapter: 9 +generator: tools/poincare_tex_extract.py +label: 2DGSS +labels: +- 2DGSS +mtref: '9.50' +ref: +- da767c71996b +sort: corollary +source: tex +src: morgan-tian +tex_file: temp2kappa +--- +- **(1)** Let $(M,g(t))$ be a two-dimensional Ricci flow satisfying all +the hypotheses of Proposition 9.50 except possible the +non-compactness hypothesis. Then $M$ is compact and for any $a>0$ +the restriction of the flow to any interval of the form +$(-\infty,-a]$ followed by a shift of time by $+a$ is a +$\kappa$-solution. +- **(2)** Any asymptotic gradient +shrinking soliton for a two-dimensional $\kappa$-solution is a shrinking family +of round surfaces. +- **(3)** Let $(M,g(t)), -\infty0$ be given and let $A\subset +{\mathcal M}\cap \mathbf{t}^{-1}(-\infty,T-\tau_1+\epsilon)$. Suppose that there +is a subset $F\subset {\mathcal M}$ on which $|\mathit{Ric}|$ and $|\nabla R|$ are +bounded and a neighborhood $\nu(A)$ of $A$ contained in $F$ with the property +that for every point $z\in \nu(A)$ there is a minimizing ${\mathcal +L}$-geodesic from $x$ to $z$ contained in $F$. Then $l_x$ is defined on all of +$\nu(A)$. Furthermore, there is a smaller neighborhood $\nu_0(A)\subset \nu(A)$ +of $A$ on which $l_x$ is a locally Lipschitz function with respect to the +Riemannian metric, denoted $\widehat G$, on ${\mathcal M}$ which is defined as +the orthogonal sum of the Riemannian metric $G$ on ${\mathcal H}T{\mathcal M}$ +and the metric $dt^2$ on the tangent line spanned by $\chi$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/db9ef2c8d5c1.md b/projects/poincare-conjecture/.astrolabe/atoms/db9ef2c8d5c1.md new file mode 100644 index 00000000..2f60a362 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/db9ef2c8d5c1.md @@ -0,0 +1,18 @@ +--- +chapter: 18 +generator: tools/poincare_tex_extract.py +label: varareaform +labels: +- varareaform +mtref: '18.12' +ref: +- db9ef2c8d5c1 +sort: claim +source: tex +src: morgan-tian +tex_file: energy1 +--- +$$ +\frac{d\mathit{Area}_{g(t)}(f(S^2))}{dt}(t_0)\le -4\pi -\frac{1}{2} +R_\mathit{min}(g(t_0))\mathit{Area}_{g(t_0)}f(S^2). +$$ diff --git a/projects/poincare-conjecture/.astrolabe/atoms/dbe4fabd0076.md b/projects/poincare-conjecture/.astrolabe/atoms/dbe4fabd0076.md new file mode 100644 index 00000000..ec7486a2 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/dbe4fabd0076.md @@ -0,0 +1,25 @@ +--- +chapter: 3 +generator: tools/poincare_tex_extract.py +label: firstshi +labels: +- firstshi +mtref: '3.27' +ref: +- dbe4fabd0076 +sort: theorem +source: tex +src: morgan-tian +tex_file: flowbasics +--- +There is a constant $C=C(n)$, depending only on the dimension $n$, +such that the following holds for every $K<\infty$, for every $T>0$, +and for every $r>0$. Suppose that $(U,g(t)),\ 0\le t\le T$, is an $n$-dimensional +Ricci flow with $|\mathit{Rm}(x,t)|\le K$ for all $x\in U$ and $t\in [0,T]$. +Suppose that $p\in U$ has the property that $B(p,0,r)$ has compact +closure in $U$. Then + +$$ +|\nabla \mathit{Rm}(p,t)|\le +CK\left(\frac{1}{r^2}+\frac{1}{t}+K\right)^{1/2}. +$$ diff --git a/projects/poincare-conjecture/.astrolabe/atoms/dbf6a3c0bae7.md b/projects/poincare-conjecture/.astrolabe/atoms/dbf6a3c0bae7.md new file mode 100644 index 00000000..4a3b75b9 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/dbf6a3c0bae7.md @@ -0,0 +1,25 @@ +--- +chapter: 9 +generator: tools/poincare_tex_extract.py +label: scaleinv2 +labels: +- scaleinv2 +mtref: '9.13' +ref: +- dbf6a3c0bae7 +sort: claim +source: tex +src: morgan-tian +tex_file: temp2kappa +--- +For each $k\ge 1$ denote by $x_k\in M_k$ the point $(p,0)\in +M_k$. Let $\widetilde V_{x_k}(\tau)=\widetilde +V_{x_k}(M_k\times\{\tau\})$ denote the reduced volume function for +the Ricci flow $(M_k,g_k(t))$ from the point $x_k$, and let +$\widetilde V_x(\tau)$ denote the reduced volume of +$M\times\{\tau\}$ for the Ricci flow $(M,g(t))$ from the point $x$. +Then + +$$ +\widetilde V_{x_k}(\tau) = \widetilde V_x (\bar \tau_{k}\tau). +$$ diff --git a/projects/poincare-conjecture/.astrolabe/atoms/dc263832416c.md b/projects/poincare-conjecture/.astrolabe/atoms/dc263832416c.md new file mode 100644 index 00000000..f0b12556 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/dc263832416c.md @@ -0,0 +1,26 @@ +--- +chapter: 2 +generator: tools/poincare_tex_extract.py +label: anglecompar +labels: +- anglecompar +mtref: '2.6' +ref: +- dc263832416c +sort: corollary +source: tex +src: morgan-tian +tex_file: prelim +--- +Let $(M,g)$ be a complete Riemannian manifold of non-negative +curvature. Let $p,a,b,c$ be four points in $M$ and let +$\alpha,\beta,\gamma$ be minimizing geodesic arcs from the point +$p$ to $a,b,c$ respectively. Let $T(a,p,b)$, $T(b,p,c)$ and +$T(c,p,a)$ be the triangles in $M$ made out of these minimizing +geodesics and minimizing geodesics between $a,b,c$. Let +$T(a',p',b')$, $T(b',p',c')$ and $T(c',p',a')$ be planar triangles +with the same side lengths. Then + +$$ +\angle_{p'}T(a',p',b')+\angle_{p'}T(b',p',c')+\angle_{p'}T(c',p',a')\le 2\pi. +$$ diff --git a/projects/poincare-conjecture/.astrolabe/atoms/dc5709757703.md b/projects/poincare-conjecture/.astrolabe/atoms/dc5709757703.md new file mode 100644 index 00000000..c224792b --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/dc5709757703.md @@ -0,0 +1,31 @@ +--- +chapter: 19 +generator: tools/poincare_tex_extract.py +label: neckcurv +labels: +- neckcurv +mtref: '19.2' +ref: +- dc5709757703 +sort: lemma +source: tex +src: morgan-tian +tex_file: canonnbhd +--- +For any $0<\alpha<1/8$ there is $\epsilon_1=\epsilon_1(\alpha)>0$ such that +the following two conditions hold for all $0<\epsilon\le \epsilon_1$. + +- **(1)** If $(N,g)$ is an $\epsilon$-neck centered at $x$ of +scale one (i.e., with $R(x)=1$) then the principal sectional +curvatures at any point of $N$ are within $\alpha/6$ of +$\{1/2,0,0\}$. In particular, for any $y\in N$ we have + +$$ +(1-\alpha)\le R(y)\le (1+\alpha). +$$ + +- **(2)** There is unique two-plane of maximal sectional curvature at +every point of an $\epsilon$-neck, and the angle between the +distribution of two-planes of maximal sectional curvature and the +two-plane field tangent to the family of two-spheres of the +$\epsilon$-neck structure is everywhere less than $\alpha$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/dc7704c76958.md b/projects/poincare-conjecture/.astrolabe/atoms/dc7704c76958.md new file mode 100644 index 00000000..f00de533 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/dc7704c76958.md @@ -0,0 +1,18 @@ +--- +chapter: 2 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '2.9' +ref: +- dc7704c76958 +sort: lemma +source: tex +src: morgan-tian +tex_file: prelim +--- +Let $(M,g)$ be a complete, non-compact Riemannian manifold of +non-negative sectional curvature and let $p\in M$. For every +$\epsilon>0$ there is a compact subset $K=K(p,\epsilon)\subset M$ +such that for all points $q\notin K$, if $\gamma$ and $\mu$ are +minimizing geodesics from $p$ to $q$, then the angle that $\gamma$ +and $\mu$ make at $q$ is less than $\epsilon$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/dda8a7c7acb6.md b/projects/poincare-conjecture/.astrolabe/atoms/dda8a7c7acb6.md new file mode 100644 index 00000000..e2adef9d --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/dda8a7c7acb6.md @@ -0,0 +1,15 @@ +--- +chapter: 10 +generator: tools/poincare_tex_extract.py +label: morethanone +labels: +- morethanone +mtref: '10.18' +ref: +- dda8a7c7acb6 +sort: lemma +source: tex +src: morgan-tian +tex_file: bddcurvbdddist +--- +There is more than one direction at ${\mathcal E}$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/ddf1aed4b3bb.md b/projects/poincare-conjecture/.astrolabe/atoms/ddf1aed4b3bb.md new file mode 100644 index 00000000..33b4de2d --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/ddf1aed4b3bb.md @@ -0,0 +1,18 @@ +--- +chapter: 10 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '10.22' +ref: +- ddf1aed4b3bb +sort: lemma +source: tex +src: morgan-tian +tex_file: bddcurvbdddist +--- +If $\gamma,\mu,\nu$ are minimizing geodesic rays limiting to +${\mathcal E}$, then + +$$ +\theta(\gamma,\mu)+\theta(\mu,\nu)\ge \theta(\gamma,\nu). +$$ diff --git a/projects/poincare-conjecture/.astrolabe/atoms/de62de36e983.md b/projects/poincare-conjecture/.astrolabe/atoms/de62de36e983.md new file mode 100644 index 00000000..63b9a244 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/de62de36e983.md @@ -0,0 +1,29 @@ +--- +chapter: 1 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '1.7' +ref: +- de62de36e983 +sort: definition +source: tex +src: morgan-tian +tex_file: prelim +--- +Using the metric, one can replace the Riemann curvature tensor +${\mathcal R}$ by a symmetric bilinear form $\mathit{Rm}$ on +$\wedge^2TM$. In local coordinates let +$\varphi=\varphi^{ij}\partial_i\wedge\partial _j$ and +$\psi=\psi^{kl}\partial _k\wedge\partial_l$ be local sections of +$\wedge^2TM$. The formula for $\mathit{Rm}$ is + +$$ +\mathit{Rm}(\varphi, \psi)=R_{ijkl}\varphi^{ij}\psi^{kl}. +$$ + +We call $\mathit{Rm}$ the *curvature +operator*. We say $(M,g)$ has +*positive curvature operator* if $\mathit{Rm}(\varphi,\varphi)>0$ for any nonzero 2-form $\varphi = +\varphi^{ij}\partial_{i}\wedge +\partial_{j}$ and has *nonnegative curvature operator* if +$\mathit{Rm}(\varphi,\varphi)\geq0$ for any $\varphi\in \wedge^2TM$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/de785906ce83.md b/projects/poincare-conjecture/.astrolabe/atoms/de785906ce83.md new file mode 100644 index 00000000..111617b0 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/de785906ce83.md @@ -0,0 +1,21 @@ +--- +chapter: 7 +generator: tools/poincare_tex_extract.py +label: wineq +labels: +- wineq +mtref: '7.22' +ref: +- de785906ce83 +sort: claim +source: tex +src: morgan-tian +tex_file: newcomp2 +--- +Let $\phi\colon B_{(q,t)}'\to \Ar$ be a non-negative, smooth +function with compact support. Then + +$$ +\int_{B_{(q,t)}'}\beta_{(q,t)}\triangle \phi d\mathit{vol}(g(t))\le \int_{(B_{(q,t)}')^*}\phi +\triangle \beta_{(q,t)} d\mathit{vol}(g(t)). +$$ diff --git a/projects/poincare-conjecture/.astrolabe/atoms/deb9b83d83f2.md b/projects/poincare-conjecture/.astrolabe/atoms/deb9b83d83f2.md new file mode 100644 index 00000000..397c3799 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/deb9b83d83f2.md @@ -0,0 +1,28 @@ +--- +chapter: 6 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '6.10' +ref: +- deb9b83d83f2 +sort: lemma +source: tex +src: morgan-tian +tex_file: newcompar +--- +$Y(\tau)$ satisfies the *${\mathcal L}$-Jacobi +equation*: + +$$ +\nabla_X\nabla_XY+{\mathcal +R}(Y,X)X-\frac{1}{2}\nabla_Y(\nabla +R)+\frac{1}{2\tau}\nabla_XY+2(\nabla_Y\mathit{Ric})(X,\cdot)^*+2\mathit{Ric}(\nabla_XY,\cdot)^*=0. +$$ + + This is a second-order +linear equation for $Y$. Supposing that $\tau_1>0$, there is a +unique horizontal vector field $Y$ along $\gamma$ solving this +equation vanishing at $\tau_1$ with a given first-order derivative +along $\gamma$ at $\tau_1$. Similarly, there is a unique solution +$Y$ to this equation vanishing at $\tau_2$ and with a given +first-order derivative at $\tau_2$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/df842ef2ab9f.md b/projects/poincare-conjecture/.astrolabe/atoms/df842ef2ab9f.md new file mode 100644 index 00000000..afb6c1f6 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/df842ef2ab9f.md @@ -0,0 +1,34 @@ +--- +chapter: 6 +generator: tools/poincare_tex_extract.py +label: defnD +labels: +- defnD +mtref: '6.17' +ref: +- df842ef2ab9f +sort: definition +source: tex +src: morgan-tian +tex_file: newcompar +--- +We define *the domain of definition of* ${\mathcal L}\mathit{exp}_x$, denoted ${\mathcal D}_x$, to be the +subset of $T_xM_{T-\tau_1}\times (\tau_1,\infty)$ consisting of all pairs +$(Z,\tau)$ for which $\tau> \tau_1$ is in the maximal domain of definition of +$\gamma_Z$. Then we define $\mathcal{L}\mathit{exp}_{x}\colon {\mathcal D}_x\to +{\mathcal M}$ by setting $\mathcal{L}\mathit{exp}_{x}(Z,\tau) = \gamma_{Z}(\tau)$ +for all $(Z,\tau)\in {\mathcal D}_x$. (See Fig. 6.1.) We +define the map $\widetilde L\colon {\mathcal D}_x\to \Ar$ +by $\widetilde L(Z,\tau)={\mathcal L}\left(\gamma_Z|_{[\tau_1,\tau]}\right)$. +Lastly, for any $\tau> \tau_1$ we denote by ${\mathcal L}\mathit{exp}_x^\tau$ the restriction of +${\mathcal L}\mathit{exp}_x$ to the slice + +$$ +{\mathcal +D}^\tau_x={\mathcal D}_x\cap \left(T_xM_{T-\tau_1}\times \{\tau\}\right), +$$ + +which is *the domain of definition of* ${\mathcal L}\mathit{exp}_x^\tau$. We +also denote by $\widetilde L^\tau$ the +restriction of $\widetilde L$ to this slice. We will implicitly identify +${\mathcal D}_x^\tau$ with a subset of $T_xM_{T-\tau_1}$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/dff170186455.md b/projects/poincare-conjecture/.astrolabe/atoms/dff170186455.md new file mode 100644 index 00000000..7d4cfd26 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/dff170186455.md @@ -0,0 +1,16 @@ +--- +chapter: 11 +generator: tools/poincare_tex_extract.py +label: hyp +labels: +- hyp +mtref: '11.32' +ref: +- dff170186455 +sort: claim +source: tex +src: morgan-tian +tex_file: singlimit2 +--- +The sequence $(\widehat{\mathcal M}_n,\widehat G_n,x_n)$ satisfies +the five hypothesis of Theorem 11.1. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/e07da18ab307.md b/projects/poincare-conjecture/.astrolabe/atoms/e07da18ab307.md new file mode 100644 index 00000000..0520d767 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/e07da18ab307.md @@ -0,0 +1,14 @@ +--- +chapter: 1 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '1.18' +ref: +- e07da18ab307 +sort: theorem +source: tex +src: morgan-tian +tex_file: prelim +--- +(Hopf-Rinow) If $(M,g)$ is complete as a metric space, then every +geodesic extends to a geodesic defined for all time. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/e08895f2b0d8.md b/projects/poincare-conjecture/.astrolabe/atoms/e08895f2b0d8.md new file mode 100644 index 00000000..d22990e0 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/e08895f2b0d8.md @@ -0,0 +1,20 @@ +--- +chapter: 9 +generator: tools/poincare_tex_extract.py +label: redvolconst +labels: +- redvolconst +mtref: '9.14' +ref: +- e08895f2b0d8 +sort: corollary +source: tex +src: morgan-tian +tex_file: temp2kappa +--- +There is a non-negative constant $V_\infty<(4\pi)^{n/2}$ such that +for all $\tau\in(0,\infty)$, we have + +$$ +\mathit{lim}_{k\rightarrow\infty}\widetilde V_{x_k}(\tau)=V_\infty. +$$ diff --git a/projects/poincare-conjecture/.astrolabe/atoms/e0b6a1f646a7.md b/projects/poincare-conjecture/.astrolabe/atoms/e0b6a1f646a7.md new file mode 100644 index 00000000..0698a20d --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/e0b6a1f646a7.md @@ -0,0 +1,47 @@ +--- +chapter: 3 +generator: tools/poincare_tex_extract.py +label: shiw/deriv +labels: +- shiw/deriv +mtref: '3.29' +ref: +- e0b6a1f646a7 +sort: theorem +source: tex +src: morgan-tian +tex_file: flowbasics +--- +Fix the dimension $n$ of the Ricci flows under consideration. Let +$K<\infty$ and $\alpha>0$ be given positive constants. Fix an +integer $l\ge 0$. Then for each integer $k\ge 0$ and for each $r>0$ +there is a constant $C'_{k,l}=C'_{k,l}(K,\alpha,r,n)$ such that the +following holds. Let $(U,g(t)),\ 0\le t\le T$, be a Ricci flow with +$T\le \alpha/K$. Fix $p\in U$ and suppose that the metric ball +$B(p,0,r)$ has compact closure in $U$. Suppose that + +$$ +\begin{aligned} +\left\vert \operatorname*{Rm}\left( x,t\right) \right\vert & +\leq K\ \ \text{ for all }x\in U\text{ and all}\ t\in[0,T],\\ +\left\vert \nabla^{\beta}\operatorname*{Rm}\left( x,0\right) +\right\vert & \leq K\ \ \text{ for all }x\in U\ \ \text{ and all }\ +\ \beta\leq l. +\end{aligned} +$$ + +Then + +$$ +\left\vert \nabla^{k}\operatorname*{Rm}\left( y,t\right) +\right\vert +\leq\frac{C'_{k,l}}{t^{\max\left\{ k-l,0\right\} /2}} +$$ + +for all $y\in$ $B(p,0,r/2) $ and all $t\in(0,T].$ In particular if +$k\leq l$, then for $y\in B(p,0,r/2)$ and $t\in (0,T]$ we have + +$$ +\left\vert \nabla^{k}\operatorname*{Rm}\left( y,t\right) +\right\vert \leq C_{k,l}'. +$$ diff --git a/projects/poincare-conjecture/.astrolabe/atoms/e1277342360a.md b/projects/poincare-conjecture/.astrolabe/atoms/e1277342360a.md new file mode 100644 index 00000000..d5b93572 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/e1277342360a.md @@ -0,0 +1,33 @@ +--- +chapter: 6 +generator: tools/poincare_tex_extract.py +label: adaptform +labels: +- adaptform +mtref: '6.36' +ref: +- e1277342360a +sort: lemma +source: tex +src: morgan-tian +tex_file: newcompar +--- +Suppose that $Y(\tau)$ is an adapted vector field along $\gamma$. + Then + +$$ +\begin{aligned} +\frac{d}{d\tau}\langle Y(\tau),Y(\tau)\rangle & = & +2\mathit{Ric}(Y(\tau),Y(\tau)) + 2\langle \nabla_{X}Y(\tau),Y(\tau)\rangle \\ +& = & \frac{1}{\sqrt{\tau}(\sqrt{\tau}-\sqrt{\tau_1})}\langle +Y(\tau),Y(\tau)\rangle\nonumber. +\end{aligned} +$$ + + It follows that + +$$ +|Y(\tau)|^2 = C\frac{(\sqrt{\tau}-\sqrt{\tau_1})^2}{(\sqrt{\bar\tau}-\sqrt{\tau_1})^2}, +$$ + +where $C=|Y(\bar\tau)|^2$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/e27e5f5d2bf7.md b/projects/poincare-conjecture/.astrolabe/atoms/e27e5f5d2bf7.md new file mode 100644 index 00000000..e46d28be --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/e27e5f5d2bf7.md @@ -0,0 +1,15 @@ +--- +chapter: 6 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '6.65' +ref: +- e27e5f5d2bf7 +sort: claim +source: tex +src: morgan-tian +tex_file: newcompar +--- +For $\delta$ sufficiently small (how small depending on $\delta_2$ +and $\delta'_2$) we have $B(z,t',\epsilon/4)\subset +B(y,t_0,\epsilon)$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/e2e64cdb0b75.md b/projects/poincare-conjecture/.astrolabe/atoms/e2e64cdb0b75.md new file mode 100644 index 00000000..52c29ba8 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/e2e64cdb0b75.md @@ -0,0 +1,35 @@ +--- +chapter: 10 +generator: tools/poincare_tex_extract.py +label: Uinfty +labels: +- Uinfty +mtref: '10.7' +ref: +- e2e64cdb0b75 +sort: proposition +source: tex +src: morgan-tian +tex_file: bddcurvbdddist +--- +The geometric limit $(U_\infty,g_\infty,z_\infty)$ is an incomplete +Riemannian $3$-manifold of finite diameter. There is a +diffeomorphism $\psi\colon U_\infty\to S^2\times (0,1)$. There is a +$2\epsilon$-neck centered at $z_\infty$ whose central $2$-sphere +$S^2(z_\infty)$ maps under $\psi$ to a $2$-sphere isotopic to a +$2$-sphere factor in the product decomposition. The scalar +curvature is bounded at one end of $U_\infty$ but tends to infinity +at the other end, the latter end which is denoted ${\mathcal E}$. +Let ${\mathcal U}_\infty\subset U_\infty\times(-\infty,0]$ be the +open subset consisting of all $(x,t)$ for which $-R(x)^{-1}0$ depending on $(M,g_0)$ and a Ricci flow +$(M,g(t)),\ 0\le t0$ is sufficiently small, the smallest +eigenvalue of $\mathit{Rm}_{\hat h}$ is greater than the smallest eigenvalue +of $\mathit{Rm}_h$ at the same point. Consequently, at any point where $h$ +has non-negative curvature so does $\hat h$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/e724298e5bc3.md b/projects/poincare-conjecture/.astrolabe/atoms/e724298e5bc3.md new file mode 100644 index 00000000..d2a3cf71 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/e724298e5bc3.md @@ -0,0 +1,24 @@ +--- +chapter: 9 +generator: tools/poincare_tex_extract.py +label: limitcannbhd +labels: +- limitcannbhd +mtref: '9.95' +ref: +- e724298e5bc3 +sort: corollary +source: tex +src: morgan-tian +tex_file: temp2kappa +--- +Fix $0<\epsilon\le \bar\epsilon'$, and let $C(\epsilon)$ be as in +the last corollary. Suppose that $({\mathcal M}_n,G_n,x_n)$ is a +sequence of based, generalized Ricci flows with $\mathbf{t}(x_n)=0$ for +all $n$. Suppose that none of the time-slices of the ${\mathcal +M}_n$ contain embedded $\Ar P^2$'s with trivial normal bundle. +Suppose also that there is a smooth limiting flow +$(M_\infty,g_\infty(t),(x_\infty,0))$ defined for $-\infty0 \\ 0 & \text{if }n=0 \\ -1& +\text{if }n<0.\end{cases} +$$ diff --git a/projects/poincare-conjecture/.astrolabe/atoms/e800ffebd2ce.md b/projects/poincare-conjecture/.astrolabe/atoms/e800ffebd2ce.md new file mode 100644 index 00000000..e371b50d --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/e800ffebd2ce.md @@ -0,0 +1,25 @@ +--- +chapter: 18 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '18.15' +ref: +- e800ffebd2ce +sort: remark +source: tex +src: morgan-tian +tex_file: energy1 +--- +Notice that it follows from the list of disappearing components that +the only ones with non-trivial $\pi_2$ are those based on the +geometry $S^2\times \Ar$; that is to say, $2$-sphere bundles over +$S^1$ and $\Ar P^3\#\Ar P^3$. Thus, once we have reached the level +$T_0$ after which all $2$-sphere surgeries are performed on +homotopically trivial $2$-spheres the only components that can have +non-trivial $\pi_2$ are components of these types. Thus, for example +if the original manifold has no $\Ar P^3$ prime factors and no +non-separating $2$-spheres, then when we reach time $T_0$ we have +done a connected sum decomposition into components each of which has +trivial $\pi_2$. Each of these components is either covered by a +contractible $3$-manifold or by a homotopy $3$-sphere, depending on +whether its fundamental group has infinite or finite order. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/e8f087a9cbc6.md b/projects/poincare-conjecture/.astrolabe/atoms/e8f087a9cbc6.md new file mode 100644 index 00000000..ed35cb6f --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/e8f087a9cbc6.md @@ -0,0 +1,21 @@ +--- +chapter: 9 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '9.35' +ref: +- e8f087a9cbc6 +sort: claim +source: tex +src: morgan-tian +tex_file: temp2kappa +--- +The following equality holds: + +$$ +\begin{aligned} +\triangle(|\nabla f|^2) & = & +2\langle\nabla(\triangle f),\nabla f\rangle+2\mathit{Ric}(\nabla +f,\nabla f)+2|\mathit{Hess}(f)|^2, +\end{aligned} +$$ diff --git a/projects/poincare-conjecture/.astrolabe/atoms/e94b6de05447.md b/projects/poincare-conjecture/.astrolabe/atoms/e94b6de05447.md new file mode 100644 index 00000000..b3f4113c --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/e94b6de05447.md @@ -0,0 +1,15 @@ +--- +chapter: 18 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '18.71' +ref: +- e94b6de05447 +sort: claim +source: tex +src: morgan-tian +tex_file: energy1 +--- +$$ +\frac{dl}{dt}(t)\le \mathit{max}_{\psi\in E}h(\psi)\int_{\lambda(t)}k_\mathit{geod}ds. +$$ diff --git a/projects/poincare-conjecture/.astrolabe/atoms/e967f00df0a8.md b/projects/poincare-conjecture/.astrolabe/atoms/e967f00df0a8.md new file mode 100644 index 00000000..900aab03 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/e967f00df0a8.md @@ -0,0 +1,13 @@ +--- +chapter: 18 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '18.79' +ref: +- e967f00df0a8 +sort: claim +source: tex +src: morgan-tian +tex_file: energy1 +--- +The length of $X'_J$ is at least $1-2\delta-\mu$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/e98dee3d7485.md b/projects/poincare-conjecture/.astrolabe/atoms/e98dee3d7485.md new file mode 100644 index 00000000..77533e53 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/e98dee3d7485.md @@ -0,0 +1,17 @@ +--- +chapter: 10 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '10.17' +ref: +- e98dee3d7485 +sort: definition +source: tex +src: morgan-tian +tex_file: bddcurvbdddist +--- +We say that two minimizing geodesic rays limiting to ${\mathcal E}$ +are *equivalent* if one is contained in the other. From the +unique continuation of geodesics it is easy to see that this +generates an equivalence relation. An equivalence class is a *direction at ${\mathcal E}$*, and the set of equivalence classes is +the *set of directions at ${\mathcal E}$*. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/e9a9172d16e2.md b/projects/poincare-conjecture/.astrolabe/atoms/e9a9172d16e2.md new file mode 100644 index 00000000..c334f54d --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/e9a9172d16e2.md @@ -0,0 +1,28 @@ +--- +chapter: 5 +generator: tools/poincare_tex_extract.py +label: topsplit +labels: +- topsplit +mtref: '5.35' +ref: +- e9a9172d16e2 +sort: theorem +source: tex +src: morgan-tian +tex_file: converge2 +--- +Let $(M,g)$ be a complete, connected manifold of non-negative +sectional curvature. Let $\{x_n\}$ be a sequence of points going off +to infinity, and suppose that we can find scaling factors +$\lambda_n>0$ such that the based Riemannian manifolds +$(M,\lambda_ng,x_n)$ have a geometric limit +$(M_\infty,g_\infty,x_\infty)$. Suppose that there is a point $p\in +M$ such that $\lambda_nd^2(p,x_n)\rightarrow \infty$ as +$n\rightarrow\infty$. Then, after passing to a subsequence, +minimizing geodesic arcs $\gamma_n$ from $x_n$ to $p$ converge to a +minimizing geodesic ray in $M_\infty$. This minimizing geodesic ray +is part of a minimizing geodesic line $\ell$ in $M_\infty$. In +particular, there is a Riemannian product decomposition +$M_\infty=N\times \Ar$ with the property that $\ell$ is $\{x\}\times +\Ar$ for some $x\in N$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/ea6d9419271a.md b/projects/poincare-conjecture/.astrolabe/atoms/ea6d9419271a.md new file mode 100644 index 00000000..16662cce --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/ea6d9419271a.md @@ -0,0 +1,18 @@ +--- +chapter: 11 +generator: tools/poincare_tex_extract.py +label: hmonotone +labels: +- hmonotone +mtref: '11.36' +ref: +- ea6d9419271a +sort: corollary +source: tex +src: morgan-tian +tex_file: singlimit2 +--- +We can take the function $h(\rho,\delta)$ in the last lemma to be +$\le \delta\rho$, to be a weakly monotone non-decreasing function of +$\delta$ when $\rho$ is fixed, and to be a weakly monotone +non-decreasing function of $\rho$ when $\delta$ is held fixed. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/ea7c58d04fc7.md b/projects/poincare-conjecture/.astrolabe/atoms/ea7c58d04fc7.md new file mode 100644 index 00000000..51d575bf --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/ea7c58d04fc7.md @@ -0,0 +1,17 @@ +--- +chapter: 18 +generator: tools/poincare_tex_extract.py +label: avgeo +labels: +- avgeo +mtref: '18.68' +ref: +- ea7c58d04fc7 +sort: claim +source: tex +src: morgan-tian +tex_file: energy1 +--- +Let $a\in \mathit{int} \xi$. There is a shortest path in $B$ from $a$ +to $v$. This shortest path is a geodesic meeting $\partial B$ only +in its end points. It has length $\le(1/2)+\delta'$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/ea9b285e40cb.md b/projects/poincare-conjecture/.astrolabe/atoms/ea9b285e40cb.md new file mode 100644 index 00000000..cc4c9789 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/ea9b285e40cb.md @@ -0,0 +1,17 @@ +--- +chapter: 9 +generator: tools/poincare_tex_extract.py +label: 2Rbound +labels: +- 2Rbound +mtref: '9.66' +ref: +- ea9b285e40cb +sort: claim +source: tex +src: morgan-tian +tex_file: temp2kappa +--- +There is a universal (i.e., independent of the $3$-dimensional +$\kappa$-solution) upper bound $C$ for $R(q',0)/R(q,0)$ for all +$q'\in B(q,0,2d)$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/eaf31795ca89.md b/projects/poincare-conjecture/.astrolabe/atoms/eaf31795ca89.md new file mode 100644 index 00000000..a3715cd0 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/eaf31795ca89.md @@ -0,0 +1,26 @@ +--- +chapter: 18 +generator: tools/poincare_tex_extract.py +label: vclaim +labels: +- vclaim +mtref: '18.26' +ref: +- eaf31795ca89 +sort: claim +source: tex +src: morgan-tian +tex_file: energy1 +--- +We have + +$$ +w_{a,t'}(t'')=\mathit{exp}(-A(t''))\left(a-2\pi\int_{t'}^{t''}\mathit{exp}(A(t))dt\right). +$$ + + If $a'>a$, then for $t_0\le t'0$ sufficiently small, there is a $\kappa>0$ such that +the standard flow is $\kappa$-non-collapsed on all scales $\le r$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/ed0e92ce8f57.md b/projects/poincare-conjecture/.astrolabe/atoms/ed0e92ce8f57.md new file mode 100644 index 00000000..5a932685 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/ed0e92ce8f57.md @@ -0,0 +1,20 @@ +--- +chapter: 18 +generator: tools/poincare_tex_extract.py +label: shortpi2triv +labels: +- shortpi2triv +mtref: '18.27' +ref: +- ed0e92ce8f57 +sort: lemma +source: tex +src: morgan-tian +tex_file: energy1 +--- +Given a compact Riemannian manifold $(X,g)$ with $\pi_2(X)=0$. Then +there is $\zeta>0$ such that if $\xi\in \pi_3({\mathcal X})$ is +represented by a family $\Gamma\colon S^2\to\Lambda X$ with the +property that for every $c\in S^2$ the length of the loop +$\Gamma(c)$ is less than $\zeta$, then $\xi$ is the trivial homotopy +element. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/ed4b72caf9f9.md b/projects/poincare-conjecture/.astrolabe/atoms/ed4b72caf9f9.md new file mode 100644 index 00000000..aa70f680 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/ed4b72caf9f9.md @@ -0,0 +1,34 @@ +--- +chapter: 3 +generator: tools/poincare_tex_extract.py +label: shi +labels: +- shi +mtref: '3.28' +ref: +- ed4b72caf9f9 +sort: theorem +source: tex +src: morgan-tian +tex_file: flowbasics +--- +**(Shi's Derivative Estimates)** +Fix the dimension $n$ of the Ricci flows under consideration. Let +$K<\infty$ and $\alpha>0$ be positive constants. Then for each +non-negative integer $k$ and each $r>0$ there is a constant +$C_k=C_k(K,\alpha,r,n)$ such that the following holds. Let +$(U,g(t)),\ 0\le t\le T$, be a Ricci flow with $T\le \alpha/K$. +Fix $p\in U$ and suppose that the metric ball $B(p,0,r)$ has compact +closure in $U$. If + +$$ +|\mathit{Rm}(x,t)|\le K\ \ \text{for all }\ (x,t)\in P(x,0,r,T), +$$ + +then + +$$ +|\nabla^k(\mathit{Rm}(y,t))|\le \frac{C_k}{t^{k/2}} +$$ + +for all $y\in B(p,0,r/2)$ and all $t\in(0,T]$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/ee314aacd207.md b/projects/poincare-conjecture/.astrolabe/atoms/ee314aacd207.md new file mode 100644 index 00000000..7eb19874 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/ee314aacd207.md @@ -0,0 +1,18 @@ +--- +chapter: 9 +generator: tools/poincare_tex_extract.py +label: proposition +labels: +- proposition +mtref: '9.65' +ref: +- ee314aacd207 +sort: lemma +source: tex +src: morgan-tian +tex_file: temp2kappa +--- +For each $r<\infty$ there is a constant $C(r)<\infty$, such that the +following holds. Let $(M,g(t),(p,0))$ be a based $3$-dimensional +$\kappa$-solution satisfying $R(p,0)=1$. Then $R(q,0)\le C(r)$ for +all $q\in B(p,0,r)$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/ee7e22fb33f6.md b/projects/poincare-conjecture/.astrolabe/atoms/ee7e22fb33f6.md new file mode 100644 index 00000000..a3e56298 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/ee7e22fb33f6.md @@ -0,0 +1,15 @@ +--- +chapter: 11 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '11.34' +ref: +- ee7e22fb33f6 +sort: claim +source: tex +src: morgan-tian +tex_file: singlimit2 +--- +$(M_\infty,g_\infty(0))$ is isometric to the product $(S^2,h)\times +(\Ar,ds^2)$, where $h$ is a metric of non-negative curvature on +$S^2$ and $ds^2$ is the usual Euclidean metric on the real line. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/ef94451b5785.md b/projects/poincare-conjecture/.astrolabe/atoms/ef94451b5785.md new file mode 100644 index 00000000..d751f3ff --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/ef94451b5785.md @@ -0,0 +1,22 @@ +--- +chapter: 16 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '16.6' +ref: +- ef94451b5785 +sort: claim +source: tex +src: morgan-tian +tex_file: surgery +--- +Let $(N,g_N)$ be a strong $\delta'$-neck with $N_0$ its middle half. +Suppose that $({\mathcal S},g)$ is the result of doing surgery on +(the central $2$-sphere) of $N$, adding a surgery cap ${\mathcal C}$ +to $N^-$. Let $h$ be the scale of $N$. Let $({\mathcal S}_0(N),g')$ +be the union of $N^-_0\cup {\mathcal C}$ with its induced metric as +given in Section 13.1, and let $({\mathcal +S}_0(N),\widehat g_0)$ be the result of rescaling $g_0$ by $h^{-2}$. +Then for every $\ell<\infty$ there is a uniform bound to +$|\nabla^\ell \mathit{Rm}_{\widehat g_0}(x)|$ for all $x\in {\mathcal +S}_0(N)$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/f09cbffcbdce.md b/projects/poincare-conjecture/.astrolabe/atoms/f09cbffcbdce.md new file mode 100644 index 00000000..8a4f5449 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/f09cbffcbdce.md @@ -0,0 +1,15 @@ +--- +chapter: 6 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '6.9' +ref: +- f09cbffcbdce +sort: definition +source: tex +src: morgan-tian +tex_file: newcompar +--- +An ${\mathcal L}$-geodesic is said to be *minimizing* if there is no +curve parameterized by backward time with the same endpoints and +with smaller ${\mathcal L}$-length. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/f1123507a2ea.md b/projects/poincare-conjecture/.astrolabe/atoms/f1123507a2ea.md new file mode 100644 index 00000000..032448b0 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/f1123507a2ea.md @@ -0,0 +1,19 @@ +--- +chapter: 6 +generator: tools/poincare_tex_extract.py +label: Jlimit1 +labels: +- Jlimit1 +mtref: '6.79' +ref: +- f1123507a2ea +sort: corollary +source: tex +src: morgan-tian +tex_file: newcompar +--- +Let $({\mathcal M},G)$ be a generalized Ricci flow whose sectional +curvatures are bounded. For any $x\in M_T$ and any $R<\infty$ for +all $\tau>0$ sufficiently small, the ball of radius $R$ centered at +the origin in $T_xM_T$ is contained in $\widetilde {\mathcal +U}_x(\tau)$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/f1d3021d32b8.md b/projects/poincare-conjecture/.astrolabe/atoms/f1d3021d32b8.md new file mode 100644 index 00000000..9c8ad67f --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/f1d3021d32b8.md @@ -0,0 +1,15 @@ +--- +chapter: 12 +generator: tools/poincare_tex_extract.py +label: stdinit +labels: +- stdinit +mtref: '12.2' +ref: +- f1d3021d32b8 +sort: lemma +source: tex +src: morgan-tian +tex_file: stdsoln +--- +There is a standard initial metric. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/f26ee81ded19.md b/projects/poincare-conjecture/.astrolabe/atoms/f26ee81ded19.md new file mode 100644 index 00000000..6d1de33b --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/f26ee81ded19.md @@ -0,0 +1,17 @@ +--- +chapter: 12 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '12.14' +ref: +- f26ee81ded19 +sort: claim +source: tex +src: morgan-tian +tex_file: stdsoln +--- +For all $x\in \Ar^3$ and $t\in [0,t_0]$ we have + +$$ +R(x,t)\le \frac{C}{1-Ct}. +$$ diff --git a/projects/poincare-conjecture/.astrolabe/atoms/f3441849c78b.md b/projects/poincare-conjecture/.astrolabe/atoms/f3441849c78b.md new file mode 100644 index 00000000..b406c29f --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/f3441849c78b.md @@ -0,0 +1,21 @@ +--- +chapter: 4 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '4.6' +ref: +- f3441849c78b +sort: definition +source: tex +src: morgan-tian +tex_file: maxprin +--- +Let $\pi\colon {\mathcal V}\to M$ be a vector bundle and let +${\mathcal Z}\subset {\mathcal V}$ be a closed subset. We say that +${\mathcal Z}$ is *convex* if for every $x\in M$ the fiber +$Z_x$ of ${\mathcal Z}$ over $x$ is a convex subset of the vector +space fiber $V_x$ of ${\mathcal V}$ over $x$. Let $\psi$ be a +fiberwise vector field on an open neighborhood ${\mathcal U}$ of +${\mathcal Z}$ in ${\mathcal V}$. We say that $\psi$ *preserves* +${\mathcal Z}$ if for each $x\in M$ the restriction of $\psi$ to the +fiber $U_x$ of ${\mathcal U}$ over $x$ preserves $Z_x$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/f44d182ef27e.md b/projects/poincare-conjecture/.astrolabe/atoms/f44d182ef27e.md new file mode 100644 index 00000000..0a44e692 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/f44d182ef27e.md @@ -0,0 +1,38 @@ +--- +chapter: 12 +generator: tools/poincare_tex_extract.py +label: stdsoln +labels: +- stdsoln +mtref: '12.5' +ref: +- f44d182ef27e +sort: theorem +source: tex +src: morgan-tian +tex_file: stdsoln +--- +There is a standard Ricci flow defined for some +positive amount of time. Let $(\Ar^3,g(t)),\ 0\le t0$ there is a compact subset $X$ +of $\Ar^3$ such that for any $x\in \Ar ^3\setminus X$ the +restriction of the standard flow to an appropriate neighborhood of +$x$ for time $t\in [0,t_0]$ is within $\epsilon$ in the +$C^{[1/\epsilon]}$-topology of the product Ricci flow $(S^2\times +(-\epsilon^{-1},\epsilon^{-1})),h(t)\times ds^2,\ 0\le t\le t_0$, +where $h(t)$ is the round metric with scalar curvature $1/(1-t)$ on +$S^2$. +- **(6) (Non-collapsing):** There are $r>0$ and $\kappa>0$ such that $(\Ar^3,g(t)),\ 0\le +t<1$, is $\kappa$-non-collapsed on scales less than $r$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/f4523c2b6f1e.md b/projects/poincare-conjecture/.astrolabe/atoms/f4523c2b6f1e.md new file mode 100644 index 00000000..4d215837 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/f4523c2b6f1e.md @@ -0,0 +1,19 @@ +--- +chapter: 3 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '3.37' +ref: +- f4523c2b6f1e +sort: remark +source: tex +src: morgan-tian +tex_file: flowbasics +--- +Let $({\mathcal M},G)$ be a generalized Ricci flow and let $x\in +{\mathcal M}$. Pulling $G$ back to the local coordinates $V\times +J$ defined near any point gives a one-parameter family of metrics +$(V,g(t)),\ t\in J$, satisfying the usual Ricci flow equation. It +follows that all the usual evolution formulas for Riemannian +curvature, Ricci curvature, and scalar curvature hold in this more +general context. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/f48558c60ca2.md b/projects/poincare-conjecture/.astrolabe/atoms/f48558c60ca2.md new file mode 100644 index 00000000..6fc6257b --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/f48558c60ca2.md @@ -0,0 +1,37 @@ +--- +chapter: 6 +generator: tools/poincare_tex_extract.py +label: 2ndvari +labels: +- 2ndvari +mtref: '6.14' +ref: +- f48558c60ca2 +sort: lemma +source: tex +src: morgan-tian +tex_file: newcompar +--- +Let $\gamma$ be an ${\mathcal L}$-geodesic defined on $ +[\tau_1,\tau_2]$, and let $Y_1$ and $Y_2$ be horizontal vector +fields along $\gamma$ vanishing at $\tau_1$. Suppose that +$\gamma_{u_1,u_2}$ is any family of curves parameterized by backward +time with the property that $\gamma_{0,0}=\gamma$ and the derivative +of the family in the $u_i$-direction at $u_1=u_2=0$ is $Y_i$. Let +$\widetilde Y_i$ be the image of $\partial/\partial u_i$ under +$\gamma_{u_1,u_2}$ and let $\widetilde X$ be the image of the +horizontal projection of $\partial/\partial \tau$ under this same +map, so that the restrictions of these three vector fields to the +curve $\gamma_{0,0}=\gamma$ are $Y_1,Y_2$ and $X$ respectively. Then +we have + +$$ +\begin{aligned} +\frac{\partial}{\partial +u_1}\frac{\partial}{\partial u_2}{\mathcal +L}(\gamma_{u_1,u_2})|_{u_1=u_2=0} & = & 2\sqrt{\tau_2} +Y_1(\tau_2)\langle \widetilde Y_2(\tau_2,u_1,0),\widetilde +X(\tau_2,u_1,0)\rangle|_{u_1=0}\\ +& & -\int_{\tau_1}^{\tau_2}2\sqrt{\tau}\langle \mathit{Jac}(Y_1),Y_2\rangle d\tau. +\end{aligned} +$$ diff --git a/projects/poincare-conjecture/.astrolabe/atoms/f4ee075394ef.md b/projects/poincare-conjecture/.astrolabe/atoms/f4ee075394ef.md new file mode 100644 index 00000000..46ed86d3 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/f4ee075394ef.md @@ -0,0 +1,22 @@ +--- +chapter: 10 +generator: tools/poincare_tex_extract.py +label: defntheta +labels: +- defntheta +mtref: '10.20' +ref: +- f4ee075394ef +sort: definition +source: tex +src: morgan-tian +tex_file: bddcurvbdddist +--- +Let $\gamma$ and $\mu$ be minimizing geodesic rays limiting to +${\mathcal E}$, of lengths $a$ and $b$, parameterized by the +distance from the end. For $00$. Then +$s({\mathcal T}(x,t))> 0$ for all $(x,t)\in \mathit{int}(\overline +U)\times(0,T]$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/f7b6896c9e7d.md b/projects/poincare-conjecture/.astrolabe/atoms/f7b6896c9e7d.md new file mode 100644 index 00000000..3466e929 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/f7b6896c9e7d.md @@ -0,0 +1,23 @@ +--- +chapter: 18 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '18.90' +ref: +- f7b6896c9e7d +sort: corollary +source: tex +src: morgan-tian +tex_file: energy1 +--- +For any $t\in [t',t_2]$ and any sub-arc $\gamma_{t'}$ of length $r$ +we have + +$$ +\int_{\gamma_t}\varphi kds +\le D_4\sqrt{\delta} +$$ + + for a constant $D_4$ that depends only on +the sectional curvature bound of the ambient Ricci flow and +$\Theta$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/f7e1d840a49f.md b/projects/poincare-conjecture/.astrolabe/atoms/f7e1d840a49f.md new file mode 100644 index 00000000..b0a23307 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/f7e1d840a49f.md @@ -0,0 +1,21 @@ +--- +chapter: 10 +generator: tools/poincare_tex_extract.py +label: deltanet +labels: +- deltanet +mtref: '10.28' +ref: +- f7e1d840a49f +sort: corollary +source: tex +src: morgan-tian +tex_file: bddcurvbdddist +--- +The metric completion $\bar X({\mathcal E})$ of the space of +directions at ${\mathcal E}$ is a compact space. For every +$\delta>0$ this space has a $\delta$-net consisting of realized +directions. For every $00$. There is a +$C^2$-family $\hat c(x,t)$ of immersions within $\delta$ in the +$C^2$-topology to $c(x,t)$ defined on the interval $[t^-,t^+]$ such +that + +$$ +A(t^+)\le \psi_{\delta,\hat c}(t^+) +$$ + +where $\psi_{\delta,\hat c}$ is the solution of the ODE + +$$ +\psi_{\delta,\hat c}'(t)=-2\pi+2\delta L_{\hat c}(t)-\frac{1}{2}R_\mathit{min}(t)\psi_{\delta,\hat c}(t) +$$ + + with value $A(\hat c(t^-))$ at +$t^-$, and where $L_{\hat c}(t)$ denotes the length of the loop +$\hat c(\cdot,t)$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/f8afc1c61977.md b/projects/poincare-conjecture/.astrolabe/atoms/f8afc1c61977.md new file mode 100644 index 00000000..6daab465 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/f8afc1c61977.md @@ -0,0 +1,17 @@ +--- +chapter: 4 +generator: tools/poincare_tex_extract.py +label: Ric>0 +labels: +- Ric>0 +mtref: '4.15' +ref: +- f8afc1c61977 +sort: corollary +source: tex +src: morgan-tian +tex_file: maxprin +--- +Suppose that $(M,g(t)),\ 0\le t\le T$, is a Ricci flow with $M$ a +compact, connected $3$-manifold with $\mathit{Ric}(x,0)\ge 0$ for all +$x\in M$. Then $\mathit{Ric}(x,t)\ge 0$ for all $t>0$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/f8ff249efb81.md b/projects/poincare-conjecture/.astrolabe/atoms/f8ff249efb81.md new file mode 100644 index 00000000..f3d37813 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/f8ff249efb81.md @@ -0,0 +1,21 @@ +--- +chapter: 9 +generator: tools/poincare_tex_extract.py +label: 3Dinfty +labels: +- 3Dinfty +mtref: '9.52' +ref: +- f8ff249efb81 +sort: corollary +source: tex +src: morgan-tian +tex_file: temp2kappa +--- +Let $(M,g(t))$ be a three-dimensional Ricci flow satisfying the hypotheses of +Proposition 9.39. Then the limit constructed in that proposition +splits as a product of a shrinking family of compact round surfaces with a +line. In particular, for any non-compact gradient shrinking soliton of a +three-dimensional $\kappa$-solution the limit constructed in +Proposition 9.39 is the product of a shrinking family of round +surfaces and the real line. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/f90cce212e88.md b/projects/poincare-conjecture/.astrolabe/atoms/f90cce212e88.md new file mode 100644 index 00000000..7c9d6ab7 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/f90cce212e88.md @@ -0,0 +1,20 @@ +--- +chapter: 1 +generator: tools/poincare_tex_extract.py +label: volinj +labels: +- volinj +mtref: '1.36' +ref: +- f90cce212e88 +sort: theorem +source: tex +src: morgan-tian +tex_file: prelim +--- +Fix an integer $n>0$. For every $\epsilon>0$ there is $\delta>0$ +depending on $n$ and $\epsilon$ such that the following holds. +Suppose that $(M^n,g)$ is a complete Riemannian manifold of +dimension $n$ and that $p\in M$. Suppose that $|\mathit{Rm}(x)|\le +r^{-2}$ for all $x\in B(p,r)$. If $Vol(B(p,r))\ge \epsilon r^n$ then +the injectivity radius of $M$ at $p$ is at least $\delta r$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/f917cf2b1576.md b/projects/poincare-conjecture/.astrolabe/atoms/f917cf2b1576.md new file mode 100644 index 00000000..57139392 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/f917cf2b1576.md @@ -0,0 +1,14 @@ +--- +chapter: 12 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '12.20' +ref: +- f917cf2b1576 +sort: claim +source: tex +src: morgan-tian +tex_file: stdsoln +--- +$f(r,t)$ is a smooth function defined for $r\ge 0$. It is an odd +function of $r$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/f93a1c4b1057.md b/projects/poincare-conjecture/.astrolabe/atoms/f93a1c4b1057.md new file mode 100644 index 00000000..7e892b30 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/f93a1c4b1057.md @@ -0,0 +1,14 @@ +--- +chapter: 7 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '7.23' +ref: +- f93a1c4b1057 +sort: remark +source: tex +src: morgan-tian +tex_file: newcomp2 +--- +Here $\triangle$ denotes the Laplacian with respect to the metric +$g(t)$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/f98ccf7ea70d.md b/projects/poincare-conjecture/.astrolabe/atoms/f98ccf7ea70d.md new file mode 100644 index 00000000..0c2b6301 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/f98ccf7ea70d.md @@ -0,0 +1,26 @@ +--- +chapter: 18 +generator: tools/poincare_tex_extract.py +label: embcase +labels: +- embcase +mtref: '18.30' +ref: +- f98ccf7ea70d +sort: lemma +source: tex +src: morgan-tian +tex_file: energy1 +--- +Suppose that $c\in \Lambda M$ is a homotopically trivial, embedded +$C^2$-loop. and suppose that there is a curve-shrinking flow +$c(x,t)$ defined for all $t\in [t_0,t_1]$ with each $c(\cdot,t)$ +being an embedded smooth curve. Consider the function $A(t)$ which +assigns to $t$ the minimal area of a spanning disk for $c(\cdot,t)$. +Then $A(t)$ is a continuous function of $t$ and + +$$ +\frac{dA}{dt}(t)\le -2\pi-\frac{1}{2}R_\mathit{min}(t)A(t) +$$ + +in the sense of forward difference quotients. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/f9aa8c556cab.md b/projects/poincare-conjecture/.astrolabe/atoms/f9aa8c556cab.md new file mode 100644 index 00000000..43117ed6 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/f9aa8c556cab.md @@ -0,0 +1,66 @@ +--- +chapter: 12 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '12.10' +ref: +- f9aa8c556cab +sort: remark +source: tex +src: morgan-tian +tex_file: stdsoln +--- +The covariant derivative acts on one-forms $\omega$ in such a way +that the following equation holds: + +$$ +\langle\nabla(\omega),\xi\rangle=\langle +\omega,\nabla(\xi)\rangle +$$ + + for every vector field $\xi$. This means +that in local coordinates we have + +$$ +\nabla_{\partial_r}(dx^k)=-\Gamma_{rl}^kdx^l. +$$ + + Similarly, the Riemann +curvature acts on one-forms $\omega$ satisfying + +$$ +\mathit{Rm}(\xi_1,\xi_2)(\omega)(\xi)=-\omega\left(\mathit{Rm}(\xi_1,\xi_2)(\xi)\right). +$$ + +Recall that in local coordinates + +$$ +R_{ijkl}=\langle \mathit{Rm}(\partial_i,\partial_j)(\partial_l),\partial_k\rangle. +$$ + + Thus, we +have + +$$ +\mathit{Rm}(\partial_i,\partial_j)(dx^k)=-g^{ka}R_{ijal}dx^l=-{{R_{ij}}^k}_ldx^l, +$$ + +where as usual we use the inverse metric tensor to raise the index. + +Also, notice that $\Delta X_{i}-\mathit{Ric}_{ik}X^{k}=-\Delta_{d}X_{i\text{ }}$, where by $\Delta_d$ we +mean the Laplacian associated to the operator $d$ from vector fields +to one-forms with values in the vector field. Since + +$$ +\begin{aligned} +-\left( d\delta+\delta d\right) X_{i} +& =-\nabla_{i}\left( -\nabla^{k}X_{k}\right) -\left( +-\nabla^{k}\right) +\left( \nabla_{k}X_{i}-\nabla_{i}X_{k}\right) \\ +& +=\nabla_{i}\nabla^{k}X_{k}+\nabla^{k}\nabla_{k}X_{i}-\nabla^{k}\nabla +_{i}X_{k}\\ +& ={{{R_{i}}^k}_k}^jX_{j}+\nabla^{k}\nabla_{k}X_{i}=\Delta +X_{i}-\mathit{Ric}_i^jX_{j}. +\end{aligned} +$$ diff --git a/projects/poincare-conjecture/.astrolabe/atoms/f9cff326f854.md b/projects/poincare-conjecture/.astrolabe/atoms/f9cff326f854.md new file mode 100644 index 00000000..fbec9fed --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/f9cff326f854.md @@ -0,0 +1,16 @@ +--- +chapter: 4 +generator: tools/poincare_tex_extract.py +label: flowextend +labels: +- flowextend +mtref: '4.12' +ref: +- f9cff326f854 +sort: proposition +source: tex +src: morgan-tian +tex_file: maxprin +--- +Let $(M,g(t)),\ 0\le tT$ or $|\mathit{Rm}|$ is unbounded on $M\times[0,T)$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/fa3234930898.md b/projects/poincare-conjecture/.astrolabe/atoms/fa3234930898.md new file mode 100644 index 00000000..30fa4c12 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/fa3234930898.md @@ -0,0 +1,24 @@ +--- +chapter: 6 +generator: tools/poincare_tex_extract.py +label: backwards +labels: +- backwards +mtref: '6.24' +ref: +- fa3234930898 +sort: remark +source: tex +src: morgan-tian +tex_file: newcompar +--- +When $\tau_1>0$ it is possible to consider the ${\mathcal L}\mathit{exp}^{\tau}_x$ defined for $0<\tau<\tau_1$. In this case, the curves +are moving backward in $\tau$ and hence are moving forward with +respect to the time parameter $\mathbf{t}$. Two comments are in order. +First of all, for $\tau<\tau_1$, the gradient of ${\widetilde +L}^{\tau}_x$ is $-2\sqrt{\tau}X(\tau)$. The reason for the sign +reversal is that the length is given by the integral from $\tau$ to +$\tau_1$ and hence its derivative with respect to $\tau$ is the +negative of the integrand. The second thing to remark is that +Lemma 6.23 is true for $\tau<\tau_1$ with $\tau$ +sufficiently close to $\tau_1$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/fa8d9a43f34a.md b/projects/poincare-conjecture/.astrolabe/atoms/fa8d9a43f34a.md new file mode 100644 index 00000000..87257a53 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/fa8d9a43f34a.md @@ -0,0 +1,30 @@ +--- +chapter: 19 +generator: tools/poincare_tex_extract.py +label: epstopology +labels: +- epstopology +mtref: '19.25' +ref: +- fa8d9a43f34a +sort: proposition +source: tex +src: morgan-tian +tex_file: canonnbhd +--- +For all $\epsilon>0$ sufficiently small the following holds. Suppose +that $(M,g)$ is a connected Riemannian manifold such that every +point is either contained in the core of a $(C,\epsilon)$-cap in $M$ +or is the center of an $\epsilon$-neck in $M$. Then one of the +following holds: + +- **(1)** $M$ is diffeomorphic to $S^3$, $\Ar P^3$ or $\Ar P^3\#\Ar +P^3$, +and $M$ is either a double $C$-capped $\epsilon$-tube or is the +union of two $(C,\epsilon)$-caps. +- **(2)** $M$ is diffeomorphic to $\Ar^3$ or $\Ar P^3\setminus \{\mathit{point}\}$, +and $M$ is either a $(C,\epsilon$-cap or a $C$-capped +$\epsilon$-tube. +- **(3)** $M$ is diffeomorphic to $S^2\times \Ar$ and is an +$\epsilon$-tube. +- **(4)** $M$ is diffeomorphic to an $S^2$-bundle over $S^1$ and is an $\epsilon$-fibration. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/fac15ebeba31.md b/projects/poincare-conjecture/.astrolabe/atoms/fac15ebeba31.md new file mode 100644 index 00000000..dd0a9828 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/fac15ebeba31.md @@ -0,0 +1,14 @@ +--- +chapter: 12 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '12.35' +ref: +- fac15ebeba31 +sort: remark +source: tex +src: morgan-tian +tex_file: stdsoln +--- +Recall that $p_0$ is the origin in $\Ar^3$ and hence is the tip of +the surgery cap. Also, $A_0$ is defined in Lemma 12.3. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/faca28b0bd43.md b/projects/poincare-conjecture/.astrolabe/atoms/faca28b0bd43.md new file mode 100644 index 00000000..b4990b78 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/faca28b0bd43.md @@ -0,0 +1,27 @@ +--- +chapter: 6 +generator: tools/poincare_tex_extract.py +label: posform +labels: +- posform +mtref: '6.21' +ref: +- faca28b0bd43 +sort: corollary +source: tex +src: morgan-tian +tex_file: newcompar +--- +Let $Z\in T_xM_{T-\tau_1}$. Suppose that the associated ${\mathcal +L}$-geodesic $\gamma_Z$ minimizes ${\mathcal L}$-length between its +endpoints, $x$ and $\gamma_Z(\bar\tau)$, and that $d_Z{\mathcal +L}\mathit{exp}^{\bar\tau}_x$ is an isomorphism. Then for any family +$\gamma_u$ of curves parameterized by backward time with +$Y=(\partial \gamma/\partial u)|_{u=0}$ vanishing at both endpoints, +we have + +$$ +\frac{d^2}{du^2}{\mathcal L}(\gamma_u)|_{u=0}\ge 0, +$$ + +with equality if and only if $Y=0$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/faf16f1a175d.md b/projects/poincare-conjecture/.astrolabe/atoms/faf16f1a175d.md new file mode 100644 index 00000000..d7420c75 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/faf16f1a175d.md @@ -0,0 +1,34 @@ +--- +chapter: 3 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '3.38' +ref: +- faf16f1a175d +sort: definition +source: tex +src: morgan-tian +tex_file: flowbasics +--- +Let ${\mathcal M}$ be a space-time. Given a space $C$ and an +interval $I\subset \Ar$ we say that an embedding $C\times I\to +{\mathcal M}$ is *compatible with* the time and the vector field +if: (i) the restriction of $\mathbf{t}$ to the image agrees with the +projection onto the second factor and (ii) for each $c\in C$ the +image of $\{c\}\times I$ is the integral curve for the vector field +$\chi$. + If in addition $C$ is a subset of $M_{t}$ +we require that $t\in I$ and that the map $C\times\{t\}\to M_{t}$ be +the identity. Clearly, by the uniqueness of integral curves for vector fields, + two such embeddings agree on their common interval of definition, so that, +given $C\subset M_{t}$ there is a maximal interval $I_C$ containing +$t$ such that such an embedding, compatible with time and the vector field, + is defined on $C\times I$. In the +special case when $C=\{x\}$ for a point $x\in M_{t}$ we say that +such an embedding is *the flow line* through $x$. The embedding +of the maximal interval through $x$ compatible with time and the vector +field $\chi$ is called *the domain of definition* of the flow +line through $x$. For a more general subset $C\subset M_{t}$ there +is an embedding $C\times I$ compatible with time and the vector +field $\chi$ if an only if for every $x\in C$, $I$ is contained in +the domain of definition of the flow line through $x$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/fb6be46de671.md b/projects/poincare-conjecture/.astrolabe/atoms/fb6be46de671.md new file mode 100644 index 00000000..4ca5be9b --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/fb6be46de671.md @@ -0,0 +1,14 @@ +--- +chapter: 18 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '18.81' +ref: +- fb6be46de671 +sort: lemma +source: tex +src: morgan-tian +tex_file: energy1 +--- +For any $x\in X$ there is no sub-geodesic of $D_x$ that is an +embedded loop in $A$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/fba418d1df15.md b/projects/poincare-conjecture/.astrolabe/atoms/fba418d1df15.md new file mode 100644 index 00000000..712c1a87 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/fba418d1df15.md @@ -0,0 +1,13 @@ +--- +chapter: 18 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '18.25' +ref: +- fba418d1df15 +sort: definition +source: tex +src: morgan-tian +tex_file: energy1 +--- +Let $A(t)=\int_{t'}^t\frac{1}{2}R_\mathit{min}(s)ds$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/fbf53047d0c1.md b/projects/poincare-conjecture/.astrolabe/atoms/fbf53047d0c1.md new file mode 100644 index 00000000..7aaeec3a --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/fbf53047d0c1.md @@ -0,0 +1,15 @@ +--- +chapter: 3 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '3.4' +ref: +- fbf53047d0c1 +sort: example +source: tex +src: morgan-tian +tex_file: flowbasics +--- +$\mathbb{C}P^{n}$ equipped with the Fubini-Study metric, which is induced +from the standard metric of $S^{2n+1}$ under the Hopf fibration with the fibers +of great circles, is Einstein. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/fc2d2036a96c.md b/projects/poincare-conjecture/.astrolabe/atoms/fc2d2036a96c.md new file mode 100644 index 00000000..7b966d8f --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/fc2d2036a96c.md @@ -0,0 +1,18 @@ +--- +chapter: 9 +generator: tools/poincare_tex_extract.py +label: kappacannbhd +labels: +- kappacannbhd +mtref: '9.94' +ref: +- fc2d2036a96c +sort: corollary +source: tex +src: morgan-tian +tex_file: temp2kappa +--- +For every $0<\epsilon\le\bar\epsilon'$ there is +$C=C(\epsilon)<\infty$ such that every point in a $\kappa$-solution +has a strong $(C,\epsilon)$-canonical neighborhood unless the +$\kappa$-solution is a product $\Ar P^2\times \Ar$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/fc44d8653f35.md b/projects/poincare-conjecture/.astrolabe/atoms/fc44d8653f35.md new file mode 100644 index 00000000..4e378389 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/fc44d8653f35.md @@ -0,0 +1,39 @@ +--- +chapter: 4 +generator: tools/poincare_tex_extract.py +label: pincha +labels: +- pincha +mtref: '4.32' +ref: +- fc44d8653f35 +sort: theorem +source: tex +src: morgan-tian +tex_file: maxprin +--- +Fix $a\ge 0$. Let $(M,g(t)),\ a\le t0$. Then for all $a\le t 0$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/fc5c13ed2e67.md b/projects/poincare-conjecture/.astrolabe/atoms/fc5c13ed2e67.md new file mode 100644 index 00000000..21d15cf1 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/fc5c13ed2e67.md @@ -0,0 +1,15 @@ +--- +chapter: 16 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '16.25' +ref: +- fc5c13ed2e67 +sort: claim +source: tex +src: morgan-tian +tex_file: surgery +--- +The subset $Z'=\{z\in Z|{\mathcal L}_x(z)\le L/2\}$ has the property +that for any compact interval $I\subset [T_{i-1},T)$ the +intersection $\mathbf{t}^{-1}(I)\cap Z'$ is compact. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/fc6c6ae605d2.md b/projects/poincare-conjecture/.astrolabe/atoms/fc6c6ae605d2.md new file mode 100644 index 00000000..f686faa2 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/fc6c6ae605d2.md @@ -0,0 +1,25 @@ +--- +chapter: 19 +generator: tools/poincare_tex_extract.py +label: curveshort +labels: +- curveshort +mtref: '19.1' +ref: +- fc6c6ae605d2 +sort: lemma +source: tex +src: morgan-tian +tex_file: canonnbhd +--- +The following holds for all $\epsilon>0$ sufficiently small. Suppose +that $(M,g)$ is a Riemannian manifold and that $N\subset M$ is an +$\epsilon$-neck centered at $x$. Let $S(x)$ be the central +two-sphere of this neck and suppose that $S(x)$ separates $M$. Let +$y\in M$. Orient $s$ so that $y$ lies in the closure of the positive +side of $S(x)$. Let $\gamma\colon [0,a]\to M$ be a rectifiable curve +from $x$ to $y$. If $\gamma$ contains a point of +$s^{-1}(-\epsilon^{-1},-\epsilon^{-1}/2)$ then there is a +rectifiable curve from $x$ to $y$ contained in the closure of the +positive side of $S(x)$ whose length is at most the length of +$\gamma$ minus $\frac{1}{2}\epsilon^{-1}R(x)^{-1/2}$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/fd11f1f8def3.md b/projects/poincare-conjecture/.astrolabe/atoms/fd11f1f8def3.md new file mode 100644 index 00000000..985d78ab --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/fd11f1f8def3.md @@ -0,0 +1,24 @@ +--- +chapter: 9 +generator: tools/poincare_tex_extract.py +label: compact2Dsol +labels: +- compact2Dsol +mtref: '9.49' +ref: +- fd11f1f8def3 +sort: corollary +source: tex +src: morgan-tian +tex_file: temp2kappa +--- +There is no non-compact, two-dimensional Riemannian manifold $(M,g)$ +satisfying the hypotheses of Theorem 9.42. For any +non-compact three-manifold $(M,g)$ of positive curvature satisfying +the hypotheses of Theorem 9.42, there is a sequence of +points $q_i\in M$ tending to infinity such that $\mathit{lim}_{i\rightarrow \infty}R_g(q_i)=\mathit{sup}_{p\in M}$ such that the +based Ricci flows $(M,G(t),(q_i,-1))$ converge to a Ricci flow +$(M_\infty,G_\infty(t),(q_\infty,-1))$ defined for $-\infty0$. Then there +is a continuous family $\widetilde\Gamma(t),\ t_0\le t\le t_1$, of +maps $S^2\to \Lambda M$ whose image consists of homotopically +trivial loops with $[\widetilde\Gamma(t_0)]=[\Gamma]$ in +$\pi_3(M,*)$ such that for each $c\in S^2$ we have +$|A(\widetilde\Gamma(t_0)(c))-A(\Gamma(c))|<\zeta$ and furthermore, +one of the following two alternatives holds: + +- **(i)** The length of $\widetilde\Gamma(t_1)(c)$ is less than $\zeta$. +- **(ii)** $A(\widetilde\Gamma(t_1)(c))\le w_{A(\widetilde\Gamma(t_0)(c))}(t_1)+\zeta$. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/ff5d51a483d9.md b/projects/poincare-conjecture/.astrolabe/atoms/ff5d51a483d9.md new file mode 100644 index 00000000..d50eb8db --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/ff5d51a483d9.md @@ -0,0 +1,29 @@ +--- +chapter: 18 +generator: tools/poincare_tex_extract.py +label: csshi1 +labels: +- csshi1 +mtref: '18.86' +ref: +- ff5d51a483d9 +sort: lemma +source: tex +src: morgan-tian +tex_file: energy1 +--- +Let $(W,h(t)),\ t_0\le t\le t_1$, be a Ricci flow and fix +$\Theta<\infty$. Then there exist constants $\delta>0$ and $00$ and any $q$ we have + +$$ +|\nabla l_\infty(q,\tau)|\le +\frac{3}{\tau}d_{g_\infty}(-\tau)(q_\infty,q)+\sqrt{\frac{3}{\tau}}C_1(\tau), +$$ + +where $C_1(\tau)$ is the continuous function from +Corollary 9.16. diff --git a/projects/poincare-conjecture/.astrolabe/atoms/ffae4504318a.md b/projects/poincare-conjecture/.astrolabe/atoms/ffae4504318a.md new file mode 100644 index 00000000..b144c10c --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/ffae4504318a.md @@ -0,0 +1,20 @@ +--- +chapter: 12 +generator: tools/poincare_tex_extract.py +label: Restim +labels: +- Restim +mtref: '12.31' +ref: +- ffae4504318a +sort: proposition +source: tex +src: morgan-tian +tex_file: stdsoln +--- +There is a constant $c>0$ such that for all $(p,t)$ in the standard +solution we have + +$$ +R(p,t)\ge \frac{c}{1-t}. +$$ diff --git a/projects/poincare-conjecture/.astrolabe/atoms/ffb86c646793.md b/projects/poincare-conjecture/.astrolabe/atoms/ffb86c646793.md new file mode 100644 index 00000000..f0e16e72 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/atoms/ffb86c646793.md @@ -0,0 +1,14 @@ +--- +chapter: 10 +generator: tools/poincare_tex_extract.py +labels: [] +mtref: '10.13' +ref: +- ffb86c646793 +sort: claim +source: tex +src: morgan-tian +tex_file: bddcurvbdddist +--- +There is a minimizing geodesic ray to ${\mathcal E}$ from each $x\in +U_\infty$ with $R(x)\ge 2Q_\infty$. diff --git a/projects/poincare-conjecture/.astrolabe/docs/01-intro.mdx b/projects/poincare-conjecture/.astrolabe/docs-src/01-introduction.mdx similarity index 82% rename from projects/poincare-conjecture/.astrolabe/docs/01-intro.mdx rename to projects/poincare-conjecture/.astrolabe/docs-src/01-introduction.mdx index 1539cd7e..1a4124f6 100644 --- a/projects/poincare-conjecture/.astrolabe/docs/01-intro.mdx +++ b/projects/poincare-conjecture/.astrolabe/docs-src/01-introduction.mdx @@ -1,65 +1,70 @@ -# Introduction + + +# Chapter 0 -- Introduction In this book we present a complete and detailed proof of the -\begin{quote}{\bf Poincar\'e Conjecture: -a closed, smooth, simply connected $3$-manifold is -diffeomorphic}\footnote{Every topological $3$-manifold admits a -differentiable structure and every homeomorphism between smooth -$3$-manifolds can be approximated by a diffeomorphism. Thus, -classification results about topological $3$-manifolds up to -homeomorphism and about smooth $3$-manifolds up to diffeomorphism -are equivalent. In this book `manifold' means `smooth manifold.'} -**to $S^3$.** -\end{quote} - -This conjecture was formulated by Henri Poincar\'e +> **Poincaré Conjecture: +> a closed, smooth, simply connected $3$-manifold is +> diffeomorphic to $S^3$.** (Every topological $3$-manifold admits a +> differentiable structure and every homeomorphism between smooth +> $3$-manifolds can be approximated by a diffeomorphism. Thus, +> classification results about topological $3$-manifolds up to +> homeomorphism and about smooth $3$-manifolds up to diffeomorphism +> are equivalent. In this book 'manifold' means 'smooth manifold.') + +This conjecture was formulated by Henri Poincaré [Poincare] in 1904 and has remained open until the recent work of Perelman. The arguments we give here are a detailed version of those that appear -in Perelman's three preprints~. +in Perelman's three preprints [Perelman1, Perelman2, Perelman3]. Perelman's arguments rest on a foundation built by Richard Hamilton with his study of the Ricci flow equation for Riemannian metrics. Indeed, Hamilton believed that Ricci flow could be used to establish -the Poincar\'e Conjecture and more general topological +the Poincaré Conjecture and more general topological classification results in dimension $3$, and laid out a program to accomplish this. The difficulty was to deal with singularities in the Ricci flow. Perelman's breakthrough was to understand the qualitative nature of the singularities sufficiently to allow him to -prove the Poincar\'e Conjecture (and Theorem~ below -which implies the Poincar\'e Conjecture). For -a detailed history of the Poincar\'e Conjecture, see Milnor's survey -article . +prove the Poincaré Conjecture (and \entryref{9999efa2be2c} below +which implies the Poincaré Conjecture). For +a detailed history of the Poincaré Conjecture, see Milnor's survey +article [Milnorsurvey]. -A class of examples closely related to the $3$-sphere are *the +i̱gskip +A class of examples closely related to the $3$-sphere are *the $3$-dimensional spherical space-forms*, i.e., the quotients of $S^3$ by free, linear actions of finite subgroups of the -orthogonal group $O(4)$. There is a generalization of the Poincar\'e +orthogonal group $O(4)$. There is a generalization of the Poincaré Conjecture, called the **$3$-dimensional spherical space-form conjecture**, which conjectures that any closed $3$-manifold with finite fundamental group is diffeomorphic to a $3$-dimensional spherical space-form. Clearly, a special case of the -$3$-dimensional spherical space-form conjecture is the Poincar\'e Conjecture. +$3$-dimensional spherical space-form conjecture is the Poincaré Conjecture. - As indicated in Remark 1.4 of , the arguments we present - here not only prove the Poincar\'e Conjecture, they prove the $3$-dimensional space-form conjecture. - In fact, the purpose of this book is to prove the following + As indicated in Remark 1.4 of [Perelman3], the arguments we present + here not only prove the Poincaré Conjecture, they prove the $3$-dimensional space-form conjecture. + In fact, the purpose of this book is to prove the following more general theorem. -**Theorem.** Let $M$ be a closed, connected $3$-manifold and + +**Theorem.** Let $M$ be a closed, connected $3$-manifold and suppose that the fundamental group of $M$ is a free product of finite groups and infinite cyclic groups. Then $M$ is diffeomorphic to a connected sum of spherical space-forms, copies of $S^2\times S^1$, and copies of the unique (up to diffeomorphism) non-orientable $2$-sphere bundle over $S^1$. + -This immediately implies an affirmative resolution of the Poincar\'e +This immediately implies an affirmative resolution of the Poincaré Conjecture and of the $3$-dimensional spherical space-form conjecture. + **Corollary.** (a) A closed, simply connected $3$-manifold is diffeomorphic to -$S^3$. (b) A closed $3$-manifold with finite fundamental +$S^3$. (b) A closed $3$-manifold with finite fundamental group is diffeomorphic to a $3$-dimensional spherical space-form. + Before launching into a more detailed description of the contents of this book, one remark on the style of the exposition is in order. @@ -79,19 +84,23 @@ Riemannian geometry show that the universal cover of a closed manifold of constant positive curvature is diffeomorphic to the sphere and that the fundamental group is identified with a finite subgroup of the orthogonal group acting linearly and freely on the universal cover. Thus, one can approach the -Poincar\'e Conjecture and the more general $3$-dimensional spherical space-form +Poincaré Conjecture and the more general $3$-dimensional spherical space-form problem by asking the following question. Making the appropriate fundamental group assumptions on $3$-manifold $M$, how does one establish the existence of a metric of constant Ricci curvature on $M$? The essential ingredient in producing such a metric is the Ricci flow equation -introduced by Richard Hamilton in : -$$\frac{\partial g(t)}{\partial t}=-2Ric(g(t)),$$ -where $Ric(g(t))$ is the Ricci curvature of the metric $g(t)$. +introduced by Richard Hamilton in [Hamilton3MPRC]: + +$$ +\frac{\partial g(t)}{\partial t}=-2\mathit{Ric}(g(t)), +$$ + +where $\mathit{Ric}(g(t))$ is the Ricci curvature of the metric $g(t)$. The fixed points (up to rescaling) of this equation are the Riemannian metrics of constant Ricci curvature. For a general introduction to the subject of the Ricci flow see Hamilton's survey -paper , the book by Chow-Knopf -, or the book by Chow, Lu, and Ni . +paper [Hamiltonsurvey], the book by Chow-Knopf +[ChowKnopf], or the book by Chow, Lu, and Ni [ChowLuNi]. The Ricci flow equation is a (weakly) parabolic partial differential flow equation for Riemannian metrics on a smooth manifold. Following Hamilton, one defines a Ricci flow to be a family @@ -109,7 +118,7 @@ flow equation to obtain the constant curvature metric for which one is searching. There are two examples where things work in exactly this way, both due to Hamilton. (i) If the initial metric has positive Ricci curvature, Hamilton proved over twenty years ago, -, that under the Ricci flow the manifold shrinks +[Hamilton3MPRC], that under the Ricci flow the manifold shrinks to a point in finite time, that is to say, there is a finite-time singularity, and, as we approach the singular time, the diameter of the manifold tends to zero and the curvature blows up at every @@ -117,7 +126,7 @@ point. Hamilton went on to show that, in this case, rescaling by a time-dependent function so that the diameter is constant produces a one-parameter family of metrics converging smoothly to a metric of constant positive curvature. (ii) At the other extreme, in - Hamilton showed that if the Ricci flow exists +[HamiltonNSRF3M] Hamilton showed that if the Ricci flow exists for all time and if there is an appropriate curvature bound together with another geometric bound, then as $t\rightarrow\infty$, after rescaling to have a fixed diameter, the metric converges to a metric @@ -125,8 +134,8 @@ of constant negative curvature. The results in the general case are much more complicated to formulate and much more difficult to establish. While Hamilton -established that the Ricci flow equation has short-term existence -properties, i.e., one can define $g(t)$ for $t$ in some interval +established that the Ricci flow equation has short-term existence +properties, i.e., one can define $g(t)$ for $t$ in some interval $[0,T)$ where $T$ depends on the initial metric, it turns out that if the topology of the manifold is sufficiently complicated, say it is a non-trivial connected sum, then no matter what the initial @@ -143,7 +152,7 @@ the analysis the first time a singularity arises in the Ricci flow. One is led to study a more general evolution process called *Ricci flow with surgery*, first introduced by Hamilton in the context - of four-manifolds, . + of four-manifolds, [Hamilton4MPIC]. This evolution process is still parameterized by an interval in time, so that for each $t$ in the interval of definition there is a compact Riemannian $3$-manifold @@ -154,10 +163,10 @@ intervals to the singular times, the evolution is the usual Ricci flow, though, because of the surgeries, the topological type of the manifold $M_t$ changes as $t$ moves from one complementary interval to the next. From an analytic point of view, the surgeries at the -discontinuity times are introduced in order to `cut away' a +discontinuity times are introduced in order to 'cut away' a neighborhood of the singularities as they develop and insert by -hand, in place of the `cut away' regions, geometrically nice -regions. This allows one to continue the Ricci flow (or more +hand, in place of the 'cut away' regions, geometrically nice +regions. This allows one to continue the Ricci flow (or more precisely, restart the Ricci flow with the new metric constructed at the discontinuity time). Of course, the surgery process also changes the topology. To be able to say anything useful topologically about @@ -181,11 +190,12 @@ The bulk of this book (Chapters 1-17 and the Appendix) concerns the establishment of the following long-time existence result for Ricci flow with surgery. -**Theorem.** Let $(M,g_0)$ be a closed Riemannian $3$-manifold. + +**Theorem.** Let $(M,g_0)$ be a closed Riemannian $3$-manifold. Suppose that there is no embedded, locally separating $\Ar P^2$ -contained\footnote{I.e., no embedded $\Ar P^2$ in $M$ with trivial +contained (I.e., no embedded $\Ar P^2$ in $M$ with trivial normal bundle. Clearly, all orientable manifolds satisfy this -condition.} in $M$. Then there is a Ricci flow with surgery defined +condition.) in $M$. Then there is a Ricci flow with surgery defined for all $t\in [0,\infty)$ with initial metric $(M,g_0)$. The set of discontinuity times for this Ricci flow with surgery is a discrete subset of $[0,\infty)$. The topological change in the $3$-manifold @@ -194,43 +204,45 @@ together with removal of connected components, each of which is diffeomorphic to one of $S^2\times S^1$, $\Ar P^3\#\Ar P^3$, the non-orientable $2$-sphere bundle over $S^1$, or a manifold admitting a metric of constant positive curvature. + -While Theorem~ is central for all applications of Ricci +While \entryref{847af3ecd4a8} is central for all applications of Ricci flow to the topology of three-dimensional manifolds, the argument -for the $3$-manifolds described in Theorem~ is -simplified, and avoids all references to the nature of the flow as +for the $3$-manifolds described in \entryref{9999efa2be2c} is +simplified, and avoids all references to the nature of the flow as time goes to infinity, because of the following finite-time extinction result. -**Theorem.** -Let $M$ be a closed $3$-manifold whose fundamental group is a free -product of finite groups and infinite cyclic groups\footnote{In - Perelman states the result for $3$-manifolds + +**Theorem.** Let $M$ be a closed $3$-manifold whose fundamental group is a free +product of finite groups and infinite cyclic groups (In +[Perelman3] Perelman states the result for $3$-manifolds without prime factors that are acyclic. It is a standard exercise in $3$-manifold topology to show that Perelman's condition is equivalent to the group theory hypothesis stated here; see -Corollary~.}. Let $g_0$ be any Riemannian metric on +\entryref{7078952cb9e4}.). Let $g_0$ be any Riemannian metric on $M$. Then $M$ admits no locally separating $\Ar P^2$, so that there is a Ricci flow with surgery defined for all positive time with -$(M,g_0)$ as initial metric as described in Theorem~. +$(M,g_0)$ as initial metric as described in \entryref{847af3ecd4a8}. This Ricci flow with surgery becomes extinct after some time $T<\infty$, in the sense that the manifolds $M_t$ are empty for all $t\ge T$. + This result is established in Chapter 18 following the argument -given by Perelman in , see also -. +given by Perelman in [Perelman3], see also +[ColdingMinicozzi]. -We immediately deduce Theorem~ from -Theorems~ and~ as follows: Let $M$ be a -$3$-manifold satisfying the hypothesis of Theorem~. +We immediately deduce \entryref{9999efa2be2c} from +Theorems \entryref{847af3ecd4a8} and \entryref{127db1dc9315} as follows: Let $M$ be a +$3$-manifold satisfying the hypothesis of \entryref{9999efa2be2c}. Then there is a finite sequence $M=M_0,M_1,\ldots,M_k=\emptyset$ such that for each $i,\ 1\le i\le k$, $M_i$ is obtained from -$M_{i-1}$ by a connected sum decomposition or $M_i$ is obtained -from $M_{i-1}$ by removing a component diffeomorphic to one of +$M_{i-1}$ by a connected sum decomposition or $M_i$ is obtained +from $M_{i-1}$ by removing a component diffeomorphic to one of $S^2\times S^1$, $\Ar P^3\#\Ar P^3$, a non-orientable $2$-sphere bundle over $S^1$, or a $3$-dimensional spherical space-form. -Clearly, it follows by downward induction on $i$ that each +Clearly, it follows by downward induction on $i$ that each connected component of $M_i$ is diffeomorphic to a connected sum of $3$-dimensional spherical space-forms, copies of $S^2\times S^1$, and copies of the non-orientable $2$-sphere bundle over $S^1$. In @@ -238,35 +250,37 @@ particular, $M=M_0$ has this form. Since $M$ is connected by hypothesis, this proves the theorem. In fact, this argument proves the following: -**Corollary.** -Let $(M_0,g_0)$ a connected Riemannian manifold with no locally + +**Corollary.** Let $(M_0,g_0)$ a connected Riemannian manifold with no locally separating $\Ar P^2$. Let $({\mathcal M},G)$ be a Ricci flow with surgery defined for $0\le t<\infty$ with $(M_0,g_0)$ as initial manifold. Then the following four conditions are equivalent: -\begin{enumerate} \item[(1)] $({\mathcal M},G)$ becomes extinct after a + +- **(1)** $({\mathcal M},G)$ becomes extinct after a finite time, i.e., $M_T=\emptyset$ for all $T$ sufficiently large, -\item[(2)] $M_0$ is diffeomorphic to a connected sum of three-dimensional -spherical space-forms and $S^2$-bundles over $S^1$,\item[(3)] the +- **(2)** $M_0$ is diffeomorphic to a connected sum of three-dimensional +spherical space-forms and $S^2$-bundles over $S^1$, +- **(3)** the fundamental group of $M_0$ is a free product of finite groups and infinite cyclic groups, -\item[(4)] no prime\footnote{A three-manifold $P$ is prime -if every separating two-sphere in $P$ bounds a three-ball in $P$. +- **(4)** no prime (A three-manifold $P$ is prime +if every separating two-sphere in $P$ bounds a three-ball in $P$. Equivalently, $P$ is prime if it admits no non-trivial connected sum decomposition. Every closed three-manifold decomposes as a connected sum of prime factors with the decomposition being unique up to -diffeomorphism of the factors and the order of the factors.} factor +diffeomorphism of the factors and the order of the factors.) factor of $M_0$ is acyclic, i.e., every prime factor of $M_0$ has either non-trivial $\pi_2$ or non-trivial $\pi_3$. -\end{enumerate} + -**Proof.** Repeated application of Theorem~ shows that (1) implies +**Proof.** Repeated application of \entryref{847af3ecd4a8} shows that (1) implies (2). The implication (2) implies (3) is immediate from van Kampen's -theorem. The fact that (3) implies (1) is Theorem~. +theorem. The fact that (3) implies (1) is \entryref{127db1dc9315}. This shows that (1), (2) and (3) are all equivalent. Since three-dimensional spherical space-forms and $S^2$-bundles over $S^1$ -are easily seen to be prime, (2) implies (4). Thus, it remains +are easily seen to be prime, (2) implies (4). Thus, it remains only to see that (4) implies (3). We consider a manifold $M$ -satisfying (4), a prime factor $P$ of $M$, and universal covering +satisfying (4), a prime factor $P$ of $M$, and universal covering $\widetilde P$ of $P$. First suppose that $\pi_2(P)=\pi_2(\widetilde P)$ is trivial. Then, by hypothesis $\pi_3(P)=\pi_3(\widetilde P)$ is non-trivial. By the Hurewicz theorem this means that @@ -283,6 +297,7 @@ two-spheres. Since $P_0$ is prime, it follows that $P_0$ is diffeomorphic to $S^2\times I$ and hence $P$ is diffeomorphic to a two-sphere bundle over the circle. + **Remark.** (i) The use of the sphere theorem is unnecessary in the above argument for what we actually prove is that if every prime factor of $M$ has non-trivial $\pi_2$ or non-trivial $\pi_3$, then the Ricci @@ -290,7 +305,7 @@ flow with surgery with $(M,g_0)$ as initial metric becomes extinct after a finite time. In fact, the sphere theorem for closed three-manifolds follows from the results here. - (ii) If the initial manifold is simpler then all the + (ii) If the initial manifold is simpler then all the time-slices are simpler: If $({\mathcal M},G)$ is a Ricci flow with surgery whose initial manifold is prime, then every time-slice is a disjoint union of connected components, all but at most one being @@ -306,7 +321,7 @@ connected or has the same fundamental group as the initial manifold. (iii) The conclusion of this result is a natural generalization of Hamilton's conclusion in analyzing the Ricci flow -on manifolds of positive Ricci curvature in . +on manifolds of positive Ricci curvature in [Hamilton3MPRC]. Namely, under appropriate hypotheses, during the evolution process of Ricci flow with surgery the manifold breaks into components each of which disappears in finite time. As a component disappears at @@ -316,7 +331,7 @@ non-flat, homogeneous Riemannian metric of non-negative sectional curvature, i.e., a metric locally isometric to either a round $S^3$ or to a product of a round $S^2$ with the usual metric on $ \Ar$. The existence of such a metric on a component immediately gives the -topological conclusion of Theorem~ for that component, +topological conclusion of \entryref{9999efa2be2c} for that component, i.e., that it is diffeomorphic to a $3$-dimensional spherical space-form, to $S^2\times S^1$ to a non-orientable $2$-sphere bundle over $S^1$, or to $\Ar P^3\#\Ar P^3$. The biggest difference between @@ -325,60 +340,61 @@ these two results is that Hamilton's hypothesis is geometric (information about the fundamental group). (iv) It is also worth pointing out that it follows from -Corollary~ that the manifolds that satisfy the four equivalent +\entryref{7078952cb9e4} that the manifolds that satisfy the four equivalent conditions in that corollary are exactly the closed, connected, three-manifolds that admit a Riemannian metric of positive scalar curvature, cf, - and \cite {GromovLawson}. +[SchoenYau2] and [GromovLawson]. + One can use Ricci flow in a more general study of three-manifolds than the one we carry out here. There is a conjecture due to -Thurston, see , known as Thurston's +Thurston, see [Thurstongeom], known as Thurston's Geometrization Conjecture or simply as the Geometrization Conjecture for three-manifolds. It conjectures that every $3$-manifold without locally separating $\Ar P^2$'s (in particular every orientable $3$-manifold) is a connected sum of prime $3$-manifolds each of -which admits a decomposition along incompressible\footnote{I.e., -embedded by a map that is injective on $\pi_1$.} tori into pieces -that admit locally homogeneous geometries of finite volume. Modulo +which admits a decomposition along incompressible (I.e., +embedded by a map that is injective on $\pi_1$.) tori into pieces +that admit locally homogeneous geometries of finite volume. Modulo questions about cofinite-volume lattices in $SL_2(\Cee)$, proving this conjecture leads to a complete classification - of $3$-manifolds without locally separating $\Ar P^2$'s, + of $3$-manifolds without locally separating $\Ar P^2$'s, and in particular to a complete classification of all orientable -$3$-manifolds. (See Peter Scott's survey article .) By +$3$-manifolds. (See Peter Scott's survey article [Scott].) By passing to the orientation double cover and working equivariantly, these results can be extended to all $3$-manifolds. Perelman in - has stated results which imply a positive +[Perelman2] has stated results which imply a positive resolution of Thurston's Geometrization conjecture. Perelman's proposed proof of Thurston's Geometrization Conjecture relies in an -essential way on Theorem~, namely the existence of +essential way on \entryref{847af3ecd4a8}, namely the existence of Ricci flow with surgery for all positive time. But it also involves a further analysis of the limits of these Ricci flows as time goes to infinity. This further analysis involves analytic arguments which are exposed in Sections 6 and 7 of Perelman's second paper -(), following earlier work of Hamilton -() in a simpler case of bounded curvature. They -also involve a result (Theorem 7.4 from ) from the +([Perelman2]), following earlier work of Hamilton +([HamiltonNSRF3M]) in a simpler case of bounded curvature. They +also involve a result (Theorem 7.4 from [Perelman2]) from the theory of manifolds with curvature locally bounded below that are -collapsed, related to results of Shioya-Yamaguchi . The +collapsed, related to results of Shioya-Yamaguchi [SY]. The Shioya-Yamaguchi results in turn rely on an earlier, unpublished -work of Perelman proving the so-called `Stability Theorem.' -Recently, Kapovich, has put a preprint on the archive +work of Perelman proving the so-called 'Stability Theorem.' +Recently, Kapovich, [Kap] has put a preprint on the archive giving a proof of the stability result. We have been examining -another approach, one suggested by Perelman in , -avoiding the stability theorem, cf, and -. It is our view that the collapsing results +another approach, one suggested by Perelman in [Perelman2], +avoiding the stability theorem, cf, [KleinerLott2] and +[MorganTian2]. It is our view that the collapsing results needed for the Geometrization Conjecture are in place, but that before a definitive statement that the Geometrization Conjecture has been resolved can be made these arguments must be subjected to the -same close scrutiny that the arguments proving the Poincar\'e +same close scrutiny that the arguments proving the Poincaré Conjecture have received. This process is underway. In this book we do not attempt to explicate any of the results -beyond Theorem~ described in the previous paragraph +beyond \entryref{847af3ecd4a8} described in the previous paragraph that are needed for the Geometrization Conjecture. Rather, we -content ourselves with presenting a proof of Theorem~ +content ourselves with presenting a proof of \entryref{9999efa2be2c} above which, as we have indicated, concerns initial Riemannian manifolds for which the Ricci flow with surgery becomes extinct after finite time. We are currently preparing a detailed proof, @@ -387,7 +403,7 @@ will complete the proof of the Geometrization Conjecture. As should be clear from the above overview, Perelman's argument did not arise in a vacuum. Firstly, it resides in a context provided by -the general theory of Riemannian manifolds. In particular, various +the general theory of Riemannian manifolds. In particular, various notions of convergence of sequences of manifolds play a crucial role. The most important is geometric convergence (smooth convergence on compact subsets). Even more importantly, Perelman's @@ -395,7 +411,7 @@ argument resides in the context of the theory of the Ricci flow equation, introduced by Richard Hamilton and extensively studied by him and others. Perelman makes use of almost every previously established result for $3$-dimensional Ricci flows. One exception is -Hamilton's proposed classification results for three-dimensional +Hamilton's proposed classification results for three-dimensional singularities. These are replaced by Perelman's strong qualitative description of singularity development for Ricci flows on compact three-manifolds. @@ -404,11 +420,11 @@ The first five chapters of the book review the necessary background material from these two subjects. Chapters 6 through 11 then explain Perelman's advances. In Chapter 12 we introduce the standard solution, which is the manifold -constructed by hand that one `glues in' in doing surgery. Chapters +constructed by hand that one 'glues in' in doing surgery. Chapters 13 through 17 describe in great detail the surgery process and prove the main analytic and topological estimates that are needed to show that one can continue the process for all positive time. At the end -of Chapter 17 we have established Theorem~. Chapter 18 +of Chapter 17 we have established \entryref{847af3ecd4a8}. Chapter 18 discusses the finite-time extinction result. Chapter 19 is an appendix on some topological results that were needed in the surgery analysis in Chapters 13-17. @@ -422,22 +438,22 @@ of a manifold being non-collapsed at a point. Suppose that we have a point $x$ in a complete Riemannian $n$-manifold. Then we say that the manifold is *$\kappa$-non-collapsed* at $x$ provided that the following holds: For any $r$ such that the norm of -the Riemannian curvature tensor, $|Rm|$, is $\le r^{-2}$ at +the Riemannian curvature tensor, $|\mathit{Rm}|$, is $\le r^{-2}$ at all points of the metric ball, $B(x,r)$, of radius $r$ centered at -$x$, we have $Vol\, B(x,r)\ge \kappa r^n$. There is a +$x$, we have $\mathit{Vol} B(x,r)\ge \kappa r^n$. There is a relationship between this notion and the injectivity -radius of $M$ at $x$. Namely, if $|Rm|\le r^{-2}$ on $B(x,r)$ and if $B(x,r)$ is +radius of $M$ at $x$. Namely, if $|\mathit{Rm}|\le r^{-2}$ on $B(x,r)$ and if $B(x,r)$ is $\kappa$-non-collapsed then the injectivity radius of $M$ at $x$ is -greater than or equal to a positive constant that depends only on +greater than or equal to a positive constant that depends only on $r$ and $\kappa$. The advantage of working with the volume non-collapsing condition is that, unlike for the injectivity radius, there is a simple equation for the evolution of volume under Ricci flow. Another important general result is the Bishop-Gromov volume -comparison} result that says that if the +comparison result that says that if the Ricci curvature of a complete Riemannian $n$-manifold $M$ is bounded -below by a constant $(n-1)K$ then for any $x\in M$ the ratio of the +below by a constant $(n-1)K$ then for any $x\in M$ the ratio of the volume of $B(x,r)$ to the volume of the ball of radius $r$ in the space of constant curvature $K$ is a non-increasing function whose limit as $r\rightarrow 0$ is $1$. @@ -503,10 +519,10 @@ some point $x$ in the cap, have bounded geometry (bounded diameter, bounded ratio of the curvatures at any two points, and bounded volume). If $C$ represents the bound for these quantities, then we call the cap an $(C,\epsilon)$-cap. See -Fig.~. +Fig. 0.1. An $\epsilon$-tube in $M$ is a submanifold of -$M$ diffeomorphic to $S^2\times (0,1)$ which is a union of +$M$ diffeomorphic to $S^2\times (0,1)$ which is a union of $\epsilon$-necks and with the property that each point of the $\epsilon$-tube is the center of an $\epsilon$-neck in $M$. @@ -515,8 +531,8 @@ $3$-manifolds -- (i) a $C$-component and (ii) an $\epsilon$-round component. The $C$-component is a compact, connected Riemannian manifold of positive sectional curvature diffeomorphic to either $S^3$ or $\Ar -P^3$ with the property that rescaling the metric by $R(x)$ for any -$x$ in the component produces a Riemannian manifold whose diameter +P^3$ with the property that rescaling the metric by $R(x)$ for any +$x$ in the component produces a Riemannian manifold whose diameter is at most $C$, whose sectional curvature at any point and in any $2$-plane direction is between $C^{-1}$ and $C$, and whose volume is between $C^{-1}$ and $C$. An $\epsilon$-round component is a @@ -535,9 +551,9 @@ neighborhood of the boundaries) their product structures almost line up, so that the two $\epsilon$-necks can be glued together to form a manifold fibered by $S^2$'s. Using this idea we show that, for $\epsilon>0$ sufficiently small, if a connected manifold is a union -of $\epsilon$-tubes and $\epsilon$-caps then it is diffeomorphic to -$\Ar^3$, $S^2\times \Ar$, $S^3$, $S^2\times S^1$, $\Ar P^3\#\Ar -P^3$, the total space of a line bundle over $\Ar P^2$, or the +of $\epsilon$-tubes and $\epsilon$-caps then it is diffeomorphic to +$\Ar^3$, $S^2\times \Ar$, $S^3$, $S^2\times S^1$, $\Ar P^3\#\Ar +P^3$, the total space of a line bundle over $\Ar P^2$, or the non-orientable $2$-sphere bundle over $S^1$. This topological result is proved in the appendix at the end of the book. **We shall fix $\epsilon>0$ sufficiently small so that these results hold.** @@ -556,12 +572,16 @@ curvature are recalled in the second chapter. ## Background material from Ricci flow -Hamilton introduced the Ricci flow +Hamilton [Hamilton3MPRC] introduced the Ricci flow equation, -$$\frac{\partial g(t)}{\partial t}=-2Ric(g(t)).$$ + +$$ +\frac{\partial g(t)}{\partial t}=-2\mathit{Ric}(g(t)). +$$ + This is an evolution equation for a one-parameter family of Riemannian metrics $g(t)$ on a smooth manifold $M$. The Ricci flow equation is weakly parabolic -and is strictly parabolic modulo the `gauge group', which is the group of +and is strictly parabolic modulo the 'gauge group', which is the group of diffeomorphisms of the underlying smooth manifold. One should view this equation as a non-linear, tensor version of the heat equation. From it, one can derive the evolution equation for the Riemannian metric tensor, the Ricci @@ -569,9 +589,10 @@ tensor, and the scalar curvature function. These are all parabolic equations. For example, the evolution equation for scalar curvature $R(x,t)$ is $$ - \frac{\partial R}{\partial t}(x,t)=\triangle -R(x,t)+2|Ric(x,t)|^2, +\frac{\partial R}{\partial t}(x,t)=\triangle +R(x,t)+2|\mathit{Ric}(x,t)|^2, $$ + illustrating the similarity with the heat equation. (Here $\triangle$ is the Laplacian with non-positive spectrum.) @@ -580,14 +601,14 @@ Laplacian with non-positive spectrum.) Of course, the first results we need are uniqueness and short-time existence for solutions to the Ricci flow equation for compact manifolds. These results -were proved by Hamilton () using the Nash-Moser inverse +were proved by Hamilton ([Hamilton3MPRC]) using the Nash-Moser inverse function theorem, - (). These results are standard for + ([HamiltonIFTNM]). These results are standard for strictly parabolic equations. By now there is a fairly standard method for -working `modulo' the gauge group (the group of diffeomorphisms) and hence +working 'modulo' the gauge group (the group of diffeomorphisms) and hence arriving at a strictly parabolic situation where the classical existence, uniqueness and smoothness results apply. The method for the Ricci flow equation -goes under the name of `DeTurck's trick.' +goes under the name of 'DeTurck's trick.' There is also a result that allows us to patch together local solutions $(U,g(t)),\ a\le t\le b$, and $(U,h(t)),\ b\le t\le c$, to form a smooth @@ -602,9 +623,18 @@ $h(t)=Qg(Q^{-1}t)$ to produce a new Ricci flow. Suppose that $(M,g)$ is a complete Riemannian manifold, and suppose that there is a constant $\lambda>0$ with the property that -$$Ric(g)=\lambda g.$$ + +$$ +\mathit{Ric}(g)=\lambda g. +$$ + In this case, it is easy to see that there is a Ricci flow given by -$$g(t)=(1-2\lambda t)g.$$ In particular, all the metrics in this flow + +$$ +g(t)=(1-2\lambda t)g. +$$ + + In particular, all the metrics in this flow differ by a constant factor depending on time and the metric is a decreasing function of time. These are called *shrinking solitons*. Examples are compact @@ -612,9 +642,13 @@ manifolds of constant positive Ricci curvature. There is a closely related, but more general, class of examples: the *gradient shrinking solitons*. Suppose that $(M,g)$ is - a complete Riemannian manifold, and suppose that there is a constant + a complete Riemannian manifold, and suppose that there is a constant $\lambda>0$ and a function $f\colon M\to \Ar$ satisfying -$$Ric(g)=\lambda g-Hess^{g}f.$$ + +$$ +\mathit{Ric}(g)=\lambda g-\mathit{Hess}^{g}f. +$$ + In this case, there is a Ricci flow which is a shrinking family after we pull back by the one-parameter family of diffeomorphisms generated by the time-dependent vector field $\frac{1}{1-2\lambda t}\nabla_{g} f$. @@ -627,7 +661,7 @@ $$Ric(g)=\lambda g-Hess^{g}f.$$ ### Controlling higher derivatives of curvature Now let us discuss the smoothness results for geometric limits. The -general result along these lines is Shi's theorem}, see . Again, this is a standard type of +general result along these lines is Shi's theorem, see [Shi1, Shi2]. Again, this is a standard type of result for parabolic equations. Of course, the situation here is complicated somewhat by the existence of the gauge group. Roughly, Shi's theorem says the following. Let us denote by $B(x,t_0,r)$ the @@ -656,34 +690,36 @@ context more general than that of Ricci flow. We choose to do this in the context of generalized Ricci flows. A generalized three-dimensional Ricci flow consists of a smooth -four-dimensional manifold ${\mathcal M}$ (space-time) with a time function $**t**\colon {\mathcal M}\to \Ar$ +four-dimensional manifold ${\mathcal M}$ (space-time) with a time function $\mathbf{t}\colon {\mathcal M}\to \Ar$ and a smooth vector field $\chi$. These are required to satisfy: -\begin{enumerate} -\item Each $x\in {\mathcal M}$ has a neighborhood of the form + +1. Each $x\in {\mathcal M}$ has a neighborhood of the form $U\times J$, where $U$ is an open subset in $\Ar^3$ and $J\subset -\Ar$ is an interval, in which $**t**$ is the projection onto $J$ +\Ar$ is an interval, in which $\mathbf{t}$ is the projection onto $J$ and $\chi$ is the unit vector field tangent to the one-dimensional foliation $\{u\}\times J$ pointing in the direction of increasing -$**t**$. We call $**t**^{-1}(t)$ the $t$ time-slice. It is a +$\mathbf{t}$. We call $\mathbf{t}^{-1}(t)$ the $t$ time-slice. It is a smooth $3$-manifold. -\item The image $**t**({\mathcal M})$ is a connected interval $I$ in +1. The image $\mathbf{t}({\mathcal M})$ is a connected interval $I$ in $\Ar$, possibly infinite. The boundary of ${\mathcal M}$ is the -pre-image under $**t**$ of the boundary of $I$. -\item The level sets $**t**^{-1}(t)$ form a codimension-one foliation of +pre-image under $\mathbf{t}$ of the boundary of $I$. +1. The level sets $\mathbf{t}^{-1}(t)$ form a codimension-one foliation of ${\mathcal M}$, called the horizontal foliation, with the boundary components of ${\mathcal M}$ being leaves. -\item There is a metric $G$ on the horizontal distribution, i.e., the distribution -tangent to the level sets of $**t**$. This metric induces a +1. There is a metric $G$ on the horizontal distribution, i.e., the distribution +tangent to the level sets of $\mathbf{t}$. This metric induces a Riemannian metric on each $t$ time-slice varying smoothly as we vary the time-slice. We define the curvature of $G$ at a point $x\in{\mathcal M}$ to be the curvature of the Riemannian metric induced by $G$ on the time-slice $M_t$ at $x$. -\item Because of the first property the integral curves of $\chi$ +1. Because of the first property the integral curves of $\chi$ preserve the horizontal foliation and hence the horizontal distribution. Thus, we can take the Lie derivative of $G$ along $\chi$. The Ricci flow equation is then -$${\mathcal L}_\chi(G)=-2Ric(G).$$ -\end{enumerate} + +$$ +{\mathcal L}_\chi(G)=-2\mathit{Ric}(G). +$$ Locally in space-time the horizontal metric is simply a smoothly varying family of Riemannian metrics on a fixed smooth manifold and the evolution equation is @@ -714,53 +750,61 @@ generalized Ricci flow. The Ricci flow equation satisfies various forms of the maximum principle. The fourth chapter explains this principle, which is due -to Hamilton (see Section 4 of ), and derives -many of its consequences, which are also due to Hamilton (cf. -). This principle and its consequences are at +to Hamilton (see Section 4 of [Hamiltonsurvey]), and derives +many of its consequences, which are also due to Hamilton (cf. +[HamiltonNSRF3M]). This principle and its consequences are at the core of all the detailed results about the nature of the flow. We illustrate the idea by considering the case of the scalar curvature. A standard scalar maximum principle argument applied to -Equation~() proves that the minimum of the scalar +Equation (0.1) proves that the minimum of the scalar curvature is a non-decreasing function of time. In addition, it shows that if the minimum of scalar curvature at time $0$ is positive then we have -$$R_min(t)\ge R_min(0)\left(\frac{1}{1-\frac{2t}{n}R_min(0)}\right),$$ and thus the equation must develop a singularity at or -before time $n/\left(2R_min(0)\right)$. + +$$ +R_\mathit{min}(t)\ge R_\mathit{min}(0)\left(\frac{1}{1-\frac{2t}{n}R_\mathit{min}(0)}\right), +$$ + + and thus the equation must develop a singularity at or +before time $n/\left(2R_\mathit{min}(0)\right)$. While the above result about the scalar curvature is important and is used repeatedly, the most significant uses of the maximum principle involve the tensor version, established by Hamilton, which applies for example to the Ricci tensor and the full curvature -tensor. These have given the most significant understanding of the +tensor. These have given the most significant understanding of the Ricci flows, and they form the core of the arguments that Perelman uses in his application of Ricci flow to $3$-dimensional topology. Here are the main results established by Hamilton: -\begin{enumerate} -\item[(1)] For $3$-dimensional flows, if the Ricci curvature is positive, +- **(1)** For $3$-dimensional flows, if the Ricci curvature is positive, then the family of metrics becomes singular at finite time and as the family becomes singular, the metric becomes closer and closer to -round; see . -\item[(2)] For $3$-dimensional flows, as the scalar curvature goes to $+\infty$ +round; see [Hamilton3MPRC]. +- **(2)** For $3$-dimensional flows, as the scalar curvature goes to $+\infty$ the ratio of the absolute value of any negative eigenvalue of the Riemannian curvature to the largest positive eigenvalue goes to -zero; see . This condition is called *pinched toward positive curvature*. -\item[(3)] Motivated by a Harnack inequality for the heat equation established -by Li-Yau , Hamilton established a Harnack +zero; see [HamiltonNSRF3M]. This condition is called *pinched toward positive curvature*. +- **(3)** Motivated by a Harnack inequality for the heat equation established +by Li-Yau [LiYau], Hamilton established a Harnack inequality for the curvature tensor under the Ricci flow for complete manifolds -$(M,g(t))$ with bounded, non-negative curvature operator; see -. In the applications to three dimensions, we +$(M,g(t))$ with bounded, non-negative curvature operator; see +[Hamiltonharnack]. In the applications to three dimensions, we shall need the following consequence for the scalar curvature: -Suppose that $(M,g(t))$ is a Ricci flow defined for all $t\in +Suppose that $(M,g(t))$ is a Ricci flow defined for all $t\in [T_0,T_1]$ of complete manifolds of non-negative curvature operator with bounded curvature. Then -$$\frac{\partial R}{\partial t}(x,t)+\frac{R(x,t)}{t-T_0}\ge 0.$$ + +$$ +\frac{\partial R}{\partial t}(x,t)+\frac{R(x,t)}{t-T_0}\ge 0. +$$ + In particular, if $(M,g(t))$ is an ancient solution (i.e., defined for all $t\le 0$) of bounded, non-negative curvature then $\partial R(x,t)/\partial t\ge 0$. -\item[(4)] If a complete $3$-dimensional Ricci flow +- **(4)** If a complete $3$-dimensional Ricci flow $(M,g(t)),\ 0\le t\le T$, has non-negative curvature, if $g(0)$ is not flat, and if there is at least one point $(x,T)$ such that the Riemannian curvature tensor of $g(T)$ has a flat direction in @@ -770,12 +814,11 @@ Riemannian product of a surface of positive curvature and a Euclidean line. Furthermore, the flow on the cover $\widetilde M$ is the product of a $2$-dimensional flow and the trivial one-dimensional Ricci flow on the line; see Sections 8 and 9 of -. -\item[(5)] In particular, there is no Ricci flow of non-negative curvature tensor +[Hamilton4MPCO]. +- **(5)** In particular, there is no Ricci flow of non-negative curvature tensor $(U,g(t)),$ defined for $\ 0\le t\le T$ with $T>0$, such that $(U,g(T))$ is isometric to an open subset in a non-flat, $3$-dimensional metric cone. -\end{enumerate} ### Geometric limits @@ -783,24 +826,24 @@ In the fifth chapter we discuss geometric limits of Riemannian manifolds and of Ricci flows. Let us review the history of these ideas. The first results about geometric limits of Riemannian manifolds go back to Cheeger in his thesis in 1967; see -. Here Cheeger obtained topological results. In - Gromov proposed that geometric limits should exist in +[Cheeger]. Here Cheeger obtained topological results. In +[Gromov] Gromov proposed that geometric limits should exist in the Lipschitz topology and suggested a result along these lines, which also was known to Cheeger. In -, Greene-Wu gave a rigorous proof of the compactness +[GreeneWu], Greene-Wu gave a rigorous proof of the compactness theorem suggested by Gromov and also enhanced the convergence to be $C^{1,\alpha}$-convergence by using harmonic coordinates; see also -. Assuming that all the derivatives of curvature are +[SPeters]. Assuming that all the derivatives of curvature are bounded, one can apply elliptic theory to the expression of curvature in harmonic coordinates and deduce $C^\infty$-convergence. -These ideas lead to various types of compactness results that go +These ideas lead to various types of compactness results that go under the name Cheeger-Gromov compactness for Riemannian manifolds. -Hamilton in extended these results to Ricci +Hamilton in [Hamiltonlimits] extended these results to Ricci flows. We shall use the compactness results for both Riemannian manifolds and for Ricci flows. In a different direction, geometric limits were extended to the non-smooth context by Gromov in - where he introduced a weaker topology, called the +[Gromov] where he introduced a weaker topology, called the Gromov-Hausdorff topology and proved a compactness theorem. Recall that a @@ -819,23 +862,25 @@ type of the manifolds in the sequence. There is a similar notion of geometric convergence for a sequence of based Ricci flows. Certainly, one of the most important consequences of Shi's results, -cited above, is that, in concert with Cheeger-Gromov compactness, +cited above, is that, in concert with Cheeger-Gromov compactness, it allows us to form smooth geometric limits of sequences of based Ricci flows. We have the following result of Hamilton's; see -: +[Hamiltonlimits]: + **Theorem.** Suppose we have a sequence of based Ricci flows $(M_n,g_n(t),(x_n,0))$ defined for $t\in (-T,0]$ with the $(M_n,g_n(t))$ being complete. Suppose that: -\begin{enumerate} -\item[(1)] There is $r>0$ and $\kappa>0$ such that for every $n$ the metric ball $B(x_n,0,r)\subset + +- **(1)** There is $r>0$ and $\kappa>0$ such that for every $n$ the metric ball $B(x_n,0,r)\subset (M_n,g_n(0))$ is $\kappa$-non-collapsed. -\item[(2)] For each $A<\infty$ there is $C=C(A)<\infty$ such that the Riemannian +- **(2)** For each $A<\infty$ there is $C=C(A)<\infty$ such that the Riemannian curvature on $B(x_n,0,A)\times (-T,0]$ is bounded by $C$. -\end{enumerate} + Then after passing to a subsequence there is a geometric limit which is a based Ricci flow $(M_\infty,g_\infty(t),(x_\infty,0))$ defined for $t\in (-T,0]$. + To emphasize, the two conditions that we must check in order to extract a geometric limit of a subsequence based at points at time @@ -859,7 +904,7 @@ Ricci flows. So far we have been discussing the results that were known before Perelman's work. They concern almost exclusively Ricci flow (though -Hamilton in had introduced the notion of +Hamilton in [Hamilton4MPIC] had introduced the notion of surgery and proved that surgery can be performed preserving the condition that the curvature is pinched toward positive, as in (2) above). Perelman extended in two essential ways the analysis of @@ -886,7 +931,7 @@ the relevant contexts from this one source. ### The reduced length function -In Chapter~ we come to +In Chapter 6 we come to the first of Perelman's major contributions. Let us first describe it in the context of an ordinary three-dimensional Ricci flow, but viewing the Ricci flow as a horizontal metric on a space-time which @@ -896,27 +941,40 @@ of the flow. Suppose that $I=[0,T)$ and fix $(x,t)\in M\times \overline\tau$, in space-time with the property that for every $\tau\le \overline\tau$ we have $\gamma(\tau)\in M\times \{t-\tau\}$ and $\gamma(0)=x$. These paths are said to be *parameterized by -backward time*. See Fig.~. The ${\mathcal +backward time*. See Fig. 0.2. The ${\mathcal L}$-*length* of such a path is given by -$${\mathcal L}(\gamma)=\int_0^{\overline -\tau}\sqrt{\tau}\left(R(\gamma(\tau))+|\gamma'(\tau)|^2\right)d\tau,$$ + +$$ +{\mathcal L}(\gamma)=\int_0^{\overline +\tau}\sqrt{\tau}\left(R(\gamma(\tau))+|\gamma'(\tau)|^2\right)d\tau, +$$ + where the derivative on $\gamma$ refers to the spatial derivative. There is also the closely related *reduced length* -$$\ell(\gamma)=\frac{{\mathcal L}(\gamma)}{2\sqrt{\overline\tau}}.$$ + +$$ +\ell(\gamma)=\frac{{\mathcal L}(\gamma)}{2\sqrt{\overline\tau}}. +$$ + There is a theory for the functional ${\mathcal L}$ analogous to the -theory for the usual energy function\footnote{Even though this +theory for the usual energy function (Even though this functional is called a length, the presence of the $|\gamma'(\tau)|^2$ in the integrand means that it behaves more like -the usual energy functional for paths in a Riemannian manifold.}. +the usual energy functional for paths in a Riemannian manifold.). In particular, there is the notion of an ${\mathcal -L}$-geodesic$-geodesic}, and the reduced length +L}$-geodesic, and the reduced length as a function on space-time $\ell_{(x,t)}\colon M\times [0,t)\to \Ar$. One establishes a crucial monotonicity for this reduced length along ${\mathcal L}$-geodesics. Then one defines the reduced volume -$$\widetilde + +$$ +\widetilde V_{(x,t)}(U\times\{\bar t\})=\int_{U\times\{\bar -t\}}\bar\tau^{-3/2}e^{-\ell_{(x,t)}(q,\bar\tau)}dvol_{g(\bar\tau}(q),$$ where, as before $\bar\tau=t-\bar t$. +t\}}\bar\tau^{-3/2}e^{-\ell_{(x,t)}(q,\bar\tau)}d\mathit{vol}_{g(\bar\tau}(q), +$$ + + where, as before $\bar\tau=t-\bar t$. Because of the monotonicity of $\ell_{(x,t)}$ along ${\mathcal L}$-geodesics, the reduced volume is also non-increasing under the flow (forward in $\bar\tau$ and hence @@ -929,7 +987,7 @@ The definitions and the analysis of the reduced length function and the reduced volume as well as the monotonicity results are valid in the context of the generalized Ricci flow. The only twist to be aware of is that in the more general context one cannot always -extend ${\mathcal L}$-geodesics; they may run `off the edge' of +extend ${\mathcal L}$-geodesics; they may run 'off the edge' of space-time. Thus, the reduced length function and reduced volume cannot be defined globally, but only on appropriate open subsets of a time-slice (those reachable by minimizing ${\mathcal @@ -947,13 +1005,13 @@ definition of the surgery process. As we indicated in the previous paragraph, one of the main applications of the reduced length function is to prove non-collapsing results for three-dimensional Ricci flows with -surgery. In order to make this argument work, one takes a weaker +surgery. In order to make this argument work, one takes a weaker notion of $\kappa$-non-collapsed by making a stronger curvature bound assumption: one considers - points $(x,t)$ and constants $r$ with the property that $|Rm|\le r^{-2}$ on + points $(x,t)$ and constants $r$ with the property that $|\mathit{Rm}|\le r^{-2}$ on $P(x,t,r,-r^2)=B(x,t,r)\times (t-r^2,t]$. The $\kappa$-non-collapsing condition applies to these balls and says -that $Vol(B(x,t,r))\ge \kappa r^3.$ The basic idea in proving +that $\mathit{Vol}(B(x,t,r))\ge \kappa r^3.$ The basic idea in proving non-collapsing is to use the fact that as we flow forward in time via minimizing ${\mathcal L}$-geodesics the reduced volume is a non-decreasing function. Hence, a lower bound of the reduced volume @@ -972,7 +1030,7 @@ bounded away from zero. One case where it is easy to do this is when we have a Ricci flow of compact manifolds or of complete manifolds of non-negative curvature. Hence, these manifolds are non-collapsed at all points -with a non-collapsing constant that depends only on the geometry of +with a non-collapsing constant that depends only on the geometry of the initial metric of the Ricci flow. Non-collapsing results are crucial and are used repeatedly in dealing with Ricci flows with surgery in Chapters 10 -- 17, for these give one of the two @@ -1009,8 +1067,8 @@ is a subsequence of based Riemannian manifolds, $(M,\frac{1}{|t_n|}g(t_n),y_n)$, with a geometric limit, and this limit is a gradient shrinking soliton. This gradient shrinking soliton is called an *asymptotic -soliton*} for the -original $\kappa$-solution, see Fig.~. +soliton* for the +original $\kappa$-solution, see Fig. 0.3. The point is that there are only two types of gradient shrinking solitons in dimension three -- (i) those finitely covered by a @@ -1029,15 +1087,17 @@ union of cores of $(C_1,\epsilon)$-caps and points that are the center points of $\epsilon$-necks. In order to prove the above results (for example the uniformity of $C_1$ as -above over all $\kappa$-solutions) one needs the following result: +above over all $\kappa$-solutions) one needs the following result: -**Theorem.** The space of based $\kappa$-solutions, based at points $(x,0)$ with + +**Theorem.** The space of based $\kappa$-solutions, based at points $(x,0)$ with $R(x,0)=1$, is compact. + This result does not generalize to ancient solutions that are not non-collapsed because, in order to prove compactness, one has to take limits of subsequences, and in doing this the non-collapsing -hypothesis is essential. See Hamilton's work +hypothesis is essential. See Hamilton's work [Hamiltonsurvey] for more on general ancient solutions (i.e., those that are not necessarily non-collapsed). @@ -1058,7 +1118,7 @@ so that the scalar curvature at a point is one, or admits an $\epsilon$-tube whose complement is either a disjoint union of the cores of two $(C_1,\epsilon)$-caps. -This gives a rough qualitative understanding of $\kappa$-solutions. Either +This gives a rough qualitative understanding of $\kappa$-solutions. Either they are round, or they are finitely covered by the product of a round surface and a line, or they are a union of $\epsilon$-tubes and cores of $(C_1,\epsilon)$-caps , or they are diffeomorphic to $S^3$ or $\Ar P^3$ and @@ -1081,16 +1141,16 @@ bounded curvature at bounded distance for blow-up limits of generalized Ricci flows. As we have alluded to several times, many steps in the argument require taking (smooth) geometric limits of a sequence of based generalized flows about points of curvature -tending to infinity. To study such a sequence we rescale each term +tending to infinity. To study such a sequence we rescale each term in the sequence so that its curvature at the base point becomes one. Nevertheless, in taking such limits we face the problem that even though the curvature at the point we are focusing on (the points we -take as base points) was originally large and has been rescaled to +take as base points) was originally large and has been rescaled to be one, there may be other points in the same time-slice of much larger curvature, which, even after the rescalings, can tend to infinity. If -these points are at uniformly bounded (rescaled) distance from the +these points are at uniformly bounded (rescaled) distance from the base points, then they would preclude the existence of a smooth geometric limit of the based, rescaled flows. In his arguments, Hamilton avoided this problem by always focusing on points of @@ -1114,10 +1174,10 @@ distance with unbounded, rescaled curvature means that there is a point at infinity at finite distance from the base point where the curvature blows up. A neighborhood of this point at infinity is cone-like in a manifold of non-negative curvature. This contradicts -Hamilton's maximum principle result (5) in Chapter~) +Hamilton's maximum principle result (5) in Chapter 0.3.5) that the result of a Ricci flow of manifolds of non-negative curvature is never an open subset of a cone. (We know that any -`blow-up limit' like this has non-negative curvature because of the +'blow-up limit' like this has non-negative curvature because of the curvature pinching result.) This contradiction establishes the result. @@ -1129,14 +1189,14 @@ Now we are ready to discuss three-dimensional Ricci flows with surgery. In preparing the way for defining the surgery process, we must construct a metric on the $3$-ball that we shall glue in when we -perform surgery. This we do in Chapter~. We fix a +perform surgery. This we do in Chapter 12. We fix a non-negatively curved, rotationally symmetric metric on $\Ar^3$ that is isometric near infinity to $S^2\times [0,\infty)$ where the metric on $S^2$ is the round metric of scalar curvature $1$, and -outside this region has positive sectional curvature, see Fig.~. Any such metric will suffice for the gluing +outside this region has positive sectional curvature, see Fig. 0.4. Any such metric will suffice for the gluing process, and we fix one and call it the *standard -metric*}. It is important to understand Ricci flow -with the standard metric as initial metric. Because of the special +metric*. It is important to understand Ricci flow +with the standard metric as initial metric. Because of the special nature of this metric (the rotational symmetry and the asymptotic nature at infinity), it is fairly elementary to show that there is a unique solution of bounded curvature on each time-slice to the Ricci @@ -1148,16 +1208,15 @@ shows that the Ricci flow is non-collapsed, and that the bounded curvature and bounded distance result applies to it. This allows one to prove that every point $(x,t)$ in this flow has one of the following types of neighborhoods: -\begin{enumerate} -\item $(x,t)$ is contained in the core of a $(C_2,\epsilon)$-cap, + +1. $(x,t)$ is contained in the core of a $(C_2,\epsilon)$-cap, where $C_2<\infty$ is a given universal constant depending only on $\epsilon$. -\item $(x,t)$ is the center of a strong $\epsilon$-neck. -\item $(x,t)$ is the center of an evolving $\epsilon$-neck whose +1. $(x,t)$ is the center of a strong $\epsilon$-neck. +1. $(x,t)$ is the center of an evolving $\epsilon$-neck whose initial slice is at time zero. -\end{enumerate} These form the second source of models for canonical neighborhoods in a Ricci -flow with surgery. Thus, we shall set $C=C(\epsilon)=max(C_1(\epsilon),C_2(\epsilon))$ and we shall find $(C,\epsilon)$-canonical +flow with surgery. Thus, we shall set $C=C(\epsilon)=\mathit{max}(C_1(\epsilon),C_2(\epsilon))$ and we shall find $(C,\epsilon)$-canonical neighborhoods in Ricci flows with surgery. ### Ricci flows with surgery @@ -1166,28 +1225,28 @@ Now it is time to introduce the notion of a Ricci flow with surgery. To do this we formulate an appropriate notion of $4$-dimensional space-time that allows for the surgery operations. We define *space-time* to be a -$4$-dimensional Hausdorff singular space with a time function $**t**$ with the property that each time-slice is a compact, smooth +$4$-dimensional Hausdorff singular space with a time function $\mathbf{t}$ with the property that each time-slice is a compact, smooth $3$-manifold, but level sets at different times are not necessarily diffeomorphic. Generically space-time is a smooth $4$-manifold, but there are *exposed regions* at a discrete -set of times. Near a point in the exposed region space-time is a +set of times. Near a point in the exposed region space-time is a $4$-manifold with boundary. The singular points of space-time are the boundaries of the exposed regions. Near these, space-time is modeled on the product of $\Ar^2$ with the square $(-1,1)\times (-1,1)$, the latter having a topology in which the open sets are, in addition to the usual open sets, open subsets of $(0,1)\times -[0,1)$, see Fig.~. There is a natural +[0,1)$, see Fig. 0.5. There is a natural notion of smooth functions on space-time. These are smooth in the usual sense on the open subset of non-singular points. Near the singular points, and in the local coordinates described above, they are required to be pull-backs from smooth functions on $\Ar^2\times (-1,1)\times (-1,1)$ under the natural map. Space-time is equipped with a smooth vector field $\chi$ -with $\chi(**t**)=1$. +with $\chi(\mathbf{t})=1$. A Ricci flow with surgery is a smooth horizontal metric $G$ on a -space-time with the property that the restriction of $G$, $**t**$ +space-time with the property that the restriction of $G$, $\mathbf{t}$ and $\chi$ to the open subset of smooth points forms a generalized -Ricci flow. We call this the *associated generalized Ricci +Ricci flow. We call this the *associated generalized Ricci flow* for the Ricci flow with surgery. ### The inductive conditions necessary for doing surgery @@ -1200,38 +1259,36 @@ to keep track of various properties as we go along to ensure that we can continue to do surgery. Here we discuss the conditions we verify at each step. -Fix $\epsilon>0$ sufficiently small and let $C=max(C_1,C_2)<\infty$, where $C_1$ is the constant associated to +Fix $\epsilon>0$ sufficiently small and let $C=\mathit{max}(C_1,C_2)<\infty$, where $C_1$ is the constant associated to $\epsilon$ for $\kappa$-solutions and $C_2$ is the constant associated to $\epsilon$ for the standard solution. We say that a point $x$ in a generalized Ricci flow has a $(C,\epsilon)$-canonical neighborhood if one of the following holds: -\begin{enumerate} -\item $x$ is contained in a connected component of a + +1. $x$ is contained in a connected component of a time-slice that is a $C$-component. -\item $x$ is contained in a connected component of its time-slice +1. $x$ is contained in a connected component of its time-slice that is within $\epsilon$ of round in the $C^{[1/\epsilon]}$-topology. -\item $x$ is contained in the core of a $(C,\epsilon)$-cap. -\item $x$ is the center of a strong $\epsilon$-neck. -\end{enumerate} +1. $x$ is contained in the core of a $(C,\epsilon)$-cap. +1. $x$ is the center of a strong $\epsilon$-neck. -We shall study Ricci flows with surgery defined for $0\le +We shall study Ricci flows with surgery defined for $0\le t0$ so that the associated generalized Ricci flow is +1. The curvature of the flow is pinched toward positive. +1. There is $\kappa>0$ so that the associated generalized Ricci flow is $\kappa$-non-collapsed on scales at most $\epsilon$, in the sense that we require only that balls of radius $r\le \epsilon$ be $\kappa$-non-collapsed. -\item There is $r_0>0$ such that any point of space-time at which +1. There is $r_0>0$ such that any point of space-time at which the scalar curvature is $\ge r_0^{-2}$ has an $(C,\epsilon)$-canonical neighborhood. -\end{enumerate} The main result is that, having a Ricci flow with surgery defined on some time interval satisfying these conditions, it is possible to @@ -1250,7 +1307,7 @@ there are infinitely many surgeries. Let us describe how we extend a Ricci flow with surgery satisfying all the conditions listed above and becoming singular at time -$T<\infty$. Fix $T^-T_0$, becomes negative at some finite $T_1$ (depending on the +$t>T_0$, becomes negative at some finite $T_1$ (depending on the initial value). This is absurd since $W_2(t)$ is the minimum of positive quantities. This contradiction shows that such a path of components with non-trivial $\pi_2$ cannot exist for all $t\ge T_0$. @@ -1490,9 +1551,9 @@ $t\ge T$. Three remarks are in order. This argument showing that eventually every component of the time-slice $t$ has trivial $\pi_2$ is not -necessary for the topological application (Theorem~), +necessary for the topological application (\entryref{127db1dc9315}), or indeed, for any other topological application. The reason is the -sphere theorem (see ), which says that if $\pi_2(M)$ is +sphere theorem (see [Hempel]), which says that if $\pi_2(M)$ is non-trivial then either $M$ is diffeomorphic to an $S^2$ bundle over $S^1$ or $M$ has a non-trivial connected sum decomposition. Thus, we can establish results for all $3$-manifolds if we can establish them @@ -1506,7 +1567,7 @@ the sphere theorem so that establishing the cutting into pieces with trivial $\pi_2$ allows us to give a different proof of this result (though admittedly one using much deeper ideas). -Let us now fix $T<\infty$ such that for all $t\ge T$ all the +Let us now fix $T<\infty$ such that for all $t\ge T$ all the time-slices $M_t$ have trivial $\pi_2$. There is a simple topological consequence of this and our assumption on the initial manifold. If $M$ is a compact $3$-manifold whose fundamental group @@ -1515,7 +1576,7 @@ then $M$ admits a homotopically non-trivial embedded $2$-sphere. Since we began with a manifold $M_0$ whose fundamental group is a free product of finite groups and infinite cyclic groups, it follows that for $t\ge T$ every component of $M_t$ has finite fundamental -group. Fix $t\ge T$. Then each component of $M_t$ has a finite +group. Fix $t\ge T$. Then each component of $M_t$ has a finite cover that is simply connected, and thus, by an elementary argument in algebraic topology, each component of $M_t$ has non-trivial $\pi_3$. The second step in the finite-time extinction argument is @@ -1525,17 +1586,17 @@ the manifolds have trivial $\pi_2$. There are two approaches to this second step: the first is due to Perelman in - and the other due to Colding-Minicozzi in -. In their approach Colding-Minicozzi +[Perelman3] and the other due to Colding-Minicozzi in +[ColdingMinicozzi]. In their approach Colding-Minicozzi associate to a non-trivial element in $\pi_3(M)$ a non-trivial -element in $\pi_1(Maps(S^2,M))$. This element is represented -by a one-parameter family of $2$-spheres (starting and ending at +element in $\pi_1(\mathit{Maps}(S^2,M))$. This element is represented +by a one-parameter family of $2$-spheres (starting and ending at the constant map) representing a non-trivial element $\xi\in \pi_3(M_0)$. They define the *width* of this homotopy class by $W(\xi,t)$ by associating to each representative the maximal energy of the $2$-spheres in the family and then minimizing over all representatives of the homotopy class. Using results of Jost -, they show that this function satisfies the same forward +[Jost], they show that this function satisfies the same forward difference inequality that $W_2$ satisfies (and has the same continuity property under Ricci flow and the same semi-continuity under surgery). Since $W(\xi,t)$ is always $\ge 0$ if it is defined, @@ -1544,8 +1605,8 @@ manifolds $M_t$ must eventually become empty. While this approach seemed completely natural to us, and while we believe that it works, we found the technical details -daunting\footnote{Colding and Minicozzi tell us they plan to give an -expanded version of their argument with a more detailed proof.} +daunting (Colding and Minicozzi tell us they plan to give an +expanded version of their argument with a more detailed proof.) (because one is forced to consider index-one critical points of the energy functional rather than minima). For this reason we chose to follow Perelman's approach. He represents a non-trivial element in @@ -1560,7 +1621,11 @@ $W(\Gamma)$. As before, this function is continuous under Ricci flow and is lower semi-continuous under surgery (unless the surgery removes the component in question). It also satisfies a forward difference quotient -$$\frac{dW(\xi)}{dt}\le -2\pi+\frac{3}{4t+1}W(\xi).$$ + +$$ +\frac{dW(\xi)}{dt}\le -2\pi+\frac{3}{4t+1}W(\xi). +$$ + The reason for the term $-2\pi$ instead of $-4\pi$ which occurs in the other cases is that we are working with minimal $2$-disks instead of minimal $2$-spheres. Once this forward difference @@ -1570,7 +1635,7 @@ with the properties we have just established cannot be non-negative for all positive time. This means the component in question, and indeed all components at later time derived from it, must disappear in finite time. Hence, under the hypothesis on the fundamental group -in Theorem~ the entire manifold must disappear at +in \entryref{127db1dc9315} the entire manifold must disappear at finite time. Because this approach uses only minima for the energy or area @@ -1579,10 +1644,10 @@ points. But one is forced to face other difficulties though -- namely boundary issues. Here, one must prescribe the deformation of the family of boundary curves before computing the forward difference quotient of the energy. The obvious choice is the -curve-shrinking flow (see ). Unfortunately, this flow can +curve-shrinking flow (see [AG]). Unfortunately, this flow can only be defined when the curve in question is immersed and even in this case the curve-shrinking flow can develop singularities even if -the Ricci flow does not. Following Perelman, or indeed , +the Ricci flow does not. Following Perelman, or indeed [AG], one uses the device of taking the product with a small circle and using loops, called *ramps*, that go around that circle once. In this context the curve-shrinking flow remains @@ -1596,7 +1661,7 @@ This compactness result holds off of a set of time of small total measure, which is sufficient for the argument. At the very end of the argument we need an elementary but complicated result on annuli, which we could not find in the literature. For more details on these -points see Chapter~. +points see Chapter 18. ## Acknowledgements @@ -1610,16 +1675,16 @@ Hamilton has also given unstintingly of his time, explaining to us his results and his ideas about Perelman's results. We have benefitted tremendously from the work of Bruce Kleiner and John Lott. They produced a lengthy set of notes filling in the details in -Perelman's arguments . We have referred to those +Perelman's arguments [KleinerLott]. We have referred to those notes countless times as we came to grips with Perelman's ideas. In late August and early September of 2004, Kleiner, Lott and the two -of us participated in a workshop at Princeton University, supported +of us participated in a workshop at Princeton University, supported by the Clay Math Institute, going through Perelman's second paper (the one on Ricci flow with surgery) in detail. This workshop played a significant role in convincing us that Perelman's arguments were complete and correct and also in convincing us to write this book. We thank all the participants of this workshop and especially -Guo-Feng Wei, Peng Lu, Yu Ding, and X.-C. Rong who, together with +Guo-Feng Wei, Peng Lu, Yu Ding, and X.-C. Rong who, together with Kleiner and Lott, made significant contributions to the workshop. Before, during, and after this workshop, we have benefitted from private conversations too numerous to count with Bruce Kleiner and @@ -1646,7 +1711,7 @@ especially about annuli of small area. The second author gave courses at Princeton University and ran seminars on this material at MIT and Princeton. Natasa Sesum and -Xiao-Dong Wang, see , wrote notes for the seminars at MIT, +Xiao-Dong Wang, see [STW], wrote notes for the seminars at MIT, and Edward Fan and Alex Subotic took notes for the seminars and courses at Princeton, and they produced preliminary manuscripts. We have borrowed freely from these manuscripts, and it is a pleasure to @@ -1676,22 +1741,22 @@ for his support during the period that he was a faculty member at MIT. For the readers' convenience we gather here references to all the closely related articles. -First and foremost are Perelman's three preprints, , -, and . The first of these +First and foremost are Perelman's three preprints, [Perelman1], +[Perelman2], and [Perelman3]. The first of these introduces the main techniques in the case of Ricci flow, the second discusses the extension of these techniques to Ricci flow with -surgery, and the last gives the short-cut to the Poincar\'e +surgery, and the last gives the short-cut to the Poincaré Conjecture and the $3$-dimensional spherical space-form conjecture, avoiding the study of the limits as time goes to infinity and collapsing space arguments. There are the detailed notes by Bruce -Kleiner and John Lott, , which greatly expand and +Kleiner and John Lott, [KleinerLott], which greatly expand and clarify Perelman's arguments from the first two preprints. There is -also a note on Perelman's second paper by Yu Ding . -There is the article by Colding-Minicozzi , +also a note on Perelman's second paper by Yu Ding [yuding]. +There is the article by Colding-Minicozzi [ColdingMinicozzi], which gives their alternate approach to the material in Perelman's third preprint. Collapsing space arguments needed for the full geometrization conjecture are discussed in Shioya-Yamaguchi -. Lastly, after we had submitted a preliminary version of +[SY]. Lastly, after we had submitted a preliminary version of this manuscript for refereeing, H.-D. Cao and X.-P. Zhu published an -article on the Poincar\'e Conjecture and Thurston's Geometrization -Conjecture; see . +article on the Poincaré Conjecture and Thurston's Geometrization +Conjecture; see [CaoZhu]. diff --git a/projects/poincare-conjecture/.astrolabe/docs-src/02-preliminaries-from-riemannian-geometry.mdx b/projects/poincare-conjecture/.astrolabe/docs-src/02-preliminaries-from-riemannian-geometry.mdx new file mode 100644 index 00000000..5543f25c --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/docs-src/02-preliminaries-from-riemannian-geometry.mdx @@ -0,0 +1,1285 @@ + + +# Chapter 1 -- Preliminaries from Riemannian geometry + +In this chapter we will recall some basic facts in Riemannian +geometry. For more details we refer the reader to [doCarmo] and +[Petersen]. Throughout, we always adopt Einstein's summation +convention on repeated indices and 'manifold' means a paracompact, +Hausdorff, smooth manifold. + +## Riemannian metrics and the Levi-Civita connection + +Let $M$ be a manifold and let $p$ be a point of $M$. Then $TM$ +denotes the tangent bundle of $M$ and $T_pM$ is the tangent space at +$p$. Similarly, $T^*M$ denotes the cotangent bundle of $M$ and +$T^*_pM$ is the cotangent space at $p$. For any vector bundle +${\mathcal V}$ over $M$ we denote by $\Gamma({\mathcal V})$ the +vector space of smooth sections of ${\mathcal V}$. + + +**Definition.** Let $M$ be an $n$-dimensional manifold. A Riemannian metric $g$ on $M$ is a smooth section of $T^*M\otimes T^*M$ defining a +positive definite symmetric bilinear form on $T_pM$ for each $p\in +M$. In local coordinates $(x^1,\cdots, x^n)$, one has a natural +local basis $\{\partial_1,\cdots,\partial_n\}$ for $TM$, where +$\partial_i=\frac{\partial}{\partial x^i}$. The metric tensor +$g=g_{ij} dx^i\otimes dx^j$ is represented by a smooth matrix-valued +function + +$$ +g_{ij}=g(\partial_i,\partial_j). +$$ + + The +pair $(M,g)$ is a *Riemannian manifold*. We denote by $(g^{ij})$ +the inverse of the matrix $(g_{ij})$. + + +Using a partition of unity one can easily see that any manifold +admits a Riemannian metric. A Riemannian metric on $M$ allows us to +measure lengths of smooth paths in $M$ and hence to define a +distance function by setting $d(p,q)$ equal to the infimum of the +lengths of smooth paths from $p$ to $q$. This makes $M$ a metric +space. For a point $p$ in a Riemannian manifold $(M,g)$ and for $r> +0$ we denote the metric ball of radius $r$ centered at $p$ in $M$ by +$B(p,r)$ or by $B_g(p,r)$ if the metric needs specifying or +emphasizing. It is defined by + +$$ +B(p,r)=\{q\in M\left|\right. d(p,q) +**Theorem.** Given a Riemannian metric $g$ on $M$, there uniquely exists a +torsion-free connection on $TM$ making $g$ parallel, i.e., there is +a unique $\mathbb{R}$-linear mapping $\nabla\colon +\Gamma(TM)\rightarrow \Gamma(T^*M\otimes TM)$ satisfying the Leibniz +formula + +$$ +\nabla(fX)=df\otimes X+f\nabla X, +$$ + +and the following two additional conditions for all vector fields +$X$ and $Y$: + +- **$\bullet$** ($g$ orthogonal) $d(g(X,Y))=g(\nabla X,Y)+g(X,\nabla +Y)$. +- **$\bullet$** (Torsion-free) $\nabla_XY-\nabla_YX-[X,Y]=0$ +(where, as is customary, we denote $\nabla Y(X)$ by $\nabla_XY$); + + +We call the +above connection the *Levi-Civita connection* of the metric and $\nabla X$ the *covariant +derivative* of $X$. On a Riemannian manifold we always use the +Levi-Civita connection. + +In local coordinates $(x^1,\ldots,x^n)$ the Levi-Civita connection +$\nabla$ is given by the $\nabla_{\partial_i}(\partial_j) = +\Gamma^{k}_{ij}\partial_k$, where the Christoffel symbols +$\Gamma^k_{ij}$ are the smooth functions + +$$ +\Gamma^{k}_{ij}= \frac{1}{2}g^{kl}(\partial_i g_{l j} +\partial_j +g_{il}-\partial_l g_{ij}). +$$ + + Note that the above two +additional conditions for the Levi-Civita connection $\nabla$ +correspond respectively to + +$\bullet$ $\Gamma^k_{ij}=\Gamma^k_{ji}$, + +$\bullet$ $\partial_k g_{ij}=g_{l j}\Gamma^l_{ki}+g_{il +}\Gamma^l_{kj}$. + +The covariant derivative extends to all tensors. In the special case +of a function $f$ we have $\nabla( f)=df$. Note that there is a +possible confusion between this and the notation in the literature +since one often sees $\nabla f$ written for the gradient of $f$, +which is the vector field dual to $df$. We always use $\nabla f$ to +mean $df$, and we will denote the gradient of $f$ by $(\nabla f)^*$, + +The covariant derivative allows us to define the Hessian of a smooth +function at any point, not just a critical point. Let $f$ be a +smooth real-valued function on $M$. We define the +Hessian of $f$, denoted $\mathit{Hess}(f)$, as +follows: + +$$ +\mathit{Hess}(f)(X,Y)=X(Y(f))-\nabla_XY(f). +$$ + + +**Lemma.** The Hessian is a contravariant, symmetric two-tensor, i.e., for + vector fields $X$ and $Y$ we have + +$$ +\mathit{Hess}(f)(X,Y)=\mathit{Hess}(f)(Y,X) +$$ + +and + +$$ +\mathit{Hess}(f)(\phi X,\psi Y)=\phi\psi \mathit{Hess}(f)(X,Y) +$$ + +for all smooth functions $\phi,\psi$. Other formulas for the Hessian +are + +$$ +\mathit{Hess}(f)(X,Y)=\langle \nabla_X(\nabla f),Y\rangle=\nabla_X(\nabla_Y(f))=\nabla^2f(X,Y). +$$ + +Also, in local coordinates we have + +$$ +\mathit{Hess}(f)_{ij}=\partial_i\partial_jf-(\partial_kf)\Gamma^k_{ij}. +$$ + + +**Proof.** The proof of symmetry is direct from the torsion-free assumption: + +$$ +\mathit{Hess}(f)(X,Y)-\mathit{Hess}(f)(Y,X)=[X,Y](f)-(\nabla_XY-\nabla_YX)(f)=0. +$$ + + The fact that +$\mathit{Hess}(f)$ is a tensor is also established by direct +computation. The equivalence of the various formulas is also +immediate: + +$$ +\begin{aligned} +\langle\nabla_X(\nabla f),Y\rangle & = & X(\langle \nabla +f,Y\rangle)-\langle\nabla f,\nabla_XY\rangle +\\ & = & X(Y(f))-\nabla_XY(f)=\mathit{Hess}(f)(X,Y).\nonumber +\end{aligned} +$$ + +Since $df=(\partial_rf) dx^r$ and +$\nabla(dx^k)=-\Gamma_{ij}^kdx^i\otimes dx^j$, it follows that + +$$ +\nabla(df)=\left(\partial_i\partial_jf-(\partial_kf)\Gamma^k_{ij}\right)dx^i\otimes +dx^j. +$$ + + It is direct from the definition that + +$$ +\mathit{Hess}(f)_{ij}=\mathit{Hess}(f)(\partial_i,\partial_j)=\partial_i\partial_jf-(\partial_kf)\Gamma_{ij}^k. +$$ + +When the metric that we are using to define the Hessian is not clear +from the context, we introduce it into the notation and write $\mathit{Hess}_g(f)$ to denote the Hessian of $f$ with respect to the metric +$g$. + +The Laplacian $\triangle f$ is defined as the +trace of the Hessian: That is to say, in local coordinates near $p$ +we have + +$$ +\triangle f(p)=\sum_{ij}g^{ij}\mathit{Hess}(f)(\partial_i,\partial_j). +$$ + +Thus, if $\{X_i\}$ is an orthonormal basis for $T_pM$ then + +$$ +\triangle f(p)=\sum_i\mathit{Hess}(f)(X_i,X_i). +$$ + + Notice that this is +the form of the Laplacian that is non-negative at a local minimum, and +consequently has a non-positive spectrum. + +## Curvature of a Riemannian manifold + +For the rest of this chapter $(M,g)$ is a Riemannian manifold. + + +**Definition.** The *Riemann curvature tensor* of $M$ is the $(1,3)$-tensor on $M$ + +$$ +{\mathcal R}(X,Y)Z = \nabla^{2}_{X,Y}Z +- \nabla^{2}_{Y,X}Z = \nabla_{X}\nabla_{Y}Z-\nabla_{Y}\nabla_{X}Z +-\nabla_{[X,Y]}Z, +$$ + + where +$\nabla^2_{X,Y}Z=\nabla_{X}\nabla_{Y}Z-\nabla_{\nabla_XY}Z$. + + +In local coordinates the curvature tensor can be represented as + +$$ +{\mathcal R}(\partial_i,\partial_j)\partial_k= +{{{R}_{ij}}^l}_k\partial_l, +$$ + + where + +$$ +{{R_{ij}}^l}_k= +\partial_i \Gamma^{l}_{jk}- \partial_j +\Gamma^{l}_{ik}+ +\Gamma^{s}_{jk}\Gamma^{l}_{is}-\Gamma^{s}_{ik}\Gamma^{l}_{js}. +$$ + +Using the metric tensor $g$, we can change ${\mathcal R}$ to a +$(0,4)$-tensor as follows: + +$$ +{\mathcal R}(X,Y,Z,W)=g({\mathcal R}(X,Y)W,Z). +$$ + +(Notice the change of order in the last two variables.) Notice that +we use the same symbol and the same name for both the $(1,3)$ tensor +and the $(0,4)$ tensor; which one we are dealing with in a given +context is indicated by the index structure or the variables to +which the tensor is applied. In local coordinates, the Riemann +curvature tensor can be represented as + +$$ +\begin{aligned} +{\mathcal R}(\partial_i,\partial_j,\partial_k,\partial_l) & = R_{ijkl}\\ +&=g_{ks}{{R_{ij}}^s}_l\\ +&=g_{ks}(\partial_i \Gamma^{s}_{jl}- \partial_j \Gamma^{s}_{il}+ +\Gamma^{t}_{jl}\Gamma^{s}_{it}-\Gamma^{t}_{il}\Gamma^{s}_{jt}). +\end{aligned} +$$ + +One can easily verify the following: + + +**Claim.** The Riemann curvature tensor ${\mathcal R}$ satisfies the following +properties: + +$\bullet$ (Symmetry) $R_{ijkl}=-R_{jikl}$, $R_{ijkl}=-R_{ijlk}$, +$R_{ijkl}=R_{klij}$, + +$\bullet$ (1st Bianchi identity) The sum of +$R_{ijkl}$ over the cyclic permutation of + + any three indices vanishes, + +$\bullet$ (2nd Bianchi identity) +$R_{ijkl,h}+R_{ijlh,k}+R_{ijhk,l}=0$, where + +$$ +R_{ijkl,h}=(\nabla_{\partial_h}{\mathcal R})_{ijkl}. +$$ + + +There are many important related curvatures. + + +**Definition.** The *sectional curvature* of a +2-plane $P\subset T_pM$ is defined as + +$$ +K(P) = {\mathcal R}(X,Y,X,Y), +$$ + +where $\{X,Y\}$ is an orthonormal basis of $P$. We say that $(M,g)$ +has *positive sectional curvature* (resp., *negative +sectional curvature*) if $K(P)>0$ (resp., $K(P)<0$) for every +2-plane $P$. There are analogous notions of non-negative and +non-positive sectional curvature. + + +In local coordinates, suppose that $X = X^{i}\partial_i$ and $Y = +Y^{i}\partial_i$. Then we have + +$$ +K(P) = R_{ijkl}X^{i}Y^{j}X^{k}Y^{l}. +$$ + +A Riemannian manifold is said to have *constant sectional +curvature* if $K(P)$ is the same for all $p\in M$ and all $2$-planes +$P\subset T_pM$. One can show that a manifold $(M,g)$ has constant +sectional curvature $\lambda$ if and only if + +$$ +R_{ijkl} = +\lambda(g_{ik}g_{jl}-g_{il}g_{jk}). +$$ + + Of course, the sphere of +radius $r$ in $\Ar^n$ has constant sectional curvature $1/r^2$, +$\Ar^n$ with the Euclidean metric has constant sectional curvature +$0$, and the hyperbolic space ${\mathbb H}^n$, which, in the +Poincaré model, is given by the unit disk with the metric + +$$ +\frac{4(dx^{2}_{1}+\cdots+ +dx^{2}_{n})}{(1-\abs{x}^{2})^{2}}, +$$ + + or in the upper half-space +model with coordinates $(x^1,\ldots,x^n)$ is given by + +$$ +\frac{ds^2}{(x^n)^2} +$$ + + has constant sectional curvature $-1$. In all +three cases we denote the constant curvature metric by $g_\mathit{st}$. + + +**Definition.** Using the metric, one can replace the Riemann curvature tensor +${\mathcal R}$ by a symmetric bilinear form $\mathit{Rm}$ on +$\wedge^2TM$. In local coordinates let +$\varphi=\varphi^{ij}\partial_i\wedge\partial _j$ and +$\psi=\psi^{kl}\partial _k\wedge\partial_l$ be local sections of +$\wedge^2TM$. The formula for $\mathit{Rm}$ is + +$$ +\mathit{Rm}(\varphi, \psi)=R_{ijkl}\varphi^{ij}\psi^{kl}. +$$ + +We call $\mathit{Rm}$ the *curvature +operator*. We say $(M,g)$ has +*positive curvature operator* if $\mathit{Rm}(\varphi,\varphi)>0$ for any nonzero 2-form $\varphi = +\varphi^{ij}\partial_{i}\wedge +\partial_{j}$ and has *nonnegative curvature operator* if +$\mathit{Rm}(\varphi,\varphi)\geq0$ for any $\varphi\in \wedge^2TM$. + + +Clearly, if the curvature operator is a positive (resp., +non-negative) operator then the manifold is positively (resp., +non-negatively) curved. + + +**Definition.** The *Ricci curvature tensor*, +denoted $\mathit{Ric}$ or $\mathit{Ric}_g$ when it is necessary to specify +the metric, is a symmetric contravariant two-tensor. In local +coordinates it is defined by + +$$ +\mathit{Ric}(X,Y)=g^{kl}R(X,\partial_k,Y,\partial_l). +$$ + +The value of this tensor at a point $p\in M$ is given by +$\sum_{i=1}^nR(X(p),e_i,Y(p),e_i)$ where $\{e_{1},\cdots,e_{n}\}$ is +an orthonormal basis of $T_pM$. Clearly $\mathit{Ric}$ is a symmetric +bilinear form on $TM$, given in local coordinates by + +$$ +\mathit{Ric}=\mathit{Ric}_{ij}dx^i\otimes dx^j, +$$ + + where +$\mathit{Ric}_{ij}=\mathit{Ric}(\partial_i,\partial _j)$. The +*scalar curvature* is defined by: + +$$ +R=R_g=\mathit{tr}_g \mathit{Ric}=g^{ij}\mathit{Ric}_{ij}. +$$ + + We will say that $\mathit{Ric} \geq +k$ (or $\leq k$) if all the eigenvalues of $\mathit{Ric}$ are $\geq k$ +(or $\leq k$). + + +Clearly, the curvatures are natural in the sense that if $F\colon N +\rightarrow M$ is a diffeomorphism and if $g$ is a Riemannian metric +on $M$, then $F^*g$ is a Riemannian metric on $N$ and we have $\mathit{Rm}(F^*g)=F^*(\mathit{Rm}(g))$, $\mathit{Ric}(F^{*}g) = F^{*}(\mathit{Ric}(g))$, and $R(F^*g)=F^*(R(g))$. + +### Consequences of the Bianchi identities + +There is one consequence of the second Bianchi identity that will be +important later. For any contravariant two-tensor $\omega$ on $M$ +(such as $\mathit{Ric}$ or $\mathit{Hess}(f)$) we define the contravariant +one-tensor $\mathit{div}(\omega)$ as follows: For any vector field $X$ +we set + +$$ +\mathit{div}(\omega)(X)=\nabla^*\omega(X)=g^{rs}\nabla_r(\omega)(X,\partial_s). +$$ + + +**Lemma.** + +$$ +dR=2\mathit{div}(\mathit{Ric})=2\nabla^*\mathit{Ric}. +$$ + + +For a proof see Proposition 6 of Chapter 2 on page 40 of [Petersen]. + +We shall also need a formula relating the connection Laplacian on +contravariant one-tensors with the Ricci curvature. Recall that for +a smooth function $f$, we defined the symmetric two-tensor +$\nabla^2f$ by + +$$ +\nabla^2f(X,Y)=\nabla_X\nabla_Y(f)-\nabla_{\nabla_X(Y)}(f)=\mathit{Hess}(f)(X,Y), +$$ + +and then defined the Laplacian + +$$ +\triangle f=\mathit{tr}\nabla^2f=g^{ij}(\nabla^2f)_{ij}. +$$ + + These operators extend to tensors of any +rank. Suppose that $\omega$ is a contravariant tensor of rank $k$. Then we +define $\nabla^2\omega$ to be a contravariant tensor of rank $k+2$ given by + +$$ +\nabla^2\omega(\cdot,X,Y)=(\nabla_{X}\nabla_{Y}\omega)(\cdot) +-\nabla_{\nabla_X(Y)}\omega(\cdot). +$$ + + This expression is not +symmetric in the vector fields $X,Y$ but the commutator is given by +evaluating the curvature operator ${\mathcal R}(X,Y)$ on $\omega$. +We define the connection Laplacian on the tensor $\omega$ to be + +$$ +\triangle \omega=g^{ij}\nabla^2(\omega)(\partial_i,\partial_j). +$$ + +Direct computation gives the standard Bochner formula relating these +Laplacians with the Ricci curvature; see for example Proposition +4.36 on page 168 of [Gallot]. + + +**Lemma.** Let $f$ be a smooth function on a Riemannian manifold. +Then we have the following formula for contravariant one-tensors: + +$$ +\triangle df=d\triangle f+\mathit{Ric}((\nabla f)^*,\cdot). +$$ + + +### First examples + +The most homogeneous Riemannian manifolds are those of constant +sectional curvature. These are easy to classify; see Corollary 10 of +Chapter 5 on page 147 of [Petersen]. + + +**Theorem.** **(Uniformization Theorem)** If +$(M^n,g)$ is a complete, simply-connected Riemannian manifold of +constant sectional curvature $\lambda$, then: + +1. If $\lambda=0$, then $M^n$ is isometric to Euclidean $n$-space. +1. If $\lambda>0$ there is a diffeomorphism $\phi \colon M \rightarrow S^{n}$ + such that g = $\lambda^{-1}\phi^{*}(g_\mathit{st})$ where $g_{st}$ is the usual metric on the unit sphere in $\Ar^{n+1}$. +1. If $\lambda<0$ there is a diffeomorphism $\phi \colon M \rightarrow {\mathbb H}^{n}$ + such that g = $\abs{\lambda}^{-1}\phi^{*}(g_\mathit{st})$ where $g_{st}$ is the Poincaré metric of constant curvature +$-1$ on ${\mathbb H}^n$. + + +Of course, if $(M^n,g)$ is a complete manifold of constant sectional +curvature then its universal covering satisfies the hypothesis of +the theorem and hence is one of $S^n, \Ar^n$, or ${\mathbb H}^n$, up +to a constant scale factor. This implies that $(M,g)$ is isometric +to a quotient of one of these simply connected spaces of constant +curvature by the free action of a discrete group of isometries. Such +a Riemannian manifold is called a +*space-form*. + + +**Definition.** The Riemannian manifold +$(M,g)$ is said to be an *Einstein manifold with Einstein +constant* $\lambda$ if $\mathit{Ric}(g)=\lambda g$. + + + +**Example.** Let $M$ be an $n$-dimensional manifold with $n$ being either $2$ or $3$. +If $(M,g)$ is Einstein with Einstein constant $\lambda$, one can +easily show that $M$ has constant sectional curvature +$\frac{\lambda}{n-1}$, so that in fact $M$ is a space-form. + + +### Cones + +Another class of examples that will play an important role in our +study of the Ricci flow is that of cones. + + +**Definition.** Let $(N,g)$ be a Riemannian manifold. We define the *open cone* +over $(N,g)$ to be the manifold $N\times (0,\infty)$ with the metric +$\tilde g$ defined as follows: For any $(x,s)\in N\times (0,\infty)$ +we have + +$$ +\tilde g(x,s)=s^2g(x)+ds^2. +$$ + + +Fix local coordinates $(x^1,\ldots,x^n)$ on $N$. Let +$\Gamma_{ij}^k;\ 1\le i,j,k\le n$, be the Christoffel symbols for +the Levi-Civita connection on $N$. Set $x^0=s$. In the local +coordinates $(x^0,x^1,\ldots,x^n)$ for the cone we have the +Christoffel symbols $\widetilde \Gamma_{ij}^k;\ 0\le i,j,k\le n$, +for $\tilde g$. The relation between the metrics gives the following +relations between the two sets of Christoffel symbols: + +$$ +\begin{aligned} +\widetilde +\Gamma_{ij}^k & = & \Gamma_{ij}^k; \ \ \ \ 1\le i,j,k\le n \\ +\widetilde \Gamma_{ij}^0 & = & -sg_{ij}; \ \ \ \ 1\le i,j\le n \\ +\widetilde\Gamma_{i0}^j & = & \widetilde\Gamma_{0i}^j = +s^{-1}\delta^j_i; \ \ \ \ 1\le i,j\le n \\ +\widetilde \Gamma_{i0}^0 & = & 0;\ \ \ \ 0\le i\le n \\ +\widetilde \Gamma_{00}^i & = & 0;\ \ \ \ 0\le i\le n. +\end{aligned} +$$ + +Denote by ${\mathcal R}_g$ the curvature tensor for $g$ and by +${\mathcal R}_{\tilde g}$ the curvature tensor for $\tilde g$. Then +the above formulas lead directly to: + +$$ +\begin{aligned} +{\mathcal R}_{\tilde g}(\partial_i,\partial_j)(\partial_0) & = & 0;\ \ \ \ 0\le i,j\le n \\ +{\mathcal R}_{\tilde g}(\partial_i,\partial_j)(\partial_i) & = & +{\mathcal +R}_g(\partial_i,\partial_j)(\partial_i)+g_{ii}\partial_j-g_{ji}\partial_i;\ +\ \ \ 1\le i,j\le n \\ +\end{aligned} +$$ + +This allows us to compute the Riemann curvatures of the cone in +terms of those of $N$. + + +**Proposition.** Let $N$ be a Riemannian manifold of dimension $n-1$. Fix $(x,s)\in +c(N)=N\times (0,\infty)$. With respect to the coordinates +$(x^0,\ldots,x^n)$ the curvature operator ${\mathit{Rm}}_{\tilde +g}(p,s)$ of the cone decomposes as + +$$ +\begin{pmatrix} 0 & 0 \\ s^2(\mathit{Rm}_g(p)-\wedge^2g(p)) & 0 +\end{pmatrix}, +$$ + +where $\wedge^2g(p)$ is the symmetric form on $\wedge^2T_pN$ induced +by $g$. + + + +**Corollary.** For any $p\in N$ let $\lambda_1,\ldots,\lambda_{(n-1)(n-2)/2}$ be +the eigenvalues of $\mathit{Rm}_g(p)$. Then for any $s>0$ there are +$(n-1)$ zero eigenvalues of $\mathit{Rm}_{\tilde g}(p,s)$. The other +$(n-1)(n-2)/2$ eigenvalues of $\mathit{Rm}_{\tilde g}(p,s)$ are +$s^{-2}(\lambda_i-1)$. + + +**Proof.** Clearly from \entryref{1b517dfd4bd7}, we see that under the +orthogonal decomposition $\wedge^2T_{(p,s)}c(N)=\wedge^2T_pN\oplus +T_pN$ the second subspace is contained in the null space of $ \mathit{Rm}_{\tilde g}(p,s)$, and hence contributes $(n-1)$ zero +eigenvalues. Likewise, from this proposition we see that the +eigenvalues of the restriction of $\mathit{Rm}_{\tilde g}(p,s)$ to the +subspace $\wedge^2T_pN$ are given by +$s^{-4}(s^2(\lambda_i-1))=s^{-2}(\lambda_i-1)$. + +## Geodesics and the exponential map + +Here we review standard material about geodesics, Jacobi fields, and +the exponential map. + +### Geodesics and the energy functional + + +**Definition.** Let $I$ be an open interval. +A smooth curve $\gamma\colon I \rightarrow M$ is called a +*geodesic* if $\nabla_{\dot \gamma}\dot +\gamma =0$. + + +In local coordinates, we write $\gamma(t)=(x^1(t),\ldots,x^n(t))$ +and this equation becomes + +$$ +0=\nabla_{\dot \gamma}\dot \gamma(t) = +\left(\sum\limits_k\left(\ddot x^k(t) +\dot x^i(t)\dot +x^j(t)\Gamma^{k}_{ij}(\gamma(t))\right)\partial_k\right). +$$ + + This is +a system of $2^{nd}$ order ODE's. The local existence, uniqueness +and smoothness of a geodesic through any point $p\in M$ with initial +velocity vector $v\in T_pM$ follow from the classical ODE theory. +Given any two points in a complete manifold, a standard limiting +argument shows that there is a rectifiable curve of minimal length +between these points. Any such curve is a geodesic. We call +geodesics that minimize the length between their endpoints *minimizing geodesics*. + +We have the classical theorem showing that on a complete manifold +all geodesics are defined for all time (see Theorem 16 of Chapter 5 +on p. 137 of [Petersen]). + + +**Theorem.** (Hopf-Rinow) If $(M,g)$ is complete as a metric space, then every +geodesic extends to a geodesic defined for all time. + + +Geodesics are critical points of the energy functional. Let $(M,g)$ +be a complete Riemannian manifold. Consider the space of $C^1$-paths +in $M$ parameterized by the unit interval. On this space we have the +energy functional + +$$ +E(\gamma)=\frac{1}{2}\int_{0}^{1}\langle \gamma'(t),\gamma'(t)\rangle dt. +$$ + + Suppose that we have a one-parameter family of paths parameterized by + $[0,1]$, + all having the same initial point $p$ and the same final point $q$. + By this we mean that we have a surface $\tilde\gamma(t,u)$ with the property that for each + $u$ the path $\gamma_u=\tilde\gamma(\cdot,u)$ is a path from $p$ to $q$ parameterized by $[0,1]$. + Let $\widetilde X=\partial\tilde \gamma/\partial t$ and + $\widetilde Y=\partial\tilde \gamma/\partial u$ be the corresponding vector + fields along the surface swept out by $\tilde \gamma$, and denote by $X$ and $Y$ the restriction + of these vector fields along $\gamma_0$. + We compute + +$$ +\begin{aligned} +\frac{dE(\gamma_u)}{du}\Bigl|_{u=0}\Bigr.& = & \left( \int_0^1\langle \nabla_{\widetilde Y}\widetilde X,\widetilde + X\rangle dt\right)\left|_{u=0}\right. \\ + & = & \left(\int_0^1\langle\nabla_{\widetilde X}\widetilde Y,\widetilde + X\rangle dt \right)\left|_{u=0}\right.\\ + & = & -\left( \int_0^1\langle\nabla_{\widetilde X}\widetilde X,\widetilde + Y\rangle dt\right)\left|_{u=0}\right.=-\int_0^1\langle\nabla_XX,Y\rangle, +\end{aligned} +$$ + +where the first equality in the last line comes from integration by +parts and the fact that $\widetilde Y$ vanishes at the endpoints. +Given any vector field $Y$ along $\gamma_0$ there is a one-parameter +family $\tilde \gamma(t,u)$ of paths from $p$ to $q$ with $\tilde +\gamma(t,0)=\gamma_0$ and with $\widetilde Y(t,0)=Y$. Thus, from the +above expression we see that $\gamma_0$ is a critical point for the +energy functional on the space of paths from $p$ to $q$ +parameterized by the interval $[0,1]$ if and only if $\gamma_0$ is a +geodesic. + +Notice that it follows immediately from the geodesic equation that +the length of a tangent vector along a geodesic is constant. Thus, +if a geodesic is parameterized by $[0,1]$ we have + +$$ +E(\gamma)=\frac{1}{2}L(\gamma)^2. +$$ + +It is immediate from the Cauchy-Schwarz inequality that for any +curve $\mu$ parameterized by $[0,1]$ we have + +$$ +E(\mu)\ge \frac{1}{2}L(\mu)^2 +$$ + +with equality if and only if $|\mu'|$ is constant. In particular, a +curve parameterized by $[0,1]$ minimizes distance between its +endpoints if it is a minimum for the energy functional on all paths +parameterized by $[0,1]$ with the given endpoints. + +### Families of geodesics and Jacobi fields + +Consider a family of geodesics $\tilde\gamma(u,t)=\gamma_u(t)$ +parameterized by the interval $[0,1]$ with $\gamma_u(0)=p$ for all +$u$. Here, unlike the discussion above, we allow $\gamma_u(1)$ to +vary with $u$. As before define vector fields along the surface +swept out by $\tilde \gamma$: $\widetilde X=\partial +\tilde\gamma/\partial t$ and let $\tilde +Y=\partial\tilde\gamma/\partial u$. We denote by $X$ and $Y$ the +restriction of these vector fields to the geodesic +$\gamma_0=\gamma$. + Since each +$\gamma_u$ is a geodesic, we have $\nabla_{\widetilde X}{\widetilde +X}=0$. Differentiating this equation in the $\widetilde Y$-direction +yields $\nabla_{\widetilde Y}\nabla_{\widetilde X}\widetilde X=0$. +Interchanging the order of differentiation, using +$\nabla_{\widetilde X}\widetilde Y=\nabla_{\widetilde Y}{\widetilde +X}$, and then restricting to $\gamma$, we get the *Jacobi +equation*: + +$$ +\nabla_X\nabla_XY+{\mathcal R}(Y,X)X=0. +$$ + +Notice that the left-hand side of the equation depends only on the +value of $Y$ along $\gamma$, not on the entire family. We denote the +left-hand side of this equation by ${rm Jac}(Y)$, so that the Jacobi +equation now reads + +$$ +{rm Jac}(Y)=0. +$$ + + The fact that all the geodesics begin at the same point at time +$0$ means that $Y(0)=0$. A vector field $Y$ along a geodesic +$\gamma$ is said to be a *Jacobi field* if +it satisfies this equation and vanishes at the initial point $p$. A +Jacobi field is determined by its first derivative at $p$, i.e., by +$\nabla_XY(0)$. We have just seen that this is the equation +describing, to first order, variations of $\gamma$ by a family of +geodesics with the same starting point. + +Jacobi fields are also determined by the energy functional. Consider +the space of paths parameterized by $[0,1]$ starting at a given +point $p$ but free to end anywhere in the manifold. Let $\gamma$ be +a geodesic (parameterized by $[0,1]$) from $p$ to $q$. Associated to +any one-parameter family $\tilde \gamma(t,u)$ of paths parameterized +by $[0,1]$ starting at $p$ we associate the second derivative of the +energy at $u=0$. Straightforward computation gives + +$$ +\frac{d^2E(\gamma_u)}{du^2}\Bigl|_{u=0}\Bigr.=\langle \nabla_XY(1),Y(1)\rangle+\langle +X(1),\nabla_Y\widetilde Y(1,0)\rangle-\int_0^1\langle {rm +Jac}(Y),Y\rangle dt. +$$ + +Notice that the first term is a boundary term from the integration +by parts, and it depends not just on the value of $Y$ (i.e., on +$\widetilde Y$ restricted to $\gamma$) but also on the first-order +variation of $\widetilde Y$ in the $Y$ direction. There is the +associated bilinear form that comes from two-parameter families +$\tilde \gamma(t,u_1,u_2)$ whose value at $u_1=u_1=0$ is $\gamma$. +It is + +$$ +\frac{d^2E}{du_1du_2}\Bigl|_{u_1=u_2=0}\Bigr.=\langle \nabla_XY_1(1),Y_2(1)\rangle+\langle +X(1),\nabla_{Y_1}\widetilde Y_2(1,0)\rangle-\int_0^1\langle {rm +Jac}(Y_1),Y_2\rangle dt. +$$ + + Notice that restricting to the space of +vector fields that vanish at both endpoints, the second derivatives +depend only on $Y_1$ and $Y_2$ and the formula is + +$$ +\frac{d^2E}{du_1du_2}\Bigl|_{u_1=u_2=0}\Bigr.=-\int_0^1\langle {rm Jac}(Y_1),Y_2\rangle dt, +$$ + +so that this expression is symmetric in $Y_1$ and $Y_2$. The +associated quadratic form on the space of vector fields along +$\gamma$ vanishing at both endpoints + +$$ +-\int_0^1\langle {rm Jac}(Y),Y\rangle dt +$$ + +is the second derivative of the energy function at $\gamma$ for any +one-parameter family whose value at $0$ is $\gamma$ and whose first +variation is given by $Y$. + +### Minimal geodesics + + +**Definition.** Let $\gamma$ be a geodesic beginning at $p\in M$. For any $t>0$ we +say that $q=\gamma(t)$ is a *conjugate point along +$\gamma$* if there is a non-zero Jacobi field +along $\gamma$ vanishing at $\gamma(t)$. + + + +**Proposition.** Suppose that $\gamma\colon [0,1]\to M$ is a minimal geodesic. Then +for any $t<1$ the restriction of $\gamma$ to $[0,t]$ is the unique +minimal geodesic between its endpoints and there are no conjugate +points on $\gamma([0,1))$, i.e., there is no non-zero Jacobi field +along $\gamma$ vanishing at any $t\in [0,1)$. + + +We shall sketch the proof. For more details see Proposition 19 and Lemma 14 of +Chapter 5 on pp. 139 and 140 of [Petersen]. + +**Proof.** (Sketch) +Fix $0 +**Claim.** Suppose that $\gamma$ is a minimal geodesic and $Y$ is a field +vanishing at both endpoints. Let $\tilde\gamma(t,u)$ be any +one-parameter family of curves parameterized by $[0,1]$, with +$\gamma_0=\gamma$ and with $\gamma_u(0)=\gamma_0(0)$ for all $u$. +Suppose that the first-order variation of $\tilde \gamma$ at $u=0$ +is given by $Y$. Then + +$$ +\frac{d^2E(\gamma_u)}{du^2}\Bigl|_{u=0}\Bigr.=0 +$$ + +if and only if $Y$ is a Jacobi field. + + +**Proof.** Suppose that $\tilde\gamma(u,t)$ is a one-parameter family of curves +from $\gamma(0)$ to $\gamma(1)$ with $\gamma_0=\gamma$ and $Y$ is +the first-order variation of this family along $\gamma$. Since +$\gamma$ is a minimal geodesic we have + +$$ +-\int_0^1\langle {rm Jac}(Y),Y\rangle dt=\frac{d^2E(\gamma_u)}{du^2}\Bigl|_{u=0}\Bigr.\ge 0. +$$ + + The associated symmetric bilinear form is + +$$ +B_\gamma(Y_1,Y_2)=-\int_\gamma\langle {rm Jac}(Y_1),Y_2\rangle dt +$$ + +is symmetric when $Y_1$ and $Y_2$ are constrained to vanish at both +endpoints. Since the associated quadratic form is non-negative, we +see by the usual argument for symmetric bilinear forms that +$B_\gamma(Y,Y)=0$ if and only if $B_\gamma(Y,\cdot)=0$ as a linear +functional on the space of vector fields along $\gamma$ vanishing at +point endpoints. This of course occurs if and only if ${rm +Jac}(Y)=0$. + +Now let us use this claim to show that there are no conjugate points +on $\gamma|_{(0,1)}$. If for some $t_0<1$, $\gamma(t_0)$ is a +conjugate point along $\gamma$, then there is a non-zero Jacobi +field $Y(t)$ along $\gamma$ with $Y(t_0)=0$. Notice that since $Y$ +is non-trivial $\nabla_XY(t_0)\not= 0$. Extend $Y(t)$ to a vector +field $\hat Y$ along all of $\gamma$ by setting it equal to $0$ on +$\gamma|_{[t_0,1]}$. Since the restriction of $Y$ to +$\gamma([0,t_0])$ is a Jacobi field vanishing at both ends and since +$\gamma|_{[0,t_0]}$ is a minimal geodesic, the second-order +variation of length of $\gamma|_{[0,t_0]}$ in the $Y$-direction is +zero. It follows that the second-order variation of length along +$\hat Y$ vanishes. But $\hat Y$ is not smooth (at $\gamma(t_0)$) and +hence it is not a Jacobi field along $\gamma$. This contradicts the +fact discussed in the previous paragraph that for minimal geodesics +the null space of the quadratic form is exactly the space of Jacobi +fields. + +### The exponential mapping + + +**Definition.** For any $p\in M$, we can define the *exponential +map* at $p$, $\mathit{exp}_{p}$. It is +defined on an open neighborhood $O_{p}$ of the origin in $T_{p}M$ +and is defined by $\mathit{exp}_{p}(v)=\gamma_v(1)$, the endpoint of +the unique geodesic $\gamma_v\colon [0,1]\rightarrow M$ starting +from $p$ with initial velocity vector $v$. We always take +$O_{p}\subset T_{p}M$ to be the maximal domain on which $\mathit{exp}_p$ is defined, so that $O_{p}$ is a star-shaped open +neighborhood of $0\in T_pM$. By the Hopf-Rinow Theorem, if $M$ is +complete, then the exponential map is define on all of $T_pM$. + + +By the inverse function theorem there exists $r_0=r_{0}(p,M) +> 0$, such that the restriction of $\mathit{exp}_p$ to the ball +$B_{g|T_pM}(0,r_0)$ in $T_pM$ is a diffeomorphism onto +$B_g(p,r_0)$. Fix $g$-orthonormal linear coordinates on $T_pM$. +Transferring these coordinates via $\mathit{exp}_p$ to coordinates on +$B(p,r_0)$ gives us *Gaussian normal coordinates* on $B(p,r_0)\subset M$. + +Suppose now that $M$ is complete, and fix a point $p\in M$. For +every $q\in M$, there is a length-minimizing path from $p$ to $q$. +When parameterized at constant speed equal to its length, this path +is a geodesic with domain interval $[0,1]$. Consequently, $\mathit{exp}_p\colon T_pM\to M$ is onto. The differential of the exponential +mapping is given by Jacobi fields: + Let +$\gamma\colon [0,1]\to M$ be a geodesic from $p$ to $q$, and let + $X\in T_pM$ be $\gamma'(0)$. Then the exponential +mapping at $p$ is a smooth map from $T_p(M)\to M$ sending $X$ to +$q$. Fix $Z\in T_pM$. Then there is a unique Jacobi field $Y_Z$ +along $\gamma$ with $\nabla_XY_Z(0)=Z$. The association $Z\mapsto +Y_Z(1)\in T_qM$ is a linear map from $T_p(M)\to T_qM$. Under the +natural identification of $T_pM$ with the tangent plane to $T_pM$ at +the point $Z$, this linear mapping is the differential of $\mathit{exp}_p\colon T_pM\to M$ at the point $X\in T_pM$. + + +**Corollary.** Suppose that $\gamma$ is a minimal geodesic parameterized by $[0,1]$ +starting at $p$. Let $X(0)=\gamma'(0)\in T_pM$. Then for each +$t_0<1$ the restriction $\gamma|_{[0,t_0]}$ is a minimal geodesic +and $\mathit{exp}_p\colon T_pM\to M$ is a local diffeomorphism near +$t_0X(0)$. + + +**Proof.** Of course, $\mathit{exp}_p(t_0X(0))=\gamma(t_0)$. According to the +previous discussion, the kernel of the differential of the +exponential mapping at $t_0X(0)$ is identified with the space of +Jacobi fields along $\gamma$ vanishing at $\gamma(t_0)$. According +to \entryref{50646105e5e0} the only such Jacobi field is the +trivial one. Hence, the differential of $\mathit{exp}_p$ at $t_0X(0)$ +is an isomorphism, completing the proof. + + +**Definition.** There is an open neighborhood $U_p\subset T_pM$ of $0$ consisting +of all $v\in T_pM$ for which: (i) $\gamma_v$ is the unique minimal +geodesic from $p$ to $\gamma_v(1)$, and (ii) $ \mathit{exp}_p$ is a +local diffeomorphism at $v$. We set ${\mathcal C}_p\subset M$ equal +to $M\setminus \mathit{exp}_p(U_p)$. Then ${\mathcal C}_p$ is called +the *cut locus from $p$*. It is a closed subset +of measure $0$. + + +It follows from \entryref{8141585de23a} that $U\subset T_pM$ is a +star-shaped open neighborhood of $0\in T_pM$. + + +**Proposition.** The map + +$$ +\mathit{exp}_p\colon U_p\to M\setminus {\mathcal C}_p +$$ + +is a diffeomorphism. + + +For a proof see p. 139 of [Petersen]. + + +**Definition.** The *injectivity radius $\mathit{inj}_M(p)$ of $M$ at $p$* is the +supremum of the $r> 0$ for which the restriction of $\mathit{exp}_p\colon T_pM\to M$ to the ball $B(0,r)$ of radius $r$ in $T_pM$ +is a diffeomorphism into $M$. Clearly, $\mathit{inj}_M(p)$ is the +distance in $T_pM$ from $0$ to the frontier of $U_p$. It is also the +distance in $M$ from $p$ to the cut locus ${\mathcal C}_p$. + + +Suppose that $\mathit{inj}_M(p)=r$. There are two possibilities: +Either there is a broken, closed geodesic through $p$, broken only +at $p$, of length $2r$, or there is a geodesic $\gamma$ of length +$r$ emanating from $p$ whose endpoint is a conjugate point along +$\gamma$. The first case happens when the exponential mapping is not +one-to-one of the closed ball of radius $r$ in $T_pM$, and the +second happens when there is a tangent vector in $T_pM$ of length +$r$ at which $\mathit{exp}_p$ is not a local diffeomorphism. + +## Computations in Gaussian normal coordinates + +In this section we compute the metric and the Laplacian (on +functions) in local Gaussian coordinates. A direct computation shows +that in Gaussian normal coordinates on a metric ball about $p\in M$ +the metric takes the form + +$$ +\begin{aligned} +g_{ij}(x)& = & \delta_{ij} + \frac{1}{3}R_{iklj}x^kx^l + +\frac{1}{6}R_{iklj,s}x^kx^lx^s \\ +& & +(\frac{1}{20}R_{iklj,st} +\frac{2}{45}\sum_m +R_{iklm}R_{jstm})x^kx^lx^sx^t + O(r^{5}), \nonumber +\end{aligned} +$$ + +where $r$ is the distance from $p$. (See, for example Proposition +3.1 on page 41 of [Sakai], with the understanding that, with +the conventions there, the quantity $R_{ijkl}$ there differs by sign +from ours.) + +Let $\gamma$ be a geodesic in $M$ emanating from $p$ in the direction $v$. +Choose local coordinates $\theta^1,\ldots,\theta^{n-1}$ on the unit sphere in +$T_pM$ in a neighborhood of $v/|v|$. Then $(r, \theta^{1}, ... ,\theta^{n-1})$ +are local coordinates at any point of the ray emanating from the origin in the +$v$ direction (except at $p$). Transferring these via $\mathit{exp}_p$ produces +local coordinates $(r,\theta^{1}, ... ,\theta^{n-1})$ along $\gamma$. Using +Gauss's lemma (Lemma 12 of Chapter 5 on p. 133 of [Petersen]), we can +write the metric locally as + +$$ +g=dr^{2}+ +r^{2}h_{ij}(r,\theta)d\theta^{i}\otimes d\theta^{j}. +$$ + + Then the +volume form + +$$ +\begin{aligned} +dV &= \sqrt{\mathit{det}(g_{ij})}dr\wedge d\theta^{1} +\wedge\cdots\wedge +d\theta^{n-1}\\ +& = r^{n-1}\sqrt{\mathit{det}(h_{ij})}dr\wedge d\theta^{1} +\wedge\cdots\wedge d\theta^{n-1}. +\end{aligned} +$$ + + +**Lemma.** The *Laplacian operator* acting on scalar +functions on $M$ is given in local coordinates by + +$$ +\triangle = \frac{1}{\sqrt{\mathit{det}(g)}}\partial_i +\left(g^{ij}\sqrt{\mathit{det}(g)}\partial_j\right). +$$ + + +**Proof.** Let us compute the derivative at a point $p$. We have + +$$ +\frac{1}{\sqrt{\mathit{det}(g)}}\partial_i +\left(g^{ij}\sqrt{\mathit{det}(g)}\partial_j\right)f=g^{ij}\partial_i\partial_jf+\partial_ig^{ij}\partial_jf+ +\frac{1}{2}g^{ij}\partial_iTr(\tilde g)\partial_jf, +$$ + + where $\tilde +g=g(p)^{-1}g$. On the other hand from the definition of the +Laplacian, Equation (1.4), and +Equation (\entryref{98af9c99c3c2}) we have + +$$ +\triangle +f=g^{ij}\mathit{Hess}(f)(\partial_i,\partial_j)=g^{ij}\left(\partial_i\partial_j(f)- +\nabla_{\partial_i}\partial_jf\right) +=g^{ij}\partial_i\partial_jf-g^{ij}\Gamma_{ij}^k\partial_kf. +$$ + + Thus, +to prove the claim it suffices to show that + +$$ +g^{ij}\Gamma_{ij}^k=-(\partial_ig^{ik}+\frac{1}{2}g^{ik}\mathit{Tr}(\partial_i\tilde g)). +$$ + +From the definition of the Christoffel symbols we have + +$$ +g^{ij}\Gamma_{ij}^k=\frac{1}{2}g^{ij}g^{kl}(\partial_ig_{jl}+\partial_jg_{il}-\partial_lg_{ij}). +$$ + +Of course, $g^{ij}\partial_ig_{jl}=-\partial_ig^{ij}g_{jl}$, so that + $g^{ij}g^{kl}\partial_ig_{jl}=-\partial_ig^{ik}$. It follows by symmetry that $g^{ij}g^{jl}\partial_jg_{il} + =-\partial_ig^{ik}$. The last term is clearly $-\frac{1}{2}g^{ik}\mathit{Tr}(\partial_i\tilde g).$ + +Using Gaussian local coordinates near $p$, we have + +$$ +\begin{aligned} +\triangle r &= +\frac{1}{r^{n-1}\sqrt{\mathit{det}(h)}}\partial_r\left(r^{n-1}\sqrt{\mathit{det}(h)}\right)\\ +&= \frac{n-1}{r} + + \partial_r\log\left(\sqrt{\mathit{det}(h)}\right). +\end{aligned} +$$ + +From this one computes directly that + +$$ +\triangle r + = \frac{n-1}{r} - \frac{r}{3}\mathit{Ric}(v,v)+ O(r^{2}), +$$ + + where $v = \dot r(0)$, cf, p.265-268 of [Petersen]. So + +$$ +\triangle r \leq + \frac{n-1}{r}\ \ \mathit{when\ \} r \ll 1\ \ \mathit{and\} \mathit{Ric} > 0. +$$ + +This local computation has the following global analogue. + + +**Exercise.** (E.Calabi, 1958) +Let $f(x)=d(p,x)$ be the distance function from $p$. If $(M, g)$ has +$\mathit{Ric} \geq 0$, then + +$$ +\triangle f \leq \frac{n-1}{f} +$$ + + in the sense of distributions. + + +[Compare [Petersen], p. 284 Lemma 42]. + + +**Remark.** The statement that $\triangle f\leq \frac{n-1}{f}$ in *the sense +of distributions* (or equivalently *in the weak sense*) means +that for any non-negative test function $\phi$, that is to say for +any compactly supported $C^\infty$-function $\phi$, we have + +$$ +\int_Mf\triangle \phi d\mathit{vol}\le \int_M\left(\frac{n-1}{f}\right)\phi d\mathit{vol}. +$$ + +Since the triangle inequality implies that $|f(x)-f(y)|\le d(x,y)$, it follows +that $f$ is Lipschitz, and hence that the restriction of $\nabla f$ to any +compact subset of $M$ is an $L^2$ one-form. Integration by parts then shows +that + +$$ +\int_Mf\triangle +\phi d\mathit{vol}=-\int_M\langle \nabla f,\nabla\phi\rangle d\mathit{vol}. +$$ + + +Since $\abs{\nabla f} = 1$ and $\triangle f$ is the mean curvature +of the geodesic sphere $\partial B(x,r)$, $\mathit{Ric}(v,v)$ measures +the difference of the mean curvature between the standard Euclidean +sphere and the geodesic sphere in the direction $v$. Another +important geometric object is the shape operator associated to $f$, +denoted $S$. By definition it is the Hessian of $f$; i.e., +$S=\nabla^2f=\mathit{Hess}(f)$. + +## Basic curvature comparison results + +In this section we will recall some of the basic curvature +comparison results in Riemannian geometry. The reader can refer to +[Petersen], Section 1 of Chapter 9 for details. + +We fix a point $p\in M$. For any real number $k\ge 0$ let $H^n_k$ denote the +simply connected, complete Riemannian $n$-manifold of constant sectional +curvature $-k$. Fix a point $q_k\in H^n_k$, and consider the exponential map +$\mathit{exp}_{q_k}\colon T_{q_k}(H^n_k)\to H^n_k$. This map is a global +diffeomorphism. Let us consider the pullback, $\tilde h_k$, of the Riemannian +metric on $H^n_k$ to $T_{q_k}H^n_k$. A formula for this tensor is easily given +in polar coordinates on $T_{q_k}(H^n_k)$ in terms of the following function. + + +**Definition.** We define a function $\mathit{sn}_k$ as follows: + +$$ +\mathit{sn}_k(r)=\begin{cases} +r \ \ & \mathit{if\ \}k=0 \\ +\frac{1}{\sqrt{k}}\mathit{sinh}(\sqrt{k}r) \ \ & \mathit{if \ \} k>0. +\end{cases} +$$ + + +The function $\mathit{sn}_{k}(r)$ is the solution to the equation + +$$ +\begin{aligned} +\varphi'' - k\varphi &= 0,\\ +\varphi(0) &= 0,\\ +\varphi'(0) &= 1. +\end{aligned} +$$ + +We define + $\mathit{ct}_{k}(r) = \frac{\mathit{sn}_{k}^{'}(r)}{\sqrt{k}\mathit{sn}_{k}(r)}$. + +Now we can compare manifolds of varying sectional curvature with +those of constant curvature. + + +**Theorem.** (Sectional Curvature Comparison) Fix $k\ge 0$. Let $(M, g)$ be a +Riemannian manifold with the property that $-k \leq K(P)$ for every +$2$-plane $P$ in $TM$. Fix a minimizing geodesic $\gamma\colon +[0,r_0)\to M$ parameterized at unit speed with $\gamma(0)=p$. Impose +Gaussian polar coordinates $(r,\theta^1,\ldots,\theta^{n-1})$ on a +neighborhood of $\gamma$ so that $g=dr^2+g_{ij}\theta^i\otimes +\theta^j$. Then for all $0 + +There is also an analogous result for a positive upper bound to the +sectional curvature, but in fact all we shall need is the local +diffeomorphism property of the exponential mapping. + + +**Lemma.** Fix $K\ge 0$. +If $|\mathit{Rm}(x)|\le K$ for all $x\in B(p,\pi/\sqrt{K})$, then +$\mathit{exp}_p$ is a local diffeomorphism from the ball +$B(0,\pi/\sqrt{K})$ in $T_pM$ to the ball $B(p,\pi/\sqrt{K})$ in +$M$. + + +There is a crucial comparison result for volume which involves the +Ricci curvature. + + +**Theorem.** **(Ricci curvature comparison)** Fix $k\ge 0$. Assume that $(M, +g)$ satisfies $\mathit{Ric} \geq -(n-1)k$. Let $\gamma\colon [0,r_0)\to +M$ be a minimal geodesic of unit speed. Then for any $r + +Note that the inequality in \entryref{9f5db77333af} follows from this +theorem. + +The comparison result in \entryref{569be40ea88e} holds out to +every radius, a fact that will be used repeatedly in our arguments. +This result evolved over the period 1964-1980 and now is referred to +as the Bishop-Gromov inequality; see +Proposition 4.1 of [CheegerGromovTaylor] + + +**Theorem.** (Relative Volume Comparison, Bishop-Gromov 1964-1980) Suppose $(M, +g)$ is a Riemannian manifold. Fix a point $p\in M$, and suppose that +$B(p,R)$ has compact closure in $M$. Suppose that for some $k\ge 0$ +we have $\mathit{Ric} \geq -(n-1)k$ on $B(p,R)$. Recall that $H^n_k$ is +the simply connected, complete manifold of constant curvature $-k$ +and $q_k\in H^n_k$ is a point. Then + +$$ +\frac{\mathit{Vol} B(p,r)}{\mathit{Vol} B_{H^n_k}B(q_k,r)} +$$ + +is a non-increasing function of $r$ for $r + +## Local volume and the injectivity radius + +As the following results show, in the presence of bounded curvature the volume +of a ball $B(p,r)$ in $M$ is bounded away from zero if and only if the +injectivity radius of $M$ at $p$ is bounded away from zero. + + +**Proposition.** Fix an integer $n>0$. For every $\epsilon>0$ there is $\delta>0$ +depending on $n$ and $\epsilon$ such that the following holds. +Suppose that $(M^n,g)$ is a complete Riemannian manifold of +dimension $n$ and that $p\in M$. Suppose that $|\mathit{Rm}(x)|\le +r^{-2}$ for all $x\in B(p,r)$. If the injectivity radius of $M$ at +$p$ is at least $\epsilon r$, then $\mathit{Vol}(B(p,r))\ge \delta +r^n$. + + +**Proof.** Suppose that $|\mathit{Rm}(x)|\le r^{-2}$ for all $x\in B(p,r)$. +Replacing $g$ by $r^2 g$ allows us to assume that $r=1$. Without +loss of generality we can assume that $\epsilon\le 1$. The map $\mathit{exp}_p$ is a diffeomorphism on the ball $B(0,\epsilon)$ in the +tangent space, and by \entryref{95de178b529d} the volume of +$B(p,\epsilon)$ is at least that of the ball of radius $\epsilon$ in +the $n$-sphere of radius $1$. This gives a lower bound to the volume +of $B(p,\epsilon)$, and a fortiori to $B(p,1)$, in terms of $n$ and +$\epsilon$. + + We shall normally work with volume, which behaves nicely under Ricci +flow, but in order to take limits we need to bound the injectivity +radius away from zero. Thus, the more important, indeed crucial, +result for our purposes is the converse to the previous proposition; +see Theorem 4.3, especially Inequality (4.22), on page 46 of +[CheegerGromovTaylor], or see Theorem 5.8 on page 96 of +[CheegerEbin]. + + +**Theorem.** Fix an integer $n>0$. For every $\epsilon>0$ there is $\delta>0$ +depending on $n$ and $\epsilon$ such that the following holds. +Suppose that $(M^n,g)$ is a complete Riemannian manifold of +dimension $n$ and that $p\in M$. Suppose that $|\mathit{Rm}(x)|\le +r^{-2}$ for all $x\in B(p,r)$. If $Vol(B(p,r))\ge \epsilon r^n$ then +the injectivity radius of $M$ at $p$ is at least $\delta r$. + diff --git a/projects/poincare-conjecture/.astrolabe/docs-src/03-manifolds-of-non-negative-curvature.mdx b/projects/poincare-conjecture/.astrolabe/docs-src/03-manifolds-of-non-negative-curvature.mdx new file mode 100644 index 00000000..3d9d43b5 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/docs-src/03-manifolds-of-non-negative-curvature.mdx @@ -0,0 +1,782 @@ + + +# Chapter 2 -- Manifolds of non-negative curvature + +In studying singularity development in $3$-dimensional Ricci flows +one forms blow-up limits. By this we mean the following. One +considers a sequence of points $x_k$ in the flow converging to the +singularity. It will be the case that $R(x_k)$ tends to $\infty$ as +$k$ tends to $\infty$. We form a sequence of based Riemannian +manifolds labeled by $k$, where the $k^{th}$ Riemannian manifold is +obtained by taking the time-slice of $x_k$, rescaling its metric by +$R(x_k)$, and then taking $x_k$ as the base point. This creates a +sequence with the property that for each member of the sequence the +scalar curvature at the base point is one. Because of a pinching +result of Hamilton's (see Chapter 4), if there is a +geometric limit of this sequence, or of any subsequence of it, then +that limit is non-negatively curved. Hence, it is important to +understand the basic properties of Riemannian manifolds of +non-negative curvature in order to study singularity development. In +this chapter we review the properties that we shall need. We suppose +that $M$ is non-compact and of positive (resp., non-negative) +curvature. The key to understanding these manifolds is the Busemann +function associated to a minimizing geodesic ray. + +## Busemann functions + +A geodesic ray $\lambda\colon [0,\infty)\to M$ is said to be *minimizing* if the restriction of $\lambda$ to every compact +subinterval of $[0,\infty)$ is a length-minimizing geodesic arc, +i.e., a geodesic arc whose length is equal to the distance between +its endpoints. Likewise, a geodesic line $\lambda\colon +(-\infty,\infty)\to M$ is said to be *minimizing* if its +restriction to every compact sub-interval of $\Ar$ is a length +minimizing geodesic arc. + +Clearly, if a sequence of minimizing geodesic arcs $\lambda_k$ +converges to a geodesic arc, then the limiting geodesic arc is also +minimizing. More generally, if $\lambda_k$ is a sequence of length +minimizing geodesic arcs whose initial points converge and whose +lengths go to infinity, then, after passing to a subsequence, there +is a limit which is a minimizing geodesic ray. (The existence of a +limit of a subsequence is a consequence of the fact that a geodesic +ray is determined by its initial point and its initial tangent +direction.) Similarly, if $I_k$ is an sequence of compact intervals +with the property that every compact subset of $\Ar$ is contained in +$I_k$ for all sufficiently large $k$, if for each $k$ the map +$\lambda_k\colon I_k\to M$ is a minimizing geodesic arc, and if +$\mathit{lim}_{k\rightarrow \infty}\lambda_k(0)$ exists, then, after +passing to a subsequence there is a limit which is a minimizing +geodesic line. Using these facts one establishes the following +elementary lemma. + + +**Lemma.** Suppose that $M$ is a complete, connected, non-compact Riemannian +manifold and let $p$ be a point of $M$. Then $M$ has a minimizing +geodesic ray emanating from $p$. If $M$ has more than one end, then +it has a minimizing line. + + + +**Definition.** Suppose that $\lambda\colon [0,\infty)\to M$ is a minimizing +geodesic ray with initial point $p$. For each $t\ge 0$ we consider +$B_{\lambda,t}(x)=d(\lambda(t),x)-t$. This is a family of functions +satisfying $|B_{\lambda,t}(x)-B_{\lambda,t}(y)|\le d(x,y)$. Since +$\lambda$ is a minimizing geodesic, $B_{\lambda,t}(p)=0$ for all +$t$. It follows that $B_{\lambda,t}(x)\ge -d(x,p)$ for all $x\in M$. +Thus, the family of functions $B_{\lambda,t}$ is pointwise bounded +below. The triangle inequality shows that for each $x\in M$ the +function $B_{\lambda,t}(x)$ is a non-increasing function of $t$. + It follows that, for +each $x\in M$, $\mathit{lim}_{t\rightarrow \infty} B_{\lambda,t}(x)$ +exists. We denote this limit by $B_\lambda(x)$. This is the *Busemann function* for $\lambda$. + + +Clearly, $B_\lambda(x)\ge -d(x,\lambda(0))$. By equicontinuity +$B_\lambda(x)$ is a continuous function of $x$ and in fact a +Lipschitz function satisfying $|B_\lambda(x)-B_\lambda(y)|\le +d(x,y)$ for all $x,y\in X$. Clearly $B_\lambda(\lambda(s))=-s$ for +all $s\ge 0$. Since $B_\lambda$ is Lipschitz, $\nabla B_\lambda$ is +well-defined as an $L^2$-vector field. + + +**Proposition.** Suppose that $M$ is complete and of non-negative Ricci curvature. +Then, for any minimizing geodesic ray $\lambda$, the Busemann +function $B_\lambda$ satisfies $\Delta B_\lambda\le 0$ in the weak +sense. + + +**Proof.** First notice that since $B_\lambda$ is Lipschitz, $\nabla B_\lambda$ +is an $L^2$-vector field on $M$. That is to say, $B_\lambda\in +W_\mathit{loc}^{1,2}$, i.e., $B_\lambda$ locally has one derivative in +$L^2$. Hence, there is a sequence of $C^\infty$-functions $f_n$ +converging to $B_\lambda$ in $W^{1,2}_{loc}$. Let $\varphi$ be a +test function (i.e., a compactly supported $C^\infty$-function). +Integrating by parts yields + +$$ +-\int_M\langle \nabla +f_n,\nabla\varphi\rangle d\mathit{vol}=\int_Mf_n\triangle \varphi d\mathit{vol}. +$$ + + Using the fact that $f_n$ converges to $B_\lambda$ in +$W^{1,2}_{loc}$ and taking limits yields + +$$ +-\int_M\langle \nabla B_\lambda,\nabla \varphi\rangle d\mathit{vol}= \int_M +B_\lambda\triangle \varphi d\mathit{vol}. +$$ + +Thus, to prove the proposition we need only show that if $\varphi$ +is a non-negative test function, then + +$$ +-\int_M \langle\nabla B_\lambda,\nabla\varphi\rangle d\mathit{vol}\le 0. +$$ + +For a proof of this see + Proposition 1.1 and its proof on pp. 7 and 8 in [SchoenYau]. + +## Comparison results in non-negative curvature + +Let us review some elementary comparison results for manifolds of +non-negative curvature. These form the basis for Toponogov +theory, [Toponogov]. For any pair +of points $x,y$ in a complete Riemannian manifold $s_{xy}$ denotes a +minimizing geodesic from $x$ to $y$. We set $|s_{xy}|=d(x,y)$ and +call it the *length* of the side. A *triangle* in a +Riemannian manifold consists of three vertices $a,b,c$ and three +sides $s_{ab}$,$s_{ac}$,$s_{bc}$. We denote by $\angle_a$ the angle +of the triangle at $a$, i.e., the angle at $a$ between the geodesic +rays $s_{ab}$ and $s_{ac}$. + + +**Theorem.** **(Length comparison)** +Let $(M,g)$ be a manifold of non-negative curvature. Suppose that +$\triangle(a,b,c)$ is a triangle in $M$ and let +$\triangle(a',b',c')$ be a Euclidean triangle. + +1. Suppose that the corresponding +sides of $\triangle(a,b,c)$ and $\triangle(a',b',c')$ have the same +lengths. Then the angle at each vertex of the Euclidean triangle is +no larger than the corresponding angle of $\triangle(a,b,c)$. +Furthermore, for any $\alpha$ and $\beta$ less than $|s_{ab}|$ and +$|s_{ac}|$ respectively, let $x$, resp. $x'$, be the point on +$s_{ab}$, resp. $s_{a'b'}$, at distance $\alpha$ from $a$, resp. +$a'$, and let $y$, resp. $y'$, be the point on $s_{ac}$, resp. +$s_{a'c'}$, at distance $\beta$ from $a$, resp. $a'$. Then +$d(x,y)\ge d(x',y')$. +1. Suppose that $|s_{ab}|=|s_{a'b'}|$, that +$|s_{ac}|=|s_{a'c'}|$ and that $\angle_a=\angle_{a'}$. Then +$|s_{b'c'}|\ge |s_{bc}|$. + + +See Fig. 2.1. For a proof of this result see +Theorem 4.2 on page 161 of [Sakai], or Theorem 2.2 on page 42 +of [CheegerEbin]. + +One corollary is a monotonicity result. Suppose that +$\triangle(a,b,c)$ is a triangle in a complete manifold of +non-negative curvature. Define a function $EA(u,v)$ defined for +$0\le u\le |s_{ab}|$ and $0\le v\le |s_{ac}|$ as follows. For $u$ +and $v$ in the indicated ranges, let $x(u)$ be the point on $s_{ab}$ +at distance $u$ from $a$ and let $y(v)$ be the point of $s_{ac}$ at +distance $v$ from $a$. Let $EA(u,v)$ be the angle at $a'$ of the +Euclidean triangle with side lengths $|s_{a'b'}|=u$, $|s_{a'c'}|=v$ +and $|s_{b'c'}|=d(x(u),y(v))$. + + +**Corollary.** Under the assumptions of the previous theorem, $EA(u,v)$ is a +monotone non-increasing function of each variable $u$ and $v$ when +the other variable is held fixed. + + +Suppose that $\alpha, \beta,\gamma$ are three geodesics +emanating from a point $p$ in a Riemannian manifold. Let $\angle_p +(\alpha,\beta)$, $\angle_p(\beta,\gamma)$ and +$\angle_p(\alpha,\gamma)$ be the angles of these geodesics at $p$ as +measured by the Riemannian metric. Then of course + +$$ +\angle_p(\alpha,\beta)+\angle_p(\beta,\gamma)+\angle_p(\alpha,\gamma)\le +2\pi +$$ + + since this inequality holds for the angles between straight +lines in Euclidean $n$-space. There is a second corollary of +\entryref{04518fa1ac2f} which gives an analogous result for the +associated Euclidean angles. + + +**Corollary.** Let $(M,g)$ be a complete Riemannian manifold of non-negative +curvature. Let $p,a,b,c$ be four points in $M$ and let +$\alpha,\beta,\gamma$ be minimizing geodesic arcs from the point +$p$ to $a,b,c$ respectively. Let $T(a,p,b)$, $T(b,p,c)$ and +$T(c,p,a)$ be the triangles in $M$ made out of these minimizing +geodesics and minimizing geodesics between $a,b,c$. Let +$T(a',p',b')$, $T(b',p',c')$ and $T(c',p',a')$ be planar triangles +with the same side lengths. Then + +$$ +\angle_{p'}T(a',p',b')+\angle_{p'}T(b',p',c')+\angle_{p'}T(c',p',a')\le 2\pi. +$$ + + +**Proof.** Consider the sum of these angles as the geodesic arcs in $M$ are +shortened without changing their direction. By the first property of +\entryref{04518fa1ac2f} the sum of the angles of these triangles +is a monotone decreasing function of the lengths. Of course, the +limit as the lengths all go to zero is the corresponding Euclidean +angle. The result is now clear. + +## The soul theorem + +A subset $X$ of a Riemannian manifold $(M,g)$ is said to be *totally convex* if every geodesic segment with endpoints in $X$ is +contained in $X$. Thus, a point $p$ in $M$ is totally convex if and +only if there is no broken geodesic arc in $M$ broken exactly at +$x$. + + +**Theorem.** (Cheeger-Gromoll, see +[CheegerGromollbul] and [CheegerGromoll]) Suppose that +$(M,g)$ is a connected, complete, non-compact Riemannian manifold of +non-negative sectional curvature. Then $M$ contains a soul $S\subset +M$. By definition a *soul* is a compact, totally +geodesic, totally convex submanifold (automatically of positive +codimension). Furthermore, $M$ is diffeomorphic to the total space +of the normal bundle of the $S$ in $M$. If $(M,g)$ has positive +curvature, then any soul for it is a point, and consequently $M$ is +diffeomorphic to $\Ar^n$. + + + +**Remark.** We only use the soul theorem for manifolds with positive curvature +and the fact that any soul of such a manifold is a point. A proof of +this result first appears in [GromollMeyer]. + + +The rest of this section is devoted to a sketch of the proof of this +result. Our discussion follows closely that in [Petersen] +starting on p. 349. We shall need more information about complete, +non-compact manifolds of non-negative curvature, so we review a +little of their theory as we sketch the proof of the soul theorem. + + +**Lemma.** Let $(M,g)$ be a complete, non-compact Riemannian manifold of +non-negative sectional curvature and let $p\in M$. For every +$\epsilon>0$ there is a compact subset $K=K(p,\epsilon)\subset M$ +such that for all points $q\notin K$, if $\gamma$ and $\mu$ are +minimizing geodesics from $p$ to $q$, then the angle that $\gamma$ +and $\mu$ make at $q$ is less than $\epsilon$. + + +See Fig. 2.2. + +**Proof.** The proof is by contradiction. Fix $0<\epsilon<1$ sufficiently small +so that $\mathit{cos}(\epsilon/2)<1-\epsilon^2/12$. Suppose that there +is a sequence of points $q_n$ tending to infinity such that for each +$n$ there are minimizing geodesics $\gamma_n$ and $\mu_n$ from $p$ +to $q_n$ making angle at least $\epsilon$ at $q_n$. For each $n$ let +$d_n=d(p,q_n)$. By passing to a subsequence we can suppose that for +all $n$ and $m$ the cosine of the angle at $p$ between $\gamma_n$ +and $\gamma_m$ at least $1-\epsilon^2/24$, and the cosine of the +angle at $p$ between $\mu_n$ and $\mu_m$ is at least +$1-\epsilon^2/24$. We can also assume that for all $n\ge 1$ we have +$d_{n+1}\ge (100/\epsilon^2) d_n$. Let $\delta_n=d(q_n,q_{n+1})$. +Applying the first Toponogov property at $p$, we see that +$\delta_n^2\le d_n^2+d_{n+1}^2-2d_nd_{n+1}(1-\epsilon^2/24)$. +Applying the same property at $q_n$ we have + +$$ +d_{n+1}^2\le d_n^2+\delta_n^2-2d_n\delta_n\mathit{cos}(\theta), +$$ + + where +$\theta\le \pi$ is the angle at $q_n$ between $\gamma_n$ and a +minimal geodesic joining $q_n$ to $q_{n+1}$. Thus, + +$$ +\mathit{cos}(\theta)\le +\frac{d_n-d_{n+1}(1-\epsilon^2/24)}{\delta_n}. +$$ + + By the triangle +inequality (and the fact that $\epsilon<1$) we have $\delta_n\ge +(99/\epsilon)d_n$ and $\delta_n\ge d_{n+1}(1-(\epsilon^2/100))$. +Thus, + +$$ +\mathit{cos}(\theta)\le +\epsilon^2/99-(1-\epsilon^2/24)/(1-(\epsilon^2/100))<-(1-\epsilon^2/12). +$$ + +This implies that $\mathit{cos}(\pi-\theta)>(1-\epsilon^2/12)$, which +implies that $\pi-\theta<\epsilon/2$. That is to say, the angle at +$q_n$ between $\gamma_n$ and a shortest geodesic from $q_n$ to +$q_{n+1}$ is between $\pi-\epsilon/2$ and $\pi$. By symmetry, the +same is true for the angle between $\mu_n$ and the same shortest +geodesic from $q_n$ to $q_{n+1}$. Thus, the angle between $\gamma_n$ +and $\mu_n$ at $q_n$ is less than $\epsilon$, contradicting our +assumption. + + +**Corollary.** Let $(M,g)$ be a complete, non-compact manifold of non-negative +sectional curvature. Let $p\in M$ and define a function $f\colon +M\to \Ar$ by $f(q)=d(p,q)$. Then there is $R<\infty$ such that for +$R\le s + +**Proof.** Given $(M,g)$ and $p\in M$ as in the statement of the corollary, choose a +constant $R<\infty$ such that any two minimal geodesics from $p$ to a point $q$ +with $d(p,q)\ge R/2$ make an angle at most $\pi/6$ at $q$. Now following +[Petersen] p. 335, it is possible to find a smooth unit vector field $X$ +on $U=M-\overline{B(p,R/2)}$ with the property that $f(\cdot)=d(p,\cdot)$ is +increasing along any integral curve for $X$ at a rate bounded below by $\mathit{cos}(\pi/3)$. In particular, for any $s\ge R$ each integral curve of $X$ +crosses the level set $f^{-1}(s)$ in a single point. Using this vector field +we see that for any $s,s'>R$, the pre-image $f^{-1}([s,s'])$ is homeomorphic to +$f^{-1}(s)\times[s,s']$ and that the end $f^{-1}\left([s,\infty)\right)$ is +homeomorphic to $f^{-1}(s)\times [s,\infty)$. + +In a complete, non-compact $n$-manifold of positive curvature any soul is a +point. While the proof of this result uses the same ideas as discussed above, +we shall not give a proof. Rather we refer the reader to Theorem 84 of +[Petersen] on p. 349. A soul has the property that if two minimal +geodesics emanate from $p$ and end at the same point $q\not= p$, then the angle +that they make at $q$ is less than $\pi/2$. Also, of course, the exponential +mapping is a diffeomorphism sufficiently close to the soul. Applying the above +lemma and a standard compactness argument, we see that in fact there is +$\epsilon>0$ such that all such pairs of minimal geodesics from $p$ ending at +the same point make angle less than $\pi/2-\epsilon$ at that point. Hence, in +this case there is a vector field $X$ on all of $M$ vanishing only at the soul, +and agreeing with the gradient of the distance function near the soul, so that +the distance function from $p$ is strictly increasing to infinity along each +flow line of $X$ (except the fixed point). Using $X$ one establishes that $M$ +is diffeomorphic to $\Ar^n$. It also follows that all the level surfaces +$f^{-1}(s)$ for $s>0$ are homeomorphic to $S^{n-1}$ and for $0 +**Corollary.** Let $(M,g)$ be a complete, non-compact Riemannian $n$-manifold of positive +curvature. Then for any point $p\in M$ there is a constant $R=R(p)$ such that +for any $s + +**Proof.** Given $(M,g)$ and $p$ fix $R<\infty$ sufficiently large so that +\entryref{64a0f82323d3} holds. Since $M$ is diffeomorphic to $\Ar^n$ +it has only one end and hence the level sets $f^{-1}(s)$ for $s\ge +R$ are connected. Given any compact subset $K\subset M$ there is a +larger compact set $B$ (a ball) such that $M\setminus B$ has trivial +fundamental group and trivial homology groups $H_i$ for $i +**Definition.** Let $M$ be a connected manifold. Consider the inverse system of +spaces indexed by the compact, codimension-$0$ submanifolds +$K\subset M$, where the space associated to $K$ is the finite set +$\pi_0(M\setminus K)$ with the discrete topology. The inverse limit +of this inverse system is the *space of ends* of $M$. It is a +compact space. An *end* of $M$ is a +point of the space of ends. An end ${\mathcal E}$ determines a +complementary component of each compact, codimension-$0$ submanifold +$K\subset M$, called a *neighborhood* of the end. Conversely, by +definition these neighborhoods are cofinal in the set of all +neighborhoods of the end. A sequence $\{x_n\}$ in $M$ *converges +to the end* ${\mathcal E}$ if it is eventually in every neighborhood +of the end. In fact, what we are doing is defining a topology on the +union of $M$ and its space of ends that makes this union a compact, +connected Hausdorff space which is a compactification of $M$. + + +A proper map between topological manifolds induces a map on the +space of ends, and in fact induces a map on the compactifications +sending the subspace of ends of the compactification of the domain +to the subspace of ends of the compactification of the range. + +We say that a path $\gamma\colon [a,b)\to M$ is a path *to the +end ${\mathcal E}$* if it is a proper map and it sends the end +$\{b\}$ of $[a,b)$ to the end ${\mathcal E}$ of $M$. This condition +is equivalent to saying that given a neighborhood $U$ of ${\mathcal +E}$ there is a neighborhood of the end $\{b\}$ of $[a,b)$ that maps +to $U$. + +Now suppose that $M$ has a Riemannian metric $g$. Then we can +distinguish between ends at finite and infinite distance. An end is +at *finite distance* if there is a rectifiable path of finite +length to the end. Otherwise, the end is at infinite distance. If an +end is at finite distance we have the notion of the distance from a +point $x\in M$ to the end. It is the infimum of the lengths of +rectifiable paths from $x$ to the end. This distance is always +positive. Also, notice that the Riemannian manifold is complete if +and only if has no end at finite distance. + +## The splitting theorem + +In this section we give a proof of the following theorem which is +originally due to Cheeger-Gromoll [CheegerGromoll2]. The weaker +version giving the same conclusion under the stronger hypothesis of +non-negative sectional curvature (which is in fact all we need in +this work) was proved earlier by Toponogov, see [Toponogov]. + + +**Theorem.** Suppose that $M$ is complete, of +non-negative Ricci curvature and suppose that $M$ has at least two +ends. Then $M$ is isometric to a product $N\times \Ar$ where $N$ is +a compact manifold. + + +**Proof.** We begin the proof by establishing a result of independent interest, +which was formulated as the main theorem in [CheegerGromoll2].. + + +**Lemma.** Any complete Riemannian manifold $X$ of non-negative Ricci curvature +containing a minimizing line is isometric to a product $N\times \Ar$ +for some Riemannian manifold $N$. + + +**Proof.** Given a minimizing line $\lambda\colon \Ar\to X$, define +$\lambda_\pm\colon [0,\infty)\to X$ by $\lambda_+(t)=\lambda(t)$ and +$\lambda_-(t)=\lambda(-t)$. Then we have the Busemann functions +$B_+=B_{\lambda_+}$ and $B_-=B_{\lambda_-}$. +\entryref{7c06c4b22b84} applies to both $B_+$ and $B_-$ and shows +that $\Delta (B_+ +B_-)\le 0$. On the other hand, using the fact +that $\lambda$ is distance minimizing, we see that for any $s,t>0$ +and for any $x\in M$ we have $d(x,\lambda(t))+d(x,\lambda(-s))\ge +s+t$, and hence $B_+(x)+B_-(x)\ge 0$. Clearly, $B_+(x)+B_-(x)=0$ for +any $x$ in the image of $\lambda$. Thus, the function $B_++B_-$ is +everywhere $\ge 0$, vanishes at at a least one point and satisfies +$\Delta (B_++B_-)\le 0$ in the weak sense. This is exactly the +set-up for the maximum principle, cf. [Petersen], p. 279. + + +**Theorem.** **(The Maximum Principle)** +Let $f$ be a real-valued continuous function on a connected +Riemannian manifold with $\Delta f\ge 0$ in the weak sense. Then $f$ +is locally constant near any local maximum. In particular, if $f$ +achieves its maximum then it is a constant. + + +Applying this result to $-(B_++B_-)$, we see that $B_++B_-=0$, so that +$B_-=-B_+$. It now follows that $\Delta B_+=0$ in the weak sense. By standard +elliptic regularity results this implies that $B_+$ is a smooth harmonic +function. + +Next, we show that for all $x\in M$ we have $|\nabla B_+(x)|=1$. +Fix $x\in M$. Take a sequence $t_n$ tending to infinity and consider +minimizing geodesics $\mu_{+,n}$ from $x$ to $\lambda_+(t_n)$. By +passing to a subsequence we can assume that there is a limit as +$n\rightarrow \infty$. This limit is a minimizing geodesic ray +$\mu_+$ from $x$, which we think of as being `asymptotic at +infinity' to $\lambda_+$. Similarly, we construct a minimizing +geodesic ray $\mu_-$ from $x$ asymptotic at infinity to $\lambda_+$. + Since +$\mu_+$ is a minimal geodesic ray, it follows that for any $t$ the +restriction $\mu_+|_{[0,t]}$ is the unique length minimizing +geodesic from $x$ to $\mu_+(t)$ and that $\mu_+(t)$ is not a +conjugate point along $\mu_+$. It follows by symmetry that $x$ is +not a conjugate point along the reversed geodesic $-\mu_+|_{[0,t]}$ +and hence that $x\in U_{\mu_+(t)}$. This means that the function +$d(\mu_+(t),\cdot)$ is smooth at $x$ with gradient equal to the unit +tangent vector in the negative direction at $x$ to $\mu_+$, and +consequently that $B_{\mu_+,t}$ is smooth at $x$. Symmetrically, for +any $t>0$ the function $B_{\mu_-,t}$ is smooth at $x$ with the +opposite gradient. Notice that these gradients have norm one. We +have + +$$ +B_{\mu_+,t}+B_+(x)\ge B_+=-B_-\ge -(B_{\mu_-,t}+B_-(x)). +$$ + +Of course, $B_{\mu_+,t}(x)=0$ and $B_{\mu_-,t}(x)=0$, so that + +$$ +B_{\mu_+,t}(x)+B_+(x)=-(B_{\mu_-,t}(x)+B_-(x)). +$$ + +This squeezes $B_+$ between two smooth functions with the same value +and same gradient at $x$ and hence shows that $B_+$ is $C^1$ at $x$ +and $|\nabla B_+(x)|$ is of norm one. + +Thus, $B$ defines a smooth Riemannian submersion from $M\to \Ar$ +which implies that $M$ is isometric to a product of the fiber over +the origin with $\Ar$. + +This result together with \entryref{89a6041c42b9} shows that if $M$ +satisfies the hypothesis of the theorem, then it can be written as +a Riemannian product $M=N\times \Ar$. Since $M$ has at least two +ends, it follows immediately that $N$ is compact. This completes the +proof of the theorem. + +## $\epsilon$-necks + +Certain types of (incomplete) Riemannian manifolds play an +especially important role in our analysis. The purpose of this +section is to introduce these manifolds and use them to prove one +essential result in Riemannian geometry. + +For all of the following definitions we fix $0<\epsilon<1/2$. Set +$k$ equal to the greatest integer less than or equal to +$\epsilon^{-1}$. In particular, $k\ge 2$. + + +**Definition.** Suppose that we have a fixed metric $g_0$ on a manifold $M$ and an +open submanifold $X\subset M$. We say that another metric $g$ on $X$ +is *within $\epsilon$ of $g_0|_X$ in the +$C^{[1/\epsilon]}$-topology* if, setting $k=[1/\epsilon]$ we have + +$$ +\mathit{sup}_{x\in X}\left(|g(x)-g_0(x)|_{g_0}^2+ +\sum_{\ell=1}^k|\nabla_{g_0}^\ell g(x)|_{g_0}^2\right)< +\epsilon^2, +$$ + + where the covariant derivative +$\nabla^\ell_{g_0}$ is the Levi-Civita connection of $g_0$ and +norms are the pointwise $g_0$-norms on + +$$ +\mathit{Sym}^2T^*M\otimes \underbrace{T^*M\otimes\cdots\otimes +T^*M}_{\ell-\mathit{times}}. +$$ + +More generally, given two smooth families of metrics $g(t)$ and +$g_0(t)$ on $M$ defined for $t$ in some interval $I$ we say that +the family $g(t)|_X$ is within $\epsilon$ of the family $g_0(t)|_X$ +in the $C^{[1/\epsilon]}$-topology if we have + +$$ +\mathit{sup}_{(x,t)\in X\times I}\left(\left| g(x,t) +-g_0(x,t)\right|_{g_0(t)}^2 +\sum_{\ell=1}^k\left|\nabla_{g_0}^\ell +g(x,t)\right|_{g_0}^2\right)<\epsilon^2. +$$ + + + +**Remark.** Notice that if we view a one-parameter family of metrics $g(t)$ as a +curve in the space of metrics on $X$ with the +$C^{[1/\epsilon]}$-topology then this is the statement that the two +paths are pointwise within $\epsilon$ of each other. It says nothing +about the derivatives of the paths, or equivalently about the time +derivatives of the metrics and of their covariant derivatives. We +will always be considering paths of metrics satisfying the Ricci +flow equation. In this context two one-parameter families of metrics +that are close in the $C^{2k}$-topology exactly when the $r^{th}$ +time derivatives of the $s^{th}$-covariant derivatives are close for +all $r,s$ with $s+2r\le 2k$. + + +The first object of interest is one that, up to scale, is close to a +long, round cylinder. + + +**Definition.** Let $(N,g)$ be a Riemannian manifold and $x\in N$ a point. Then *an $\epsilon$-neck structure on $(N,g)$ +centered at $x$* consists of a diffeomorphism + +$$ +\varphi\colon S^2\times (-\epsilon^{-1},\epsilon^{-1})\to N, +$$ + +with $x\in \varphi(S^2\times\{0\})$, such that the metric +$R(x)\varphi^*g$ is within $\epsilon$ in the +$C^{[1/\epsilon]}$-topology of the product of the usual Euclidean +metric on the open interval with the metric of constant Gaussian +curvature $1/2$ on $S^2$. We also use the terminology *$N$ is an +$\epsilon$-neck centered at $x$*. The image under $\varphi$ of the +family of submanifolds $S^2\times \{t\}$ is called the *family +of $2$-spheres of the $\epsilon$-neck*. The submanifold +$\varphi(S^2\times \{0\})$ is called *the central $2$-sphere* of +the $\epsilon$-neck structure. We denote by $s_N\colon N\to \Ar$ +the composition $p_2\circ \varphi^{-1}$, where $p_2$ is the +projection of $S^2\times (-\epsilon^{-1},\epsilon^{-1})$ to the +second factor. There is also the vector field $\partial/\partial +s_N$ on $N$ which is $\varphi_*$ of the standard vector field in the +interval-direction of the product. We also use the terminology of +the *plus* and *minus* end of the $\epsilon$-neck in the +obvious sense. The opposite (or reversed) $\epsilon$-neck structure +is the one obtained by composing the structure map with $\mathit{Id}_{S^2}\times -1$. We define the + *positive half of the neck* to be the region +$s_N^{-1}(0,\epsilon^{-1})$ and the *negative half* to be the +region $s_N^{-1} (-\epsilon^{-1},0)$. For any other fraction, e.g., +the left-hand three-quarters, the right-hand one-quarter, there are +analogous notions, all measured with respect to $s_N\colon N\to +(-\epsilon^{-1},\epsilon^{-1})$. We also use the terminology the +middle one-half, or middle one-third of the $\epsilon$-neck; again +these regions have their obvious meaning when measured via $s_N$. + +*An $\epsilon$-neck* in a Riemannian manifold $X$ is a +codimension-zero submanifold $N$ and an $\epsilon$-structure on $N$ +centered at some point $x\in N$. + +The *scale* of an $\epsilon$-neck $N$ centered +at $x$ is $R(x)^{-1/2}$. The scale of $N$ is denoted $r_N$. Intuitively, this +is a measure of the radius of the cross-sectional $S^2$ in the neck. In fact, +the extrinsic diameter of any $S^2$ factor in the neck is close to $\sqrt{2}\pi +r_N$. See Fig. 0.1 in the introduction. + + +Here is the result that will be so important in our later arguments. + + +**Proposition.** The following +holds for any $\epsilon>0$ sufficiently small. + Let $(M,g)$ be a complete, positively curved Riemannian +$3$-manifold. Then $(M,g)$ does not contain $\epsilon$-necks of +arbitrarily small scale. + + +**Proof.** The result is obvious if $M$ is compact, so we assume that $M$ is +non-compact. Let $p\in M$ be a soul for $M$ (\entryref{2b5f18292e62}), +and let $f$ be the distance function from $p$. Then $f^{-1}(s)$ is +connected for all $s>0$. + + +**Lemma.** Suppose that $\epsilon>0$ is sufficiently small that \entryref{04dd3c3ea1c8} from +the appendix holds. Let $(M,g)$ be a non-compact $3$-manifold of positive +curvature and let $p\in M$ be a soul for it. Then for any $\epsilon$-neck $N$ +disjoint from $p$ + the central $2$-sphere of $N$ separates the soul from the end of the manifold. In particular, if + two $\epsilon$-necks $N_1$ and $N_2$ in $M$ are disjoint from each other and from $p$, then +the central $2$-spheres of $N_1$ and $N_2$ are the boundary +components of a region in $M$ diffeomorphic to $S^2\times I$. + + +**Proof.** Let $N$ be an $\epsilon$-neck disjoint from $p$. By +\entryref{04dd3c3ea1c8} for any point $z$ in the middle third of $N$, the +boundary of the metric ball $B(p,d(z,p))$ is a topological +$2$-sphere in $N$ isotopic in $N$ to the central $2$-sphere of $N$. +Hence, the central $2$-sphere separates the soul from the end of +$M$. The second statement follows immediately by applying this to +$N_1$ and $N_2$. + +Let $N_1$ and $N_2$ be disjoint $\epsilon$-necks, each disjoint from +the soul. By the previous lemma, the central $2$-spheres $S_1$ and +$S_2$ of these necks are smoothly isotopic to each other and they +are the boundary components of a region diffeomorphic to $S^2\times +I$. Reversing the indices if necessary we can assume that $N_2$ is +closer to $\infty$ than $N_1$, i.e., further from the soul. +Reversing the directions of the necks if necessary, we can arrange +that for $i=1,2$ the function $s_{N_i}$ is increasing as we go away +from the soul. We define $C^\infty$- functions $\psi_i$ on $N_i$, +functions depending only on $s_{N_i}$, as follows. The function +$\psi_1$ is zero on the negative side of the middle third of $N_1$ +and increases to be identically one on the positive side of the +middle third. The function $\psi_2$ is one on the negative side of +the middle third of $N_2$ and decreases to be zero on the positive +side. We extend $\psi_1,\psi_2$ to a function $\psi$ defined on all +of $M$ by requiring that it be identically one on the region $X$ +between $N_1$ and $N_2$ and to be identically zero on $M\setminus +(N_1\cup X\cup N_2)$. + +Let $\lambda$ be a geodesic ray from the soul of $M$ to infinity, +and $B_\lambda$ its Busemann function. Let $N$ be any +$\epsilon$-neck disjoint from the soul, with $ s_N$ direction chosen +so that it points away from the soul. At any point of the middle +third of $N$ where $B_\lambda$ is smooth, $\nabla B_\lambda$ is a +unit vector in the direction of the unique minimal geodesic ray from +the end of $\lambda$ to this point. Invoking \entryref{88a5a2233418} +from the appendix we see that at such points $\nabla B_\lambda$ is +close to $-R(x)^{1/2}\partial/\partial s_N$, where $x\in N$ is the +center of the $\epsilon$-neck. Since $\nabla B_\lambda$ is $L^2$ its +non-smooth points have measure zero and hence, the restriction of +$\nabla B_\lambda$ to the middle third of $N$ is close in the +$L^2$-sense to $-R(x)^{1/2}\partial/\partial s_N$. + +Applying this to $N_1$ and $N_2$ we see that + +$$ +\int_M\langle \nabla B_\lambda,\nabla\psi\rangle d\mathit{vol}=\left(\alpha_2R(x_2)^{-1}-\alpha_1R(x_1)^{-1}\right)\mathit{Vol}_{h_0}(S^2)), +$$ + + where $h(0)$ is the round metric of +scalar curvature $1$ and where each of $\alpha_1$ and $\alpha_2$ +limits to $1$ as $\epsilon$ goes to zero. Since $\psi\ge 0$, +\entryref{7c06c4b22b84} tells us that the left-hand side of +Equation (2.2) must be $\ge 0$. This shows that, +provided that $\epsilon$ is sufficiently small, $R(x_2)$ is bounded +above by $2R(x_1)$. This completes the proof of the proposition. + + +**Corollary.** Fix $\epsilon>0$ sufficiently small so that \entryref{04dd3c3ea1c8} holds. Then +there is a constant $C<\infty$ depending on $\epsilon$ such that the following +holds. Suppose that $M$ is a non-compact $3$-manifold of positive sectional +curvature. Suppose that $N$ is an $\epsilon$-neck in $M$ centered at a point +$x$ and disjoint from a soul $p$ of $M$. Then for any $\epsilon$-neck $N'$ that +is separated from $p$ by $N$ with center $x'$ we have $R(x')\le CR(x)$. + + +## Forward difference quotients + +Let us review quickly some standard material on forward difference quotients. + +Let $f\colon [a,b]\to \Ar$ be a continuous function on an interval. +We say that the *forward difference quotient of $f$ at a point +$t\in [a,b)$, denoted $\frac{df}{dt}(t)$, is less than +$c$* provided that + +$$ +\overline\mathit{lim}_{\triangle t\rightarrow +0^+}\frac{f(t+\triangle t)-f(t)}{\triangle t}\le c. +$$ + + We say that it +is greater than or equal to $c'$ if + +$$ +c'\le \underline\mathit{lim}_{\triangle t\rightarrow 0^+}\frac{f(t+\triangle +t)-f(t)}{\triangle t}. +$$ + + Standard comparison arguments show: + + +**Lemma.** Suppose that $f\colon [a,b]\to \Ar$ is a continuous function. Suppose that +$\psi$ is a $C^1$-function on $[a,b]\times \Ar$ and suppose that +$\frac{df}{dt}(t)\le \psi(t,f(t))$ for every $t\in [a,b)$ in the sense of +forward difference quotients. Suppose also that there is a function $G(t)$ +defined on $[a,b]$ that satisfies the differential equation +$G'(t)=\psi(t,G(t))$ and has $f(a)\le G(a)$. Then $f(t)\le G(t)$ for all $t\in +[a,b]$. + + +The application we shall make of these results is the following. + + +**Proposition.** Let $M$ be a smooth manifold with a smooth vector field $\chi$ and a +smooth function $\mathbf{t}\colon M\to [a,b]$ with $\chi(\mathbf{t})=1$. +Suppose also that $F\colon M\to \Ar$ is a smooth function with the +properties: + +1. for each $t_0\in [a,b]$ the restriction of $F$ to the level set $\mathbf{t}^{-1}(t_0)$ achieves its maximum, and +1. the subset ${\mathcal Z}$ of $M$ +consisting of all $x$ for which $F(x)\ge F(y)$ for all $y\in \mathbf{t}^{-1}(\mathbf{t}(x))$ is a compact set. + + Suppose also that at each $x\in {\mathcal Z}$ we have $\chi(F(x))\le \psi(\mathbf{t}(x),F(x))$. Set $F_\mathit{max}(t)=\mathit{max}_{x\in \mathbf{t}^{-1}(t)}F(x)$. Then $F_\mathit{max}(t)$ is a continuous function and + +$$ +\frac{dF_\mathit{max}}{dt}(t)\le \psi(t,F_\mathit{max}(t)) +$$ + +in the sense of forward difference quotients. + Suppose that $G(t)$ satisfies the +differential equation + +$$ +G'(t)=\psi(t,G(t)) +$$ + + and has initial condition $F_\mathit{max}(a)\le G(a)$. Then for all $t\in [a,b]$ we have + +$$ +F_\mathit{max}(t)\le G(t). +$$ + + +**Proof.** Under the given hypothesis it is a standard and easy exercise to +establish the statement about the forward difference quotient of +$F_\mathit{max}$. The second statement then is an immediate corollary +of the previous result. diff --git a/projects/poincare-conjecture/.astrolabe/docs/03-flowbasics.mdx b/projects/poincare-conjecture/.astrolabe/docs-src/04-basics-of-ricci-flow.mdx similarity index 68% rename from projects/poincare-conjecture/.astrolabe/docs/03-flowbasics.mdx rename to projects/poincare-conjecture/.astrolabe/docs-src/04-basics-of-ricci-flow.mdx index ecc906c5..f3f9da58 100644 --- a/projects/poincare-conjecture/.astrolabe/docs/03-flowbasics.mdx +++ b/projects/poincare-conjecture/.astrolabe/docs-src/04-basics-of-ricci-flow.mdx @@ -1,27 +1,29 @@ -# Basics of Ricci flow + + +# Chapter 3 -- Basics of Ricci flow In this chater we introduce the Ricci flow equation due to R. -Hamilton . For the basic material on the Ricci flow -equation see . +Hamilton [Hamilton3MPRC]. For the basic material on the Ricci flow +equation see [ChowKnopf]. ## The definition of the Ricci flow -\quad - + **Definition.** The *Ricci flow equation* is the following evolution equation for a Riemannian metric: $$ - -\frac{\partial g}{\partial t}= -2Ric(g). +\frac{\partial g}{\partial t}= -2\mathit{Ric}(g). $$ + A solution to this equation (or a *Ricci flow*) is a one-parameter family of metrics $g(t)$, parameterized by $t$ in a non-degenerate interval -$I$, on a smooth manifold $M$ satisfying Equation~(). If $I$ has -an initial point $t_0$ then $(M,g(t_0))$ is called *the initial condition -of* or *the initial metric for* the +$I$, on a smooth manifold $M$ satisfying Equation (3.1). If $I$ has +an initial point $t_0$ then $(M,g(t_0))$ is called *the initial condition +of* or *the initial metric for* the Ricci flow (or of the solution). + Let us give a quick indication of what the Ricci flow equation means. In harmonic coordinates $(x^1,\ldots,x^n)$ about $p$, @@ -29,31 +31,31 @@ means. In harmonic coordinates $(x^1,\ldots,x^n)$ about $p$, we have $$ -Ric_{ij}=Ric(\frac{\partial}{\partial x^i},\frac{\partial}{\partial x^j})= +\mathit{Ric}_{ij}=\mathit{Ric}(\frac{\partial}{\partial x^i},\frac{\partial}{\partial x^j})= -\frac{1}{2}\triangle g_{ij} + Q_{ij}(g^{-1},\partial g) $$ where $Q$ is a quadratic form in $g^{-1}$ and $\partial g$, and so in particular is a lower order term -in the derivatives of $g$. See Lemma 3.32 on page 92 of . +in the derivatives of $g$. See Lemma 3.32 on page 92 of [ChowKnopf]. So, in these coordinates, the Ricci flow equation is actually a heat -equation for the Riemannian metric +equation for the Riemannian metric $$ \frac{\partial}{\partial t}g = \triangle g +2Q(g^{-1},\partial g). $$ -**Definition.** -We introduce some notation that will be used throughout. Given a + +**Definition.** We introduce some notation that will be used throughout. Given a Ricci flow $(M^n,g(t))$ defined for $t$ contained in an interval -$I$, then the *space-time* for this flow is $M\times I$. The +$I$, then the *space-time* for this flow is $M\times I$. The *$t$ time-slice* of space-time is the Riemannian manifold $M\times\{t\}$ with the Riemannian metric $g(t)$. Let ${\mathcal HT}(M\times I)$ be the *horizontal tangent bundle* of space-time, i.e., the bundle of tangent vectors to the time-slices. It is a smooth, rank-$n$ subbundle the tangent bundle of space-time. The evolving -metric $g(t)$ is then a smooth section of $Sym^2{\mathcal +metric $g(t)$ is then a smooth section of $\mathit{Sym}^2{\mathcal HT}^*(M\times I)$. We denote points of space-time as pairs $(p,t)$. Given $(p,t)$ and any $r>0$ we denote by $B(p,t,r)$ the metric ball of radius $r$ centered at $(p,t)$ in the $t$ time-slice. For any @@ -64,30 +66,27 @@ $P(x,t,r,-\Delta t)$ with a time-slice other that the $t$ time-slice need not be a metric ball in that time-slice. There is the corresponding notion of a forward parabolic neighborhood $P(x,t,r,\Delta t)$ provided that $[t,t+\Delta t]\subset I$. + ## Some exact solutions to the Ricci flow ### Einstein manifolds -Let $g_0$ be an Einstein metric: $Ric(g_0) = \lambda g_0$, where $\lambda$ is a constant. Then for -any positive constant $c$, setting $g = cg_0$ we have $Ric(g) -= Ric(g_0) = \lambda g_0= \frac{\lambda}{c} g.$ Using this we +Let $g_0$ be an Einstein metric: $\mathit{Ric}(g_0) = \lambda g_0$, where $\lambda$ is a constant. Then for +any positive constant $c$, setting $g = cg_0$ we have $\mathit{Ric}(g) += \mathit{Ric}(g_0) = \lambda g_0= \frac{\lambda}{c} g.$ Using this we can construct solutions to the Ricci flow equation as follows. Consider $g(t) = u(t)g_{0}$. If this one-parameter family of metrics is a solution of the Ricci flow, then -$$ - $$ \begin{aligned} -\frac{\partial g}{\partial t}& =u'(t)g_{0}\\& = -2Ric(u(t)g_{0})\\& -= -2Ric(g_{0})\\& = -2\lambda g_{0}. +\frac{\partial g}{\partial t}& =u'(t)g_{0}\\& = -2\mathit{Ric}(u(t)g_{0})\\& += -2\mathit{Ric}(g_{0})\\& = -2\lambda g_{0}. \end{aligned} $$ -$$ - -So $u'(t)= -2\lambda$, and hence $u(t) = 1-2 \lambda t$. Thus +So $u'(t)= -2\lambda$, and hence $u(t) = 1-2 \lambda t$. Thus $g(t)= (1-2 \lambda t)g_{0}$ is a solution of the Ricci flow. The cases $\lambda >0, \lambda=0,$ and $\lambda< 0$ correspond to *shrinking*, *steady* @@ -95,8 +94,8 @@ and *expanding* solutions. Notice that in the shrinking case the solution exists for $t\in [0,\frac{1}{2\lambda})$ and goes singular at $t=\frac{1}{2\lambda}$. -\begin{exam} -The standard metric on each of $S^{n}, \Ar^{n},$ and ${\mathbb + +**Example.** The standard metric on each of $S^{n}, \Ar^{n},$ and ${\mathbb H}^{n}$ is Einstein. Ricci flow is contracting on $S^n$, constant on $\Ar^n$, and expanding on ${\mathbb H}^n$. The Ricci flow on $S^n$ has a finite-time singularity where the diameter of the manifold @@ -104,25 +103,31 @@ goes to zero and the curvature goes uniformly to $+\infty$. The Ricci flow on ${\mathbb H}^n$ exists for all $t\ge 0$ and as $t$ goes to infinity the distance between any pair of points grows without bound and the curvature goes uniformly to zero. -\end{exam} + -\begin{exam} -$\mathbb{C}P^{n}$ equipped with the Fubini-Study metric, which is induced + +**Example.** $\mathbb{C}P^{n}$ equipped with the Fubini-Study metric, which is induced from the standard metric of $S^{2n+1}$ under the Hopf fibration with the fibers of great circles, is Einstein. -\end{exam} + + + +**Example.** Let $h_0$ be the round metric on $S^2$ with constant Gausssian +curvature $1/2$. Set $h(t)=(1-t)h_0$. Then the flow + +$$ +(S^2,h(t)),\ +-\infty The standard shrinking round cylinder is a model for evolving -$\epsilon$-necks. In Chapter~ we introduced the +$\epsilon$-necks. In Chapter 1 we introduced the notion of an $\epsilon$-neck. In the case of flows in order to take smooth geometric limits, it is important to have a stronger version of this notion. In this stronger notion, the neck not only exists in @@ -132,13 +137,18 @@ round cylinder on the entire time interval. The existence of evolving necks is exploited when we study limits of Ricci flows. -**Definition.** -Let $(M,g(t))$ be a Ricci flow. An *evolving $\epsilon$-neck + +**Definition.** Let $(M,g(t))$ be a Ricci flow. An *evolving $\epsilon$-neck centered at $(x,t_0)$ and defined for rescaled time $t_1$* is an $\epsilon$-neck -$$\varphi\colon S^2\times + +$$ +\varphi\colon S^2\times (-\epsilon^{-1},\epsilon^{-1})\buildrel\cong\over\longrightarrow -N\subset (M,g(t))$$ centered at $(x,t_0)$ with the property that +N\subset (M,g(t)) +$$ + + centered at $(x,t_0)$ with the property that pull-back via $\varphi$ of the family of metrics $R(x,t_0)g(t')|_N,\ -t_1< t'\le 0$, where $t_1=R(x,t_0)^{-1}(t-t_0)$, is within $\epsilon$ in the $C^{[1/\epsilon]}$-topology of the product of the @@ -147,11 +157,12 @@ with scalar curvature $1/(1-t')$ at time $t'$. A *strong $\epsilon$-neck centered at $(x,t_0)$* in a Ricci flow is an evolving $\epsilon$-neck centered at $(x,t_0)$ and defined for -rescaled time $1$, see Fig.~. +rescaled time $1$, see Fig. 3.1. + ### Solitons -A *Ricci soliton*} +A *Ricci soliton* is a Ricci flow $(M,g(t)),\ 0\le t0$. Then, for all $t\in [0,T)$ we define a function -$$\sigma(t)=1-2\lambda t,$$ + +$$ +\sigma(t)=1-2\lambda t, +$$ + and a vector field -$$Y_t(x)=\frac{X(x)}{\sigma(t)}.$$ + +$$ +Y_t(x)=\frac{X(x)}{\sigma(t)}. +$$ + Then we define $\varphi_t$ as the one-parameter family of diffeomorphisms generated by the time-dependent vector fields $Y_t$. -**Claim.** - The flow + +**Claim.** The flow $(M,g(t)),\ 0\le t0$. + **Proof.** We check that this flow satisfies the Ricci flow equation; from that, the result follows immediately. We have -$$ - $$ \begin{aligned} \frac{\partial g(t)}{\partial t} & = & \sigma'(t)\varphi_t^*g(0)+\sigma(t)\varphi_t^*{\mathcal L}_{Y(t)}g(0)\\ & = & \varphi_t^*(-2\lambda+{\mathcal L}_X)g(0)\\ -& = & \varphi_t^*(-2Ric(g(0))) =-2Ric(\varphi_t^*(g(0))). +& = & \varphi_t^*(-2\mathit{Ric}(g(0))) =-2\mathit{Ric}(\varphi_t^*(g(0))). \end{aligned} $$ -$$ +Since $\mathit{Ric}(\alpha g)=\mathit{Ric}(g)$ for any $\alpha>0$, it follows that -Since $Ric(\alpha g)=Ric(g)$ for any $\alpha>0$, it follows that -$$\frac{\partial g(t)}{\partial t} = -2Ric(g(t)).$$ +$$ +\frac{\partial g(t)}{\partial t} = -2\mathit{Ric}(g(t)). +$$ There is one class of shrinking solitons which are of special importance to us. These are the gradient shrinking solitons. + **Definition.** A shrinking soliton $(M,g(t)),\ 0\le t -**Proposition.** - Suppose we have a complete Riemannian manifold $(M,g(0))$, a + +**Proposition.** Suppose we have a complete Riemannian manifold $(M,g(0))$, a smooth function $f\colon M\to \Ar$, and a constant $\lambda>0$ such that $$ - --Ric(g(0))=Hess(f)-\lambda g(0). +-\mathit{Ric}(g(0))=\mathit{Hess}(f)-\lambda g(0). $$ Then there is $T>0$ and a gradient shrinking soliton -$(M,g(t))$ defined for $0\le t **Proof.** Since -$${\mathcal L}_{\nabla f}g(0)=2Hess(f),$$ - Equation~() is the soliton equation, -Equation~(), with the vector field $X$ being the + +$$ +{\mathcal L}_{\nabla f}g(0)=2Hess(f), +$$ + + Equation (3.3) is the soliton equation, +Equation (3.2), with the vector field $X$ being the gradient vector field $\nabla f$. It is a shrinking soliton by assumption since $\lambda>0$. + **Definition.** In this case we say that $(M,g(0))$ and $f\colon M\to \Ar$ *generate* a gradient shrinking soliton. + ## Local existence and uniqueness The following is the first basic result in the theory -- local existence and uniqueness for Ricci flow in the case of compact manifolds. -**Theorem.** {\bf (Hamilton, cf. -.)} Let $(M,g_0)$ be a compact + +**Theorem.** **(Hamilton, cf. +[Hamilton3MPRC].)** Let $(M,g_0)$ be a compact Riemannian manifold of dimension $n$. -\begin{enumerate} -\item There is a $T>0$ depending on $(M,g_0)$ and a Ricci flow + +1. There is a $T>0$ depending on $(M,g_0)$ and a Ricci flow $(M,g(t)),\ 0\le t We remark that the Ricci flow is a weakly parabolic system where degeneracy comes from the gauge invariance of the equation under diffeomorphisms. Therefore the short-time existence does not come from general theory. R. Hamilton's original proof of the short-time existence was involved and used the Nash-Moser inverse function -theorem, . Soon after, DeTurck +theorem, [HamiltonIFTNM]. Soon after, DeTurck [DeTurck] substantially simplified the short-time existence proof by breaking the diffeomorphism invariance of the equation. For the reader's convenience, and also because in establishing the uniqueness for -Ricci flows from the standard solution in Section~ +Ricci flows from the standard solution in Section 12.4 we use a version of this idea in the non-compact case, we sketch DeTurck's argument. -**Proof.** Let's sketch the proof due to DeTurck , cf, Section -3 of Chapter 3 starting on page 78 of for more details. First, we compute the first variation +**Proof.** Let's sketch the proof due to DeTurck [DeTurck], cf, Section +3 of Chapter 3 starting on page 78 of [ChowKnopf] for more details. First, we compute the first variation at a Riemannian metric $g$ of minus twice the Ricci curvature tensor in the direction $h$: -$$ - $$ \begin{aligned} -\delta_g(-2Ric)(h)=\triangle h-Sym(\nabla V)+S +\delta_g(-2\mathit{Ric})(h)=\triangle h-\mathit{Sym}(\nabla V)+S \end{aligned} $$ +where: + +1. $V$ is the one-form given by + +$$ +V_k=\frac{1}{2}g^{pq}(\nabla_ph_{qk}+\nabla_qh_{pk}-\nabla_kh_{pq}), $$ -where: -\begin{enumerate} -\item $V$ is the one-form given by -$$V_k=\frac{1}{2}g^{pq}(\nabla_ph_{qk}+\nabla_qh_{pk}-\nabla_kh_{pq}),$$ -\item $Sym(\nabla V)$ is the symmetric two-tensor obtained by +1. $\mathit{Sym}(\nabla V)$ is the symmetric two-tensor obtained by symmetrizing the covariant derivative of $V$, and -\item $S$ is a symmetric two-tensor constructed from the inverse of the metric, +1. $S$ is a symmetric two-tensor constructed from the inverse of the metric, the Riemann curvature tensor and $h$, but involves no derivatives of $h$. -\end{enumerate} Now let $g_0$ be the initial metric. For any metric $g$ we define a one-form $\hat W$ by taking the trace, with respect to $g$, of the matrix-valued one-form that is the difference of the connections of $g$ and $g_0$. Now we form a second-order operator of $g$ by setting -$$P(g)={\mathcal L}_Wg,$$ + +$$ +P(g)={\mathcal L}_Wg, +$$ + the Lie derivative of $g$ with respect to the vector field $W$ dual to $\hat W$. Thus, in local coordinates we have $P(g)_{ij}=\nabla_i\hat W_j+\nabla_j\hat W_i$. The linearization at $g$ of the second-order operator $P$ in the direction $h$ is symmetric and is given by -$$\delta_gP(h)=Sym(\nabla V)+T$$ + +$$ +\delta_gP(h)=\mathit{Sym}(\nabla V)+T +$$ + where $T$ is a first-order operator in $h$. -Thus, defining $Q=-2Ric+P$ we have -$$\delta_g(Q)(h)=\triangle h+U$$ -where $U$ is a first-order operator in $h$. -Now we introduce the Ricci-DeTurck flow +Thus, defining $Q=-2\mathit{Ric}+P$ we have $$ +\delta_g(Q)(h)=\triangle h+U +$$ + +where $U$ is a first-order operator in $h$. +Now we introduce the Ricci-DeTurck flow $$ \begin{aligned} - -\frac{\partial g}{\partial t}&=-2Ric(g)+P. +\frac{\partial g}{\partial t}&=-2\mathit{Ric}(g)+P. \end{aligned} $$ -$$ - -The computations above show that the Ricci-DeTurck flow is strictly parabolic. - Thus, Equation~() has a +The computations above show that the Ricci-DeTurck flow is strictly parabolic. + Thus, Equation (3.4) has a short-time solution $\bar g(t)$ with $\bar g(0)=g_0$ by the standard PDE theory. Given this solution $\bar g(t)$ we define the time-dependent vector field $W(t)=W(\bar g(t),g_0)$ as above. - Let $\phi_t$ be a one-parameter family of diffeomorphisms, with $\phi_0=Id$, + Let $\phi_t$ be a one-parameter family of diffeomorphisms, with $\phi_0=\mathit{Id}$, generated by this time-dependent vector field, i.e., -$$\frac{\partial\phi_t}{\partial t}=W(t).$$ + +$$ +\frac{\partial\phi_t}{\partial t}=W(t). +$$ + Then, direct computation shows that $g(t)=\phi_t^*\bar g(t)$ solves the Ricci flow equation. In performing surgery at time $T$, we will have an open submanifold $\Omega$ of -the compact manifold with the following property. As $t$ approaches $T$ from +the compact manifold with the following property. As $t$ approaches $T$ from below, the metrics $g(t)|_\Omega$ converge smoothly to a limiting metric $g(T)$ -on $\Omega$. We will `cut away' the rest of the manifold $M\setminus \Omega$ +on $\Omega$. We will 'cut away' the rest of the manifold $M\setminus \Omega$ where the metrics are not converging and glue in a piece $E$ coming from the standard solution to form a new compact manifold $M'$. Then we extend the Riemannian metric $g(T)$ on $\Omega$ to one defined on $M'=\Omega\cup E$. The @@ -331,14 +370,15 @@ $(\Omega, \tilde g(t)),\ T\le t +**Proposition.** Suppose that $(U,g(t)),\ a\le t ## Evolution of curvatures @@ -348,52 +388,62 @@ written in local coordinates $$ \frac{\partial g_{ij}}{\partial t} = --2Ric_{ij} +-2\mathit{Ric}_{ij} $$ - implies a heat equation for the Riemann + + implies a heat equation for the Riemann curvature tensor $R_{ijkl}$ which we now derive. Various second-order derivatives of the curvature tensor are likely to differ by terms quadratic in the curvature tensors. To this end we introduce -the tensor $$B_{ijkl} = g^{pr}g^{qs}R_{ipjq}R_{krls}.$$ Note that we +the tensor + +$$ +B_{ijkl} = g^{pr}g^{qs}R_{ipjq}R_{krls}. +$$ + + Note that we have the obvious symmetries -$$B_{ijkl} = B_{jilk} = B_{klij},$$ but the other symmetries of + +$$ +B_{ijkl} = B_{jilk} = B_{klij}, +$$ + + but the other symmetries of the curvature tensor $R_{ijkl}$ may fail to hold for $B_{ijkl}$. -**Theorem.** -The curvature tensor $R_{ijkl}$, the Ricci curvature $Ric_{ij}$, the scalar -curvature $R$, and the volume form $dvol(x,t)$ satisfy the following evolution + +**Theorem.** The curvature tensor $R_{ijkl}$, the Ricci curvature $\mathit{Ric}_{ij}$, the scalar +curvature $R$, and the volume form $d\mathit{vol}(x,t)$ satisfy the following evolution equations under Ricci flow: -$$ - $$ \begin{aligned} \frac{\partial R_{ijkl}}{\partial t} &= & \triangle R_{ijkl} +2(B_{ijkl} - -B_{ijlk}-B_{iljk} + B_{ikjl})\nonumber \\& & - g^{pq}(R_{pjkl}Ric_{qi} + -R_{ipkl}Ric_{qj} + R_{ijpl}Ric_{qk} + -R_{ijkp}Ric_{ql}) \\ - \frac{\partial}{\partial t}Ric_{jk} & = & \triangle -Ric_{jk}+2g^{pq}g^{rs}R_{pjkr}Ric_{qs}-2g^{pq}Ric_{jp}Ric_{qk} \\ -\frac{\partial}{\partial t}R & = & \Delta R+2|Ric|^2 \\ - \frac{\partial}{\partial t}dvol(x,t) & = & --R(x,t)dvol(x,t). +B_{ijlk}-B_{iljk} + B_{ikjl})\nonumber \\& & - g^{pq}(R_{pjkl}\mathit{Ric}_{qi} + +R_{ipkl}\mathit{Ric}_{qj} + R_{ijpl}\mathit{Ric}_{qk} + +R_{ijkp}\mathit{Ric}_{ql}) \\ + \frac{\partial}{\partial t}\mathit{Ric}_{jk} & = & \triangle +\mathit{Ric}_{jk}+2g^{pq}g^{rs}R_{pjkr}\mathit{Ric}_{qs}-2g^{pq}\mathit{Ric}_{jp}\mathit{Ric}_{qk} \\ +\frac{\partial}{\partial t}R & = & \Delta R+2|\mathit{Ric}|^2 \\ + \frac{\partial}{\partial t}d\mathit{vol}(x,t) & = & +-R(x,t)d\mathit{vol}(x,t). \end{aligned} $$ + -$$ - -These equations are contained in Lemma 6.15 on page 179, Lemma 6.9 - on page 176, Lemma 6.7 on page 176, and Equation (6.5) on -page 175 of , respectively. +These equations are contained in Lemma 6.15 on page 179, Lemma 6.9 + on page 176, Lemma 6.7 on page 176, and Equation (6.5) on +page 175 of [ChowKnopf], respectively. Let us derive some consequences of these evolution equations. The first result is obvious from the Ricci flow equation and will be used implicitly throughout the paper. -**Lemma.** -Suppose that $(M,g(t)),\ a +**Lemma.** Suppose that $(M,g(t)),\ a **Proof.** The Ricci flow equation tells us that non-negative Ricci curvature implies that $\partial g/\partial t\le 0$. Hence, the length of any tangent vector in $M$, and consequently the length @@ -401,26 +451,37 @@ of any path in $M$, is a non-increasing function of $t$. Since the distance betw over all rectifiable paths from $x$ to $y$ of the length of the path, this function is also a non-increasing function of $t$. + **Lemma.** Suppose that $(M,g(t)),\ 0\le t\le T$, is a Ricci flow and $|Rm(x,t)|\le K$ for all $x\in M$ and all $t\in [0,T]$. Then there -are constants $A,A'$ depending on $K,T$ and the dimension such +are constants $A,A'$ depending on $K,T$ and the dimension such that: -\begin{enumerate} -\item For any non-zero tangent vector $v\in T_xM$ and any $t\le T$ we have -$$A^{-1}\langle v,v\rangle_{g(0)}\le \langle v,v\rangle_{g(t)}\le A\langle -v,v\rangle_{g(0)}.$$ -\item For any open subset $U\subset M$ and any $t\le T$ we have -$$(A')^{-1}Vol_0(U)\le Vol_t(U)\le A'Vol_0(U).$$ -\end{enumerate} + +1. For any non-zero tangent vector $v\in T_xM$ and any $t\le T$ we have + +$$ +A^{-1}\langle v,v\rangle_{g(0)}\le \langle v,v\rangle_{g(t)}\le A\langle +v,v\rangle_{g(0)}. +$$ + +1. For any open subset $U\subset M$ and any $t\le T$ we have + +$$ +(A')^{-1}\mathit{Vol}_0(U)\le \mathit{Vol}_t(U)\le A'\mathit{Vol}_0(U). +$$ + **Proof.** The Ricci flow equation yields -$$\frac{d}{dt}\left(\langle v,v\rangle_{g(t)}\right)=-2Ric(v,v).$$ -The bound on the Riemann curvature gives a bound on $Ric$. + +$$ +\frac{d}{dt}\left(\langle v,v\rangle_{g(t)}\right)=-2\mathit{Ric}(v,v). +$$ + +The bound on the Riemann curvature gives a bound on $\mathit{Ric}$. Integrating yields the result. The second statement is proved -analogously using Equation~(). +analogously using Equation (3.5). -## Curvature evolution in an evolving orthonormal -frame +## Curvature evolution in an evolving orthonormal frame It is often best to study the evolution of the representative of the tensor in an orthonormal frame $F$. Let $(M,g(t)),\ 0\le t +**Claim.** Suppose that ${\mathcal F}(0)=\{F_a\}_a$ is a local $g(0)$-orthonormal frame, + and suppose that ${\mathcal F}(t)$ evolves according to Equation (3.6). Then for all $t\in [0,T)$ the frame ${\mathcal F}(t)$ is a local $g(t)$-orthonormal frame. + -**Proof.** $$ +**Proof.** $$ \begin{aligned} @@ -466,28 +529,50 @@ $$ \frac{\partial F_a}{\partial t},F_b\rangle + \langle F_b,\frac{\partial F_b}{\partial t}\rangle + \frac {\partial g}{\partial t}(F_a,F_b) \\ - & = & Ric(F_a,F_b)+Ric(F_b,F_a)-2Ric(F_a,F_b)=0. + & = & \mathit{Ric}(F_a,F_b)+\mathit{Ric}(F_b,F_a)-2\mathit{Ric}(F_a,F_b)=0. \end{aligned} $$ -$$ - Notice that if ${\mathcal F}'(0)=\{F'_a\}_a$ is another frame related to ${\mathcal F}(0)$ by, say, -$$F'_a=A^b_aF_b$$ -then $$F_a(t)=A^b_aF_b(t).$$ This means that the evolution of frames + +$$ +F'_a=A^b_aF_b +$$ + +then + +$$ +F_a(t)=A^b_aF_b(t). +$$ + + This means that the evolution of frames actually defines a bundle automorphism -$$\Phi\colon TM|_U\times [0,T)\to TM|_U\times [0,T)$$ + +$$ +\Phi\colon TM|_U\times [0,T)\to TM|_U\times [0,T) +$$ + covering the identity map of $U\times[0,T)$ which is independent of the choice of initial frame and is the identity at time $t=0$. Of course, since the resulting bundle automorphism is independent of the initial frame it globalizes to produce a bundle isomorphism -$$\Phi\colon TM\times[0,T)\to TM\times [0,T)$$ + +$$ +\Phi\colon TM\times[0,T)\to TM\times [0,T) +$$ + covering the identity on $M\times [0,T)$. We view this as an evolving identification $\Phi_t$ of $TM$ with itself which is the -identity at $t=0$. The content of Claim~ is: +identity at $t=0$. The content of \entryref{29b84db22679} is: + + +**Corollary.** -**Corollary.** $$\Phi_t^*(g(t))=g(0).$$ +$$ +\Phi_t^*(g(t))=g(0). +$$ + Returning to the local situation of the orthonormal frame ${\mathcal F}$, we set ${\mathcal F}^*=\{F^1,\ldots, F^n\}$ equal the dual @@ -495,23 +580,30 @@ coframe to $\{F_1,\ldots,F_n\}$. In this coframe the Riemann curvature tensor is given by $R_{abcd}F^aF^bF^cF^d$ where $$ - R_{abcd} = +R_{abcd} = R_{ijkl}F^{i}_{a}F^{j}_{b}F^{k}_{c}F^{l}_{d}. $$ + One advantage of working in the evolving frame is that the evolution equation for the Riemann curvature tensor simplifies: -**Lemma.** -Suppose that the orthonormal frame ${\mathcal F}(t)$ evolves by -Formula~(). Then we have the evolution equation -$$\frac{\partial R_{abcd}}{\partial t} = \triangle R_{abcd} + -2(B_{abcd} +B_{acbd}-B_{abdc}-B_{adbc}),$$ where $B_{abcd} = + +**Lemma.** Suppose that the orthonormal frame ${\mathcal F}(t)$ evolves by +Formula (3.6). Then we have the evolution equation + +$$ +\frac{\partial R_{abcd}}{\partial t} = \triangle R_{abcd} + +2(B_{abcd} +B_{acbd}-B_{abdc}-B_{adbc}), +$$ + + where $B_{abcd} = \sum_{e,f}R_{aebf}R_{cedf}$. + -**Proof.** For a proof see Theorem 2.1 in . +**Proof.** For a proof see Theorem 2.1 in [Hamiltonharnack]. -Of course, the other way to describe all of this is to consider the +Of course, the other way to describe all of this is to consider the four-tensor $\Phi_t^*({\mathcal R}_{g(t)})=R_{abcd}F^aF^bF^cF^d$ on $M$. Since $\Phi_t$ is a bundle map but not a bundle map induced by a diffeomorphism, even though the pullback of the metric @@ -522,45 +614,56 @@ the curvature $\Phi_t^*{\mathcal R}_{g(t)}$ is constant. It simplifies the notation somewhat to work directly with a basis of $\wedge^2TM$. We chose an orthonormal basis -$$\{\varphi^{1},\ldots,\varphi^{\frac{n(n-1)}{2}}\},$$ -of $\wedge^2T^*_pM$ where we have + +$$ +\{\varphi^{1},\ldots,\varphi^{\frac{n(n-1)}{2}}\}, +$$ + +of $\wedge^2T^*_pM$ where we have + $$ \varphi^{\alpha}(F_{a},F_{b}) = \varphi^{\alpha}_{ab} $$ + and write the curvature tensor in this basis as ${\mathcal T} = ({\mathcal T}_{\alpha\beta})$ so that $$ - R_{abcd}={\mathcal T}_{\alpha\beta}\varphi^{\alpha}_{ab}\varphi^{\beta}_{cd}. $$ -**Proposition.** -The evolution of the curvature operator ${\mathcal -T}(t)=\Phi_t^*Rm(g(t))$ is given by + +**Proposition.** The evolution of the curvature operator ${\mathcal +T}(t)=\Phi_t^*\mathit{Rm}(g(t))$ is given by $$ \frac{\partial {\mathcal T}_{\alpha\beta}}{\partial t} = \triangle {\mathcal T}_{\alpha\beta} + {\mathcal T}^{2}_{\alpha\beta} + {\mathcal T}^{\sharp}_{\alpha\beta}, $$ + where ${\mathcal T}^{2}_{\alpha\beta}={\mathcal T}_{\alpha\gamma}{\mathcal T}_{\gamma\beta}$ is the operator square; ${\mathcal T}^{\sharp}_{\alpha\beta} = c_{\alpha\gamma\zeta}c_{\beta\delta\eta}{\mathcal T}_{\gamma\delta}{\mathcal T}_{\zeta\eta}$ is the Lie algebra square; and $c_{\alpha\beta\gamma} = \langle[\varphi^{\alpha},\varphi^{\beta}],\varphi^{\gamma}\rangle $ are the -structure constants of the Lie algebra $so(n)$ relative to the basis -$\{\varphi^{\alpha}\}$. The structure constants $c_{\alpha\beta\gamma}$ are +structure constants of the Lie algebra $\mathit{so}(n)$ relative to the basis +$\{\varphi^{\alpha}\}$. The structure constants $c_{\alpha\beta\gamma}$ are fully antisymmetric in the three indices. + **Proof.** We work in local coordinates that are orthonormal at -the point. By the first Bianchi identity $$R_{abcd}+ R_{acdb}+ -R_{adbc} = 0,$$ we get +the point. By the first Bianchi identity $$ +R_{abcd}+ R_{acdb}+ +R_{adbc} = 0, +$$ + + we get $$ \begin{aligned} @@ -571,12 +674,8 @@ B_{adbc}). \end{aligned} $$ -$$ - Note that -$$ - $$ \begin{aligned} \sum_{e,f}R_{abef}R_{cdef}& = @@ -589,12 +688,8 @@ T}^{2}_{\alpha\beta}\varphi^{\alpha}_{ab}\varphi^{\beta}_{cd}. \end{aligned} $$ -$$ - Also, -$$ - $$ \begin{aligned} 2(B_{acbd}-B_{adbc}) &= 2\sum_{e,f}(R_{aecf}R_{bedf} - @@ -621,10 +716,9 @@ T}^{\sharp}_{\alpha\beta}\varphi^{\alpha}_{ab}\varphi^{\beta}_{cd}. \end{aligned} $$ -$$ - So we can rewrite the equation for the evolution of the curvature -tensor given in Lemma~ as +tensor given in \entryref{d0267ffa1d98} as + $$ \frac{\partial R_{abcd} }{\partial t} = \triangle R_{abcd} + {\mathcal @@ -632,12 +726,15 @@ T}^{2}_{\alpha\beta}\varphi^{\alpha}_{ab}\varphi^{\beta}_{cd} + {\mathcal T}^{\sharp}_{\alpha\beta}\varphi^{\alpha}_{ab}\varphi^{\beta}_{cd}, $$ + or equivalently as + $$ \frac{\partial {\mathcal T}_{\alpha\beta}}{\partial t} = \triangle {\mathcal T}_{\alpha\beta} + {\mathcal T}^{2}_{\alpha\beta} + {\mathcal T}^{\sharp}_{\alpha\beta}. $$ + We abbreviate the last equation as @@ -646,8 +743,10 @@ $$ {\mathcal T}^{\sharp}. $$ + **Remark.** Notice that neither ${\mathcal T}^{2}$ nor ${\mathcal T}^{\sharp}$ satisfies the Bianchi identity, but their sum does. + ## Variation of distance under Ricci flow @@ -655,80 +754,108 @@ There is one result that we will use several times in the arguments to follow. Since it is an elementary result (though the proof is somewhat involved), we have chosen to include it here. -**Proposition.** - Let $t_0\in \Ar$ and let $(M,g(t))$ be a Ricci flow defined for $t$ in an interval containing $t_0$ - with $(M,g(t))$ complete for every $t$ in this interval. Fix a constant $K<\infty$. Let $x_0,x_1$ be two points + +**Proposition.** Let $t_0\in \Ar$ and let $(M,g(t))$ be a Ricci flow defined for $t$ in an interval containing $t_0$ + with $(M,g(t))$ complete for every $t$ in this interval. Fix a constant $K<\infty$. Let $x_0,x_1$ be two points of $M$ and let $r_0>0$ such that $d_{t_0}(x_0,x_1)\ge 2r_0$. Suppose -that $Ric(x,t_0)\le (n-1)K$ for all $x\in B(x_0,r_0,t_0)\cup +that $\mathit{Ric}(x,t_0)\le (n-1)K$ for all $x\in B(x_0,r_0,t_0)\cup B(x_1,r_0,t_0)$. Then -$$\frac{d(d_t(x_0,x_1))}{dt}\Bigl|_{t=t_0}\Bigr.\ge -2(n-1)\left(\frac{2}{3}Kr_0+r_0^{-1}\right).$$ + +$$ +\frac{d(d_t(x_0,x_1))}{dt}\Bigl|_{t=t_0}\Bigr.\ge -2(n-1)\left(\frac{2}{3}Kr_0+r_0^{-1}\right). +$$ + If the distance function $d_{t}(x_0,x_1)$ is not a differentiable function of $t$ at $t=t_0$ then this inequality is understood as an inequality for the forward difference quotient. + + **Remark.** Of course, if the distance function is differentiable at $t=t_0$ then the derivative statement is equivalent to the forward difference quotient statement. Thus, in the proof of this result we shall always work with the forward difference quotients. + **Proof.** The first step in the proof is to replace the distance function by the length of minimal geodesics. The following is standard. -**Claim.** -Suppose that for every minimal $g(t_0)$-geodesic $\gamma$ from $x_0$ + +**Claim.** Suppose that for every minimal $g(t_0)$-geodesic $\gamma$ from $x_0$ to $x_1$ the function $\ell_t(\gamma)$ which is the $g(t)$-length of $\gamma$ satisfies -$$\frac{d(\ell_t(\gamma))}{dt}\Bigl|_{t=t_0}\Bigr.\ge C.$$ + +$$ +\frac{d(\ell_t(\gamma))}{dt}\Bigl|_{t=t_0}\Bigr.\ge C. +$$ + Then -$$\frac{d(d_t(x_0,x_1))}{dt}\Bigl|_{t=t_0}\Bigr.\ge C,$$ + +$$ +\frac{d(d_t(x_0,x_1))}{dt}\Bigl|_{t=t_0}\Bigr.\ge C, +$$ + where, as in the statement of the proposition, if the distance function is not differentiable at $t_0$ then the inequality in the conclusion is interpreted by replacing the derivative on the -left-hand side with the $liminf$ of the forward difference -quotients of $d_t(x_0,x_1)$ at $t_0$. +left-hand side with the $\mathit{liminf}$ of the forward difference +quotients of $d_t(x_0,x_1)$ at $t_0$. + The second step in the proof is to estimate the time derivative of a minimal geodesic under the hypothesis of the proposition. -**Claim.** Assuming the hypothesis of the proposition, for any + +**Claim.** Assuming the hypothesis of the proposition, for any minimal $g(t_0)$-geodesic $\gamma$ from $x_0$ to $x_1$, we have -$$\frac{d(\ell_t(\gamma))}{dt}\Bigl|_{t=t_0}\Bigr.\ge -2(n-1) -\left(\frac{2}{3}Kr_0+r_0^{-1}\right).$$ -\begin{proof} -Fix a minimal $g(t_0)$-geodesic $\gamma(u)$ from $x_0$ to $x_1$, +$$ +\frac{d(\ell_t(\gamma))}{dt}\Bigl|_{t=t_0}\Bigr.\ge -2(n-1) +\left(\frac{2}{3}Kr_0+r_0^{-1}\right). +$$ + + +**Proof.** Fix a minimal $g(t_0)$-geodesic $\gamma(u)$ from $x_0$ to $x_1$, parameterized by arc length. We set $d=d_{t_0}(x_0,x_1)$, we set $X(u)=\gamma'(u)$, and we take tangent vectors $Y_1,\ldots,Y_{n-1}$ in $T_{x_0}M$ which together with $X(0)=\gamma'(0)$ form an orthonormal basis. We let $Y_i(u)$ be the parallel translation of $Y_i$ along $\gamma$. Define $f\colon [0,d]\to [0,1]$ by: -$$f(u)=\begin{cases}u/r_0 & 0\le u\le r_0 \\ 1 & r_0\le u\le d-r_0 \\ -(d-u)/r_0 & d-u\le r_0\le d, \end{cases}$$ and define $$\widetilde -Y_i(u)=f(u)Y_i(u).$$ See Fig.~. + +$$ +f(u)=\begin{cases}u/r_0 & 0\le u\le r_0 \\ 1 & r_0\le u\le d-r_0 \\ +(d-u)/r_0 & d-u\le r_0\le d, \end{cases} +$$ + + and define + +$$ +\widetilde +Y_i(u)=f(u)Y_i(u). +$$ + + See Fig. 3.2. For $1\le i\le n-1$, let $s''_{\widetilde Y_i}(\gamma)$ be the second variation of the $g(t_0)$-length of $\gamma$ along $\widetilde Y_i$. Since $\gamma$ is a minimal $g(t_0)$-geodesic, for all $i$ we have $$ - s''_{\widetilde Y_i}(\gamma)\ge 0. $$ Let us now compute $s''_{\widetilde Y_i}(\gamma)$ by taking a two-parameter family $\gamma(u,s)$ such that the curve $\gamma(u,0)$ is the original minimal geodesic and $\frac{\partial}{\partial -s}(\gamma(u,s))|_{s=0}=\widetilde Y_i(u)$. We denote by $X(u,s)$ +s}(\gamma(u,s))|_{s=0}=\widetilde Y_i(u)$. We denote by $X(u,s)$ the image $D\gamma_{(u,s)}(\partial/\partial u)$ and by $\widetilde Y_i(u,s)$ the image $D\gamma_{(u,s)}(\partial/\partial s)$. We wish to compute -$$ - $$ \begin{aligned} -\lefteqn{s''_{\widetilde Y_i}(\gamma) = -\frac{d^2}{ds^2}\left(\int_0^d\sqrt{X(u,s),X(u,s)}du\right)\Bigl|_{s=0}\Bigr.} +s''_{\widetilde Y_i}(\gamma) = +\frac{d^2}{ds^2}\left(\int_0^d\sqrt{X(u,s),X(u,s)}du\right)\Bigl|_{s=0}\Bigr. & & \nonumber \\ & = & \frac{d}{ds}\left(\int_0^d\langle X(u,s),X(u,s)\rangle^{-1/2} @@ -744,27 +871,27 @@ Y_i}X(u,0)\rangle}{\langle X(u,0),X(u,0)\rangle^{1/2}}du \nonumber. \end{aligned} $$ -$$ - Using the fact that $X$ and $\widetilde Y_i$ commute (since they are the coordinate partial derivatives of a map of a surface into $M$) and using the fact that $Y_i(u)$ is parallel along $\gamma$, meaning that $\nabla_X(Y_i)(u)=0$, we see that $\nabla_{\widetilde Y_i}X(u,0)=\nabla_X\widetilde Y_i(u,0)=f'(u)Y_i(u)$. By construction $\langle Y_i(u),X(u,0)\rangle=0$. It follows that -$$\langle \nabla_{\widetilde Y_i}X(u,0),X(u,0)\rangle=\langle \nabla_X(\widetilde -Y_i)(u,0),X(u,0)\rangle = \langle f'(u)Y_i(u),X(u,0)\rangle =0.$$ -Also, $\langle X(u,0),X(u,0)\rangle=1$, and by construction $\langle -Y_i(u,0),Y_i(u,0)\rangle=1$. Thus, Equation~() -simplifies to $$ +\langle \nabla_{\widetilde Y_i}X(u,0),X(u,0)\rangle=\langle \nabla_X(\widetilde +Y_i)(u,0),X(u,0)\rangle = \langle f'(u)Y_i(u),X(u,0)\rangle =0. +$$ + +Also, $\langle X(u,0),X(u,0)\rangle=1$, and by construction $\langle +Y_i(u,0),Y_i(u,0)\rangle=1$. Thus, Equation (3.10) +simplifies to $$ \begin{aligned} -\lefteqn{s''_{\widetilde +s''_{\widetilde Y_i}(\gamma)=\frac{d^2}{ds^2}\left(\int_0^d\sqrt{X(u,s),X(u,s)}du\right)\Bigl|_{s=0}\Bigr. - } + \\& = & \int_0^d\left((f'(u))^2\langle Y_i(u),Y_i(u)\rangle+\langle \nabla_{\widetilde Y_i}\nabla_X(\widetilde @@ -776,114 +903,114 @@ du.\nonumber \end{aligned} $$ -$$ Now we restrict to $s=0$ and for simplicity of notation we leave the variable $u$ implicit. We have -$$\langle + +$$ +\langle \nabla_X\nabla_{\widetilde Y_i}\widetilde Y_i,X\rangle=\frac{d}{du}\langle \nabla_{\widetilde Y_i}\widetilde Y_i,X\rangle-\langle \nabla_{\widetilde Y_i}\widetilde Y_i,\nabla_XX\rangle=\frac{d}{du}\langle \nabla_{\widetilde -Y_i}\widetilde Y_i,X\rangle,$$ where the last equality is a +Y_i}\widetilde Y_i,X\rangle, +$$ + + where the last equality is a consequence of the geodesic equation, $\nabla_XX=0$. It follows that -$$\int_0^d\langle + +$$ +\int_0^d\langle \nabla_X\nabla_{\widetilde Y_i}\widetilde Y_i,X\rangle du=\int_0^d\frac{d}{du}\langle \nabla_{\widetilde Y_i}\widetilde -Y_i,X\rangle=0,$$ where the last equality is a consequence of the +Y_i,X\rangle=0, +$$ + + where the last equality is a consequence of the fact that $\widetilde Y_i$ vanishes at the end points. -Consequently, plugging these into Equation~() we have +Consequently, plugging these into Equation (3.11) we have $$ - s''_{\widetilde Y_i}(\gamma) = +s''_{\widetilde Y_i}(\gamma) = \int_0^d\left(\langle R(\widetilde Y_i,X)\widetilde Y_i(u,0),X(u,0)\rangle+(f'(u))^2\right)du. $$ + Of course, it is immediate from the definition that $f'(u)^2=1/r_0^2$ for $u\in [0,r_0]$ and for $u\in [d-r_0,d]$ and is zero otherwise. Also, from the definition of the vector fields $Y_i$ we have -$$\sum_{i=1}^{n-1}\langle -R(Y_i,X)Y_i(u),X(u)\rangle=-Ric_{g(t_0)}(X(u),X(u)),$$ so that -$$\sum_{i=1}^{n-1}\langle -R(\widetilde Y_i,X)\widetilde -Y_i(u),X(u)\rangle=-f^2(u)Ric_{g(t_0)}(X(u),X(u)).$$ -Hence, summing Equalities~() for $i=1,\ldots, n-1$ and -using Equation~() gives +$$ +\sum_{i=1}^{n-1}\langle +R(Y_i,X)Y_i(u),X(u)\rangle=-\mathit{Ric}_{g(t_0)}(X(u),X(u)), +$$ + + so that $$ +\sum_{i=1}^{n-1}\langle +R(\widetilde Y_i,X)\widetilde +Y_i(u),X(u)\rangle=-f^2(u)\mathit{Ric}_{g(t_0)}(X(u),X(u)). +$$ + +Hence, summing Equalities (3.12) for $i=1,\ldots, n-1$ and +using Equation (3.9) gives $$ \begin{aligned} 0\le\sum_{i=1}^{n-1}s''_{\widetilde Y_i}(\gamma) & = & -\int_0^{r_0}\left[\frac{u^2}{r_0^2}\left(-Ric_{g(t_0)}\left(X(u),X(u)\right)\right) +\int_0^{r_0}\left[\frac{u^2}{r_0^2}\left(-\mathit{Ric}_{g(t_0)}\left(X(u),X(u)\right)\right) +\frac{n-1}{r_0^2}\right]du \\ -& & + \int_{r_0}^{d-r_0}-Ric_{g(t_0)}(X(u),X(u))du +& & + \int_{r_0}^{d-r_0}-\mathit{Ric}_{g(t_0)}(X(u),X(u))du \\ & & +\int_{d-r_0}^d\left[\frac{(d-u)^2}{r_0^2} -\left(-Ric_{g(t_0)}(X(u),X(u))\right)+\frac{n-1}{r_0^2}\right]du. +\left(-\mathit{Ric}_{g(t_0)}(X(u),X(u))\right)+\frac{n-1}{r_0^2}\right]du. \end{aligned} $$ -$$ - Rearranging the terms yields -$$ - $$ \begin{aligned} -0 & \le & -\int_0^d Ric_{g(t_0)}(X(u),X(u))du \\ +0 & \le & -\int_0^d \mathit{Ric}_{g(t_0)}(X(u),X(u))du \\ & &+\int_0^{r_0}\left[\left(1-\frac{u^2}{r_0^2}\right) -\left(Ric_{g(t_0)}(X(u),X(u))\right) +\frac{n-1}{r_0^2}\right]du\\ +\left(\mathit{Ric}_{g(t_0)}(X(u),X(u))\right) +\frac{n-1}{r_0^2}\right]du\\ & & +\int_{d-r_0}^d -\left[\left(1-\frac{(d-u)^2}{r_0^2}\right)\left(Ric_{g(t_0)}(X(u),X(u))\right) +\left[\left(1-\frac{(d-u)^2}{r_0^2}\right)\left(\mathit{Ric}_{g(t_0)}(X(u),X(u))\right) +\frac{n-1}{r_0^2}\right]du. \end{aligned} $$ -$$ - Since -$$ - $$ \begin{aligned} \frac{d(\ell_t(\gamma))}{dt}\Bigl|_{t=t_0}\Bigr. & = & \frac{d}{dt}\left[\left(\int_0^d\sqrt{\langle X(u),X(u)\rangle}dt\right)^{1/2}\right]|_{t=t_0} \\ -& = & -\int_0^d Ric_{g(t_0)}(X(u),X(u))du, +& = & -\int_0^d \mathit{Ric}_{g(t_0)}(X(u),X(u))du, \end{aligned} $$ -$$ - we have -$$ - $$ \begin{aligned} -\lefteqn{\frac{d(\ell_t(\gamma))}{dt}\Bigl|_{t=t_0}\Bigr.\ge --\left\{\int_0^{r_0}\left[\left(1-\frac{u^2}{r_0^2}\right)\left(Ric_{g(t_0)}(X(u),X(u))\right) -+\frac{n-1}{r_0^2}\right]du\right.} +\frac{d(\ell_t(\gamma))}{dt}\Bigl|_{t=t_0}\Bigr.\ge +-\left\{\int_0^{r_0}\left[\left(1-\frac{u^2}{r_0^2}\right)\left(\mathit{Ric}_{g(t_0)}(X(u),X(u))\right) ++\frac{n-1}{r_0^2}\right]du\right. \\ & & \left. +\int_{d-r_0}^d -\left[\left(1-\frac{(d-u)^2}{r_0^2}\right)\left(Ric_{g(t_0)}(X(u),X(u))\right) +\left[\left(1-\frac{(d-u)^2}{r_0^2}\right)\left(\mathit{Ric}_{g(t_0)}(X(u),X(u))\right) +\frac{n-1}{r_0^2}\right]du\right\} \end{aligned} $$ -$$ - Now, since $|X(u)|=1$, by the hypothesis of the proposition we have -the estimate $Ric_{g(t_0)}(X(u),X(u))\le (n-1)K$ on the regions of +the estimate $\mathit{Ric}_{g(t_0)}(X(u),X(u))\le (n-1)K$ on the regions of integration on the right-hand side of the above inequality. Thus, -$$ - $$ \begin{aligned} \frac{d(\ell_t(\gamma))}{dt}\Bigl|_{t=t_0}\Bigr.\ge @@ -891,58 +1018,73 @@ $$ \end{aligned} $$ -$$ - -This completes the proof of Claim~. +This completes the proof of \entryref{a43d37f014e8}. -Claims~ and~ together prove the proposition. -\end{proof} +Claims \entryref{0f76a0750b40} and \entryref{a43d37f014e8} together prove the proposition. -**Corollary.** -Let $t_0\in \Ar$ and let $(M,g(t))$ be a Ricci flow defined for $t$ in an + +**Corollary.** Let $t_0\in \Ar$ and let $(M,g(t))$ be a Ricci flow defined for $t$ in an interval containing $t_0$ and with $(M,g(t))$ complete for every $t$ in this -interval. Fix a constant $K<\infty$. Suppose that $Ric(x,t_0)\le (n-1)K$ +interval. Fix a constant $K<\infty$. Suppose that $\mathit{Ric}(x,t_0)\le (n-1)K$ for all $x\in M$. Then for any points $x_0,x_1\in M$ we have -$$\frac{d(d_t(x_0,x_1))}{dt}\Bigl|_{t=t_0}\Bigr.\ge -4(n-1)\sqrt{\frac{2K}{3}}$$ + +$$ +\frac{d(d_t(x_0,x_1))}{dt}\Bigl|_{t=t_0}\Bigr.\ge -4(n-1)\sqrt{\frac{2K}{3}} +$$ + in the sense of forward difference quotients. + **Proof.** There are two cases: Case (i): $d_{t_0}(x_0,x_1)\ge \sqrt{\frac{6}{K}}$ and Case (ii) $d_{t_0}(x_0,x_1)< -\sqrt{\frac{6}{K}}$. In Case (i) we take $r_0=\sqrt{3/2K}$ in -Proposition~, and we conclude that the $liminf$ at +\sqrt{\frac{6}{K}}$. In Case (i) we take $r_0=\sqrt{3/2K}$ in +\entryref{5d895f9c3918}, and we conclude that the $\mathit{liminf}$ at $t_0$ of the difference quotients for $d_{t}(x_0,x_1)$ is at most $-4(n-1)\sqrt{\frac{2K}{3}}$. In Case (ii) w let $\gamma(u)$ be any minimal $g(t_0)$-geodesic from $x_0$ to $x_1$ parameterized by arc length. Since -$$\frac{d}{dt}(\ell_t(\gamma))|_{t=t_0}=-\int_\gamma -Ric_{g(t_0)}(\gamma'(u),\gamma'(u))du,$$ we see that -$$\frac{d}{dt}(\ell_t(\gamma))|_{t=t_0}\ge --(n-1)K\sqrt{6/K}=-(n-1)\sqrt{6K}.$$ By Claim~, this -implies that the $liminf$ of the forward difference quotient of + +$$ +\frac{d}{dt}(\ell_t(\gamma))|_{t=t_0}=-\int_\gamma +\mathit{Ric}_{g(t_0)}(\gamma'(u),\gamma'(u))du, +$$ + + we see that + +$$ +\frac{d}{dt}(\ell_t(\gamma))|_{t=t_0}\ge +-(n-1)K\sqrt{6/K}=-(n-1)\sqrt{6K}. +$$ + + By \entryref{0f76a0750b40}, this +implies that the $\mathit{liminf}$ of the forward difference quotient of $d_t(x_0,x_1)$ at $t=t_0$ is at least $-(n-1)\sqrt{6K}\ge -4(n-1)\sqrt{2K/3}$. -**Corollary.** - Let $(M,g(t)),\ a\le t\le b$, be a Ricci flow with $(M,g(t))$ complete for every $t -\in [0,T)$. Fix a positive function $K(t)$, and suppose that $Ric_{g(t)}(x,t)\le (n-1)K(t)$ for all $x\in M$ and all $t\in [a,b]$. Let + +**Corollary.** Let $(M,g(t)),\ a\le t\le b$, be a Ricci flow with $(M,g(t))$ complete for every $t +\in [0,T)$. Fix a positive function $K(t)$, and suppose that $\mathit{Ric}_{g(t)}(x,t)\le (n-1)K(t)$ for all $x\in M$ and all $t\in [a,b]$. Let $x_0,x_1$ be two points of $M$. Then -$$d_a(x_0,x_1)\le d_b(x_0,x_1)+4(n-1)\int_a^b\sqrt{\frac{2K(t)}{3}}dt.$$ - -**Proof.** By Corollary~ we have $$ +d_a(x_0,x_1)\le d_b(x_0,x_1)+4(n-1)\int_a^b\sqrt{\frac{2K(t)}{3}}dt. +$$ + +**Proof.** By \entryref{d22d3cf47fbf} we have + +$$ \frac{d}{dt}d_t(x_0,x_1)|_{t=t'}\ge -4(n-1)\sqrt{\frac{2K(t')}{3}} $$ -in the sense of forward difference quotients. Thus, this result is -an immediate consequence of Lemma~. +in the sense of forward difference quotients. Thus, this result is +an immediate consequence of \entryref{67c378f87457}. ## Shi's derivative estimates -The last `elementary' result we discuss is Shi's result controlling all derivatives in terms of a +The last 'elementary' result we discuss is Shi's result controlling all derivatives in terms of a bound on curvature. This is a consequence of the parabolic nature of the Ricci flow equation. More precisely, we can control all derivatives of the curvature tensor at a point $p\in M$ and at a @@ -951,49 +1093,63 @@ an entire backward parabolic neighborhood of $(p,t)$ in space-time. The estimates become weaker as the parabolic neighborhood shrinks, either in the space direction or the time direction. -Recall that for any $K<\infty$ if $(M,g)$ is a Riemannian manifold -with $|Rm|\le K$ and if for some $r\le \pi/\sqrt{K}$ the metric ball +Recall that for any $K<\infty$ if $(M,g)$ is a Riemannian manifold +with $|\mathit{Rm}|\le K$ and if for some $r\le \pi/\sqrt{K}$ the metric ball $B(p,r)$ has compact closure in $M$, then the exponential mapping -$exp_p$ is defined on the ball $B(0,r)$ of radius $r$ centered -at the origin of $T_pM$ and $exp_p\colon B(0,r)\to M$ is a +$\mathit{exp}_p$ is defined on the ball $B(0,r)$ of radius $r$ centered +at the origin of $T_pM$ and $\mathit{exp}_p\colon B(0,r)\to M$ is a local diffeomorphism onto $B(p,r)$. The first of Shi's derivative estimates controls the first -derivative of $Rm$. +derivative of $\mathit{Rm}$. -**Theorem.** -There is a constant $C=C(n)$, depending only on the dimension $n$, + +**Theorem.** There is a constant $C=C(n)$, depending only on the dimension $n$, such that the following holds for every $K<\infty$, for every $T>0$, and for every $r>0$. Suppose that $(U,g(t)),\ 0\le t\le T$, is an $n$-dimensional -Ricci flow with $|Rm(x,t)|\le K$ for all $x\in U$ and $t\in [0,T]$. +Ricci flow with $|\mathit{Rm}(x,t)|\le K$ for all $x\in U$ and $t\in [0,T]$. Suppose that $p\in U$ has the property that $B(p,0,r)$ has compact closure in $U$. Then -$$|\nabla Rm(p,t)|\le -CK\left(\frac{1}{r^2}+\frac{1}{t}+K\right)^{1/2}.$$ + +$$ +|\nabla \mathit{Rm}(p,t)|\le +CK\left(\frac{1}{r^2}+\frac{1}{t}+K\right)^{1/2}. +$$ + For a proof of this result see Chapter 6.2, starting on page 212, of -. +[ChowLuNi]. We also need higher derivative estimates. These are also due to Shi, but they take a slightly different form. (See Theorem 6.9 on page -210 of .) +210 of [ChowLuNi].) + **Theorem.** **(Shi's Derivative Estimates)** Fix the dimension $n$ of the Ricci flows under consideration. Let -$K<\infty$ and $\alpha>0$ be positive constants. Then for each +$K<\infty$ and $\alpha>0$ be positive constants. Then for each non-negative integer $k$ and each $r>0$ there is a constant $C_k=C_k(K,\alpha,r,n)$ such that the following holds. Let $(U,g(t)),\ 0\le t\le T$, be a Ricci flow with $T\le \alpha/K$. Fix $p\in U$ and suppose that the metric ball $B(p,0,r)$ has compact closure in $U$. If -$$|Rm(x,t)|\le K\ \ \text{for all }\ (x,t)\in P(x,0,r,T),$$ + +$$ +|\mathit{Rm}(x,t)|\le K\ \ \text{for all }\ (x,t)\in P(x,0,r,T), +$$ + then -$$|\nabla^k(Rm(y,t))|\le \frac{C_k}{t^{k/2}}$$ + +$$ +|\nabla^k(\mathit{Rm}(y,t))|\le \frac{C_k}{t^{k/2}} +$$ + for all $y\in B(p,0,r/2)$ and all $t\in(0,T]$. + -For a proof of this result see Chapter 6.2 of where +For a proof of this result see Chapter 6.2 of [ChowLuNi] where these estimates are proved for the first and second derivatives of -$Rm$. The proofs of the higher derivatives follow similarly. Below, we +$\mathit{Rm}$. The proofs of the higher derivatives follow similarly. Below, we shall prove a stronger form of this result below including the proof for all derivatives. @@ -1005,8 +1161,8 @@ the curvature at later times. The argument is basically the same as that of the result cited above, but since there is no good reference for it we include the proof, which was shown to us by Lu Peng. -**Theorem.** -Fix the dimension $n$ of the Ricci flows under consideration. Let + +**Theorem.** Fix the dimension $n$ of the Ricci flows under consideration. Let $K<\infty$ and $\alpha>0$ be given positive constants. Fix an integer $l\ge 0$. Then for each integer $k\ge 0$ and for each $r>0$ there is a constant $C'_{k,l}=C'_{k,l}(K,\alpha,r,n)$ such that the @@ -1014,8 +1170,6 @@ following holds. Let $(U,g(t)),\ 0\le t\le T$, be a Ricci flow with $T\le \alpha/K$. Fix $p\in U$ and suppose that the metric ball $B(p,0,r)$ has compact closure in $U$. Suppose that -$$ - $$ \begin{aligned} \left\vert \operatorname*{Rm}\left( x,t\right) \right\vert & @@ -1026,8 +1180,6 @@ $$ \end{aligned} $$ -$$ - Then $$ @@ -1043,27 +1195,32 @@ $$ \left\vert \nabla^{k}\operatorname*{Rm}\left( y,t\right) \right\vert \leq C_{k,l}'. $$ + -**Remark.** Clearly, the case $l=0$ of Theorem~ is Shi's theorem -(Theorem~). + +**Remark.** Clearly, the case $l=0$ of \entryref{e0b6a1f646a7} is Shi's theorem +(\entryref{dbe4fabd0076}). + -Theorem~ leads immediately to the following: +\entryref{e0b6a1f646a7} leads immediately to the following: -**Corollary.** Suppose that $(M,g(t)),\ 0\le t\le T$, is a Ricci flow + +**Corollary.** Suppose that $(M,g(t)),\ 0\le t\le T$, is a Ricci flow with $(M,g(t))$ being complete and with $T<\infty$. Suppose that $\operatorname*{Rm}(x,0)$ is bounded in the $C^\infty$-topology independent of $x\in M$ and suppose that $\left\vert\operatorname*{Rm}(x,t)\right\vert$ is bounded -independent of $x\in M$ and $t\in[0,T]$. Then the operator -$\operatorname*{Rm}(x,t)$ is bounded in the $C^\infty$-topology +independent of $x\in M$ and $t\in[0,T]$. Then the operator +$\operatorname*{Rm}(x,t)$ is bounded in the $C^\infty$-topology independent of $(x,t)\in M\times [0,T]$. + - For a proof of Theorem~ see . We give the proof of -a stronger result, Theorem~. +For a proof of \entryref{ed4b72caf9f9} see [Shi1, Shi2]. We give the proof of +a stronger result, \entryref{e0b6a1f646a7}. -**Proof.** The first remark is that establishing Theorem~ for +**Proof.** The first remark is that establishing \entryref{e0b6a1f646a7} for one value of $r$ immediately gives it for all $r'\ge 2r$. The reason -is that for such $r'$ any point $y\in B(p,0,r'/2)$ has the property +is that for such $r'$ any point $y\in B(p,0,r'/2)$ has the property that $B(y,0,r)\subset B(p,0,r')$ so that a curvature bound on $B(p,0,r')$ will imply one on $B(y,0,r)$ and hence by the result for $r$ will imply the higher derivative bounds at $y$. @@ -1083,7 +1240,6 @@ $A_j,\ 0\le j\le m$, depending on $(l,K,\alpha,r,n)$ such that for all $(x,t)\in B(p,0,r/2)\times (0,T]$ we have $$ - \left\vert \nabla^{j}\operatorname*{Rm} \left( x,t\right) \right\vert \leq A_{j}t^{-\max\left\{ j-l,0\right\} /2}. $$ @@ -1104,7 +1260,11 @@ $$ Notice that bounding $F_m$ above by a constant $(C'_{m+1,l})^2$ will yield -$$|\nabla^{m+1}Rm(x,t)|^2\le \frac{(C'_{m+1,l})^2}{t^{max\{m+1-l,0\}}},$$ + +$$ +|\nabla^{m+1}\mathit{Rm}(x,t)|^2\le \frac{(C'_{m+1,l})^2}{t^{\mathit{max}\{m+1-l,0\}}}, +$$ + and hence will complete the proof of the result. Bounding $F_m$ @@ -1112,28 +1272,34 @@ above (assuming the inductive hypothesis) is what is accomplished in the rest of this proof. The main calculation is the proof of the following claim under the inductive hypothesis. -**Claim.** - With $F_m$ as defined above and with $C\ge \max(4A^2_m,1)$, + +**Claim.** With $F_m$ as defined above and with $C\ge \max(4A^2_m,1)$, there are constants $c_1$ and $C_0,C_1$ depending on $C$ as well as $K,\alpha,A_1,\ldots,A_m$ for which the following holds on $B(p,0,3r/4)\times (0,T]$: + $$ \left( \frac{\partial}{\partial t}-\Delta\right) -F_{m}(x,t)\leq-\frac {c_1}{t^{s\left\{ \max\left\{ +F_{m}(x,t)\leq-\frac {c_1}{t^{\mathit{s}\left\{ \max\left\{ m-l+1,0\right\} \right\} }}\left( F_{m}(x,t)-C_0\right) -^{2}+\frac{C_1}{t^{s\{\max\left\{ m-l+1,0\right\} \} }}, +^{2}+\frac{C_1}{t^{\mathit{s}\{\max\left\{ m-l+1,0\right\} \} }}, $$ + where -$$s(n)=\begin{cases} +1 & \text{if } n>0 \\ 0 & \text{if }n=0 \\ -1& -\text{if }n<0.\end{cases}$$ -Let us assume this claim and use it to prove Theorem~. We fix +$$ +\mathit{s}(n)=\begin{cases} +1 & \text{if } n>0 \\ 0 & \text{if }n=0 \\ -1& +\text{if }n<0.\end{cases} +$$ + + +Let us assume this claim and use it to prove \entryref{e0b6a1f646a7}. We fix $C=\max\{4A_m^2,1\}$, and consider the resulting function $F_m$. The -constants $c_1,C_0,C_1$ from Claim~ depend only on $K,\alpha$, and +constants $c_1,C_0,C_1$ from \entryref{e7fe06259f43} depend only on $K,\alpha$, and $A_1,\ldots,A_m$. Since $r\le \pi/2\sqrt{K}$, and $B(p,0,r)$ has compact closure in $U$, there is some $r'>r$ so that the exponential -mapping $exp_p\colon B(0,r')\to U$ is a local diffeomorphism -onto $B(p,0,r')$. Pulling back by the exponential map, we replace +mapping $\mathit{exp}_p\colon B(0,r')\to U$ is a local diffeomorphism +onto $B(p,0,r')$. Pulling back by the exponential map, we replace the Ricci flow on $U$ by a Ricci flow on $B(0,r')$ in $T_pM$. Clearly, it suffices to establish the estimates in the statement of the proposition for $B(0,r/2)$. This remark allows us to assume that @@ -1142,25 +1308,25 @@ curvature then comes into play in the following crucial proposition, which goes back to Shi. The function given in the next proposition allows us to localize the computation in the ball $B(p,0,r)$. -**Proposition.** -Fix constants $0<\alpha$ and the dimension $n$. Then there is a constant + +**Proposition.** Fix constants $0<\alpha$ and the dimension $n$. Then there is a constant $C_2'=C_2'(\alpha,n)$ and for each $r>0$ and $K<\infty$ there is a constant $C_2=C_2(K,\alpha,r,n)$ such that the following holds. Suppose that $(U,g(t)),\ -0\le t\le T$, is an $n$-dimensional Ricci flow with $T\le \alpha/K$. Suppose +0\le t\le T$, is an $n$-dimensional Ricci flow with $T\le \alpha/K$. Suppose that $p\in U$ and that $B(p,0,r)$ has compact closure in $U$ and that the exponential mapping from the ball of radius $r$ in $T_pU$ to $B(p,0,r)$ is a -diffeomorphism. Suppose that $|Rm(x,0)|\le K$ for all $x\in B(p,0,r)$. There is +diffeomorphism. Suppose that $|\mathit{Rm}(x,0)|\le K$ for all $x\in B(p,0,r)$. There is a smooth function $\eta\colon B(p,0,r)\to [0,1]$ satisfying the following for all $t\in [0,T]$: -\begin{enumerate} -\item $\eta$ has compact support in $B(p,0,r/2)$ -\item The restriction of $\eta$ to $B(p,0,r/4)$ is identically $1$. -\item $|\Delta_{g(t)} \eta|\le C_2(K,\alpha,r,n)$. -\item $\frac{|\nabla \eta|_{g(t)}^2}{\eta}\le \frac{C'_2(\alpha,n)}{r^2}$ -\end{enumerate} + +1. $\eta$ has compact support in $B(p,0,r/2)$ +1. The restriction of $\eta$ to $B(p,0,r/4)$ is identically $1$. +1. $|\Delta_{g(t)} \eta|\le C_2(K,\alpha,r,n)$. +1. $\frac{|\nabla \eta|_{g(t)}^2}{\eta}\le \frac{C'_2(\alpha,n)}{r^2}$ + For a proof of this result see Lemma 6.62 on page 225 of -. +[ChowLuNi]. We can apply this proposition to our situation, because we are assuming that $r\le \pi/2\sqrt{K}$ so that the exponential mapping @@ -1170,8 +1336,8 @@ Ricci flow back to the ball in the tangent space. Fix any $y\in B(p,0,r/2)$ and choose $\eta$ as in the previous proposition for the constants $C_2(\alpha,n)$ and $C_2'(K,\alpha,r/4,n)$. Notice that $B(y,0,r/4)\subset B(p,0,3r/4)$ -so that the conclusion of Claim~ holds for every $(z,t)$ -with $z\in B(y,0,r/4)$ and $t\in [0,T]$. We shall show that the restriction of $\eta +so that the conclusion of \entryref{e7fe06259f43} holds for every $(z,t)$ +with $z\in B(y,0,r/4)$ and $t\in [0,T]$. We shall show that the restriction of $\eta F_m$ to $P(y,0,r/4,T)$ is bounded by a constant that depends only on $K,\alpha,r,n,A_1,\ldots,A_m$. It will then follow immediately that the restriction of $F_m$ to $P(y,0,r/8,T)$ is bounded by the same @@ -1186,7 +1352,7 @@ $(C+K^2)K^2$ which is a constant depending only on $K$ and $A_m$. This, of course, immediately implies the result. Thus we can assume that the maximum is achieved at some $t>0$. When $s\left\{ \max\left\{ m+1-l,0\right\} \right\} -0,$ according to the -Claim~, we have +\entryref{e7fe06259f43}, we have $$ \left( \frac{\partial}{\partial t}-\Delta\right) @@ -1203,11 +1369,13 @@ F_m-2\nabla\eta\cdot\nabla F_m. $$ Since $(x,t)$ is a maximum point for $\eta F_m$ and since -$t>0$, a simple maximum principle argument shows that -$$\left( \frac{\partial}{\partial t}-\Delta\right)\eta F_{m}(x,t)\ge 0.$$ -Hence, in this case we conclude that +$t>0$, a simple maximum principle argument shows that $$ +\left( \frac{\partial}{\partial t}-\Delta\right)\eta F_{m}(x,t)\ge 0. +$$ + +Hence, in this case we conclude that $$ \begin{aligned} @@ -1220,14 +1388,17 @@ F_m(x,t). \end{aligned} $$ + Hence, + +$$ +c_1\eta (F_m(x,t)-C_0)^2\le \eta(x) +C_1-\Delta\eta(x)\cdot F_m(x,t)-2\nabla\eta(x)\cdot \nabla F_m(x,t). $$ - Hence, $$c_1\eta (F_m(x,t)-C_0)^2\le \eta(x) -C_1-\Delta\eta(x)\cdot F_m(x,t)-2\nabla\eta(x)\cdot \nabla F_m(x,t).$$ Since we + Since we are proving that $F_m$ is bounded, we are free to argue by contradiction and assume that $F_m(x,t)\ge 2C_0$, implying that $F_m(x,t)-C_0\ge F_m(x,t)/2$. Using this inequality yields -$$ $$ \begin{aligned} @@ -1239,41 +1410,69 @@ $$ \end{aligned} $$ -$$ - Since $(x,t)$ is a maximum for $\eta F_m$ we have -$$0=\nabla(\eta(x) F_m(x,t))=\nabla\eta(x) F_m(x,t)+\eta(x)\nabla F_m(x,t),$$ + +$$ +0=\nabla(\eta(x) F_m(x,t))=\nabla\eta(x) F_m(x,t)+\eta(x)\nabla F_m(x,t), +$$ + so that -$$\frac{\nabla\eta(x)}{\eta(x)}=-\frac{\nabla F_m(x,t)}{F_m(x,t)}.$$ + +$$ +\frac{\nabla\eta(x)}{\eta(x)}=-\frac{\nabla F_m(x,t)}{F_m(x,t)}. +$$ + Plugging this in gives -$$\eta(x) F_m(x,t)\le + +$$ +\eta(x) F_m(x,t)\le \frac{C_1}{c_1C_0}-\frac{2\Delta\eta(x)}{c_1}+4\frac{|\nabla\eta(x)|^2}{c_1\eta(x)}+\eta -C_0.$$ +C_0. +$$ Of course, the gradient and Laplacian of $\eta$ are taken at the -point $(x,t)$. Thus, because of the properties of $\eta$ given in -Proposition~, it immediately follows that $\eta F_m(x,t)$ is +point $(x,t)$. Thus, because of the properties of $\eta$ given in +\entryref{81f8de5102de}, it immediately follows that $\eta F_m(x,t)$ is bounded by a constant depending only on $K,n,\alpha,r,c_1,C_0,C_1$, and as we have already seen, $c_1,C_0,C_1$ depend only on $K,\alpha,A_1,\ldots,A_m$. Now suppose that $s\left\{ \max\left\{ m-l+1,0\right\} \right\}=1.$ Again we compute the evolution inequality for $\eta -F_{m}$. The result is $$\left( \frac{\partial}{\partial +F_{m}$. The result is + +$$ +\left( \frac{\partial}{\partial t}-\Delta\right) \left( \eta F_{m}\right) \leq\eta\left( -\frac{c_1}{t} (F_{m}-C_0)^{2}+\frac{C_1}{t}\right) -\Delta\eta\cdot -F_m-2\nabla\eta\cdot\nabla F_m.$$ Thus, using the maximum principle +F_m-2\nabla\eta\cdot\nabla F_m. +$$ + + Thus, using the maximum principle as before, we have -$$\left( \frac{\partial}{\partial t}-\Delta\right)\eta F_{m}(x,t)\ge 0.$$ + +$$ +\left( \frac{\partial}{\partial t}-\Delta\right)\eta F_{m}(x,t)\ge 0. +$$ + Hence, -$$\frac{\eta(x) c_1(F_m(x,t)-C_0)^2}{t}\le -\frac{\eta(x)C_1}{t}-\Delta\eta(x)F_m(x,t)-2\nabla\eta(x)\cdot\nabla F_m(x,t).$$ + +$$ +\frac{\eta(x) c_1(F_m(x,t)-C_0)^2}{t}\le +\frac{\eta(x)C_1}{t}-\Delta\eta(x)F_m(x,t)-2\nabla\eta(x)\cdot\nabla F_m(x,t). +$$ + Using the assumption that $F_m(x,t)\ge 2C_0$ as before, and rewriting the last term as before, we have -$$\eta F_m(x,t)\le + +$$ +\eta F_m(x,t)\le \frac{\eta(x)C_1}{c_1C_0}-\frac{2t\Delta\eta(x)} -{c_1}+\frac{4t|\nabla\eta(x)|^2}{c_1\eta(x)}+\eta C_0.$$ The right-hand side is +{c_1}+\frac{4t|\nabla\eta(x)|^2}{c_1\eta(x)}+\eta C_0. +$$ + + The right-hand side is bounded by a constant depending only on $K,n,\alpha,r,c_1,C_0,C_1$. We conclude that in all cases $\eta F_{m}$ is bounded by a constant depending only on $K,n,\alpha,r,c_1,C_0,C_1$, and hence on $K,n,\alpha,r,A_1,\ldots,A_m$. @@ -1282,19 +1481,20 @@ This proves that for any $y\in B(p,0,r/2)$, the value $\eta F_m(x,t)$ is bounded by a constant $A_{m+1}$ depending only on $(m+1,l,K,n,\alpha,r)$ for all $(x,t)\in B(y,0,r/2)\times [0,T]$. Since $\eta(y)=1$, for all $0\le t\le T$ we have -$$t^{max\{m+1-l,0\}}|\nabla^{m+1}Rm(y,t)|^2\le F_m(y,t)=\eta(y)F_m(y,t)\le A_{m+1}.$$ + +$$ +t^{\mathit{max}\{m+1-l,0\}}|\nabla^{m+1}\mathit{Rm}(y,t)|^2\le F_m(y,t)=\eta(y)F_m(y,t)\le A_{m+1}. +$$ This completes the inductive proof that the result holds for -$k=m+1$ and hence establishes Theorem~, modulo the proof of Claim~. +$k=m+1$ and hence establishes \entryref{e0b6a1f646a7}, modulo the proof of \entryref{e7fe06259f43}. -Now we turn to the proof of Claim~. +Now we turn to the proof of \entryref{e7fe06259f43}. **Proof.** In this argument we fix $(x,t)\in B(p,0,3r/4)\times (0,T]$ and we drop $(x,t)$ from the notation. Recall that by Equations (7.4a) and (7.4b) on p. 229 of - we have - -$$ +[ChowKnopf] we have $$ \begin{aligned} @@ -1308,22 +1508,18 @@ $$ \end{aligned} $$ -$$ - where the constants $c_{\ell,j}$ depend only on $\ell$ and $j$. Hence, setting $m_l= \max\left\{ m+1-l,0\right\}$ and denoting $c_{m+1,i}$ by $\tilde c_i$, we have -$$ - $$ \begin{aligned} -\lefteqn{\frac{\partial}{\partial t}\left( t^{m_l}\left\vert +\frac{\partial}{\partial t}\left( t^{m_l}\left\vert \nabla^{m+1}\operatorname*{Rm}\right\vert ^{2}\right) \leq\Delta\left( t^{m_l }\left\vert \nabla ^{m+1}\operatorname*{Rm}\right\vert ^{2}\right) -2t^{m_l } -\left\vert \nabla^{m+2}\operatorname*{Rm}\right\vert ^{2}}\\ +\left\vert \nabla^{m+2}\operatorname*{Rm}\right\vert ^{2}\\ & & +t^{m_l }\sum_{i=0}^{m+1}\tilde c_{i}\left\vert \nabla ^{i}\operatorname*{Rm}\right\vert \left\vert \nabla^{m+1-i}\operatorname*{Rm} \right\vert \left\vert @@ -1343,24 +1539,32 @@ c_{i}\left\vert \nabla \end{aligned} $$ -$$ - -Using the inductive hypothesis, Inequality~(), there is a constant +Using the inductive hypothesis, Inequality (3.14), there is a constant $A<\infty$ depending only on $K,\alpha,A_1,\ldots,A_m$ such that + $$ \sum_{i=1}^{m}\tilde c_{i}\left\vert \nabla ^{i}\operatorname*{Rm}\right\vert \left\vert \nabla^{m+1-i}\operatorname*{Rm} \right\vert \leq At^{-m_l/2}. $$ -Also, let $c=\tilde c_{0}+\tilde c_{m+1}$ and define a new constant + +Also, let $c=\tilde c_{0}+\tilde c_{m+1}$ and define a new constant $B$ by -$$B=c(\alpha+K)+m_l.$$ Then, since $t\le T\le \alpha/K$ and $m_l\ge 0$, we have -$$((\tilde c_{0}+\tilde c_{m+1})t\left\vert\operatorname*{Rm}\right\vert +m_l)t^{m_l-1}\le -\frac{Bt^{m_l}}{t^{s(m_l)}}.$$ Putting this together allows us to -rewrite Inequality~() as $$ +B=c(\alpha+K)+m_l. +$$ + + Then, since $t\le T\le \alpha/K$ and $m_l\ge 0$, we have + +$$ +((\tilde c_{0}+\tilde c_{m+1})t\left\vert\operatorname*{Rm}\right\vert +m_l)t^{m_l-1}\le +\frac{Bt^{m_l}}{t^{s(m_l)}}. +$$ + + Putting this together allows us to +rewrite Inequality (3.16) as $$ \begin{aligned} @@ -1383,12 +1587,8 @@ t^{m_l -1}\left\vert \nabla^{m+1}\operatorname*{Rm}\right\vert \end{aligned} $$ -$$ - Completing the square gives -$$ - $$ \begin{aligned} \frac{\partial}{\partial t}\left( t^{m_l}\left\vert @@ -1402,14 +1602,10 @@ $$ \end{aligned} $$ -$$ - Let $\hat{m}_l = \max\left\{ m-l,0\right\} $. From -() and the induction hypothesis, there is a constant +(3.15) and the induction hypothesis, there is a constant $D$, depending on $K,\alpha,A_1,\ldots,A_m$ such that -$$ - $$ \begin{aligned} \frac{\partial}{\partial t}\left( t^{\hat{m}_l }\left\vert @@ -1423,20 +1619,16 @@ $$ \end{aligned} $$ -$$ - Now, defining new constants $\widetilde B=B+1$ and $\widetilde A=A^2/4$ we have -$$ - $$ \begin{aligned} -\lefteqn{ \left( \frac{\partial}{\partial t}-\Delta\right)F_m= + \left( \frac{\partial}{\partial t}-\Delta\right)F_m= \left( \frac{\partial}{\partial t}-\Delta\right) \left[ \left( C+t^{\hat{m}_l }\left\vert \nabla^{m}\operatorname*{Rm} \right\vert ^{2}\right) t^{m_l }\left\vert -\nabla^{m+1}\operatorname*{Rm}\right\vert ^{2}\right] \leq} \\ +\nabla^{m+1}\operatorname*{Rm}\right\vert ^{2}\right] \leq \\ & & \left( C+t^{\hat{m}_l }\left\vert \nabla^{m} \operatorname*{Rm}\right\vert ^{2}\right) \left( -2t^{m_l }\left\vert \nabla^{m+2} \operatorname*{Rm} \right\vert ^{2} @@ -1454,19 +1646,15 @@ l} \left\vert \nabla^{m+1} \operatorname*{Rm}\right\vert ^{2}\\ \end{aligned} $$ -$$ - Since $C\ge 4t^{\hat m_l}\left\vert\nabla^m\operatorname*{Rm}\right\vert^2$, this implies -$$ - $$ \begin{aligned} -\lefteqn{ \left( \frac{\partial}{\partial t}-\Delta\right) F_m \leq + \left( \frac{\partial}{\partial t}-\Delta\right) F_m \leq -10t^{\hat{m}_l +m_l} \left\vert \nabla^{m}\operatorname*{Rm}\right\vert ^{2}\left\vert -\nabla^{m+2}\operatorname*{Rm}\right\vert ^{2}}\\ +\nabla^{m+2}\operatorname*{Rm}\right\vert ^{2}\\ & & -8t^{\hat{m}_l +m_l }\left\vert \nabla^{m}\operatorname*{Rm}\right\vert \left\vert \nabla^{m+1} \operatorname*{Rm}\right\vert ^{2}\left\vert \nabla^{m+2}\operatorname*{Rm} @@ -1483,12 +1671,11 @@ Bt^{m_l-s(m_l) }\left\vert \nabla^{m+1}\operatorname*{Rm}\right\vert \end{aligned} $$ -$$ - Now we can write the first three terms on the right-hand side of -Inequality~() as +Inequality (3.17) as + $$ - -t^{\hat +-t^{\hat m_l+m_l}\left(\sqrt{10}\left\vert\nabla^{m+2}\operatorname*{Rm}\right\vert \left\vert\nabla^m\operatorname*{Rm}\right\vert+\frac{4}{\sqrt{10}}\left\vert\nabla^{m+1} \operatorname*{Rm}\right\vert^2\right)^2-\frac{2}{5}t^{\hat @@ -1498,7 +1685,6 @@ $$ In addition we have $$ - C+t^{\hat m_l}\left\vert \nabla^{m} \operatorname*{Rm}\right\vert ^{2} \leq C+A_m^2. $$ @@ -1506,29 +1692,30 @@ $$ Let us set $\widetilde D=\max(\alpha/K,1)D$. If $\hat m_l=0$, then $$ - \hat m_lt^{\hat m_l-1}\left\vert\nabla^m\operatorname*{Rm}\right\vert^2+D =D\le \frac{\widetilde D}{t^{s(m_l)}}=\hat m_lA_m^2+D\le \frac{\hat m_lA_m^2+\widetilde D}{t^{s(m_l)}}. $$ - On the other + + On the other hand, if $\hat m_l>0$, then $s(\hat m_l)=s(m_l)=1$ and hence -$$\hat m_lt^{\hat m_l-1}\left\vert\nabla^m\operatorname*{Rm}\right\vert^2+D\le + +$$ +\hat m_lt^{\hat m_l-1}\left\vert\nabla^m\operatorname*{Rm}\right\vert^2+D\le \frac{1}{t^{s(m_l)}}\hat m_lA_m^2+D\le \frac{\hat -m_lA_m^2+\widetilde D}{t^{s(m_l)}}.$$ +m_lA_m^2+\widetilde D}{t^{s(m_l)}}. +$$ Since $\hat m_l =m_l-s(m_l)$, -Inequalities~(),~(), and~() then -allow us rewrite Inequality~() as - -$$ +Inequalities (3.18), (3.19), and (3.20) then +allow us rewrite Inequality (3.17) as $$ \begin{aligned} -\lefteqn{ \left( \frac{\partial}{\partial t}-\Delta\right)F_m + \left( \frac{\partial}{\partial t}-\Delta\right)F_m \leq -\frac{2}{5t^{s( m_l) } - }t^{2m_l}\left\vert\nabla^{m+1}\operatorname*{Rm}\right\vert^4}\\ + }t^{2m_l}\left\vert\nabla^{m+1}\operatorname*{Rm}\right\vert^4\\ & & +(C+A_m^2)\left(\frac{\widetilde B}{t^{s(m_l)}} t^{m_l}\left\vert\nabla^{m+1}\operatorname*{Rm}\right\vert^2+\widetilde At^{s(m_l)}\right) +\frac{\hat m_lA_m^2+\widetilde @@ -1537,13 +1724,13 @@ D}{t^{s(m_l)}}t^{m_l}\left\vert\nabla^{m+1}\operatorname*{Rm} \end{aligned} $$ -$$ - Setting -$$B'=(C+A_m^2)\widetilde B+(\hat m_lA_m^2+\widetilde D),$$ -and $A'=\widetilde A(C+A_m^2)$ we have $$ +B'=(C+A_m^2)\widetilde B+(\hat m_lA_m^2+\widetilde D), +$$ + +and $A'=\widetilde A(C+A_m^2)$ we have $$ \begin{aligned} @@ -1556,12 +1743,8 @@ $$ \end{aligned} $$ -$$ - We rewrite this as -$$ - $$ \begin{aligned} \left( \frac{\partial}{\partial t}-\Delta\right) F_m & \leq & @@ -1571,8 +1754,6 @@ $$ \end{aligned} $$ -$$ - and hence $$ @@ -1582,12 +1763,27 @@ $$ $$ where the constants $B''$ and $A''$ are defined by $ B''=5 B'/4$ and -$$A''=(\max\{\alpha/K,1\})^2+5( B')^2/8.$$ (Recall that $t\le T\le + +$$ +A''=(\max\{\alpha/K,1\})^2+5( B')^2/8. +$$ + + (Recall that $t\le T\le \alpha/K$.) Let -$$Y=(C+t^{\hat -m_l}\left\vert\nabla^m\operatorname*{Rm}\right\vert^2).$$ (Notice + +$$ +Y=(C+t^{\hat +m_l}\left\vert\nabla^m\operatorname*{Rm}\right\vert^2). +$$ + + (Notice that $Y$ is not a constant.) Of course, by definition -$$F_m=Yt^{m_l}|\nabla^{m+1}Rm|^2.$$ Then the previous inequality + +$$ +F_m=Yt^{m_l}|\nabla^{m+1}\mathit{Rm}|^2. +$$ + + Then the previous inequality becomes $$ @@ -1599,8 +1795,6 @@ $$ Since $C\le Y\le 5C/4$ we have -$$ - $$ \begin{aligned} \left( \frac{\partial}{\partial t}-\Delta\right) F_m \leq @@ -1609,20 +1803,30 @@ Y\right)^2+\frac{A''}{t^{s(m_l)}} \end{aligned} $$ +At any point where $F_m\ge 5CB''/4$, the last inequality gives + +$$ +\left( \frac{\partial}{\partial t}-\Delta\right) F_m \leq +-\frac{32}{125t^{s(m_l)}C^2}\left(F_m-5CB''/4\right)^2+\frac{A''}{t^{s(m_l)}}. $$ -At any point where $F_m\ge 5CB''/4$, the last inequality gives -$$\left( \frac{\partial}{\partial t}-\Delta\right) F_m \leq --\frac{32}{125t^{s(m_l)}C^2}\left(F_m-5CB''/4\right)^2+\frac{A''}{t^{s(m_l)}}.$$ -At any point where $F_m\le 5CB''/4$, since $F_m\ge 0$ and $0\le +At any point where $F_m\le 5CB''/4$, since $F_m\ge 0$ and $0\le B''Y\le 5CB''/4$, we have $(F_m-B''Y)^2\le 25C^2(B'')^2/16$, so that -$$-\frac{32}{125t^{s(m_l)}C^2}\left(F_m-5CB''/4\right)^2\ge --2(B'')^2/5t^{s(m_l)}.$$ Thus, in this case we have -$$\left(\frac{\partial}{\partial t}-\Delta\right) F_m \leq + +$$ +-\frac{32}{125t^{s(m_l)}C^2}\left(F_m-5CB''/4\right)^2\ge +-2(B'')^2/5t^{s(m_l)}. +$$ + + Thus, in this case we have + +$$ +\left(\frac{\partial}{\partial t}-\Delta\right) F_m \leq \frac{A''}{t^{s(m_l)}}\le --\frac{32}{125t^{s(m_l)}C^2}\left(F_m-5CB''/4\right)^2+\frac{A''+2(B'')^2/5}{t^{s(m_l)}}.$$ +-\frac{32}{125t^{s(m_l)}C^2}\left(F_m-5CB''/4\right)^2+\frac{A''+2(B'')^2/5}{t^{s(m_l)}}. +$$ -These two cases together prove Claim~. +These two cases together prove \entryref{e7fe06259f43}. ## Generalized Ricci flows @@ -1642,37 +1846,40 @@ horizontal slices are the *time-slices*. In defining the generalized Ricci flow, it is the second approach that we generalize. + **Definition.** By *space-time* we mean a smooth $(n+1)$-dimensional manifold ${\mathcal M}$ (possibly with boundary), equipped with a smooth -function $**t**\colon {\mathcal M}\to \Ar$, called *time* and +function $\mathbf{t}\colon {\mathcal M}\to \Ar$, called *time* and a smooth vector field $\chi$ subject to the following axioms: -\begin{enumerate} -\item The image of $**t**$ is an interval $I$ (possibly infinite) and -the boundary of ${\mathcal M}$ is the preimage under $**t**$ of + +1. The image of $\mathbf{t}$ is an interval $I$ (possibly infinite) and +the boundary of ${\mathcal M}$ is the preimage under $\mathbf{t}$ of $\partial I$. -\item For each $x\in {\mathcal M}$ there is an open neighborhood $U\subset +1. For each $x\in {\mathcal M}$ there is an open neighborhood $U\subset {\mathcal M}$ of $x$ and a diffeomorphism $f\colon V\times J\to U$, where $V$ is an open subset in $\Ar ^n$ and $J$ is an interval with -the property that (i) $**t**$ is the composition of $f^{-1}$ +the property that (i) $\mathbf{t}$ is the composition of $f^{-1}$ followed by the projection onto the interval $J$ and (ii) $\chi$ is the image under $f$ of the unit vector field in the positive direction tangent to the foliation by the lines $\{v\}\times J$ of $V\times J$. -\end{enumerate} + - Notice that it follows -that $\chi(**t**)=1$. +Notice that it follows +that $\chi(\mathbf{t})=1$. + **Definition.** The *time-slices* -of space-time are the level sets $**t**$. These form a +of space-time are the level sets $\mathbf{t}$. These form a codimension-one foliation of ${\mathcal M}$. For each $t\in I$ we denote by $M_t\subset {\mathcal M}$ the $t$ time-slice, that is to -say $**t**^{-1}(t)$. Notice that each boundary component of +say $\mathbf{t}^{-1}(t)$. Notice that each boundary component of ${\mathcal M}$ is contained in a single time-slice. The *horizontal distribution*, ${\mathcal H}T{\mathcal M}$ is the distribution tangent to this foliation. A *horizontal metric* on space-time is a smoothly varying positive definite inner product on ${\mathcal H}T{\mathcal M}$. + Notice that a horizontal metric on space-time induces an ordinary Riemannian metric on each time-slice. Conversely, given a Riemannian @@ -1680,11 +1887,11 @@ metric on each time-slice $M_t$, the condition that they fit together to form a horizontal metric on space-time is that they vary smoothly on space-time. We define the curvature of a horizontal metric $G$ to be the section of the dual of the symmetric square of -$\wedge^2{\mathcal H}T{\mathcal M}$ whose value at each point $x$ with $**t**(x)=t$ is the usual Riemannian curvature tensor of the induced -metric on $M_t$ at the point $x$. This is a smooth section of $Sym^2(\wedge^2{\mathcal H}T^*{\mathcal M})$. +$\wedge^2{\mathcal H}T{\mathcal M}$ whose value at each point $x$ with $\mathbf{t}(x)=t$ is the usual Riemannian curvature tensor of the induced +metric on $M_t$ at the point $x$. This is a smooth section of $\mathit{Sym}^2(\wedge^2{\mathcal H}T^*{\mathcal M})$. The Ricci curvature and the scalar curvature of a horizontal metric are given in the usual way from its Riemannian curvature. - The Ricci curvature is a smooth section of $Sym^2({\mathcal + The Ricci curvature is a smooth section of $\mathit{Sym}^2({\mathcal H}T^*{\mathcal M})$ while the scalar curvature is a smooth function on ${\mathcal M}$. ### The generalized Ricci flow equation @@ -1694,18 +1901,25 @@ vector field $\chi$ preserves the horizontal foliation and hence the horizontal distribution. Thus, we can form the Lie derivative of a horizontal metric with respect to $\chi$. + **Definition.** An *$n$-dimensional generalized Ricci flow* consists of a space-time ${\mathcal M}$ that is $(n+1)$-dimensional and a horizontal metric $G$ satisfying the generalized Ricci flow equation: -$${\mathcal L}_\chi(G)=-2Ric(G).$$ +$$ +{\mathcal L}_\chi(G)=-2\mathit{Ric}(G). +$$ + + + **Remark.** Let $({\mathcal M},G)$ be a generalized Ricci flow and let $x\in -{\mathcal M}$. Pulling $G$ back to the local coordinates $V\times +{\mathcal M}$. Pulling $G$ back to the local coordinates $V\times J$ defined near any point gives a one-parameter family of metrics $(V,g(t)),\ t\in J$, satisfying the usual Ricci flow equation. It follows that all the usual evolution formulas for Riemannian curvature, Ricci curvature, and scalar curvature hold in this more general context. + Of course, any ordinary Ricci flow is a generalized Ricci flow where space-time is a product $M\times I$ with time being the projection @@ -1714,10 +1928,11 @@ $I$-direction. ### More definitions for generalized Ricci flows + **Definition.** Let ${\mathcal M}$ be a space-time. Given a space $C$ and an -interval $I\subset \Ar$ we say that an embedding $C\times I\to +interval $I\subset \Ar$ we say that an embedding $C\times I\to {\mathcal M}$ is *compatible with* the time and the vector field -if: (i) the restriction of $**t**$ to the image agrees with the +if: (i) the restriction of $\mathbf{t}$ to the image agrees with the projection onto the second factor and (ii) for each $c\in C$ the image of $\{c\}\times I$ is the integral curve for the vector field $\chi$. @@ -1736,33 +1951,39 @@ line through $x$. For a more general subset $C\subset M_{t}$ there is an embedding $C\times I$ compatible with time and the vector field $\chi$ if an only if for every $x\in C$, $I$ is contained in the domain of definition of the flow line through $x$. + + **Definition.** We say that $t$ is a *regular time* if there is $\epsilon>0$ and -a diffeomorphism $M_{t}\times (t-\epsilon,t+\epsilon)\to **t**^{-1}((t-\epsilon,t+\epsilon))$ compatible with time +a diffeomorphism $M_{t}\times (t-\epsilon,t+\epsilon)\to \mathbf{t}^{-1}((t-\epsilon,t+\epsilon))$ compatible with time and the vector field. A time is *singular* if it is not regular. Notice that if all times are regular, then space-time is a product -$M_{t}\times I$ with $**t**$ and $\chi$ coming from the second -factor. If the image $**t**({\mathcal M})$ is an interval $I$ +$M_{t}\times I$ with $\mathbf{t}$ and $\chi$ coming from the second +factor. If the image $\mathbf{t}({\mathcal M})$ is an interval $I$ bounded below, then the *initial time* for the flow is the greatest lower bound for $I$. If $I$ includes $(-\infty,A]$ for some $A$, then the initial time for the generalized Ricci flow is $-\infty$. + + **Definition.** Suppose that $({\mathcal M},G)$ is a generalized Ricci flow and that $Q>0$ is a positive constant. Then we can define a new generalized -Ricci flow by setting $G'=QG$, $**t'**=Q**t**$ and +Ricci flow by setting $G'=QG$, $\mathbf{t'}=Q\mathbf{t}$ and $\chi'=Q^{-1}\chi$. It is easy to see that the result still satisfies the generalized Ricci flow equation. We denote this new generalized Ricci flow by $(Q{\mathcal M},QG)$ where the changes in -$**t**$ and $\chi$ are denoted by the factor of $Q$ in front of +$\mathbf{t}$ and $\chi$ are denoted by the factor of $Q$ in front of ${\mathcal M}$. It is also possible to translate a generalized solution $({\mathcal M},G)$ by -replacing the time function $**t**$ by $**t'**=**t**+a$ for any constant +replacing the time function $\mathbf{t}$ by $\mathbf{t'}=\mathbf{t}+a$ for any constant $a$, leaving $G$ and $\chi$ unchanged. + + **Definition.** Let $({\mathcal M},G)$ be a generalized Ricci flow and let $x$ be a -point of space-time. Set $t=**t**(x)$. For any $r>0$ we define +point of space-time. Set $t=\mathbf{t}(x)$. For any $r>0$ we define $B(x,t,r)\subset M_t$ to be the metric ball of radius $r$ centered at $x$ in the Riemannian manifold $(M_t,g(t))$. For any $\Delta t>0$ we say that $P(x,t,r,\Delta t)$, @@ -1773,4 +1994,5 @@ $B(x,t,r)\times [t,t+\Delta t]$, respectively, $B(x,t,r)\times vector field. When this embedding exists, its image is defined to be the *forward parabolic neighborhood* $P(x,t,r,\Delta t)$, respectively the *backward parabolic neighborhood* $P(x,t,r,-\Delta t)$. -See Fig.~. +See Fig. 3.3. + diff --git a/projects/poincare-conjecture/.astrolabe/docs/04-maxprin.mdx b/projects/poincare-conjecture/.astrolabe/docs-src/05-the-maximum-principle.mdx similarity index 61% rename from projects/poincare-conjecture/.astrolabe/docs/04-maxprin.mdx rename to projects/poincare-conjecture/.astrolabe/docs-src/05-the-maximum-principle.mdx index ea64ae99..5a629cf2 100644 --- a/projects/poincare-conjecture/.astrolabe/docs/04-maxprin.mdx +++ b/projects/poincare-conjecture/.astrolabe/docs-src/05-the-maximum-principle.mdx @@ -1,7 +1,15 @@ -# The maximum principle + - Recall that the maximum principle for the heat equation says that if $h$ is a solution to -the heat equation $$\frac{\partial h}{\partial t}=\Delta h$$ on a +# Chapter 4 -- The maximum principle + + Recall that the maximum principle for the heat equation says that if $h$ is a solution to +the heat equation + +$$ +\frac{\partial h}{\partial t}=\Delta h +$$ + + on a compact manifold and if $h(x,0)\ge 0$ for all $x\in M$, then $h(x,t)\ge 0$ for all $(x,t)$. In this chapter we discuss analogues of this result for the scalar curvature, the Ricci curvature, and @@ -12,7 +20,7 @@ in particular show that at zero curvature they have the appropriate sign. Also, in the latter two cases we are working with tensors rather than with scalars and hence we require a tensor version of the maximum principle, which was established by Hamilton in -. +[Hamilton4MPIC]. As further applications of these results beyond just establishing non-negativity, we indicate Hamilton's result that if the initial @@ -41,124 +49,159 @@ easiest evolution equation, that for the scalar curvature, where the argument uses only the (non-linear) version of the maximum principle. This result is valid in all dimensions: -**Proposition.** -Let $(M,g(t)),\ 0\le t +**Proposition.** Let $(M,g(t)),\ 0\le t + +**Proof.** According to Equation (3.5), the evolution equation for $R$ is -$$\frac{\partial}{\partial t}R(x,t)=\Delta -R(x,t)+2|Ric(x,t)|^2.$$ Since $M$ is compact, the function -$R_min(t)$ is continuous but may not be $C^1$ at points where + +$$ +\frac{\partial}{\partial t}R(x,t)=\Delta +R(x,t)+2|\mathit{Ric}(x,t)|^2. +$$ + + Since $M$ is compact, the function +$R_\mathit{min}(t)$ is continuous but may not be $C^1$ at points where the minimum of the scalar curvature is achieved at more than one point. The first thing to notice is the following: -**Claim.** If $R(x,t)=R_min(t)$ then $(\partial R/\partial t)(x,t)\ge \frac{2}{n}R^2(x,t)$. + +**Claim.** If $R(x,t)=R_\mathit{min}(t)$ then $(\partial R/\partial t)(x,t)\ge \frac{2}{n}R^2(x,t)$. + -\begin{proof} -This is immediate from the evolution equation for $R$, the fact -that if $R(x,t)=R_min(t)$, then $\Delta R(x,t)\ge 0$, and the -fact that $R$ is the trace of $Ric$ which implies by the -Cauchy-Schwarz inequality that $|R|^2\le n|Ric|^2$. +**Proof.** This is immediate from the evolution equation for $R$, the fact +that if $R(x,t)=R_\mathit{min}(t)$, then $\Delta R(x,t)\ge 0$, and the +fact that $R$ is the trace of $\mathit{Ric}$ which implies by the +Cauchy-Schwarz inequality that $|R|^2\le n|\mathit{Ric}|^2$. Now it follows that: + **Claim.** -$$\frac{d}{d t}(R_min(t))\ge \frac{2}{n}R_min^2(t),$$ -where, at times $t$ where $R_min(t)$ is not smooth, this + +$$ +\frac{d}{d t}(R_\mathit{min}(t))\ge \frac{2}{n}R_\mathit{min}^2(t), +$$ + +where, at times $t$ where $R_\mathit{min}(t)$ is not smooth, this inequality is interpreted as an inequality for the forward difference quotients. + **Proof.** This is immediate from the first statement in -Proposition~. +\entryref{37d7d059a526}. -If follows immediately from Claim~ and -Lemma~ that $R_min(t)$ is a non-decreasing +If follows immediately from \entryref{84e573c0f2b2} and +\entryref{67c378f87457} that $R_\mathit{min}(t)$ is a non-decreasing function of $t$. This establishes the first item and also the second -item in the case when $R_min(0)=0$. +item in the case when $R_\mathit{min}(0)=0$. -Suppose that $R_min(0)\not= 0$. +Suppose that $R_\mathit{min}(0)\not= 0$. Consider the function -$$S(t)=\frac{-1}{R_min(t)}-\frac{2t}{n}+\frac{1}{R_min(0)}.$$ + +$$ +S(t)=\frac{-1}{R_\mathit{min}(t)}-\frac{2t}{n}+\frac{1}{R_\mathit{min}(0)}. +$$ + Clearly, $S(0)=0$ and $S'(t)\ge 0$ (in the sense of forward difference quotients), - so that by Lemma~ we have $S(t)\ge 0$ for all $t$. This means that + so that by \entryref{67c378f87457} we have $S(t)\ge 0$ for all $t$. This means that $$ - \frac{1}{R_min(t)}\le -\frac{1}{R_min(0)}-\frac{2t}{n} +\frac{1}{R_\mathit{min}(t)}\le +\frac{1}{R_\mathit{min}(0)}-\frac{2t}{n} $$ + provided that -$R_min$ is not ever zero on the interval $[0,t]$. If $R_min(0)>0$, then by the first item, $R_min(t)>0$ for all $t$ +$R_\mathit{min}$ is not ever zero on the interval $[0,t]$. If $R_\mathit{min}(0)>0$, then by the first item, $R_\mathit{min}(t)>0$ for all $t$ for which the flow is defined, and the inequality in the second item -of the proposition is immediate from Equation~(). The +of the proposition is immediate from Equation (4.1). The third inequality in the proposition also follows easily from -Equation~() when $R_min(t)<0$. But if $R_min(t)\ge 0$, then the third item is obvious. -\end{proof} +Equation (4.1) when $R_\mathit{min}(t)<0$. But if $R_\mathit{min}(t)\ge 0$, then the third item is obvious. ## The maximum principle for tensors For the applications to the Ricci curvature and the curvature tensor we need a version of the maximum principle for tensors that is due to Hamilton; see -. +[Hamilton4MPCO]. Suppose that $V$ is a finite-dimensional real vector space and $Z\subset V$ is a closed convex set. For each $z$ in the frontier of $ Z$ we define the *tangent cone to* $Z$ at $z$, denoted $T_zZ$, to be the intersection of all closed half-spaces $H$ of $V$ such that $z\in -\partial H$ and $Z\subset H$. For $z\in int\, Z$ we define +\partial H$ and $Z\subset H$. For $z\in \mathit{int} Z$ we define $T_zZ=V$. Notice that $v\notin T_zZ$ if and only if there is a affine linear function $\ell$ vanishing at $z$ non-positive on $Z$ and positive on $v$. + **Definition.** Let $Z$ be a closed convex subset of a finite-dimensional real vector space $V$. We say that a smooth vector field $\psi$ defined on an open neighborhood $U$ of $Z$ in $V$ *preserves* $Z$ if for every $z\in Z$ we have $\psi(z)\in T_zZ$. + It is an easy exercise to show the following; see Lemma 4.1 on page -183 of : +183 of [Hamilton4MPCO]: + **Lemma.** Let $Z$ be a closed convex subset in a finite dimensional real vector space $V$. Let $\psi$ be a smooth vector field defined on an open neighborhood of $Z$ in $V$. Then $\psi$ preserves $Z$ if and -only if every integral curve $\gamma\colon [0,a)\to V$ for $\psi$ +only if every integral curve $\gamma\colon [0,a)\to V$ for $\psi$ with $\gamma(0)\in Z$ has $\gamma(t)\in Z$ for all $t\in [0,a)$. Said more informally, $\psi$ preserves $Z$ if and only if every integral curve for $\psi$ that starts in $Z$ remains in $Z$. + -### The global -version +### The global version The maximum principle for tensors generalizes this to tensor flows evolving by parabolic equations. First we introduce a generalization of the notion of a vector field preserving a closed convex set to the context of vector bundles. + **Definition.** Let $\pi\colon {\mathcal V}\to M$ be a vector bundle and let -${\mathcal Z}\subset {\mathcal V}$ be a closed subset. We say that -${\mathcal Z}$ is *convex* if for every $x\in M$ the fiber +${\mathcal Z}\subset {\mathcal V}$ be a closed subset. We say that +${\mathcal Z}$ is *convex* if for every $x\in M$ the fiber $Z_x$ of ${\mathcal Z}$ over $x$ is a convex subset of the vector space fiber $V_x$ of ${\mathcal V}$ over $x$. Let $\psi$ be a fiberwise vector field on an open neighborhood ${\mathcal U}$ of ${\mathcal Z}$ in ${\mathcal V}$. We say that $\psi$ *preserves* ${\mathcal Z}$ if for each $x\in M$ the restriction of $\psi$ to the fiber $U_x$ of ${\mathcal U}$ over $x$ preserves $Z_x$. + The following global version of the maximum principle for tensors is -Theorem 4.2 of . +Theorem 4.2 of [Hamilton4MPCO]. + **Theorem.** **(The maximum principle for tensors)** Let $(M,g)$ be a compact Riemannian manifold. Let ${\mathcal V}\to M$ be a tensor bundle and let ${\mathcal Z}\subset {\mathcal V}$ be a closed, convex @@ -167,24 +210,30 @@ connection. Suppose that $\psi$ is a fiberwise vector field defined on an open neighborhood of ${\mathcal Z}$ in ${\mathcal V}$ that preserves ${\mathcal Z}$. Suppose that ${\mathcal T}(x,t),\ 0\le t\le T$, is a one-parameter family of sections of ${\mathcal V}$ that evolves according to the parabolic equation -$$\frac{\partial {\mathcal T}}{\partial t}=\Delta {\mathcal T} -+\psi({\mathcal T}).$$ If ${\mathcal T}(x,0)$ is contained in + +$$ +\frac{\partial {\mathcal T}}{\partial t}=\Delta {\mathcal T} ++\psi({\mathcal T}). +$$ + + If ${\mathcal T}(x,0)$ is contained in ${\mathcal Z}$ for all $x\in M$, then ${\mathcal T}(x,t)$ is contained in ${\mathcal Z}$ for all $x\in M$ and for all $0\le t\le T$. + -For a proof we refer the reader to Theorem 4.3 and its proof (and -the related Theorem 4.2 and its proof) in . +For a proof we refer the reader to Theorem 4.3 and its proof (and +the related Theorem 4.2 and its proof) in [Hamilton4MPCO]. There is a slight improvement of this result where the convex set ${\mathcal Z}$ is allowed to vary with $t$. It is proved by the same argument; see Theorem -4.8 on page 101 of . +4.8 on page 101 of [ChowKnopf]. -**Theorem.** -Let $(M,g)$ be a compact Riemannian manifold. Let ${\mathcal V}\to M$ be a + +**Theorem.** Let $(M,g)$ be a compact Riemannian manifold. Let ${\mathcal V}\to M$ be a tensor bundle and let ${\mathcal Z}\subset {\mathcal V}\times [0,T]$ be a closed subset with the property that for each $t\in [0,T]$ the time-slice -${\mathcal Z}(t)$ is a convex subset of ${\mathcal V}\times\{t\}$ invariant +${\mathcal Z}(t)$ is a convex subset of ${\mathcal V}\times\{t\}$ invariant under the parallel translation induced by the Levi-Civita connection. Suppose that $\psi$ is a fiberwise vector field defined on an open neighborhood of ${\mathcal Z}$ in ${\mathcal V}\times [0,T]$ that preserves the family @@ -193,19 +242,24 @@ t_1$, for $\psi$ with the property that $\gamma(t_0)\in {\mathcal Z}(t_0)$ has $\gamma(t)\in {\mathcal Z}(t)$ for every $t\in [t_0,t_1]$. Suppose that ${\mathcal T}(x,t),\ 0\le t\le T$, is a one-parameter family of sections of ${\mathcal V}$ that evolves according to the parabolic equation -$$\frac{\partial {\mathcal T}}{\partial t}=\Delta {\mathcal T}+\psi({\mathcal T}).$$ + +$$ +\frac{\partial {\mathcal T}}{\partial t}=\Delta {\mathcal T}+\psi({\mathcal T}). +$$ + If ${\mathcal T}(x,0)$ is contained in ${\mathcal Z}(0)$ for all $x\in M$, then ${\mathcal T}(x,t)$ is contained in ${\mathcal Z}(t)$ for all $x\in M$ and for all $0\le t\le T$. + ### The local version Here is the local result. It is proved by the same argument as given -in the proof of Theorem 4.3 in . +in the proof of Theorem 4.3 in [Hamilton4MPCO]. -**Theorem.** -Let $(M,g)$ be a Riemannian manifold. Let $\overline U\subset M$ be a compact, + +**Theorem.** Let $(M,g)$ be a Riemannian manifold. Let $\overline U\subset M$ be a compact, smooth, connected, codimension-$0$ submanifold. Let ${\mathcal V}\to M$ be a tensor bundle and let ${\mathcal Z}\subset {\mathcal V}$ be a closed, convex subset. Suppose that $\psi$ is a fiberwise vector field defined on an open @@ -214,12 +268,18 @@ Suppose that ${\mathcal Z}$ is invariant under the parallel translation induced by the Levi-Civita connection. Suppose that ${\mathcal T}(x,t),\ 0\le t\le T$, is a one-parameter family of sections of ${\mathcal V}$ that evolves according to the parabolic equation -$$\frac{\partial {\mathcal T}}{\partial t}=\Delta {\mathcal T} -+\psi({\mathcal T}).$$ If ${\mathcal T}(x,0)$ is contained in + +$$ +\frac{\partial {\mathcal T}}{\partial t}=\Delta {\mathcal T} ++\psi({\mathcal T}). +$$ + + If ${\mathcal T}(x,0)$ is contained in ${\mathcal Z}$ for all $x\in \overline U$ and if ${\mathcal T}(x,t)\in {\mathcal Z}$ for all $x\in \partial \overline U$ and all $0\le t\le T$, then ${\mathcal T}(x,t)$ is contained in ${\mathcal Z}$ for all $x\in \overline U$ and all $0\le t\le T$. + ## Applications of the maximum principle @@ -237,19 +297,20 @@ make the constants absolute we impose scale fixing (or rather scale bounding) conditions on the initial metrics of the flows that we shall consider. The following definition makes precise the exact conditions that we shall use. -**Definition.** -We say that a that a Ricci flow $(M,g(t))$ has *normalized + +**Definition.** We say that a that a Ricci flow $(M,g(t))$ has *normalized initial conditions* if $0$ is the initial time for the flow and if the compact Riemannian manifold $(M^n,g(0))$ satisfies: -\begin{enumerate} -\item $|Rm(x,0)|\le 1$ for all $x\in M$. -\item Let $\omega_n$ be the volume of the ball of radius $1$ in $n$ -dimensional Euclidean -space. Then $Vol(B(x,0,r))\ge (\omega_n/2)r^n$ for any $p\in + +1. $|\mathit{Rm}(x,0)|\le 1$ for all $x\in M$. +1. Let $\omega_n$ be the volume of the ball of radius $1$ in $n$ -dimensional Euclidean +space. Then $\mathit{Vol}(B(x,0,r))\ge (\omega_n/2)r^n$ for any $p\in M$ and any $r\le 1$. -\end{enumerate} + We also use the terminology $(M,g(0))$ *is normalized* to indicate that it satisfies these two conditions. + The evolution equation for the Riemann curvature and a standard maximum principle argument show that if $(M,g(0))$ has an upper bound on the Riemann @@ -258,36 +319,42 @@ flow has Riemann curvature bounded above and volumes of balls bounded below on a fixed time interval. Here is the result in the context of normalized initial condition. -**Proposition.** - There is $\kappa_0>0$ depending only on the dimension $n$ + +**Proposition.** There is $\kappa_0>0$ depending only on the dimension $n$ such that the following holds. Let $(M^n,g(t)),\ 0\le t\le T$, be a Ricci flow with bounded curvature, with each $(M,g(t))$ being complete, and with normalized initial conditions. Then -$|Rm(x,t)|\le 2$ for all $x\in M$ and all $t\in [0,min(T,2^{-4})]$. -Furthermore, for any $t\in [0,min(T,2^{-4})]$ and any $x\in M$ and any -$r\le 1$ we have $Vol\,B(x,t,r)\ge \kappa_0r^n$. +$|\mathit{Rm}(x,t)|\le 2$ for all $x\in M$ and all $t\in [0,\mathit{min}(T,2^{-4})]$. +Furthermore, for any $t\in [0,\mathit{min}(T,2^{-4})]$ and any $x\in M$ and any +$r\le 1$ we have $\mathit{Vol} B(x,t,r)\ge \kappa_0r^n$. + **Proof.** The bound on the Riemann curvature follows directly from Lemma 6.1 -on page 207 of and the definition of normalized +on page 207 of [ChowLuNi] and the definition of normalized initial conditions. Once we know that the Riemann curvature is bounded by $2$ on $[0,2^{-4}]$, there is an $0T$ or $|Rm|$ is unbounded on $M\times[0,T)$. + +**Proposition.** Let $(M,g(t)),\ 0\le tT$ or $|\mathit{Rm}|$ is unbounded on $M\times[0,T)$. + ### Non-negative curvature is preserved @@ -295,36 +362,54 @@ We need to consider the tensor versions of the maximum principle when the tensor in question is the Riemann or Ricci curvature and the evolution equation is that induced by the Ricci flow. This part of the discussion is valid in dimension three only. We begin by -evaluating the expressions in Equation~() in the +evaluating the expressions in Equation (\entryref{883dba87d14c}) in the $3$-dimensional case. Fix a symmetric bilinear form ${\mathcal S}$ on a $3$-dimensional real vector space $V$ with a positive definite inner product. The inner product determines an identification of $\wedge^2V$ with $V^*$. Hence, $\wedge^2{\mathcal S}^*$ is -identified with a symmetric automorphism of $V$, denoted by +identified with a symmetric automorphism of $V$, denoted by ${\mathcal S}^\sharp$. -**Lemma.** -Let $(M,g)$ be a Riemannian $3$-manifold. Let ${\mathcal T}\in Sym^2(\wedge^2T_x^*M)$ be the curvature operator written with -respect to the evolving frame as in Proposition~. Then -the evolution equation given in Proposition~ is: -$$\frac{\partial {\mathcal T}}{\partial t}=\triangle {\mathcal T}+\psi({\mathcal T})$$ + +**Lemma.** Let $(M,g)$ be a Riemannian $3$-manifold. Let ${\mathcal T}\in \mathit{Sym}^2(\wedge^2T_x^*M)$ be the curvature operator written with +respect to the evolving frame as in \entryref{883dba87d14c}. Then +the evolution equation given in \entryref{883dba87d14c} is: + +$$ +\frac{\partial {\mathcal T}}{\partial t}=\triangle {\mathcal T}+\psi({\mathcal T}) +$$ + where -$$\psi({\mathcal T})={\mathcal T}^2+{\mathcal T}^\sharp.$$ + +$$ +\psi({\mathcal T})={\mathcal T}^2+{\mathcal T}^\sharp. +$$ + In particular, in an orthonormal basis in which -$${\mathcal T}=\begin{pmatrix} \lambda & 0 & 0 \\ 0 & \mu & 0 \\ 0 & -0 & \nu\end{pmatrix}$$ with $\lambda\ge \mu\ge \nu$, the vector + +$$ +{\mathcal T}=\begin{pmatrix} \lambda & 0 & 0 \\ 0 & \mu & 0 \\ 0 & +0 & \nu\end{pmatrix} +$$ + + with $\lambda\ge \mu\ge \nu$, the vector field is given by -$$\psi({\mathcal T})={\mathcal T}^2+{\mathcal T}^\sharp=\begin{pmatrix} + +$$ +\psi({\mathcal T})={\mathcal T}^2+{\mathcal T}^\sharp=\begin{pmatrix} \lambda^2+\mu\nu & 0 & 0 \\ 0 & \mu^2+\lambda\nu & 0 \\ 0 & 0 & -\nu^2+\lambda\mu\end{pmatrix}.$$ +\nu^2+\lambda\mu\end{pmatrix}. +$$ + -**Corollary.** -Let $(M,g(t)),\ 0\le t\le T$, be a Ricci flow with $M$ a compact, -connected $3$-manifold. Suppose that $Rm(x,0)\ge 0$ for all -$x\in M$. Then $Rm(x,t)\ge 0$ for all $x\in M$ and all $t\in + +**Corollary.** Let $(M,g(t)),\ 0\le t\le T$, be a Ricci flow with $M$ a compact, +connected $3$-manifold. Suppose that $\mathit{Rm}(x,0)\ge 0$ for all +$x\in M$. Then $\mathit{Rm}(x,t)\ge 0$ for all $x\in M$ and all $t\in [0,T]$. + -**Proof.** Let $\nu_x\colon Sym^2(\wedge^2T^*_xM)\to \Ar$ associate to +**Proof.** Let $\nu_x\colon \mathit{Sym}^2(\wedge^2T^*_xM)\to \Ar$ associate to each endomorphism its smallest eigenvalue. Then $\nu_x({\mathcal T})$ is the minimum over all lines in $\wedge^2T_xM$ of the trace of the restriction of ${\mathcal T}$ to that line. As a minimum of @@ -333,123 +418,148 @@ $Z_x=\nu_x^{-1}([0,\infty))$ is a convex subset. We let ${\mathcal Z}$ be the union over all $x$ of $Z_x$. Clearly, ${\mathcal Z}$ is a closed convex subset of the tensor bundle. Since parallel translation is orthogonal, ${\mathcal Z}$ is invariant under -parallel translation. The expressions in Lemma~ show that +parallel translation. The expressions in \entryref{0c451aa3e5cc} show that if ${\mathcal T}$ is an endomorphism of $\wedge^2T^*_xM$ with $\nu({\mathcal T})\ge 0$, then the symmetric matrix $\psi({\mathcal T})$ is non-negative. This implies that $\nu_x$ is non-decreasing in the direction $\psi({\mathcal T})$ at the point ${\mathcal T}$. That is to say, for each $x\in M$, the vector field $\psi({\mathcal T})$ preserves the set $\{\nu_x^{-1}([c,\infty))\}$ for any $c\ge 0$. The -hypothesis that $Rm(x,0)\ge 0$ means that $Rm(x,0)\in -{\mathcal Z}$ for all $x\in M$. Applying Theorem~ proves the +hypothesis that $\mathit{Rm}(x,0)\ge 0$ means that $\mathit{Rm}(x,0)\in +{\mathcal Z}$ for all $x\in M$. Applying \entryref{c954e95b5d89} proves the result. -**Corollary.** -Suppose that $(M,g(t)),\ 0\le t\le T$, is a Ricci flow with $M$ a -compact, connected $3$-manifold with $Ric(x,0)\ge 0$ for all -$x\in M$. Then $Ric(x,t)\ge 0$ for all $t>0$. + +**Corollary.** Suppose that $(M,g(t)),\ 0\le t\le T$, is a Ricci flow with $M$ a +compact, connected $3$-manifold with $\mathit{Ric}(x,0)\ge 0$ for all +$x\in M$. Then $\mathit{Ric}(x,t)\ge 0$ for all $t>0$. + -**Proof.** The statement that $Ric(x,t)\ge 0$ is equivalent to the statement that +**Proof.** The statement that $\mathit{Ric}(x,t)\ge 0$ is equivalent to the statement that for every two-plane in $\wedge^2 T_xM$ the trace of the Riemann curvature -operator on this plane is $\ge 0$. For ${\mathcal T}\in Sym^2(\wedge^2T_x^*M)$, we define $s({\mathcal T})$ as the minimum over all +operator on this plane is $\ge 0$. For ${\mathcal T}\in \mathit{Sym}^2(\wedge^2T_x^*M)$, we define $s({\mathcal T})$ as the minimum over all two-planes $P$ in $\wedge^2TM$ of the trace of ${\mathcal T}$ on $P$. The restriction $s_x$ of $s$ to the fiber over $x$ is the minimum of a collection of linear functions and hence is convex. Thus, the subset ${\mathcal S}=s^{-1}([0,\infty))$ is convex. Clearly, $s$ is preserved by orthogonal isomorphisms, so ${\mathcal S}$ is invariant under parallel translation. Let $\lambda\ge\mu\ge\nu$ be the eigenvalues of ${\mathcal T}$. According to -Lemma~ the derivative of $s_x$ at ${\mathcal T}$ in the +\entryref{0c451aa3e5cc} the derivative of $s_x$ at ${\mathcal T}$ in the $\psi({\mathcal T})$-direction is $(\mu^2+\lambda\nu)+(\nu^2+\lambda\mu)=(\mu^2+\nu^2)+\lambda(\mu+\nu)$. The condition that $s({\mathcal T})\ge 0$, is the condition that $\nu+\mu\ge 0$, and hence $\mu\ge 0$, implying that $\lambda\ge 0$. Thus, if $s({\mathcal T})\ge 0$, it is also the case that the derivatifve of $s_x$ in the $\psi({\mathcal T})$-direction is non-negative. This implies that $\psi$ -preserves ${\mathcal S}$. Applying Theorem~ gives the result. +preserves ${\mathcal S}$. Applying \entryref{c954e95b5d89} gives the result. ## The strong maximum principle for curvature First let us state the strong maximum principle for the heat equation. -**Theorem.** -Let $\overline U$ be a compact, connected manifold, possibly with boundary. Let + +**Theorem.** Let $\overline U$ be a compact, connected manifold, possibly with boundary. Let $h(x,t),\ 0\le t\le T$, be a solution to the heat equation -$$\frac{\partial h(x,t)}{\partial t}=\Delta h(x,t).$$ - Suppose that $h$ has Dirichlet boundary conditions in the sense + +$$ +\frac{\partial h(x,t)}{\partial t}=\Delta h(x,t). +$$ + + Suppose that $h$ has Dirichlet boundary conditions in the sense that $h(x,t)=0$ for all $(x,t)\in\partial \overline U\times [0,T]$. If $h(x,0)\ge 0$ for all $x\in \overline U$, then $h(x,t)\ge 0$ for all $(x,t)\in \overline U\times [0,T]$. If, in addition, there is $y\in \overline U$ with $h(y,0)>0$, then $h(x,t)>0$ for all -$(x,t)\in int(\overline U)\times(0,T]$. +$(x,t)\in \mathit{int}(\overline U)\times(0,T]$. + We shall use this strong maximum principle to establish an analogous result for the curvature tensors. The hypotheses are in some ways more restrictive -- they are set up to apply to the Riemann and Ricci curvature. -**Proposition.** -Let $(M,g)$ be a Riemannian manifold and let ${\mathcal V}$ be a + +**Proposition.** Let $(M,g)$ be a Riemannian manifold and let ${\mathcal V}$ be a tensor bundle. Suppose that $\overline U$ is a compact, connected, smooth codimension-$0$ submanifold of $M$. Consider a one-parameter family of sections ${\mathcal T}(x,t),\ 0\le t\le T$, of ${\mathcal V}$. Suppose that ${\mathcal T}$ evolves according to the equation -$$\frac{\partial {\mathcal T}}{\partial t}=\Delta {\mathcal T}+\psi({\mathcal T})$$ + +$$ +\frac{\partial {\mathcal T}}{\partial t}=\Delta {\mathcal T}+\psi({\mathcal T}) +$$ + for some smooth, fiberwise vector field $\psi({\mathcal T})$ defined on ${\mathcal V}$. Suppose that $s\colon {\mathcal V}\to \Ar$ is a function satisfying the following properties: -\begin{enumerate} -\item For each $x\in M$ the restriction $s_x$ to the fiber $V_x$ of -${\mathcal V}$ over $x$ is a convex function. -\item For any $A$ satisfying $s_x(A)\ge 0$ the vector $\psi(A)$ is contained + +1. For each $x\in M$ the restriction $s_x$ to the fiber $V_x$ of +${\mathcal V}$ over $x$ is a convex function. +1. For any $A$ satisfying $s_x(A)\ge 0$ the vector $\psi(A)$ is contained in the tangent cone of the convex set $\{y|s_x(y)\ge s_x(A)$ at the point $A$. -\item $s$ is invariant under parallel translation. -\end{enumerate} +1. $s$ is invariant under parallel translation. + Suppose that $s({\mathcal T}(x,0))\ge 0$ for all $x\in \overline U$ and that $s({\mathcal T}(x,t))\ge 0$ for all $x\in \partial\overline -U$ and all $t\in [0,T]$. Suppose also that there is $x_0\in int(\overline U)$ with $s({\mathcal T}(x_0,0))>0$. Then -$s({\mathcal T}(x,t))> 0$ for all $(x,t)\in int(\overline +U$ and all $t\in [0,T]$. Suppose also that there is $x_0\in \mathit{int}(\overline U)$ with $s({\mathcal T}(x_0,0))>0$. Then +$s({\mathcal T}(x,t))> 0$ for all $(x,t)\in \mathit{int}(\overline U)\times(0,T]$. + **Proof.** Let $h\colon \overline U\times\{0\}\to \Ar$ be a smooth function with $h(x,0)=0$ for all $x\in \partial \overline U$ and with $s({\mathcal T}(x,0))\ge h(x,0)\ge 0$ for all $x\in \overline U$. We -choose $h$ so that $h(x_0,0)>0$. Let $h(x,t),\ 0\le t<\infty$, be +choose $h$ so that $h(x_0,0)>0$. Let $h(x,t),\ 0\le t<\infty$, be the solution to the heat equation on $\overline U$ -$$\frac{\partial h}{\partial t}=\Delta h$$ + +$$ +\frac{\partial h}{\partial t}=\Delta h +$$ + with Dirichlet boundary conditions $h(x,t)=0$ for all $x\in \partial \overline U$ and all $t\ge 0$ and with the given initial conditions. Consider the tensor bundle ${\mathcal V}\oplus \Ar$ over $M$. We define -$$Z_x=\left\{({\mathcal T},h)\in V_x\oplus -\Ar\bigl|\bigr. s_x({\mathcal T})\ge h\ge 0\right\}.$$ The union + +$$ +Z_x=\left\{({\mathcal T},h)\in V_x\oplus +\Ar\bigl|\bigr. s_x({\mathcal T})\ge h\ge 0\right\}. +$$ + + The union over all $x\in M$ of the $Z_x$ defines a closed convex subset ${\mathcal Z}\subset {\mathcal V}\oplus \Ar$ which is invariant under parallel translation since $s$ is. We consider the family of sections $({\mathcal T}(x,t),h(x,t)),\ 0\le t\le T$, of ${\mathcal V}\oplus \Ar$. These evolve by -$$\frac{d\left({\mathcal T}(x,t),h(x,t)\right)}{dt}=\left(\Delta {\mathcal T}(x,t),\Delta -h(x,t)\right)+\widetilde \psi\left({\mathcal T}(x,t),h(x,t)\right)$$ + +$$ +\frac{d\left({\mathcal T}(x,t),h(x,t)\right)}{dt}=\left(\Delta {\mathcal T}(x,t),\Delta +h(x,t)\right)+\widetilde \psi\left({\mathcal T}(x,t),h(x,t)\right) +$$ + where $\widetilde \psi({\mathcal T},h)=\left(\psi({\mathcal T}),0\right)$. Clearly, by our hypotheses, the vector field $\widetilde \psi$ preserves the convex set ${\mathcal Z}$. Applying -the local version of the maximum principle (Theorem~), +the local version of the maximum principle (\entryref{a89be7fdb431}), we conclude that ${\mathcal T}(x,t)\ge h(x,t)$ for all $(x,t)\in \overline U\times [0,T]$. -The result then follows immediately from Theorem~. +The result then follows immediately from \entryref{5b296315fdf9}. ### Applications of the strong maximum principle We have the following applications of the strong maximum principle. -**Theorem.** -Let $(U,g(t)),\ 0\le t\le T$, be a $3$-dimensional Ricci flow with + +**Theorem.** Let $(U,g(t)),\ 0\le t\le T$, be a $3$-dimensional Ricci flow with non-negative sectional curvature with $U$ connected but not necessarily complete and with $T>0$. If $R(p,T)=0$ for some $p\in U$, then $(U,g(t))$ is flat for every $t\in [0,T]$. + **Proof.** We suppose that there is $p\in U$ with $R(p,T)=0$. Since all the metrics in the flow are of non-negative sectional curvature, if the @@ -462,17 +572,21 @@ $q$ and $p$. Let $h(y,0)$ be a smooth non-negative function with support in $V$, positive at $q$, such that $R(y,0)\ge h(y,0)$ for all $y\in V$. Let $h(y,t)$ be the solution to the heat equation on $V\times [0,T]$ that vanishes on $\partial V$. Of course, $h(y,T)>0$ -for all $y\in int(V)$. Also, from Equation~() we +for all $y\in \mathit{int}(V)$. Also, from Equation (3.5) we have -$$\frac{\partial}{\partial t}(R-h)=\triangle (R-h)+2|Ric|^2,$$ + +$$ +\frac{\partial}{\partial t}(R-h)=\triangle (R-h)+2|\mathit{Ric}|^2, +$$ + so that $(R-h)(y,0)\ge 0$ on $(V\times \{0\})\cup (\partial V\times [0,T])$. It -follows from the maximum principle that $(R-h)\ge 0$ on all of $V\times +follows from the maximum principle that $(R-h)\ge 0$ on all of $V\times [0,T]$. In particular, $R(p,T)\ge h(p,T)>0$. This is a contradiction, establishing the theorem. -**Corollary.** -Fix $T>0$. Suppose that $(U,g(t)),\ 0\le t\le T$, is a Ricci flow -such that for each $t$, the Riemannian manifold $(U,g(t))$ is a + +**Corollary.** Fix $T>0$. Suppose that $(U,g(t)),\ 0\le t\le T$, is a Ricci flow +such that for each $t$, the Riemannian manifold $(U,g(t))$ is a (not necessarily complete) connected, $3$-manifold of non-negative sectional curvature. Suppose that $(U,g(0))$ is not flat and that for some $p\in M$ the Ricci curvature at $(p,T)$ has a zero @@ -481,35 +595,36 @@ $(U,g(t))$ splits locally as a product of a surface of positive curvature and a line, and under this local splitting the flow is locally the product of a Ricci flow on the surface and the trivial flow on the line. + -**Proof.** First notice that it follows from Theorem~ that +**Proof.** First notice that it follows from \entryref{016cf36686b2} that because $(U,g(0))$ is not flat, we have $R(y,t)>0$ for every $(y,t)\in U\times (0,T]$. We consider the function $s$ on -$Sym^2(\wedge^2T^*_yU)$ that associates to each endomorphism +$\mathit{Sym}^2(\wedge^2T^*_yU)$ that associates to each endomorphism the sum of the smallest $2$ eigenvalues. Then $s_y$ is the minimum -of the traces on $2$-dimensional subsets in $\wedge^2T_yU$. Thus, +of the traces on $2$-dimensional subsets in $\wedge^2T_yU$. Thus, $s$ is a convex function, and the subset ${\mathcal S}=s^{-1}([0,\infty))$ is a convex subset. Clearly, this subset is invariant under parallel translation. By the computations in the -proof of Corollary~ it is invariant under the vector +proof of \entryref{f8afc1c61977} it is invariant under the vector field $\psi({\mathcal T})$. The hypothesis of the corollary tells us that $s(p,T)=0$. Suppose that $s(q,t)>0$ for some $(q,t)\in U\times [0,T]$. Of course, by continuity we can take $t< T$. Shift the time parameter so that $t=0$, and fix a compact connected, codimension-$0$ submanifold $V$ containing $p,q$ in its interior. -Then by Theorem~ $s(y,T)>0$ for all $y\in int(V)$ +Then by \entryref{f696c0fa9db3} $s(y,T)>0$ for all $y\in \mathit{int}(V)$ and in particular $s(p,T)>0$. This is a contradiction, and we conclude that $s(q,t)=0$ for all $(q,t)\in U\times [0,T]$. Since we have already established that each $R(y,t)>0$ for all $(y,t)\in -U\times (0,T]$, so that $Rm(y,t)$ is not identically zero, this means -that for all $y\in U$ and all $t\in (0,T]$ that the null space of the operator -$Rm(y,t)$ is a $2$-dimensional subspace of $\wedge^2T_yU$. This +U\times (0,T]$, so that $\mathit{Rm}(y,t)$ is not identically zero, this means +that for all $y\in U$ and all $t\in (0,T]$ that the null space of the operator +$\mathit{Rm}(y,t)$ is a $2$-dimensional subspace of $\wedge^2T_yU$. This $2$-dimensional subspace is dual to a line in $T_xM$. Thus, we have a one-dimensional distribution (a line bundle in the tangent bundle) ${\mathcal -D}$ in $U\times (0,T]$ with the property that the sectional curvature $Rm(y,t)$ vanishes on any $2$-plane containing the line ${\mathcal D}(y,t)$. +D}$ in $U\times (0,T]$ with the property that the sectional curvature $\mathit{Rm}(y,t)$ vanishes on any $2$-plane containing the line ${\mathcal D}(y,t)$. The fact that the sectional curvature of $g(t)$ vanishes on all two-planes in $T_yM$ containing ${\mathcal D}(y,t)$ means that its eigenvalues are $\{\lambda,0,0\}$ where $\lambda>0$ is the sectional curvature of the @@ -519,7 +634,7 @@ R}(V(y,t),\cdot,\cdot,\cdot)=0$. Locally in space and time, there is a unique (up to sign) vector field $V(y,t)$ that generates ${\mathcal D}$ and satisfies $|V(y,t)|^2_{g(t)}=1$. We wish to show that this local vector field is invariant under parallel translation and -time translation; cf. Lemma 8.2 in . Fix a point $x\in M$, +time translation; cf. Lemma 8.2 in [Hamilton4MPCO]. Fix a point $x\in M$, a direction $X$ at $x$, and a time $t$. Let $\tilde V(y,t)$ be a parallel extension of $V(x,t)$ along a curve $C$ passing through $x$ in the $X$-direction, and let $\tilde W(y,t)$ be an arbitrary parallel vector field @@ -527,11 +642,21 @@ along $C$. Since the sectional curvature is non-negative, we have ${\mathcal R}(\tilde V,\tilde W,\tilde V,\tilde W)(y)\ge 0$ for all $y\in C$; furthermore, this expression vanishes at $x$. Hence, its first variation vanishes at $x$. That is to say -$$\nabla \left({\mathcal R}(\tilde V,\tilde W,\tilde V,\tilde W)\right)(x,t)=(\nabla{\mathcal R})(\tilde -V,\tilde W,\tilde V,\tilde W)$$ vanishes at $(x,t)$. Since this is true for all + +$$ +\nabla \left({\mathcal R}(\tilde V,\tilde W,\tilde V,\tilde W)\right)(x,t)=(\nabla{\mathcal R})(\tilde +V,\tilde W,\tilde V,\tilde W) +$$ + + vanishes at $(x,t)$. Since this is true for all $\tilde W$, it follows that the null space of the quadratic form $\nabla{\mathcal R}(x,t)$ contains the null space of ${\mathcal R}(x,t)$, and -thus $$(\nabla {\mathcal R})(V(x,t),\cdot,\cdot,\cdot)=0.$$ +thus + +$$ +(\nabla {\mathcal R})(V(x,t),\cdot,\cdot,\cdot)=0. +$$ + Now let us consider three parallel vector fields $\tilde W_1,\tilde W_2,$ and $\tilde W_3$ along $C$. We compute $0=\nabla_X\left({\mathcal R}( V(y,t),\tilde @@ -539,8 +664,13 @@ W_1(y,t),\tilde W_2(y,t),\tilde W_3(y,t))\right)$. (Notice that while the $\tilde W_i$ are parallel along $C$, $V(y,t)$ is defined to be the vector field spanning ${\mathcal D}(y,t)$ rather than a parallel extension of $V(x,t)$.) Given the above result we find that -$$0=2{\mathcal R}(\nabla_X V(x,t),\tilde W_1(x,t),\tilde W_2(x,t), -\tilde W_3(x,t)).$$ Since this is true for all triples of vector + +$$ +0=2{\mathcal R}(\nabla_X V(x,t),\tilde W_1(x,t),\tilde W_2(x,t), +\tilde W_3(x,t)). +$$ + + Since this is true for all triples of vector fields $\tilde W_i(x,t)$, it follows that $\nabla_XV(x,t)$ is a real multiple of $V(x,t)$. But since $|V(y,t)|^2_{g(t)}=1$, we see that $\nabla_XV(x,t)$ is orthogonal to $V(x,t)$. We conclude that @@ -551,7 +681,7 @@ translation associated to the metric $g(t)$. It follows that locally $(M,g(t))$ is a Riemannian product of a surface of positive curvature with a line. Under this product decomposition, the curvature is the pullback of the curvature of the surface. Hence, by -Equation~(), under Ricci flow on the $3$-manifold, the time +Equation (3.5), under Ricci flow on the $3$-manifold, the time derivative of the curvature at time $t$ also decomposes as the pullback of the time derivative of the curvature of the surface under Ricci flow on the surface. In particular, $(\partial {\mathcal R}/\partial @@ -565,59 +695,67 @@ and a line, and this splitting is invariant under the Ricci flow. This completes the proof of the corollary. In the complete case, this local product decomposition globalizes in -some cover; see Lemma 9.1 in . +some cover; see Lemma 9.1 in [Hamilton4MPCO]. -**Corollary.** -Suppose that $(M,g(t)),\ 0\le t\le T$, is a Ricci flow of complete, connected -Riemannian $3$-manifolds with $Rm(x,t)\ge 0$ for all $(x,t)$ and with -$T>0$. Suppose that $(M,g(0))$ is not flat and that for some $x\in M$ the -endomorphism $Rm(x,T)$ has a zero eigenvalue. Then $M$ has a cover + +**Corollary.** Suppose that $(M,g(t)),\ 0\le t\le T$, is a Ricci flow of complete, connected +Riemannian $3$-manifolds with $\mathit{Rm}(x,t)\ge 0$ for all $(x,t)$ and with +$T>0$. Suppose that $(M,g(0))$ is not flat and that for some $x\in M$ the +endomorphism $\mathit{Rm}(x,T)$ has a zero eigenvalue. Then $M$ has a cover $\widetilde M$ such that, denoting the induced family of metrics on this cover by $\tilde g(t)$, we have that $(\widetilde M,\tilde g(t))$ splits as a product -$$(N,h(t))\times (\Ar,ds^2)$$ + +$$ +(N,h(t))\times (\Ar,ds^2) +$$ + where $(N,h(t))$ is a surface of positive curvature for all $0 + **Remark.** Notice that there are only four possibilities for the cover required by the corollary. It can be trivial, or a normal $\Zee$-cover or it can be a two-sheeted cover or a normal infinite dihedral group cover. In the first two cases, there is a unit vector field on $M$ parallel under $g(t)$ for all $t$ spanning the null direction of -$Ric$. In the last two cases, there is no such vector field, +$\mathit{Ric}$. In the last two cases, there is no such vector field, only a non-orientable line field. + - Let $(N,g)$ be a Riemannian manifold. Recall from Definition~ - that the open cone +Let $(N,g)$ be a Riemannian manifold. Recall from \entryref{d4891987758a} + that the open cone on $(N,g)$ is the space $N\times (0,\infty)$ with the Riemannian metric $\tilde g(x,s)=s^2g(x)+ds^2$. An extremely important result for us is that open pieces in non-flat cones cannot arise as the result of Ricci flow with non-negative curvature. -**Proposition.** -Suppose that $(U,g(t)),\ 0\le t\le T$, is a $3$-dimensional Ricci + +**Proposition.** Suppose that $(U,g(t)),\ 0\le t\le T$, is a $3$-dimensional Ricci flow with non-negative sectional curvature, with $U$ being connected but not necessarily complete and $T>0$. Suppose that $(U,g(T))$ is isometric to a non-empty open subset of a cone over a Riemannian manifold. Then $(U,g(t))$ is flat for every $t\in [0,T]$. + -**Proof.** If $(U,g(T))$ is flat, then by Theorem~ +**Proof.** If $(U,g(T))$ is flat, then by \entryref{016cf36686b2} for every $t\in [0,T]$ the Riemannian manifold $(U,g(t))$ is flat. -We must rule out the possibility that $(U,g(T))$ is non-flat. Suppose that +We must rule out the possibility that $(U,g(T))$ is non-flat. Suppose that $(U,g(T))$ is an open subset in a non-flat cone. According to -Proposition~, for each $x\in U$ the Riemann curvature tensor of +\entryref{1b517dfd4bd7}, for each $x\in U$ the Riemann curvature tensor of $(U,g(T))$ at $x$ has a $2$-dimensional null space in $\wedge^2T_xU$. Since we are assuming that $(U,g(T))$ is not flat, the third eigenvalue of the Riemann curvature tensor is not identically zero. Restricting to a smaller open subset if necessary, we can assume that the third eigenvalue is never zero. By the -computations in Proposition~ the non-zero eigenvalue is not +computations in \entryref{1b517dfd4bd7} the non-zero eigenvalue is not constant, and in fact it scales by $s^{-2}$ in the terminology of that proposition, as we move along the cone lines. Of course, the $2$-dimensional null-space for the Riemann curvature tensor at each point is equivalent to a line field in the tangent bundle of the manifold. Clearly, that line field is -the line field along the cone lines. Corollary~ says that since +the line field along the cone lines. \entryref{8ec855a2aab5} says that since the Riemann curvature of $(U,g(T))$ has a $2$-dimensional null-space in $\wedge^2T_xU$ at every point $x\in U$, the Riemannian manifold $(U,g(T))$ locally splits as a Riemannian product of a line with a surface of positive @@ -627,22 +765,27 @@ Along these lines the non-zero eigenvalue of the curvature is constant. This is a contradiction and establishes the result. Lastly, we have Hamilton's result (Theorem 15.1 in -) that compact $3$-manifolds of non-negative +[Hamilton3MPRC]) that compact $3$-manifolds of non-negative Ricci curvature become round under Ricci flow: -**Theorem.** -Suppose that $(M,g(t)),\ 0\le t0$ for all $(x,t)\in M\times -(0,T)$ or $Ric(x,t)=0$ for all $(x,t)\in M\times[0,T)$. -Suppose that $Ric(x,t)>0$ for some $(x,t)$ and that the flow + +**Theorem.** Suppose that $(M,g(t)),\ 0\le t0$ for all $(x,t)\in M\times +(0,T)$ or $\mathit{Ric}(x,t)=0$ for all $(x,t)\in M\times[0,T)$. +Suppose that $\mathit{Ric}(x,t)>0$ for some $(x,t)$ and that the flow is maximal in the sense that there is no $T'>T$ and an extension of -the given flow to a flow defined on the time interval $[0,T')$. For +the given flow to a flow defined on the time interval $[0,T')$. For each $(x,t)$, let $\lambda(x,t)$, resp. $\nu(x,t)$, denote the -largest, resp. smallest, eigenvalue of $Rm(x,t)$ on +largest, resp. smallest, eigenvalue of $\mathit{Rm}(x,t)$ on $\wedge^2T_xM$. Then as $t$ tends to $T$ the Riemannian manifolds $(M,g(t))$ are becoming round in the sense that -$$lim_{t\rightarrow T}\frac{max_{x\in M}\lambda(x,t)}{min_{x\in M}\nu(x,t)}=1.$$ Furthermore, for any $x\in M$ the + +$$ +\mathit{lim}_{t\rightarrow T}\frac{\mathit{max}_{x\in M}\lambda(x,t)}{\mathit{min}_{x\in M}\nu(x,t)}=1. +$$ + + Furthermore, for any $x\in M$ the largest eigenvalue $\lambda(x,t)$ tends to $\infty$ as $t$ tends to $T$, and rescaling $(M,g(t))$ by $\lambda(x,t)$ produces a family of Riemannian manifolds converging smoothly as $t$ goes to $T$ to a @@ -650,8 +793,9 @@ compact round manifold. In particular, the underlying smooth manifold supports a Riemannian metric of constant positive curvature so that the manifold is diffeomorphic to a $3$-dimensional spherical space-form. + -Hamilton's proof in uses the maximum principle +Hamilton's proof in [Hamilton3MPRC] uses the maximum principle and Shi's derivative estimates. ### Solitons of positive curvature @@ -659,16 +803,17 @@ and Shi's derivative estimates. One nice application of this pinching result is the following theorem. -**Theorem.** -Let $(M,g)$ be a compact $3$-dimensional soliton of positive Ricci + +**Theorem.** Let $(M,g)$ be a compact $3$-dimensional soliton of positive Ricci curvature. Then $(M,g)$ is round. In particular, $(M,g)$ is the quotient of $S^3$ with a round metric by a finite subgroup of $O(4)$ acting freely; that is to say, $M$ is a $3$-dimensional spherical space-form. + **Proof.** Let $(M,g(t)),\ 0\le t0$ for all $x\in M$, it -follows from Theorem~ that $T<\infty$ and that as +manifold $(M,g)$. Since $\mathit{Ric}(x,0)>0$ for all $x\in M$, it +follows from \entryref{86e5a5766bda} that $T<\infty$ and that as $t$ tends to $T$ the metrics $g(t)$ converge smoothly to a round metric. Since all the manifolds $(M,g(t))$ are isometric up to diffeomorphism and a constant conformal factor, this implies that @@ -677,17 +822,18 @@ all the $g(t)$ are of constant positive curvature. The last statement is a standard consequence of the fact that the manifold has constant positive curvature. + **Remark.** After we give a stronger pinching result in the next section, we shall improve this result, replacing the positive Ricci curvature assumption by the *a priori* weaker assumption that the soliton is a shrinking soliton. + -## Pinching toward positive -curvature +## Pinching toward positive curvature As the last application of the maximum principle for tensors we give -a theorem due to R. Hamilton (Theorem 4.1 in ) -and T. Ivey which shows that, in dimension three, as the +a theorem due to R. Hamilton (Theorem 4.1 in [HamiltonNSRF3M]) +and T. Ivey [Ivey] which shows that, in dimension three, as the scalar curvature gets large, the sectional curvatures pinch toward the positive. Of course, if the sectional curvatures are non-negative, then the results in the previous section apply. Here, @@ -700,50 +846,59 @@ the scalar curvature grows. This result is central in the analysis of singularity development in finite time for a $3$-dimensional Ricci flow. + **Theorem.** **(Pinching toward positive curvature)** Let $(M,g(t)),\ 0\le t For any fixed $t$, the limit as $X$ goes to $0$ from above of -$X(log(X)+log(1+t)-3)$ is zero, so that it is natural to +$X(\mathit{log}(X)+\mathit{log}(1+t)-3)$ is zero, so that it is natural to interpret this expression to be zero when $X=0$. Of course, when -$X(x,t)=0$ all the eigenvalues of $Rm(x,t)$ are non-negative +$X(x,t)=0$ all the eigenvalues of $\mathit{Rm}(x,t)$ are non-negative so that $R(x,t)\ge 0$ as well. Thus, with this interpretation of the expression in Part 2 of the theorem, it remains valid even when $X(x,t)=0$. + **Remark.** This theorem tells us, among other things, that as the scalar -curvature goes to infinity then absolute values of all the -negative eigenvalues (if any) of $Rm$ are arbitrarily small +curvature goes to infinity then absolute values of all the +negative eigenvalues (if any) of $\mathit{Rm}$ are arbitrarily small with respect to the scalar curvature. + We proof we give below follows Hamilton's original proof in - very closely. +[HamiltonNSRF3M] very closely. -**Proof.** First note that by Proposition~, if $R_min(0)\ge 0$, then the same is true for $R_min(t)$ for every +**Proof.** First note that by \entryref{95cc838cfc6d}, if $R_\mathit{min}(0)\ge 0$, then the same is true for $R_\mathit{min}(t)$ for every $t>0$ and thus the first inequality stated in the theorem is clearly -true. If $R_min(0)<0$, the first inequality stated in the +true. If $R_\mathit{min}(0)<0$, the first inequality stated in the theorem follows easily from the last inequality in -Proposition~. +\entryref{95cc838cfc6d}. We turn now to the second inequality in the statement of the -theorem. Consider the tensor bundle ${\mathcal V}=Sym^2(\wedge^2T^*M)$. Then the curvature operator written in the +theorem. Consider the tensor bundle ${\mathcal V}=\mathit{Sym}^2(\wedge^2T^*M)$. Then the curvature operator written in the evolving frame, ${\mathcal T}(x,t)$, is a one-parameter family of smooth sections of this bundle, evolving by -$$\frac{\partial{\mathcal T}}{\partial t}= -\Delta{\mathcal T}+\psi({\mathcal T}).$$ We consider two subsets of ${\mathcal -V}$. There are two solutions to $x(log(x)+(log(1+t)-3)=-3/(1+t)$. -One is $x=1/(1+t)$; let $\xi(t)>1/(1+t)$ be the other. We set $S({\mathcal -T})=tr({\mathcal T})$, so that $R=2S$, and we set $X({\mathcal T})=max(-\nu({\mathcal T}),0)$. Define $$ +\frac{\partial{\mathcal T}}{\partial t}= +\Delta{\mathcal T}+\psi({\mathcal T}). +$$ + + We consider two subsets of ${\mathcal +V}$. There are two solutions to $x(\mathit{log}(x)+(\mathit{log}(1+t)-3)=-3/(1+t)$. +One is $x=1/(1+t)$; let $\xi(t)>1/(1+t)$ be the other. We set $S({\mathcal +T})=\mathit{tr}({\mathcal T})$, so that $R=2S$, and we set $X({\mathcal T})=\mathit{max}(-\nu({\mathcal T}),0)$. Define $$ \begin{aligned} @@ -755,72 +910,90 @@ V}\bigl|\bigr.S({\mathcal T})\geq f_t(X({\mathcal T})),\ \ \ \end{aligned} $$ -$$ +where $f_t(x)=x(\mathit{log}x+\mathit{log}(1+t)-3)$. Then we define -where $f_t(x)=x(logx+log(1+t)-3)$. Then we define -$${\mathcal Z}(t)={\mathcal Z}_1(t)\cap {\mathcal Z}_2(t).$$ +$$ +{\mathcal Z}(t)={\mathcal Z}_1(t)\cap {\mathcal Z}_2(t). +$$ + **Claim.** For each $x\in M$ and each $t\ge 0$, the fiber $Z(x,t)$ of -${\mathcal Z}(t)$ over $x$ is a convex subset of $Sym^2(\wedge^2T^*M)$. +${\mathcal Z}(t)$ over $x$ is a convex subset of $\mathit{Sym}^2(\wedge^2T^*M)$. + -\begin{proof} -First consider the function $f_t(x)=x(log(x)+log(1+t)-3)$ on the interval $[\xi(t),\infty)$. Direct computation +**Proof.** First consider the function $f_t(x)=x(\mathit{log}(x)+\mathit{log}(1+t)-3)$ on the interval $[\xi(t),\infty)$. Direct computation shows that $f'(x)>0$ and $f''(x)>0$ on this interval. Hence, for every $t\ge 0$ the region ${\mathcal C}(t)$ in the $S$-$X$ plane defined by $S\ge -3/(1+t)$ and $S\ge f_t(X)$ when $X\ge \xi(t)$ is convex and has the property that if $(S,X)\in {\mathcal C}(t)$ then so is $(S,X')$ for all -$X'\le X$. (See Fig.~). By definition an element +$X'\le X$. (See Fig. 4.1). By definition an element ${\mathcal T}\in {\mathcal V}$ is contained in ${\mathcal Z}(t)$ if and only if $(S({\mathcal T}),X({\mathcal T})\in {\mathcal C}(t)$. Now fix $t\ge 0$ and suppose that ${\mathcal T}_1$ and ${\mathcal -T}_2$ are elements of $Sym^2(\wedge^2T^*M_x)$ such that -setting $S_i=tr({\mathcal T}_i)$ and $X_i=X({\mathcal T}_i)$ +T}_2$ are elements of $\mathit{Sym}^2(\wedge^2T^*M_x)$ such that +setting $S_i=\mathit{tr}({\mathcal T}_i)$ and $X_i=X({\mathcal T}_i)$ we have $(S_i,X_i)\in {\mathcal C}(t)$ for $i=1,2$. Then we consider ${\mathcal T}=s{\mathcal T}_1+(1-s){\mathcal T}_2$ for some $s\in -[0,1]$. Let $S=tr({\mathcal T})$ and $X=X({\mathcal T})$. +[0,1]$. Let $S=\mathit{tr}({\mathcal T})$ and $X=X({\mathcal T})$. Since ${\mathcal C}(t)$ is convex, we know that $(sS_1+(1-s)S_2,sX_1+(1-s)X_2)\in{\mathcal C}(t)$, so that ${\mathcal T}\in {\mathcal Z}(t)$. Clearly, $S=sS_1+(1-s)S_2$, so that we conclude that $(S,(sX_1+(1-s)X_2))\in{\mathcal -C}(t)$. But since $\nu$ is a convex function, $X$ is a concave function, i.e., +C}(t)$. But since $\nu$ is a convex function, $X$ is a concave function, i.e., $X\le sX_1+(1-s)X_2$. Hence $(S,X)\in {\mathcal C}(t)$. + **Claim.** ${\mathcal T}(x,0)\in Z(x,0)$ for all $x\in M$. + **Proof.** Note that by the hypothesis of the theorem we have -$$\nu(x,0)+\mu(x,0)+\lambda(x,0)\geq -3$$ so $(S(x,0),X(x,0))\in + +$$ +\nu(x,0)+\mu(x,0)+\lambda(x,0)\geq -3 +$$ + + so $(S(x,0),X(x,0))\in {\mathcal C}(0)$ for all $x\in M$. On the other hand, if $0< X(x,0)$, then since $X(x,0)\le 1$ we have $S(x,0)\geq-3X(x,0)\geq -X(logX-3)$. This completes the proof that ${\mathcal +X(\mathit{log}X-3)$. This completes the proof that ${\mathcal T}(x,0)\in {\mathcal C}(0)$ for all $x\in M$. + **Claim.** The vector field $\psi({\mathcal T})={\mathcal T}^2+{\mathcal T}^\sharp$ preserves the family ${\mathcal Z}(t)$ of convex sets. + **Proof.** Fix $x\in M$ and suppose that we have an integral curve $\gamma(t),\ t_0\le t\le T$, for $\psi$ with $\gamma(t_0)\in Z(x,t_0)$. We wish to show that $\gamma(t)\in Z(x,t)$ for all $t\in [t_0,T]$. The function $S(t)=S(\gamma(t))$ satisfies -$$\frac{dS}{dt}=\lambda^2+\mu^2+\nu^2+\lambda\mu+\lambda\nu+\mu\nu=\frac{1}{2}\left((\lambda+\mu)^2+ -(\lambda+\nu)^2+(\mu+\nu)^2\right).$$ By Cauchy-Schwarz we have -$$(\lambda+\mu)^2+ -(\lambda+\nu)^2+(\mu+\nu)^2)\ge \frac{4S^2}{3}\ge \frac{2S^2}{3}.$$ Since -$\gamma(t_0)\in Z(x,t_0)$ we have $S(t_0)\ge -3/(1+t_0)$. It then follows that $$ +\frac{dS}{dt}=\lambda^2+\mu^2+\nu^2+\lambda\mu+\lambda\nu+\mu\nu=\frac{1}{2}\left((\lambda+\mu)^2+ +(\lambda+\nu)^2+(\mu+\nu)^2\right). +$$ + + By Cauchy-Schwarz we have + +$$ +(\lambda+\mu)^2+ +(\lambda+\nu)^2+(\mu+\nu)^2)\ge \frac{4S^2}{3}\ge \frac{2S^2}{3}. +$$ + + Since +$\gamma(t_0)\in Z(x,t_0)$ we have $S(t_0)\ge -3/(1+t_0)$. It then follows that -S(t)\ge -3/(1+t)\ \ \ for\ all\ \ \ t\ge t_0. +$$ +S(t)\ge -3/(1+t)\ \ \ \mathit{for\ all}\ \ \ t\ge t_0. $$ Now let us consider the evolution of $X(t)=X(\gamma(t))$. Assume that we are at a point $t$ for which $X(t)>0$. For this computation we set $Y=-\mu$. -$$ - $$ \begin{aligned} \frac{dX}{dt}&=-\frac{d\nu}{dt}=-\nu^2-\mu\lambda =-X^2+Y\lambda,\\ @@ -830,26 +1003,23 @@ $$ \end{aligned} $$ -$$ - Putting this together yields $$ - X\frac{dS}{dt}-(S+X)\frac{dX}{dt} =X^3+I, $$ where $I=XY^2+\lambda Y(Y-X)+\lambda^2(X-Y)$. + **Claim.** $I\geq 0$. + -\begin{proof} - - First we consider the case when - $Y\leq 0$. This means that $\mu\geq 0$ and +**Proof.** First we consider the case when + $Y\leq 0$. This means that $\mu\geq 0$ and hence that $\lambda\geq0$. Since by definition $X\ge 0$, we have $X\ge Y$. -This immediately gives $I\geq 0$. Now let us consider the case when +This immediately gives $I\geq 0$. Now let us consider the case when $Y>0$ which means that $\nu\le \mu<0$. In this case, we have $$ @@ -859,20 +1029,18 @@ $$ since $X\geq Y$ and $\lambda^2-\lambda Y+Y^2=(\lambda-\frac{Y}{2})^2+\frac{3Y^2}{4}>0$. -The above claim and Equation~() immediately imply that +The above claim and Equation (4.3) immediately imply that $$ - - X\frac{dS}{dt}-(S+X)\frac{dX}{dt}\geq +X\frac{dS}{dt}-(S+X)\frac{dX}{dt}\geq X^3. $$ - Set $W=\frac{S}{X}-log\,X$, then rewriting - Equation~() + Set $W=\frac{S}{X}-\mathit{log} X$, then rewriting + Equation (4.4) in terms of $W$ gives $$ - \frac{dW}{dt}\geq X. $$ @@ -882,30 +1050,29 @@ Z(x,t)$ for all $t_0\le t\le t_1$. Of course, $\gamma(t_1)\in \partial Z(x,t_1)$ which implies that $(S(t_1),X(t_1))\in \partial {\mathcal C}(t_1)$. There are two possibilities: either $S(t_1)=-3/(1+t_1)$ and $X(t_1)<\xi(t_1)$ or $X(t_1)\ge -\xi(t_1)>1/(1+t_1)$ and $S(t_1)=f_{t_1}(X(t_1))$. But -Equation~() implies that $S(t)\ge -3/(1+t)$ for all $t$. +\xi(t_1)>1/(1+t_1)$ and $S(t_1)=f_{t_1}(X(t_1))$. But +Equation (4.2) implies that $S(t)\ge -3/(1+t)$ for all $t$. Hence, if the first case holds then $\gamma(t)\in Z(x,t)$ for $t$ in some interval $[t_0,t_1']$ with $t_1'>t_1$. This contradicts the maximality of $t_1$. Thus, it must be the case that $X(t_1)\ge \xi(t_1)$. But then $X(t)> \frac{1}{1+t}$ for all $t$ sufficiently -close to $t_1$. Hence, by Equation~() we have +close to $t_1$. Hence, by Equation (4.5) we have $$ \frac{dW}{dt}(t)\geq X(t)>\frac{1}{1+t}, $$ for all $t$ sufficiently close to $t_1$. Also, since -$S(t_1)=f_{t_1}(X(t_1))$, we have $W(t_1)=(log(1+t_1)-3)$. It -follows immediately that $W(t)\ge (log(1+t)-3)$ for all +$S(t_1)=f_{t_1}(X(t_1))$, we have $W(t_1)=(\mathit{log}(1+t_1)-3)$. It +follows immediately that $W(t)\ge (\mathit{log}(1+t)-3)$ for all $t>t_1$ sufficiently close to $t_1$. This proves that $S(t)\ge f_t(X(t))$ for all $t\ge t_1$ sufficiently close to $t_1$, again contradicting the maximality of $t_1$. This contradiction proves that $\psi$ preserves the family ${\mathcal Z}(t)$. -\end{proof} -By Theorem~, the previous three claims imply that ${\mathcal +By \entryref{8f737f2b0f7d}, the previous three claims imply that ${\mathcal T}(x,t)\in {\mathcal Z}(t)$ for all $x\in M$ and all $t\in [0,T)$. That is to say, $S(x,t)\ge -3/(1+t)$ and $S(x,t)\ge f_t(X(x,t))$ whenever $X(x,t)\ge \xi(t)$. For $X\in [1/(1+t),\xi(t)]$ we have $f_t(X)\le -3/(1+t)$, and thus in @@ -914,65 +1081,83 @@ if $0< X(x,t)\le 1/(1+t)$ then $f_t(X(x,t))< -3X(x,t)\le S(x,t)$. On the other hand, since $X(x,t)$ is the negative of the smallest eigenvalue of ${\mathcal T}(x,t)$ and $S(x,t)$ is the trace of this matrix, we have $S(x,t)\ge -3X(x,t)$. Thus, $S(x,t)\ge f_t(X(x,t))$ in this case as well. This completes -the proof of Theorem~. -\end{proof} +the proof of \entryref{92827a66b5dd}. Actually, the proof establishes a stronger result which we shall need. -**Theorem.** -Fix $a\ge 0$. Let $(M,g(t)),\ a\le t0$. Then for all $a\le t +**Theorem.** Fix $a\ge 0$. Let $(M,g(t)),\ a\le t0$. Then for all $a\le t 0$. +whenever $X(x,t)> 0$. + Once again it is natural to interpret the right-hand side of the -inequalities relating $R$ and $X$ to be zero when $X(x,t)=0$. With +inequalities relating $R$ and $X$ to be zero when $X(x,t)=0$. With this convention the result remains true even when $X(x,t)=0$. + **Corollary.** Fix $a\ge 0$. Suppose that $(M,g(t)),\ a\le t **Proof.** Fix $R_0\ge e^4$ sufficiently large, and suppose that $R(x,t)\le -R_0$. If $X(x,t)=0$, then $|Rm(x,t)|\le R(x,t)/2$. If -$X(x,t)>0$, then by Theorem~ it is bounded by $R_0$. +R_0$. If $X(x,t)=0$, then $|\mathit{Rm}(x,t)|\le R(x,t)/2$. If +$X(x,t)>0$, then by \entryref{fc44d8653f35} it is bounded by $R_0$. Thus, $\lambda(x,t)\le 3R_0$. Thus, we have an upper bound on $\lambda(x,t)$ and a lower bound on $\nu(x,t)$ in terms of $R_0$. This theorem leads to a definition. -**Definition.** -Let $({\mathcal M},G)$ be a generalized Ricci flow whose domain of definition + +**Definition.** Let $({\mathcal M},G)$ be a generalized Ricci flow whose domain of definition is contained in $[0,\infty)$. Then we say that $({\mathcal M},G)$ has *curvature pinched toward positive* if for every $x\in {\mathcal M}$ the following two conditions hold: -\begin{enumerate} -\item $$R(x)\ge \frac{-6}{4**t**(x)+1}$$ -\item $$R(x)\geq 2X(x)\left(logX(x)+log(1+**t**(x))-3\right), -$$ whenever + +1. + +$$ +R(x)\ge \frac{-6}{4\mathbf{t}(x)+1} +$$ + +1. + +$$ +R(x)\geq 2X(x)\left(\mathit{log}X(x)+\mathit{log}(1+\mathbf{t}(x))-3\right), +$$ + + whenever $0 + +The content of \entryref{fc44d8653f35} is that if $(M,g(t)),\ 0\le a\le t **Theorem.** Let $(M,g)$ be a compact $3$-dimensional shrinking soliton, i.e., there is a Ricci flow $(M,g(t)),\ 0\le t -**Proof.** By rescaling we can assume that for all $x\in M$ all the eigenvalues of $Rm(x,0)$ have absolute value $\le 1$. This implies that $(M,g(0))$ satisfies -the hypothesis of Theorem~. Our first goal is to show that $Rm(x,0)\ge 0$ for all $x\in M$. Suppose that this is not true; then there is a +**Proof.** By rescaling we can assume that for all $x\in M$ all the eigenvalues of $\mathit{Rm}(x,0)$ have absolute value $\le 1$. This implies that $(M,g(0))$ satisfies +the hypothesis of \entryref{92827a66b5dd}. Our first goal is to show that $\mathit{Rm}(x,0)\ge 0$ for all $x\in M$. Suppose that this is not true; then there is a point $x$ with $X(x,0)>0$. Consider $A=X(x,0)/R(x,0)$. For each $t0$ for all $(x,t)\in M\times +\entryref{92827a66b5dd}. Now, according to \entryref{0db415314964} either all +$(M,g(t))$ are flat or $\mathit{Rm}(x,t)>0$ for all $(x,t)\in M\times (0,T)$. But if the $(M,g(t))$ are all flat, then the flow is trivial and hence the diameters of the $(M,g(t))$ do not go to zero as $t$ -approaches $T$, contradicting the hypothesis. Hence, $Rm(x,t)>0$ for all $(x,t)\in M\times (0,T)$. According to -Theorem~ this means that as the singularity +approaches $T$, contradicting the hypothesis. Hence, $\mathit{Rm}(x,t)>0$ for all $(x,t)\in M\times (0,T)$. According to +\entryref{86e5a5766bda} this means that as the singularity develops the metrics are converging to round. By the shrinking soliton hypothesis, this implies that all the metrics $(M,g(t)),\ 0 +**Theorem.** Any 3-dimensional compact Ricci soliton $g_0$ is Einstein. + Since we do not need this result, we do not include a proof. ### The Harnack inequality The last consequence of the maximum principle that we need is -Hamilton's version of the Harnack inequality for Ricci +Hamilton's version of the Harnack inequality for Ricci flows, see Theorem 1.1 -and Corollary 1.2 of . +and Corollary 1.2 of [Hamiltonharnack]. -**Theorem.** -Suppose that $(M,g(t))$ is a Ricci flow defined for $(T_0,T_1)$ with + +**Theorem.** Suppose that $(M,g(t))$ is a Ricci flow defined for $(T_0,T_1)$ with $(M,g(t))$ a complete manifold of non-negative curvature operator -with bounded curvature for each $t\in (T_0,T_1)$. Then for any +with bounded curvature for each $t\in (T_0,T_1)$. Then for any time-dependent vector field $\chi(x,t)$ on $M$ we have: -$$\frac{\partial R(x,t)}{\partial t}+\frac{R(x,t)}{t-T_0}+2\langle -\chi(x,t),\nabla R(x,t)\rangle +2Ric(x,t)(\chi(x,t),\chi(x,t))\ge 0.$$ In particular, we have -$$\frac{\partial R(x,t)}{\partial t}+\frac{R(x,t)}{t-T_0}\ge 0.$$ +$$ +\frac{\partial R(x,t)}{\partial t}+\frac{R(x,t)}{t-T_0}+2\langle +\chi(x,t),\nabla R(x,t)\rangle +2\mathit{Ric}(x,t)(\chi(x,t),\chi(x,t))\ge 0. +$$ + + In particular, we have + +$$ +\frac{\partial R(x,t)}{\partial t}+\frac{R(x,t)}{t-T_0}\ge 0. +$$ + + + **Remark.** Notice that the second result follows from the first by taking $\chi=0$. + -**Corollary.** -If $(M,g(t))$ is a Ricci flow defined for $-\infty +**Corollary.** If $(M,g(t))$ is a Ricci flow defined for $-\infty **Proof.** Apply the above theorem with $\chi(x,t)=0$ for all $(x,t)$ and for a sequence of $T_0\rightarrow -\infty$. The above is the differential form of Hamilton's Harnack inequality. There is also the integrated version, also due to Hamilton; see -Corollary 1.3 of . +Corollary 1.3 of [Hamiltonharnack]. -**Theorem.** -Suppose that $(M,g(t))$ is a Ricci flow defined for $t_1\le t\le + +**Theorem.** Suppose that $(M,g(t))$ is a Ricci flow defined for $t_1\le t\le t_2$ with $(M,g(t))$ a complete manifold of non-negative, bounded curvature operator for all $t\in [t_1,t_2]$. Let $x_1$ and $x_2$ be -two points of $M$. Then $$log\left(\frac{R(x_2,t_2)}{R(x_1,t_1)}\right)\ge --\frac{1}{2}\frac{d^2_{t_1}(x_2,x_1)}{(t_2-t_1)}.$$ +two points of $M$. Then + +$$ +\mathit{log}\left(\frac{R(x_2,t_2)}{R(x_1,t_1)}\right)\ge +-\frac{1}{2}\frac{d^2_{t_1}(x_2,x_1)}{(t_2-t_1)}. +$$ + **Proof.** Apply the differential form of the Harnack inequality to -$\chi=-\nabla(log R)/2=-\nabla R/2R$, and divide by $R$. The +$\chi=-\nabla(\mathit{log} R)/2=-\nabla R/2R$, and divide by $R$. The result is -$$R^{-1}(\partial R/\partial t)-|\nabla (log R)|^2+\frac{Ric(\nabla(log R),\nabla(log R))}{2R}\ge 0.$$ Since $Ric(A,A)/R\le |A|^2$, it follows + +$$ +R^{-1}(\partial R/\partial t)-|\nabla (\mathit{log} R)|^2+\frac{\mathit{Ric}(\nabla(\mathit{log} R),\nabla(\mathit{log} R))}{2R}\ge 0. +$$ + + Since $\mathit{Ric}(A,A)/R\le |A|^2$, it follows that -$$\frac{\partial}{\partial t}(log R)-\frac{|\nabla(log R)|^2}{2}\ge 0.$$ + +$$ +\frac{\partial}{\partial t}(\mathit{log} R)-\frac{|\nabla(\mathit{log} R)|^2}{2}\ge 0. +$$ Let $d$ be the $g(t_1)$-distance from $x_1$ to $x_2$ and let $\gamma\colon [t_1,t_2]\to M$ be a $g(t_1)$-geodesic from $x_1$ @@ -1067,25 +1283,29 @@ Let $d$ be the $g(t_1)$-distance from $x_1$ to $x_2$ and let let $\mu(t)=(\gamma(t),t)$ be a path in space-time. We compute -$$ - $$ \begin{aligned} -log\left(\frac{R(x_2,t_2)}{R(x_1,t_1)}\right) & = & \int_{t_1}^{t_2} - \frac{d}{dt}log(R(\mu(t))dt \\ - & = & \int_{t_1}^{t_2}\frac{\frac{\partial R}{\partial t}(\mu(t))}{R(\mu(t))}+\langle\nabla(log R)(\mu(t)),\frac{d\mu}{dt}(\mu(t))dt \\ - & \ge & \int_{t_1}^{t_2}\frac{1}{2}|\nabla (log R)(\mu(t))|^2-|\nabla (log R)(\mu(t))|\cdot \left|\frac{d\gamma}{dt}\right|dt \\ +\mathit{log}\left(\frac{R(x_2,t_2)}{R(x_1,t_1)}\right) & = & \int_{t_1}^{t_2} + \frac{d}{dt}\mathit{log}(R(\mu(t))dt \\ + & = & \int_{t_1}^{t_2}\frac{\frac{\partial R}{\partial t}(\mu(t))}{R(\mu(t))}+\langle\nabla(\mathit{log} R)(\mu(t)),\frac{d\mu}{dt}(\mu(t))dt \\ + & \ge & \int_{t_1}^{t_2}\frac{1}{2}|\nabla (\mathit{log} R)(\mu(t))|^2-|\nabla (\mathit{log} R)(\mu(t))|\cdot \left|\frac{d\gamma}{dt}\right|dt \\ & \ge & -\frac{1}{2}\int_{t_1}^{t_2}\left|\frac {d\gamma}{dt}\right|^2dt, \end{aligned} $$ -$$ where the last inequality comes form completing the square. - Since $Ric(x,t)\ge 0$, + Since $\mathit{Ric}(x,t)\ge 0$, $|d\gamma/dt|_{g(t)}\le |d\gamma/dt|_{g(t_1)}$, thus -$$log\left(\frac{R(x_2,t_2)}{R(x_1,t_1)}\right)\ge --\frac{1}{2}\int_{t_1}^{t_2}\left|\frac{d\gamma}{dt}\right|^2_{g(t_1)}dt.$$ + +$$ +\mathit{log}\left(\frac{R(x_2,t_2)}{R(x_1,t_1)}\right)\ge +-\frac{1}{2}\int_{t_1}^{t_2}\left|\frac{d\gamma}{dt}\right|^2_{g(t_1)}dt. +$$ + Since $\gamma$ is a $g(t_1)$-geodesic, this latter integral is -$$-\frac {1}{2}\frac{d_{g(t_1)}^2(x_1,x_2)}{t_2-t_1}.$$ + +$$ +-\frac {1}{2}\frac{d_{g(t_1)}^2(x_1,x_2)}{t_2-t_1}. +$$ diff --git a/projects/poincare-conjecture/.astrolabe/docs/05-converge2.mdx b/projects/poincare-conjecture/.astrolabe/docs-src/06-convergence-results-for-ricci-flow.mdx similarity index 71% rename from projects/poincare-conjecture/.astrolabe/docs/05-converge2.mdx rename to projects/poincare-conjecture/.astrolabe/docs-src/06-convergence-results-for-ricci-flow.mdx index 94ccb659..a81ebfed 100644 --- a/projects/poincare-conjecture/.astrolabe/docs/05-converge2.mdx +++ b/projects/poincare-conjecture/.astrolabe/docs-src/06-convergence-results-for-ricci-flow.mdx @@ -1,4 +1,6 @@ -# Convergence results for Ricci flow + + +# Chapter 5 -- Convergence results for Ricci flow The most obvious notion of smooth convergence of Riemannian manifolds is the $C^\infty$-version of Cheeger-Gromov compactness: @@ -26,28 +28,31 @@ each of the manifolds in the sequence. ## Geometric convergence of Riemannian manifolds -Above we referred to filling up `most' of the manifold. The measure of most of +Above we referred to filling up 'most' of the manifold. The measure of most of the manifold is in terms of the $\delta$-regular points as defined below. -**Definition.** -Let $(U,g)$ be a Riemannian manifold. Let $\delta>0$ be given. We + +**Definition.** Let $(U,g)$ be a Riemannian manifold. Let $\delta>0$ be given. We say that $p\in U$ is a *$\delta$-regular* point if for every $r'<\delta$ the metric ball $B(p,r')$ has compact closure in $U$. Equivalently, $p$ is $\delta$-regular if the exponential mapping at $p$ is defined on the open ball of radius $\delta$ centered at the origin in $T_pU$, i.e., if each geodesic ray emanating from $p$ extends to a geodesic -defined on $[0,\delta)$. We denote by $Reg_\delta(U,g)$ the -subset of $\delta$-regular points in $(U,g)$. For any $x\in Reg_\delta(U,g)$ we denote by $Reg_\delta(U,g,x)$ the -connected component of $Reg_\delta(U,g)$ containing $x$. +defined on $[0,\delta)$. We denote by $\mathit{Reg}_\delta(U,g)$ the +subset of $\delta$-regular points in $(U,g)$. For any $x\in \mathit{Reg}_\delta(U,g)$ we denote by $\mathit{Reg}_\delta(U,g,x)$ the +connected component of $\mathit{Reg}_\delta(U,g)$ containing $x$. + Intuitively, the $\delta$-regular points of $(U,g)$ are at distance at least $\delta$ from the boundary on $U$. -**Lemma.** $Reg_\delta(U,g)$ is a closed subset of $U$. + +**Lemma.** $\mathit{Reg}_\delta(U,g)$ is a closed subset of $U$. + **Proof.** Suppose that $p_n$ converges to $p$ as $n$ tends to $\infty$ and -suppose that $p_n\in Reg_\delta(U,g)$ for all $n$. Fix +suppose that $p_n\in \mathit{Reg}_\delta(U,g)$ for all $n$. Fix $r'<\delta$ and consider the ball $B(p,r')$. For all $n$ sufficiently large, this ball is contained in $B(p_n,(\delta+r')/2)$, and hence has compact @@ -56,29 +61,30 @@ closure. Now we are ready for the basic definition of geometric convergence of Riemannian manifolds. -**Definition.** For each $k$ + +**Definition.** For each $k$ let $(U_k,g_k,x_k)$ be a based, connected Riemannian manifold. A *geometric limit* of the sequence $\{U_k,g_k,x_k\}_{k=0}^\infty$ is a based, connected Riemannian manifold $(U_\infty,g_\infty,x_\infty)$ with the extra data: -\begin{enumerate} -\item[(1)] An increasing sequence $V_k\subset U_\infty$ of connected open subsets + +- **(1)** An increasing sequence $V_k\subset U_\infty$ of connected open subsets of $U_\infty$ whose union is $U_\infty$ and which satisfy the following for all $k$: -\begin{enumerate} -\item the closure $\overline V_k$ is compact, \item -$\overline V_k\subset V_{k+1}$, \item $V_k$ contains $x_\infty$. -\end{enumerate} -\item For each $k\ge 0$ a smooth embedding $\varphi_k\colon (V_k,x_\infty)\to -(U_k,x_k)$ with the properties that: \begin{enumerate} \item $lim_{k\rightarrow \infty}\varphi_k^*g_k=g_\infty$, where the limit + +1. the closure $\overline V_k$ is compact, +1. $\overline V_k\subset V_{k+1}$, +1. $V_k$ contains $x_\infty$. +1. For each $k\ge 0$ a smooth embedding $\varphi_k\colon (V_k,x_\infty)\to +(U_k,x_k)$ with the properties that: + +1. $\mathit{lim}_{k\rightarrow \infty}\varphi_k^*g_k=g_\infty$, where the limit is in the uniform $C^\infty$-topology on compact subsets of $U_\infty$. -\item For any $\delta>0$ and any $R<\infty$ for all $k$ sufficiently large, -$x_k\in Reg_\delta(U_k,g_k)$ and for any $\ell\ge k$ the image -$\varphi_\ell(V_k)$ contains $B(x_\ell,R)\capReg_\delta(U_\ell,g_\ell,x_\ell)$. -\end{enumerate} -\end{enumerate} +1. For any $\delta>0$ and any $R<\infty$ for all $k$ sufficiently large, +$x_k\in \mathit{Reg}_\delta(U_k,g_k)$ and for any $\ell\ge k$ the image +$\varphi_\ell(V_k)$ contains $B(x_\ell,R)\cap\mathit{Reg}_\delta(U_\ell,g_\ell,x_\ell)$. We also say that the sequence *converges geometrically to* $(U_\infty,g_\infty,x_\infty)$ if @@ -90,19 +96,24 @@ subsets and $\{V_k\}_{k=1}^\infty$ satisfying (1) above, and smooth maps $\varphi_k\colon V_k\to U_k$ satisfying (2a) above, we say that $(U_\infty,g_\infty,x_\infty)$ is a *partial geometric limit* of the sequence. + + **Remark.** Conditions (1) and (2a) in the definition above also appear in the definition in the case of complete limits. It is Condition (2b) that is extra in this incomplete case. It says that once $k$ is sufficiently large then the image $\varphi_\ell(V_k)$ contains all points satisfying two conditions: they are at most a given bounded distance from $x_\ell$, and also they are at least a fixed distance from the boundary of $U_\ell$. + Notice that if the $(U_k,g_k)$ have uniformly bounded volume by, say, $V$, then any geometric limit has volume $\le V$. + **Lemma.** The geometric limit of a sequence $(U_k,g_k,x_k)$ is unique up to based isometry. + **Proof.** Suppose that we have two geometric limits $(U_\infty,g_\infty,x_\infty)$ and @@ -112,10 +123,10 @@ required by the definition of the limit. Fix $k$. Since $V_k$ is connected and has compact closure, there are $R<\infty$ and $\delta>0$ such that $V_k\subset B(x_\infty,R)\cap -Reg_\delta(U_\infty,g_\infty,x_\infty)$. Let $x$ be contained +\mathit{Reg}_\delta(U_\infty,g_\infty,x_\infty)$. Let $x$ be contained in the closure of $V_k$. Then by the triangle inequality the closed ball $\overline{B(x,\delta/3)}$ is contained in -$B(x_\infty,R+\delta)\capReg_{\delta/2}(U_\infty,g_\infty,x_\infty)$. Since the union of +$B(x_\infty,R+\delta)\cap\mathit{Reg}_{\delta/2}(U_\infty,g_\infty,x_\infty)$. Since the union of these closed balls as $x$ ranges over $\overline V_k$ is a compact set, for all $\ell$ sufficiently large, the restriction of $\varphi_\ell^*g_\ell$ to the union of these balls is close to the @@ -123,7 +134,7 @@ restriction of $g_\infty$ to the same subset. In particular, for all $\ell$ sufficiently large and any $x\in V_k$ we see that $\varphi_\ell\left(B(x,\delta/3)\right)$ contains $B(\varphi_\ell(x),\delta/4)$. Thus, for all $\ell$ sufficiently -large $\varphi_\ell(V_k)\subset B(x_\ell,R+2\delta)\cap Reg_{\delta/4}(U_\ell,g_\ell,x_\ell)$. This implies that, for given +large $\varphi_\ell(V_k)\subset B(x_\ell,R+2\delta)\cap \mathit{Reg}_{\delta/4}(U_\ell,g_\ell,x_\ell)$. This implies that, for given $k$, for all $\ell$ sufficiently large $\varphi_\ell(V_k)\subset \varphi'_\ell(V'_\ell)$. Of course, $(\varphi'_\ell)^{-1}\circ \varphi_\ell(x_\infty)=x_\infty'$. Fix $k$ and pass to a subsequence @@ -140,14 +151,14 @@ The last thing we need to see is that the embedding of $U_\infty$ into $U'_\infty$ constructed in the previous paragraph is onto. For each $n$ we have $\overline V'_n\subset V'_{n+1}$. Since $\overline V'_n$ is compact and connected, it follows that there are $R<\infty$ and $\delta>0$ (depending on -$n$) such that $\overline V'_n\subset B(x'_\infty,R)\cap Reg_\delta(V_{n+1},g'_\infty,x_\infty')$. Since $V'_{n+1}$ has compact closure +$n$) such that $\overline V'_n\subset B(x'_\infty,R)\cap \mathit{Reg}_\delta(V_{n+1},g'_\infty,x_\infty')$. Since $V'_{n+1}$ has compact closure in $U'_\infty$, as $\ell$ tends to $\infty$ the metrics $(\varphi'_\ell)^*g_\ell$ converge uniformly on $V_{n+1}$ to $g'_\infty|_{V_{n+1}}$. This means that there are $R'<\infty$ and $\delta'>0$ (depending on $n)$ such that for all $\ell$ sufficiently large, -$\varphi'_\ell(V_n)\subset B(x_k,R')\cap Reg_{\delta'}(U_\ell,g_\ell,x_\ell)$. This implies that for all $k$ +$\varphi'_\ell(V_n)\subset B(x_k,R')\cap \mathit{Reg}_{\delta'}(U_\ell,g_\ell,x_\ell)$. This implies that for all $k$ sufficiently large and any $\ell\ge k$ the image $\varphi'_\ell(V'_n)$ is -contained in the image of $\varphi_\ell(V_k)$. Hence, for all $k$ sufficiently +contained in the image of $\varphi_\ell(V_k)$. Hence, for all $k$ sufficiently large and any $\ell\ge k$ we have $V'_n\subset(\varphi'_\ell)^{-1}(\varphi_\ell(V_k))$. Hence, the isometric embedding $U_\infty\to U'_\infty$ constructed above contains $V'_n$. Since this @@ -156,39 +167,44 @@ isometry $U_\infty\to U'_\infty$. Here is the basic existence result. -**Theorem.** -Suppose that $\{(U_k,g_k,x_k)\}_{k=1}^\infty$ is a sequence of + +**Theorem.** Suppose that $\{(U_k,g_k,x_k)\}_{k=1}^\infty$ is a sequence of based, connected, $n$-dimensional Riemannian manifolds. In addition, suppose the following: -\begin{enumerate} -\item[(1)] There is $\delta>0$ such that $x_k\in Reg_\delta(U_k,g_k)$ for all $k$. -\item[(2)] For each $R<\infty$ and $\delta>0$ there is a constant $V(R,\delta)<\infty$ such -that $Vol(B(x_k,R)\cap Reg_\delta(U_k,x_k))\le +- **(1)** There is $\delta>0$ such that $x_k\in \mathit{Reg}_\delta(U_k,g_k)$ for all $k$. +- **(2)** For each $R<\infty$ and $\delta>0$ there is a constant $V(R,\delta)<\infty$ such +that $\mathit{Vol}(B(x_k,R)\cap \mathit{Reg}_\delta(U_k,x_k))\le V(R,\delta)$ for all $k$ sufficiently large. -\item[(3)] For each non-negative integer $\ell$, each $\delta>0$, and each $R<\infty$, +- **(3)** For each non-negative integer $\ell$, each $\delta>0$, and each $R<\infty$, there is a constant $C(\ell,\delta,R)$ such that for every $k$ sufficiently large we have -$$|\nabla^\ell Rm(g_k)|\le C(\ell,\delta,R)$$ -on all of $B(x_k,R)\capReg_\delta(U_k,g_k)$. -\item[(4)] For every $R<\infty$ there are $r_0>0$ and $\kappa>0$ such that + +$$ +|\nabla^\ell \mathit{Rm}(g_k)|\le C(\ell,\delta,R) +$$ + +on all of $B(x_k,R)\cap\mathit{Reg}_\delta(U_k,g_k)$. +- **(4)** For every $R<\infty$ there are $r_0>0$ and $\kappa>0$ such that for every $k$ sufficiently large, for every $\delta\le r_0$ and -every $x\in B(x_k,R)\cap Reg_\delta(U_k,g_k,x_k)$ the volume +every $x\in B(x_k,R)\cap \mathit{Reg}_\delta(U_k,g_k,x_k)$ the volume of the metric ball $B(x,\delta)\subset U_k$ is at least $\kappa\delta^n$. -\end{enumerate} + Then, after passing to a subsequence, there exists a based Riemannian manifold $(U_\infty,g_\infty,x_\infty)$ that is a geometric limit of the sequence $\{(U_k,g_k,x_k)\}_{k=1}^\infty$. + Before giving the proof of this result, we begin with a standard lemma. + **Lemma.** Suppose that we have a sequence of $n$-dimensional balls $(B_k,h_k)$ of radius $r$ in Riemannian $n$-manifolds. Suppose that for each $\ell$ there is a constant $C(\ell)$ such that for every $k$, we -have $|\nabla ^\ell Rm(h_k)|\le C(\ell)$ throughout $B_k$. +have $|\nabla ^\ell \mathit{Rm}(h_k)|\le C(\ell)$ throughout $B_k$. Suppose also that for each $n$ the exponential mapping from the tangent space at the center of $B_k$ induces a diffeomorphism from a ball in the tangent space onto $B_k$. Then choosing an isometric @@ -197,33 +213,38 @@ $B_k$ with $\Ar^n$ and pulling back the metrics $h_k$ via the exponential mapping to metrics $\tilde h_k$ on the ball $B$ of radius $r$ in $\Ar^n$ gives us a family of metrics on $B$ that, after passing to a subsequence, converge in the $C^\infty$-topology, -uniformly on compact subsets of $B$, to a limit. +uniformly on compact subsets of $B$, to a limit. + -The basic point in proving this lemma is to `find the right gauge,' which in +The basic point in proving this lemma is to 'find the right gauge,' which in this case means find local coordinates so that the metric tensor is controlled by the curvature. The correct local coordinates are the Gaussian coordinates centered at the center of the ball. **Proof.** (of the theorem). Fix $R<\infty$ and $\delta>0$. Let -$$X(\delta,R)=B(x_k,R)\cap -Reg_{2\delta}(U_k,g_k,x_k).$$ + +$$ +X(\delta,R)=B(x_k,R)\cap +\mathit{Reg}_{2\delta}(U_k,g_k,x_k). +$$ + From the non-collapsing assumption and the curvature bound assumption if follows from -Theorem~ that there is a uniform positive lower bound +\entryref{f90cce212e88} that there is a uniform positive lower bound (independent of $k$) to the injectivity radius of every point in -$X(\delta,R)$. Fix $0<\delta'\le min(r_0,\delta/2)$ much less +$X(\delta,R)$. Fix $0<\delta'\le \mathit{min}(r_0,\delta/2)$ much less than this injectivity radius. We also choose $\delta'>0$ sufficiently small so that any ball of radius $2\delta'$ in -$B(x_k,R+\delta)\cap Reg_\delta(U_k,g_k,x_k)$ is geodesically -convex. (This is possible because of the curvature bound.) We cover +$B(x_k,R+\delta)\cap \mathit{Reg}_\delta(U_k,g_k,x_k)$ is geodesically +convex. (This is possible because of the curvature bound.) We cover $X(\delta,R)$ by balls $B_1',\ldots,B_N'$ of radii $\delta'/2$ centered at points of $X(\delta,R)$ with the property that the sub-balls of radius $\delta'/4$ are disjoint. We denote by $B_i'\subset B_i\subset \widetilde B_i$ the metric balls with the same center and radii $\delta'/2$, $\delta'$, and $2\delta'$ respectively. - Notice that each of the balls $\widetilde B_i$ is contained in $B(x_k,R+\delta)\cap Reg_{\delta}(U_k,g_k,x_k)$. Because $\delta'\le r_0$, because $Vol\,B(x_k,R+\delta)$ is bounded independent of $k$, and because the + Notice that each of the balls $\widetilde B_i$ is contained in $B(x_k,R+\delta)\cap \mathit{Reg}_{\delta}(U_k,g_k,x_k)$. Because $\delta'\le r_0$, because $\mathit{Vol} B(x_k,R+\delta)$ is bounded independent of $k$, and because the concentric balls of radius $\delta'/4$ are disjoint, there is a uniform bound (independent of $k$) to the number of such balls. Passing to a subsequence we can assume that the number of balls in these coverings is the same for all $k$. @@ -248,25 +269,38 @@ sub-balls $B_i\subset \widetilde B_i$ of radii $\delta'$. A standard argument using partitions of unity and the geodesic convexity of the balls $\widetilde B_i$ shows that, for all $k$ sufficiently large, there are smooth embeddings $\varphi_k(\delta,R)\colon -U_\infty(\delta,R)\to B(x_k,R+\delta)\cap Reg_\delta(U_k,g_k,x_k)$ sending $x_\infty(\delta,R)$ to $x_k$ and +U_\infty(\delta,R)\to B(x_k,R+\delta)\cap \mathit{Reg}_\delta(U_k,g_k,x_k)$ sending $x_\infty(\delta,R)$ to $x_k$ and converging as $k\rightarrow \infty$, uniformly in the $C^\infty$-topology on each $B_i$, to the identity. Furthermore, the -images of each of these maps contains $B(x_k,R)\cap Reg_{2\delta}(U_k,g_k,x_k)$; compare . Also, the pull +images of each of these maps contains $B(x_k,R)\cap \mathit{Reg}_{2\delta}(U_k,g_k,x_k)$; compare [Cheeger]. Also, the pull backs under these embeddings of the metrics $g_k$ converge uniformly to $g_\infty(\delta,R)$. Repeat the process with $R$ replaced by $2R$ and $\delta=\delta_1$ replaced by $\delta_2\le \delta_1/2$. This produces -$$\left(U_\infty(\delta_2,2R),g_\infty(\delta_2,2R),x_\infty(\delta_2,2R)\right)$$ + +$$ +\left(U_\infty(\delta_2,2R),g_\infty(\delta_2,2R),x_\infty(\delta_2,2R)\right) +$$ + and, for all $k$ sufficiently large, embeddings $\varphi_k(\delta_2,2R)$ of this manifold into -$$B(x_k,2R+\delta_2)\cap Reg_{\delta_2}(U_k,g_k,x_k).$$ Hence, + +$$ +B(x_k,2R+\delta_2)\cap \mathit{Reg}_{\delta_2}(U_k,g_k,x_k). +$$ + + Hence, the image of these embeddings contains the images of the original embeddings. The compositions $(\varphi_k(\delta_2,2R))^{-1}\circ \varphi_k(\delta,R)$ converge to an isometric embedding -$$\left(U_\infty(\delta,R),g_\infty(\delta,R),x_\infty(\delta,R)\right)\to -\left(U_\infty(\delta_2,2R),g_\infty(\delta_2,2R),x_\infty(\delta_2,2R)\right).$$ -Repeating this construction infinitely often produces a manifold + +$$ +\left(U_\infty(\delta,R),g_\infty(\delta,R),x_\infty(\delta,R)\right)\to +\left(U_\infty(\delta_2,2R),g_\infty(\delta_2,2R),x_\infty(\delta_2,2R)\right). +$$ + +Repeating this construction infinitely often produces a manifold $(U_\infty,g_\infty,x_\infty)$ which is written as an increasing union of open subsets $V_k=U_\infty(\delta_k,2^kR)$, where the $\delta_k$ tend to zero as $k$ tends to $\infty$. For each $k$ the open subset $V_k$ has compact closure @@ -278,112 +312,132 @@ Now let us turn to complete Riemannian manifolds, where the result is the $C^\infty$-version of the classical Cheeger-Gromov compactness. + **Lemma.** Suppose that $(U_k,g_k,x_k)$ is a sequence of based Riemannian manifolds and that there is a partial geometric limit $(U_\infty,g_\infty,x_\infty)$ that is a complete Riemannian manifold. Then this partial geometric limit is a geometric limit. + **Proof.** Since the balls $B(x_\infty,R)$ have compact closure in $U_\infty$ and since -$$Reg_\delta(U_\infty,g_\infty,x_\infty)=U_\infty$$ for + +$$ +\mathit{Reg}_\delta(U_\infty,g_\infty,x_\infty)=U_\infty +$$ + + for every $\delta>0$, it is easy to see that the extra condition, (2b), in -Definition~ is automatic in this case. +\entryref{467f4b920031} is automatic in this case. -Now as an immediate corollary of Theorem~ we have the +Now as an immediate corollary of \entryref{74c7682a68ed} we have the following. -**Theorem.** -Let $\{(M_k,g_k,x_k)\}_{k=1}^\infty$ be a sequence of connected, + +**Theorem.** Let $\{(M_k,g_k,x_k)\}_{k=1}^\infty$ be a sequence of connected, based Riemannian manifolds. Suppose that: -\begin{enumerate} -\item[(1)] For every $A<\infty$ the ball + +- **(1)** For every $A<\infty$ the ball $B(x_k,A)$ has compact closure in $M_k$ for all $k$ sufficiently large. -\item[(2)] For each integer $\ell\ge 0$ and each $A<\infty$ there is a constant +- **(2)** For each integer $\ell\ge 0$ and each $A<\infty$ there is a constant $C=C(\ell,A)$ such that for each $y_k\in B(x_k,A)$ we have -$$\left|\nabla^\ellRm(g_k)(y_k)\right|\le C$$ + +$$ +\left|\nabla^\ell\mathit{Rm}(g_k)(y_k)\right|\le C +$$ + for all $k$ sufficiently large. -\item[(3)] Suppose also that there is a constant $\delta>0$ such that $inj_{(M_k,g_k)}(x_k)\ge \delta$ for all $k$ sufficiently large. -\end{enumerate} +- **(3)** Suppose also that there is a constant $\delta>0$ such that $\mathit{inj}_{(M_k,g_k)}(x_k)\ge \delta$ for all $k$ sufficiently large. + Then after passing to a subsequence there is a geometric limit which is a complete Riemannian manifold. + **Proof.** By the curvature bounds, it follows from the Bishop-Gromov theorem -(Theorem~) that for each $A<\infty$ there is a uniform bound +(\entryref{1a1b2c4fdc9b}) that for each $A<\infty$ there is a uniform bound to the volumes of the balls $B(x_k,A)$ for all $k$ sufficiently large. It also -follows from the same result that the uniform lower bound on the injectivity +follows from the same result that the uniform lower bound on the injectivity radius at the central point implies that for each $A<\infty$ there is a uniform lower bound for the injectivity radius on the entire ball $B(x_k,A)$, again for $k$ sufficiently large. Given these two facts, it follows immediately from -Theorem~ that there is a geometric limit. +\entryref{74c7682a68ed} that there is a geometric limit. Since, for every $A<\infty$, the $B(x_k,A)$ have compact closure in $M_k$ for all $k$ sufficiently large, it follows that for every $A<\infty$ the ball $B(x_\infty,A)$ has compact closure in $M_\infty$. This means that $(M_\infty,g_\infty)$ is complete. -**Corollary.** -Suppose that $\{(M_k,g_k,x_k)\}_{k=1}^\infty$ is a sequence of based, connected + +**Corollary.** Suppose that $\{(M_k,g_k,x_k)\}_{k=1}^\infty$ is a sequence of based, connected Riemannian manifolds. Suppose that the first two conditions in -Theorem~ hold and suppose also that there are constants $\kappa>0$ -and $\delta>0$ such that $Vol_{g_k}B(x_k,\delta)\ge \kappa\delta^n$ for +\entryref{c81f82f6f8a1} hold and suppose also that there are constants $\kappa>0$ +and $\delta>0$ such that $\mathit{Vol}_{g_k}B(x_k,\delta)\ge \kappa\delta^n$ for all $k$. Then after passing to a subsequence there is a geometric limit which is a complete Riemannian manifold. + -**Proof.** Let $A=max(\delta^{-2},C(0,\delta))$, where $C(0,\delta)$ is the constant -given in the second condition in Theorem~. Rescale, replacing the +**Proof.** Let $A=\mathit{max}(\delta^{-2},C(0,\delta))$, where $C(0,\delta)$ is the constant +given in the second condition in \entryref{c81f82f6f8a1}. Rescale, replacing the Riemannian metric $g_k$ by $Ag_k$. Of course, the first condition of -Theorem~ still holds as does the second with different constants, -and we have $\left|Rm_{Ag_k}(y_k)\right|\le 1$ for all $y_k\in -B_{Ag_k}(x_k,\sqrt{A}\delta)$. Also, $Vol\,B_{Ag_k}(x_k,\sqrt{A}\delta)\ge \kappa (\sqrt{A}\delta)^n$. Thus, by the -Bishop-Gromov inequality (Theorem~), we have $Vol_{Ag_k}B(x_k,1)\ge \kappa/\Omega$ where -$$\Omega=\frac{V(\sqrt{A}\delta)}{(\sqrt{A}\delta)^nV(1)},$$ +\entryref{c81f82f6f8a1} still holds as does the second with different constants, +and we have $\left|\mathit{Rm}_{Ag_k}(y_k)\right|\le 1$ for all $y_k\in +B_{Ag_k}(x_k,\sqrt{A}\delta)$. Also, $\mathit{Vol} B_{Ag_k}(x_k,\sqrt{A}\delta)\ge \kappa (\sqrt{A}\delta)^n$. Thus, by the +Bishop-Gromov inequality (\entryref{1a1b2c4fdc9b}), we have $\mathit{Vol}_{Ag_k}B(x_k,1)\ge \kappa/\Omega$ where + +$$ +\Omega=\frac{V(\sqrt{A}\delta)}{(\sqrt{A}\delta)^nV(1)}, +$$ + where $V(a)$ is the volume of the ball of radius $a$ in hyperbolic $n$-space (the simply connected $n$-manifold of constant curvature $-1$). Since $\sqrt{A}\delta\ge 1$, this proves that for the rescaled manifolds the absolute values of the sectional curvatures on $B_{Ag_k}(x_k,1)$ are bounded by $1$ and -the $Vol_{Ag_k}B(x_k,1)$ are bounded below by a positive constant +the $\mathit{Vol}_{Ag_k}B(x_k,1)$ are bounded below by a positive constant independent of $k$. - According to Theorem~ the lower bound on the volume of the ball + According to \entryref{f90cce212e88} the lower bound on the volume of the ball of radius $1$ and the curvature bound on the ball of radius $1$ yield a uniform positive lower bound $r>0$ for the injectivity radius of the rescaled manifolds at $x_k$. Hence, the injectivity radii at the base points of the original sequence are bounded below by $\delta/\sqrt{A}$. This means that the original -sequence of manifolds satisfies the third condition in Theorem~. +sequence of manifolds satisfies the third condition in \entryref{c81f82f6f8a1}. Invoking this theorem gives the result. -### Geometric convergence of manifolds in the case of Ricci -flow +### Geometric convergence of manifolds in the case of Ricci flow As the next theorem shows, because of Shi's theorem, it is much easier to establish the geometric convergence manifolds in the context of Ricci flows than in general. -**Theorem.** -Suppose that $({\mathcal M}_k,G_k,x_k)$ is a sequence of based -generalized $n$-dimensional Ricci flows with $**t**(x_k)=0$. Let + +**Theorem.** Suppose that $({\mathcal M}_k,G_k,x_k)$ is a sequence of based +generalized $n$-dimensional Ricci flows with $\mathbf{t}(x_k)=0$. Let $(M_k,g_k)$ be the $0$ time-slice of $({\mathcal M}_k,G_k)$. Suppose that for each $A<\infty$ there are constants $C(A)<\infty$ and $\delta(A)>0$ such that for all $k$ sufficiently large the following hold: -\begin{enumerate} -\item[(1)] the ball $B(x_k,0,A)$ has compact closure in $M_k$, -\item[(2)] there is an embedding $B(x_k,0,A)\times (-\delta(A),0]\to {\mathcal M}_k$ compatible with -the time function and with the vector field, \item[(3)] $|Rm|\le C(A)$ -on the image of the embedding in the Item (2), and \item[(4)] there is $r_0>0$ -and $\kappa>0$ such that $Vol\,B(x_k,0,r_0)\ge \kappa r_0^n$ for all $k$ + +- **(1)** the ball $B(x_k,0,A)$ has compact closure in $M_k$, +- **(2)** there is an embedding $B(x_k,0,A)\times (-\delta(A),0]\to {\mathcal M}_k$ compatible with +the time function and with the vector field, +- **(3)** $|\mathit{Rm}|\le C(A)$ +on the image of the embedding in the Item (2), and +- **(4)** there is $r_0>0$ +and $\kappa>0$ such that $\mathit{Vol} B(x_k,0,r_0)\ge \kappa r_0^n$ for all $k$ sufficiently large. -\end{enumerate} Then, after passing to a + + Then, after passing to a subsequence, there is a geometric limit $(M_\infty,g_\infty,x_\infty)$ of the $0$ time-slices $(M_k,g_k,x_k)$. This limit is a complete Riemannian manifold. + -**Proof.** The first condition in Theorem~ holds by our first -assumption. It is immediate from Shi's theorem (Theorem~) -that the second condition of Theorem~ holds. The result -is then immediate from Corollary~. +**Proof.** The first condition in \entryref{c81f82f6f8a1} holds by our first +assumption. It is immediate from Shi's theorem (\entryref{ed4b72caf9f9}) +that the second condition of \entryref{c81f82f6f8a1} holds. The result +is then immediate from \entryref{b45765a3070e}. ## Geometric convergence of Ricci flows @@ -392,48 +446,56 @@ is then immediate from Corollary~. Ricci flows. Then we give Hamilton's theorem about the existence of such geometric limits. + **Definition.** Let $\{({\mathcal M}_k,G_k,x_k)\}_{k=1}^\infty$ be a sequence of based -generalized Ricci flows. We suppose that $**t**(x_k)=0$ for all $k$ and we +generalized Ricci flows. We suppose that $\mathbf{t}(x_k)=0$ for all $k$ and we denote by $(M_k,g_k)$ the time-slice of $({\mathcal M}_k,G_k)$. For some $0 Notice that the restriction to the $0$ time-slices of a partial geometric limit of generalized Ricci flows is a partial geometric limit of the $0$ time-slices. + **Definition.** For $0 Again Shi's theorem, together with a computation of Hamilton, allows us to form geometric limits of generalized Ricci flows. We have the following result due -originally to Hamilton . +originally to Hamilton [Hamiltonlimits]. -**Proposition.** -Fix constants $-\infty\le T'\le 0\le T\le \infty$ and suppose that $T' +**Proposition.** Fix constants $-\infty\le T'\le 0\le T\le \infty$ and suppose that $T' **Proof.** Suppose that we have a partial geometric limit of the time-zero slices as stated in the proposition. Fix a compact subset $K\subset @@ -454,7 +517,7 @@ course satisfy the Ricci flow equation on $K\times I$. Furthermore, by assumption the flows $g_k(K,I)(t)$ have uniformly bounded curvature. Then under these hypothesis, Shi's theorem can be used to show that the curvatures of the $g_k(K,I)$ are uniformly bounded $C^\infty$-topology. The basic computation - done by Hamilton in shows that after passing + done by Hamilton in [Hamiltonlimits] shows that after passing to a further subsequence, the Ricci flows $g_k(K,I)$ converge uniformly in the $C^\infty$-topology to a limit flow on $K\times I$. A standard diagonalization argument allows us to pass to a further @@ -463,46 +526,50 @@ converge uniformly in the $C^\infty$-topology on every compact subset of $M_\infty\times (T',T)$. Of course, the limit satisfies the Ricci flow equation. -This `local' result leads immediately to the following result for +This 'local' result leads immediately to the following result for complete limits. -**Theorem.** -Fix $-\infty\le T'\le 0\le T\le \infty$ with $T' +**Theorem.** Fix $-\infty\le T'\le 0\le T\le \infty$ with $T'0$ and $\kappa>0$ with $Vol\,B(x_k,0,r_0)\ge \kappa r_0^n$ +- **(3)** the norms of the Riemann curvature of $G_k$ on the image of the embedding in the +previous item are bounded by $C(A,I)$, and +- **(4)** there is $r_0>0$ and $\kappa>0$ with $\mathit{Vol} B(x_k,0,r_0)\ge \kappa r_0^n$ for all $k$ sufficiently large. -\end{enumerate} + Then after passing to a subsequence there is a -flow $(M_\infty,g_\infty(t),(x_\infty,0))$ which is the geometric +flow $(M_\infty,g_\infty(t),(x_\infty,0))$ which is the geometric limit. It is a solution to the Ricci flow equation defined for $t\in (T',T)$. For every $t\in (T',T)$ the Riemannian manifold $(M_\infty,g_\infty(t))$ is complete. + -**Proof.** By Theorem~ there is a geometric limit $(M_\infty,g_\infty(0))$ -of the $0$ time-slices, and the limit is a complete Riemannian manifold. Then -by Proposition~ there is a geometric limit flow defined +**Proof.** By \entryref{d9f3522dec12} there is a geometric limit $(M_\infty,g_\infty(0))$ +of the $0$ time-slices, and the limit is a complete Riemannian manifold. Then +by \entryref{4673eb9f4424} there is a geometric limit flow defined on the time interval $(T',T)$. Since for every $t\in (T',T)$ there is a compact interval $I$ containing $0$ and $t$, it follows that the Riemann curvature of the limit is bounded on $M_\infty\times I$. This means that the metrics $g_\infty(0)$ and $g_\infty(t)$ are commensurable with each other. Since $g_\infty(0)$ is complete so is $g_\infty(t)$. -**Corollary.** Suppose that $(U,g(t)),\ 0\le t +**Corollary.** Suppose that $(U,g(t)),\ 0\le t **Proof.** Take a sequence $t_n\rightarrow T$ and consider the sequence of Riemannian manifolds $(V,g(t_n))$. By Shi's theorem and the fact @@ -510,13 +577,13 @@ that $V$ has compact closure in $U$, the restriction of this sequence of metrics to $V$ has uniformly bounded curvature derivatives. Hence, this sequence has a convergent subsequence with limit $(V,g_\infty)$, where the convergence is uniform in the $C^\infty$-topology. Now by Hamilton's result - it follows that, passing to a further subsequence, the +[Hamiltonlimits] it follows that, passing to a further subsequence, the flows $(V,g(T+t-t_n),(p,0))$ converge to a flow $(V,g_\infty(t),(p,0))$ defined on $(0,T]$. Clearly, for any $0 T$ using the local existence results. The extension to $[T,T_1)$ fits together -smoothly with the flow on $[0,T]$ by Proposition~. +smoothly with the flow on $[0,T]$ by \entryref{b6bc786f7f44}. ## Gromov-Hausdorff convergence @@ -539,13 +606,14 @@ To see that $D_{GH}$ is a distance function we must establish the triangle inequality. For this it is convenient to introduce $\delta$-nets in metric spaces. -**Definition.** -A $\delta$-net in $(X,x)$ is a subset $L$ of $X$ containing $x$ + +**Definition.** A $\delta$-net in $(X,x)$ is a subset $L$ of $X$ containing $x$ whose $\delta$-neighborhood covers $X$ and for which there is some $\delta'>0$ with $d(\ell_1,\ell_2)\ge \delta'$ for all $\ell_1\not= \ell_2$ in $L$. + - Clearly, the Gromov-Hausdorff distance from a based +Clearly, the Gromov-Hausdorff distance from a based metric space $(X,x)$ to a $\delta$-net $(L,x)$ contained in it is at most $\delta$. Furthermore, for every $\delta>0$ the based space $(X,x)$ has a $\delta$-net: Consider subsets $L\subset X$ containing @@ -553,7 +621,9 @@ $x$ with the property that the $\delta/2$-balls centered at the points of $L$ are disjoint. Any maximal such subset (with respect to the inclusion relation) is a $\delta$-net in $X$. + **Lemma.** The Gromov-Hausdorff distance satisfies the triangle inequality. + **Proof.** Suppose that $D_{GH}((X,x),(Y,y))=a$ and $D_{GH}((Y,y),(Z,z))=b$. Fix any $\delta>0$. Then there is a metric $d_1$ on $X\vee Y$ such @@ -562,9 +632,13 @@ $(a+\delta)$-neighborhood of $X$ is all of $X\vee Y$ as is the $(a+\delta)$-neighborhood of $Y$. Similarly, there is a metric $d_2$ on $Y\vee Z$ with the analogous properties (with $b$ replacing $a$). Take a $\delta$-net $(L,y)\subset (Y,y)$, and define -$$d(x',z')=inf_{\ell\in L}d(x',\ell)+d(\ell,z').$$ + +$$ +d(x',z')=\mathit{inf}_{\ell\in L}d(x',\ell)+d(\ell,z'). +$$ + We claim that $d(x',z')>0$ unless $x'=z'$ is the common base point. The reason -is that if $inf_{\ell\in L}d(x',\ell)=0$, then by the triangle +is that if $\mathit{inf}_{\ell\in L}d(x',\ell)=0$, then by the triangle inequality, any sequence of $\ell_n\in L$ with $d(x',\ell_n)$ converging to zero is a Cauchy sequence, and hence is eventually constant. This means that for all $n$ sufficiently large, $x'=\ell_n\in L\cap X$ and hence $x'$ is the @@ -582,33 +656,44 @@ spaces of bounded diameter is a metric. We shall not establish this result, though we prove below closely related results about the uniqueness of Gromov-Hausdorff limits. -**Definition.** -We say that a sequence of based metric spaces $(X_k,x_k)$ of + +**Definition.** We say that a sequence of based metric spaces $(X_k,x_k)$ of uniformly bounded diameter *converges in the Gromov-Hausdorff -sense*} to a based metric space $(Y,y)$ of finite +sense* to a based metric space $(Y,y)$ of finite diameter if -$$lim_{k\rightarrow\infty}D_{GH}((X_k,x_k),(Y,y))=0.$$ + +$$ +\mathit{lim}_{k\rightarrow\infty}D_{GH}((X_k,x_k),(Y,y))=0. +$$ + Thus, a based metric space $(X,x)$ of bounded diameter is the limit of a sequence of $\delta_n$-nets $L_n\subset X$ provided that $\delta_n\rightarrow 0$ as $n\rightarrow\infty$. + -\begin{exam} -A sequence of compact $n$-manifolds of diameter tending to zero has -a point as Gromov-Hausdorff limit. -\end{exam} + +**Example.** A sequence of compact $n$-manifolds of diameter tending to zero has +a point as Gromov-Hausdorff limit. + -**Definition.** -Suppose that $\{(X_k,x_k)\}_k$ converges in the Gromov-Hausdorff sense to + +**Definition.** Suppose that $\{(X_k,x_k)\}_k$ converges in the Gromov-Hausdorff sense to $(Y,y)$. Then a *realization sequence* for this convergence is a sequence of triples $((Z_k,z_k),f_k,g_k)$ where, for each $k$, the pair $(Z_k,z_k)$ is a based metric space, -$$f_k\colon (X_k,x_k)\to -(Z_k,z_k) \ \ \ \ and \ \ \ g_k\colon (Y,y)\to (Z_k,z_k)$$ are -isometric embeddings and $D_{GH}(f_k(X_k),g_k(Y))\rightarrow 0$ as + +$$ +f_k\colon (X_k,x_k)\to +(Z_k,z_k) \ \ \ \ \mathit{and} \ \ \ g_k\colon (Y,y)\to (Z_k,z_k) +$$ + + are +isometric embeddings and $D_{GH}(f_k(X_k),g_k(Y))\rightarrow 0$ as $k\rightarrow\infty$. Given a realization sequence for the convergence, we say that a sequence $\ell_k\in X_k$ converges to $\ell\in Y$ (relative to the given realization sequence) if $d(f_k(\ell_k),g_k(\ell))\rightarrow 0$ as $i\rightarrow\infty$. + Notice that, with a different realization sequence for the convergence, a sequence $\ell_k\in X_k$ can converge to a different @@ -620,6 +705,7 @@ converging to a point of $Y$. Lastly, notice that under any realization sequence for the convergence, the base points $x_k$ converge to the base point $y$. + **Lemma.** Let $(X_k,x_k)$ be a sequence of metric spaces whose diameters are uniformly bounded. Then the $(X_k,x_k)$ converge in the Gromov-Hausdorff sense to $(X,x)$ if and only if the @@ -629,13 +715,16 @@ $(\delta+\eta)$-net $L_k\subset X_k$ and a bijection $L_k\to L$ sending $x_k$ to $x$ so that the push forward of the metric on $L_k$ induced from that of $X_k$ is $(1+\eta)$-bi-Lipschitz equivalent to the metric on $L$ induced from $X$. + -For a proof see Proposition 3.5 on page 36 of . +For a proof see Proposition 3.5 on page 36 of [Gromov]. -**Lemma.** Let $(X_k,x_k)$ be a sequence of based metric spaces whose diameters are + +**Lemma.** Let $(X_k,x_k)$ be a sequence of based metric spaces whose diameters are uniformly bounded. Suppose that $(Y,y)$ and $(Y',y')$ are limits in the Gromov-Hausdorff sense of this sequence and each of $Y$ and $Y'$ are compact. Then $(Y,y)$ is isometric to $(Y',y')$. + **Proof.** By the triangle inequality for Gromov-Hausdorff distance, it follows from the hypothesis of the lemma that $D_{GH}((Y,y),(Y',y'))=0$. Fix @@ -646,7 +735,7 @@ is a $2/n$-net in $Y'$ and such that the map from $L_n$ to its image is a $(1+\delta)$-bi-Lipschitz homeomorphism. Clearly, we can suppose that in addition the $L_n$ are nested: $L_n\subset L_{n+1}\subset \cdots$. Since $Y'$ is compact and $L_n$ is finite, -and we can pass to a subsequence so that $lim_{k\rightarrow\infty}\varphi_k|_{L_n}$ converges to a map +and we can pass to a subsequence so that $\mathit{lim}_{k\rightarrow\infty}\varphi_k|_{L_n}$ converges to a map $\psi_n\colon L_n\to Y'$ which is a $(1+\delta)$-bi-Lipschitz map onto its image which is a $2/n$ net in $Y'$. By a standard diagonalization argument, we can arrange that @@ -655,10 +744,10 @@ define an embedding $\cup_n L_n\to Y'$ that is a $(1+\delta)$-bi-Lipschitz map onto its image which is a dense subset of $Y'$. Clearly, using the compactness of $Y'$ this map extends to a $(1+\delta)$-bi-Lipschitz embedding $\psi_\delta\colon (Y,y)\to -(Y',y')$ onto a dense subset of $Y'$. Since $Y$ is also compact, +(Y',y')$ onto a dense subset of $Y'$. Since $Y$ is also compact, this image is in fact all of $Y'$. That is to say, $\psi_\delta$ is a $(1+\delta)$-bi-Lipschitz homeomorphism $(Y,y)\to (Y',y')$. Now -perform this construction for a sequence of $\delta_n\rightarrow 0$ +perform this construction for a sequence of $\delta_n\rightarrow 0$ and $(1+\delta_n)$-bi-Lipschitz homeomorphisms $\psi_{\delta_n}\colon (Y,y)\to (Y',y')$. These form an equicontinuous family so that by passing to a subsequence we can @@ -669,25 +758,29 @@ Now let us consider the more general case of spaces of not necessarily bounded diameter. It turns out that the above definition is too restrictive when applied to such spaces. Rather one takes: -**Definition.** For based metric spaces $(X_k,x_k)$ (not necessarily of finite + +**Definition.** For based metric spaces $(X_k,x_k)$ (not necessarily of finite diameter) to *converge in the Gromov-Hausdorff sense to a based metric space* $(Y,y)$ means that for each $r>0$ there is a sequence $\delta_k\rightarrow 0$ such that the sequence of balls $B(x_k,r+\delta_k)$ in $(X_k,x_k)$ converges in the Gromov-Hausdorff sense to the ball $B(y,r)$ in $Y$. + Thus, a sequence of cylinders $S^{n-1}\times \Ar$ with any base points and with the radii of the cylinders going to zero has the real line as Gromov-Hausdorff limit. + **Lemma.** Let $(X_k,x_k)$ be a sequence of locally compact metric spaces. Suppose that $(Y,y)$ and $(Y',y')$ are complete, locally compact, based metric spaces that are limits of the sequence in the Gromov-Hausdorff sense. Then there is an isometry $(Y,y)\to (Y',y')$. + **Proof.** We show that for each $r<\infty$ there is an isometry between the closed balls $\overline{B(y,r)}$ and $\overline{B(y',r)}$. By the -local compactness and completeness, these closed balls are compact. +local compactness and completeness, these closed balls are compact. Each is the limit in the Gromov-Hausdorff sense of a sequence $B(x_k,r+\delta_k)$ for some $\delta_k\rightarrow 0$ as $k\rightarrow\infty$. Thus, invoking the previous lemma we see that @@ -704,29 +797,31 @@ $\ell\in Y$ under one realization sequence for the convergence and to $\ell'\in Y$ under another, then there is an isometry of $(Y,y)$ to itself carrying $\ell$ to $\ell'$. -\begin{exam} -Let $(M_n,g_n,x_n)$ be a sequence of based Riemannian manifolds + +**Example.** Let $(M_n,g_n,x_n)$ be a sequence of based Riemannian manifolds converging geometrically to $(M_\infty,g_\infty,x_\infty)$. Then the sequence also converges in the Gromov-Hausdorff sense to the same limit. -\end{exam} + ### Precompactness There is a fundamental compactness result due to Gromov. We begin with a definition. + **Definition.** A *length space* is a connected metric space $(X,d)$ such that for any two points $x,y$ there is a rectifiable arc $\gamma$ with endpoints $x$ and $y$ and with the length of $\gamma$ equal to $d(x,y)$. + For any based metric space $(X,x)$ and constants $\delta>0$ and $R<\infty$ let $N(\delta,R,X)$ be the maximal number of disjoint $\delta$-balls in $X$ that can be contained in $B(x,R)$. -**Theorem.** -Suppose that $(X_k,x_k)$ is a sequence of based length spaces. Then + +**Theorem.** Suppose that $(X_k,x_k)$ is a sequence of based length spaces. Then there is a based length space $(X,x)$ that is the limit in the Gromov-Hausdorff sense of a subsequence of the $(X_k,x_k)$ if for every $\delta>0$ and $R<\infty$ there is an $N<\infty$ such that @@ -734,9 +829,10 @@ $N(\delta,R,X_k)\le N$ for all $k$. On the other hand, if the sequence $(X_k,x_k)$ has a Gromov-Hausdorff limit, then for every $\delta>0$ and $R<\infty$ the $N(\delta,R,X_k)$ are bounded independent of $k$. + For a proof of this result see Proposition 5.2 on page 63 of -. +[Gromov]. ### The Tits cone @@ -744,29 +840,37 @@ Let $(M,g)$ be a complete, non-compact Riemannian manifold of non-negative sectional curvature. Fix a point $p\in M$, and let $\gamma$ and $\mu$ be minimal geodesic rays emanating from $p$. For each $r>0$ let $\gamma(r)$ and $\mu(r)$ be the points along these -geodesic rays at distance $r$ from $p$. Then by Part 1 of -Theorem~ we see that -$$\ell(\gamma,\mu,r)=\frac{d(\gamma(r),\mu(r))}{r}$$ +geodesic rays at distance $r$ from $p$. Then by Part 1 of +\entryref{04518fa1ac2f} we see that + +$$ +\ell(\gamma,\mu,r)=\frac{d(\gamma(r),\mu(r))}{r} +$$ + is a non-increasing function of $r$. Hence, there is a limit $\ell(\gamma,\mu)\ge 0$ of $\ell(\gamma,\mu,r)$ as $r\rightarrow\infty$. We define the *angle at infinity* between $\gamma$ and $\mu$, $0\le\theta_\infty(\gamma,\mu)\le \pi$, to be the angle at $b$ of the Euclidean triangle $a,b,c$ with side lengths -$|ab|=|bc|=1$ and $|bc|=\ell(\gamma,\mu)$, see Fig.~. If $\nu$ is a third geodesic ray +$|ab|=|bc|=1$ and $|bc|=\ell(\gamma,\mu)$, see Fig. 5.1. If $\nu$ is a third geodesic ray emanating from $p$, then clearly, $\theta_\infty(\gamma,\mu)+\theta_\infty(\mu,\nu)\ge \theta_\infty(\gamma,\nu)$. + **Definition.** Now we define a metric space whose underlying space is the quotient space of the equivalence classes of minimal geodesic rays emanating from $p$, with two rays equivalent if and only if the angle at infinity between them is zero. The pseudo-distance function $\theta_\infty$ descends to a metric on this space. This space is a - length space . Notice that the distance between + length space [BGP]. Notice that the distance between any two points in this metric space is at most $\pi$. We denote this space by $S_\infty(M,p)$. + + **Claim.** $S_\infty(M,p)$ is a compact space. + **Proof.** Let $\{[\gamma_n]\}_n$ be a sequence of points in $S_\infty(M,p)$. We show that there is a subsequence with a limit point. By passing to a subsequence we can @@ -782,27 +886,32 @@ subsequence of the $\{[\gamma_n]\}$ We define the *Tits cone* of $M$ at $p$, denoted ${\mathcal T}(M,p)$, to be the cone over -$S_\infty(M,p)$, i.e., the quotient of the space $S_\infty(M,p)\times +$S_\infty(M,p)$, i.e., the quotient of the space $S_\infty(M,p)\times [0,\infty)$ where all points $(x,0)$ are identified together (to become the cone point). The cone metric on this space is given as follows: Let $(x_1,a_1)$ and $(x_2,a_2)$ be points of $S_\infty(M,p)\times [0,\infty)$. Then the distance between their images in the cone is determined by -$$d^2([x_1,a_1],[x_2,a_2])=a_1^2+a_2^2-2a_1a_2cos(\theta_\infty(x_1,x_2)).$$ + +$$ +d^2([x_1,a_1],[x_2,a_2])=a_1^2+a_2^2-2a_1a_2\mathit{cos}(\theta_\infty(x_1,x_2)). +$$ + It is an easy exercise to show that the Tits cone of $M$ at $p$ is in fact independent of the choice of $p$. From the previous claim, it follows that the Tits cone of $M$ is locally compact and complete. -**Proposition.** -Let $(M,g)$ be a non-negatively curved, complete, non-compact + +**Proposition.** Let $(M,g)$ be a non-negatively curved, complete, non-compact Riemannian manifold of dimension $k$. Fix a point $p\in M$ and let $\{x_n\}_{n=1}^\infty$ be a sequence tending to infinity in $M$. Let $\lambda_n=d^2(p,x_n)$ and consider the sequence of based Riemannian manifolds $(M,g_n,p)$, where $g_n=\lambda_n^{-1}g$. Then there is a subsequence converging in the Gromov-Hausdorff sense. Any Gromov-Hausdorff limit - $(X,g_\infty,x_\infty)$ of a + $(X,g_\infty,x_\infty)$ of a subsequence $(X,g_\infty)$ is isometric to the Tits cone ${\mathcal T}(M,p)$ with base point the cone point. + **Proof.** Let $c$ be the cone point of ${\mathcal T}(M,p)$, and denote by $d$ the distance function on ${\mathcal T}(M,p)$. Consider the ball $B(c,R)\subset @@ -813,7 +922,7 @@ with points of the form $([\gamma],t)$ where $\gamma$ is a minimal geodesic ray emanating from $p$ and $t>0$. We define a map from $\psi_n\colon L \to (M,g_n)$ by sending the cone point to $p$ and sending $([\gamma],t)$ to the point at $g_n$-distance $t$ from $p$ along $\gamma$. Clearly, $\psi_n(L)$ is contained -in $B_{g_n}(p,R)$. From the second item of Theorem~ and the +in $B_{g_n}(p,R)$. From the second item of \entryref{04518fa1ac2f} and the monotonicity of angles it follows that the map $\psi_n\colon L\to (M,g_n)$ is a distance non-decreasing map; i.e., $\psi_n^*(g_n|\psi_n(L))\ge d|_L$. On the other hand, by the monotonicity, $\psi_{n+1}^*(g_{n+1}|\psi_{n+1}(L))\le @@ -827,13 +936,13 @@ passing to a subsequence, for each $n$ we have a point $x_n\in B_{g_n}(p,R)$ whose distance from $\psi_n(L)$ is at least $\delta+\eta$. In particular, $d_{g_n}(x_n,p)\ge \delta$. Consider a sequence of minimal geodesic rays $\mu_n$ connecting $p$ to the -$x_n$. Since the $g$-length of $\mu_n$ is at least $n\delta$, by +$x_n$. Since the $g$-length of $\mu_n$ is at least $n\delta$, by passing to a further subsequence, we can arrange that the $\mu_n$ converge to a minimal geodesic ray $\gamma$ emanating from $p$. By passing to a further subsequence if necessary, we arrange that $d_{g_n}(x_n,p)$ converges to $r>0$. Now consider the points $\tilde x_n$ on $\gamma$ at $g$-distance $\sqrt{\lambda_n}r$ from $p$. -Clearly, from the second item of Theorem~ and the +Clearly, from the second item of \entryref{04518fa1ac2f} and the fact that the angle at $p$ between the $\mu_n$ and $\mu$ tends to zero as $n\rightarrow\infty$ we have $d_{g_n}(x_n,\tilde x_n)\rightarrow 0$ as $n\rightarrow\infty$. Hence, it suffices to @@ -859,19 +968,20 @@ T}(M,p),c))$. ## Blow-up limits -Here we introduce a type of geometric limit. These were originally -introduced and studied by Hamilton in , where, -among other things, he showed that $3$-dimensional blow-up limits +Here we introduce a type of geometric limit. These were originally +introduced and studied by Hamilton in [Hamiltonsurvey], where, +among other things, he showed that $3$-dimensional blow-up limits have non-negative sectional curvature. We shall use repeatedly blow-up limits and the positive curvature result in the arguments in the later sections. + **Definition.** Let $({\mathcal M}_k,G_k,x_k)$ be a sequence of based -generalized Ricci flows. We suppose that $**t**(x_k)=0$ for all +generalized Ricci flows. We suppose that $\mathbf{t}(x_k)=0$ for all $n$. We set $Q_k$ equal to $R(x_k)$. We denote by $(Q_k{\mathcal M}_k,Q_kG_k,x_k)$ the family of generalized flows that have been rescaled so that $R_{Q_kG_k}(x_k)=1$. - Suppose that $lim_{k\rightarrow\infty}Q_k=\infty$ and that after passing to a + Suppose that $\mathit{lim}_{k\rightarrow\infty}Q_k=\infty$ and that after passing to a subsequence there is a geometric limit of the sequence $(Q_k{\mathcal M}_k,Q_kG_k,x_k)$ which is a Ricci flow defined for $-T The significance of the condition that the generalized Ricci flows have curvature pinched toward positive is that, as Hamilton -originally established in , the latter +originally established in [Hamiltonsurvey], the latter condition implies that any blow-up limit has non-negative curvature. -**Theorem.** -Let $({\mathcal M}_k,G_k,x_k)$ be a sequence of generalized $3$-dimensional + +**Theorem.** Let $({\mathcal M}_k,G_k,x_k)$ be a sequence of generalized $3$-dimensional Ricci flows, each of which has time interval of definition contained in $[0,\infty)$ and each of which has curvature pinched toward positive. Suppose -that $Q_k=R(x_k)$ tends to infinity as $k$ tends to infinity. Let $t_k=**t**(x_k)$ and let $({\mathcal M}_k',G_k',x_k)$ be the result of shifting time by -$-t_k$ so that $**t**'(x_k)=0$. Then any blow-up limit of the sequence +that $Q_k=R(x_k)$ tends to infinity as $k$ tends to infinity. Let $t_k=\mathbf{t}(x_k)$ and let $({\mathcal M}_k',G_k',x_k)$ be the result of shifting time by +$-t_k$ so that $\mathbf{t}'(x_k)=0$. Then any blow-up limit of the sequence $({\mathcal M}_k,G_k',x_k)$ has non-negative Riemann curvature. Similarly, any blow-up limit of the zero time-slices of this sequence has non-negative curvature. + **Proof.** Let us consider the case of the geometric limit of the zero time-slice first. Let $( M_\infty,g_\infty(0),x_\infty)$ be a blow-up limit of the zero @@ -905,28 +1017,24 @@ eigenvalues of the Riemann curvature operator for $g_\infty$ at $y$. Let $\{y_k\}$ be a sequence in $Q_k{\mathcal M}_k'$ converging to $y$, in the sense that $y_k=\varphi_k(y)$ for all $k$ sufficiently large. Then -$$ - $$ \begin{aligned} -\lambda(y) & = & lim_{n\rightarrow\infty}Q_k^{-1}\lambda(y_k) \\ -\mu(y) & = & lim_{n\rightarrow\infty}Q_k^{-1}\mu(y_k) \\ -\nu(y) & = & lim_{n\rightarrow\infty}Q_k^{-1}\nu(y_k) +\lambda(y) & = & \mathit{lim}_{n\rightarrow\infty}Q_k^{-1}\lambda(y_k) \\ +\mu(y) & = & \mathit{lim}_{n\rightarrow\infty}Q_k^{-1}\mu(y_k) \\ +\nu(y) & = & \mathit{lim}_{n\rightarrow\infty}Q_k^{-1}\nu(y_k) \end{aligned} $$ -$$ - -Since by Equation~() we have $R(y_k)\ge -6$ for all $k$ and since +Since by Equation (4.6) we have $R(y_k)\ge -6$ for all $k$ and since by hypothesis $Q_k$ tends to infinity as $n$ does, it follows that $R(y)\ge 0$. -Thus if $\lambda(y)=0$, then $Rm(y)=0$ and the result is established at +Thus if $\lambda(y)=0$, then $\mathit{Rm}(y)=0$ and the result is established at $y$. Hence, we may assume that $\lambda(y)>0$, which means that $\lambda(y_k)$ tends to infinity as $k$ does. If $\nu(y_k)$ remains bounded below as $k$ tends to infinity, then $Q_k^{-1}\nu(y_k)$ converges to a limit which is $\ge 0$, and consequently $Q_k^{-1}\mu(y_k)\ge Q_k^{-1}\nu(y_k)$ has a non-negative limit. Thus, in this case the Riemann curvature of $g_\infty$ at $y$ is non-negative. On the other hand, if $\nu(y_k)$ goes to $-\infty$ as $k$ does, then according -to Equation~() the ratio of $X(y_k)/R(y_k)$ goes to zero. Since +to Equation (4.6) the ratio of $X(y_k)/R(y_k)$ goes to zero. Since $Q_k^{-1}R(y_k)$ converges to the finite limit $R(y)$, the product $Q_k^{-1}X(y_k)$ converges to zero as $k$ goes to infinity. This means that $\nu(y)=0$ and consequently that $\mu(y)\ge 0$. Thus, once again we have @@ -934,12 +1042,14 @@ non-negative curvature for $g_\infty$ at $y$. The argument in the case of a geometric limit flow is identical. + **Corollary.** Suppose that $(M_k,g_k(t))$ is a sequence of Ricci flows each of which has time interval of definition contained in $[0,\infty)$ with each $M_k$ being a -compact $3$-manifold. Suppose further that, for each $k$, we have $|Rm(p_k,0)|\le 1$ for all $p_k\in M_k$. Then any blow-up limit of this sequence +compact $3$-manifold. Suppose further that, for each $k$, we have $|\mathit{Rm}(p_k,0)|\le 1$ for all $p_k\in M_k$. Then any blow-up limit of this sequence of Ricci flows has non-negative curvature. + -**Proof.** According to Theorem~ the hypotheses imply that for every $k$ the +**Proof.** According to \entryref{92827a66b5dd} the hypotheses imply that for every $k$ the Ricci flow $(M_k,g_k(t))$ has curvature pinched toward positive. From this, the corollary follows immediately from the previous theorem. @@ -949,26 +1059,27 @@ In our later arguments we shall need a splitting result at infinity in the non-negative curvature case. Assuming that a geometric limit exists, the splitting result is quite elementary. For this reason we present it here, though it will not be used until -Chapter~. +Chapter 9. The main result of this section gives a condition under which a geometric limit -automatically splits off a line; see Fig.~. +automatically splits off a line; see Fig. 5.2. -**Theorem.** -Let $(M,g)$ be a complete, connected manifold of non-negative + +**Theorem.** Let $(M,g)$ be a complete, connected manifold of non-negative sectional curvature. Let $\{x_n\}$ be a sequence of points going off to infinity, and suppose that we can find scaling factors $\lambda_n>0$ such that the based Riemannian manifolds $(M,\lambda_ng,x_n)$ have a geometric limit $(M_\infty,g_\infty,x_\infty)$. Suppose that there is a point $p\in M$ such that $\lambda_nd^2(p,x_n)\rightarrow \infty$ as -$n\rightarrow\infty$. Then, after passing to a subsequence, +$n\rightarrow\infty$. Then, after passing to a subsequence, minimizing geodesic arcs $\gamma_n$ from $x_n$ to $p$ converge to a minimizing geodesic ray in $M_\infty$. This minimizing geodesic ray is part of a minimizing geodesic line $\ell$ in $M_\infty$. In particular, there is a Riemannian product decomposition $M_\infty=N\times \Ar$ with the property that $\ell$ is $\{x\}\times \Ar$ for some $x\in N$. + **Proof.** Let $d_n$ be the distance from $p$ to $x_n$. Consider minimizing geodesic arcs $\gamma_n$ from $p$ to $x_n$. By passing to a @@ -980,35 +1091,45 @@ that $d_{m(n))}\ge d_n(1+1/\delta)$. Let $\mu_n$ be a minimizing geodesic from $x_n$ to $x_{m(n)}$. Now applying the Toponogov comparison -(first part of Theorem~) and the usual law of +(first part of \entryref{04518fa1ac2f}) and the usual law of cosines in Euclidean space, we see that the distance $d$ from $x_n$ to $x_{m(n)}$ satisfies -$$d_{m(n)}-d_n\le d\le \sqrt{d_n^2+d_{m(n)}^2-2d_nd_{m(n)}cos(\delta)}.$$ Let $\theta_n=\angle_{x_n'}$ of the Euclidean triangle + +$$ +d_{m(n)}-d_n\le d\le \sqrt{d_n^2+d_{m(n)}^2-2d_nd_{m(n)}\mathit{cos}(\delta)}. +$$ + + Let $\theta_n=\angle_{x_n'}$ of the Euclidean triangle $\triangle(x_n',p',x_{m(n)}')$ with $|s_{x_n'p'}|=d_n,|s_{x_n'x_{m(n)}'}|=d$ and $|s_{p'x_{m(n)}'}|=d_{m(n)}$. Then for any $\alpha0$, for all $n$ sufficiently large, $1+cos(\theta_n)<\delta$. +given any $\delta>0$, for all $n$ sufficiently large, $1+\mathit{cos}(\theta_n)<\delta$. We are assuming that the based Riemannian manifolds $\{(M,\lambda_ng,x_n)\}_{n=1}^\infty$ converge to a geometric limit @@ -1023,10 +1144,10 @@ arcs converges to a minimizing geodesic arc, which we denote $\tilde \gamma$ and $\tilde \mu$, respectively, in $M_\infty$ emanating from $x_\infty$. The above computation shows that the angle between these arcs is $\pi$ and hence that their union is a geodesic, say $\ell$. -The same computation shows that $\ell $ is minimizing. +The same computation shows that $\ell $ is minimizing. The existence of the minimizing geodesic line $\ell$ together with the fact that the sectional curvatures of the limit are $\ge 0$ -implies by Lemma~ that the limit manifold is a Riemannian +implies by \entryref{006e42ef5bb5} that the limit manifold is a Riemannian product $N\times \Ar$ in such a way that $\ell$ is of the form $\{x\}\times \Ar$ for some $x\in N$. diff --git a/projects/poincare-conjecture/.astrolabe/docs/06-newcompar.mdx b/projects/poincare-conjecture/.astrolabe/docs-src/07-a-comparison-geometry-approach-to-the-ricci-flow.mdx similarity index 59% rename from projects/poincare-conjecture/.astrolabe/docs/06-newcompar.mdx rename to projects/poincare-conjecture/.astrolabe/docs-src/07-a-comparison-geometry-approach-to-the-ricci-flow.mdx index 121372aa..a25678de 100644 --- a/projects/poincare-conjecture/.astrolabe/docs/06-newcompar.mdx +++ b/projects/poincare-conjecture/.astrolabe/docs-src/07-a-comparison-geometry-approach-to-the-ricci-flow.mdx @@ -1,19 +1,19 @@ -# A comparison geometry approach to the Ricci -flow + + +# Chapter 6 -- A comparison geometry approach to the Ricci flow In this section we discuss Perelman's notions, introduced in -, of the ${\mathcal L}$-length$-length} in the context of generalized Ricci flows. This is a +[Perelman1], of the ${\mathcal L}$-length in the context of generalized Ricci flows. This is a functional defined on paths in space-time parameterized by backward time, denoted $\tau$. The ${\mathcal L}$-length is the analogue in this context of the energy for paths in a Riemannian manifold. We derive the associated Euler-Lagrange equation for the ${\mathcal L}$-length; the critical paths are then ${\mathcal -L}$-geodesics$-geodesic}. Using ${\mathcal +L}$-geodesics. Using ${\mathcal L}$-geodesics we define the ${\mathcal L}$-exponential -mapping$-exponential map}. We derive the -${\mathcal L}$-Jacobi equation$-Jacobi equation} -and relate ${\mathcal L}$-Jacobi fields$-Jacobi -field} to the differential of the ${\mathcal L}$-exponential +mapping. We derive the +${\mathcal L}$-Jacobi equation +and relate ${\mathcal L}$-Jacobi fields to the differential of the ${\mathcal L}$-exponential mapping. There is the analogue of the interior of the cut locus. It is the open subset, depending on the parameter $\tau$, of the tangent space of initial vectors for ${\mathcal L}$-geodesics which @@ -23,8 +23,8 @@ difference between this situation and that of geodesics in a Riemannian manifold is that there is such an open set in the tangent space for each positive $\tau$. The analogue of the fact that, for ordinary geodesics, the interior of the cut locus in the tangent -space is star-shaped from the origin is that the open set of `good' -initial conditions at $\tau$ is contained the open subset of `good' +space is star-shaped from the origin is that the open set of 'good' +initial conditions at $\tau$ is contained the open subset of 'good' initial conditions at time $\tau'$ for any $\tau'<\tau$. All of these results are local and are established in the context of generalized Ricci flows. In the next section we consider the case of @@ -32,19 +32,18 @@ ordinary Ricci flows, where we are able to extend our results over the entire manifold. There are two applications of this theory in our study. In -Section~ we use the theory of ${\mathcal L}$-geodesics and the +Section 8 we use the theory of ${\mathcal L}$-geodesics and the associated notion of reduced volume to establish non-collapsing results. These are crucial when we wish to take blow-up limits in studying singularities in Ricci flows and Ricci flows with surgery. The second application will be in -Section~ to $\kappa$-solutions (ancient, $\kappa$-non-collapsed +Section 9 to $\kappa$-solutions (ancient, $\kappa$-non-collapsed solutions of bounded non-negative curvature). Here the second-order inequalities on the length function that we establish in this section are used to prove the existence of an asymptotic soliton for any $\kappa$-solution. This asymptotic soliton is important for giving qualitative results on $\kappa$-solutions. -## ${\mathcal L -$-length and ${\mathcal L}$-geodesics} +## ${\mathcal L}$-length and ${\mathcal L}$-geodesics The running assumption throughout this section is that we have an $n$-dimensional generalized Ricci flow $({\mathcal M},G)$. In @@ -56,28 +55,35 @@ $\chi$ and the horizontal tangent bundle, denoted ${\mathcal HT}{\mathcal M}$. We also fix a time $T$ in the time interval of definition of the flow distinct from the initial time. + **Definition.** Let $0\le \tau_1<\tau_2$ be given and let $\gamma\colon [\tau_1,\tau_2]\to {\mathcal M}$ be a continuous map. We say that $\gamma$ is *parameterized by backward time* provided that $\gamma(\tau)\in M_{T-\tau}$ for all $\tau\in [\tau_1,\tau_2]$ + Throughout this section the paths $\gamma$ that we consider shall be parameterized by backward time. We begin with the definition of ${\mathcal L}$-length of such a path. -**Definition.** -Let $\gamma\colon [\tau_{1},\tau_{2}]\rightarrow {\mathcal M},\ 0 -\le \tau_{1} < \tau_{2}$, be a $C^{1}$-path parameterized by + +**Definition.** Let $\gamma\colon [\tau_{1},\tau_{2}]\rightarrow {\mathcal M},\ 0 +\le \tau_{1} < \tau_{2}$, be a $C^{1}$-path parameterized by backward time. We define $X_\gamma(\tau)$ to be the horizontal projection of the tangent vector $d\gamma(\tau)/d\tau$, so that $d\gamma/d\tau=-\chi+X_\gamma(\tau)$ with $X_\gamma(\tau)\in -{\mathcal HT}{\mathcal M}$. We define the $\mathcal{L}$-*length*$-length|ii} of $\gamma$ to be: -$$\mathcal{L}(\gamma) = \int_{\tau_{1}}^{\tau_{2}} +{\mathcal HT}{\mathcal M}$. We define the $\mathcal{L}$-*length* of $\gamma$ to be: + +$$ +\mathcal{L}(\gamma) = \int_{\tau_{1}}^{\tau_{2}} \sqrt{\tau}\left(R(\gamma(\tau)) + -\abs{X_\gamma(\tau)}^{2}\right)d\tau,$|ii}$$ +\abs{X_\gamma(\tau)}^{2}\right)d\tau, +$$ + where the norm of $X_\gamma(\tau)$ is measured using the metric $G_{T-\tau}$ on ${\mathcal HT}{\mathcal M}$. When $\gamma$ is clear -from the context, we write $X$ for $X_\gamma$; see Fig.~ from the Introduction. +from the context, we write $X$ for $X_\gamma$; see Fig. 0.2 from the Introduction. + With a view toward better understanding the properties of the paths that are critical points of this functional, the so-called @@ -88,15 +94,14 @@ derivative of $\gamma$ with respect to the variable $s$. One sees immediately by the chain rule that $$ - -A(s^2) = 2s X(s^2)\ \ or \ \ +A(s^2) = 2s X(s^2)\ \ \mathit{or} \ \ A(\tau)=2\sqrt{\tau}X(\tau). $$ + With respect to the variable $s$, the ${\mathcal L}$-functional is $$ - \mathcal{L}(\gamma)= \int_{\sqrt{\tau_1}}^{\sqrt{\tau_2}}\left(\frac{1}{2}|A(s)|^{2} + 2R(\gamma(s))s^{2}\right)ds. @@ -104,54 +109,65 @@ $$ Let's consider the simplest example. -\begin{exam} -Suppose that our generalized Ricci flow is a constant family of + +**Example.** Suppose that our generalized Ricci flow is a constant family of Euclidean metrics on $\Ar^n\times [0,T]$. That is to say, $g(t) = g_{0}$ is the usual Euclidean metric. Then we have $R(\gamma(\tau))\equiv 0$. Using the change of variables $s=\sqrt{\tau}$, we have -$$\mathcal{L}(\gamma) = \frac{1}{2}\int_{\sqrt{\tau_1}}^{\sqrt{\tau_2}} \abs{A(s)}^{2}ds, -$$ which is the standard energy functional in Riemannian geometry + +$$ +\mathcal{L}(\gamma) = \frac{1}{2}\int_{\sqrt{\tau_1}}^{\sqrt{\tau_2}} \abs{A(s)}^{2}ds, +$$ + + which is the standard energy functional in Riemannian geometry for the path $\gamma(s)$. The minimizers for this functional are the maps $s\mapsto (\alpha(s),T-s^2)$ where $\alpha(s)$ is a straight line in $\Ar^n$ parameterized at constant speed. Written in the $\tau$ variables the minimizers are -$$\gamma(\tau)=(x+\sqrt{\tau}v,T-\tau),$$ + +$$ +\gamma(\tau)=(x+\sqrt{\tau}v,T-\tau), +$$ + straight lines parameterized at speed varying linearly with $\sqrt{\tau}$. + -\end{exam} +### ${\mathcal L}$-geodesics -### ${\mathcal L -$-geodesics} - -**Lemma.** The -Euler-Lagrange equation$-length!Euler Lagrange -equation} for critical paths for the $\mathcal{L}$-length is + +**Lemma.** The +Euler-Lagrange equation for critical paths for the $\mathcal{L}$-length is $$ - - \nabla_{X}X - \frac{1}{2}\nabla R -+ \frac{1}{2\tau}X + 2Ric(X,\cdot)^* = 0. +\nabla_{X}X - \frac{1}{2}\nabla R ++ \frac{1}{2\tau}X + 2\mathit{Ric}(X,\cdot)^* = 0. $$ + -**Remark.** $Ric(X,\cdot)$ is a horizontal one-form along $\gamma$ and its -dual $Ric(X,\cdot)^*$ is a horizontal tangent vector field + +**Remark.** $\mathit{Ric}(X,\cdot)$ is a horizontal one-form along $\gamma$ and its +dual $\mathit{Ric}(X,\cdot)^*$ is a horizontal tangent vector field along $\gamma$. + **Proof.** First, let us suppose that the generalized Ricci flow is an ordinary Ricci flow $(M,g(t))$. Let $\gamma_u(\tau)=\gamma(\tau,u)$ be a family of curves parameterized by backward time. Let -$$\widetilde Y(\tau,u)=\frac{\partial\gamma}{\partial u}.$$ Then $\widetilde X(\tau,u)=X_{\gamma_u}(\tau,u)$ + +$$ +\widetilde Y(\tau,u)=\frac{\partial\gamma}{\partial u}. +$$ + + Then $\widetilde X(\tau,u)=X_{\gamma_u}(\tau,u)$ and $\widetilde Y(\tau,u)$ -are the coordinate vector fields along the surface obtained by taking the +are the coordinate vector fields along the surface obtained by taking the projection of $\gamma(\tau,u)$ into $M$. Thus, $[\widetilde X,\widetilde Y] =0$. We denote by $X$ and $Y$ the restrictions of $\widetilde X$ and $\widetilde Y$, respectively to $\gamma_0$. We have -$$ - $$ \begin{aligned} \frac{d}{du}\mathcal{L}(\gamma_{u})\bigl|_{u=0}\bigr. & = @@ -163,29 +179,22 @@ X(\tau,u)}^{2})d\tau\right)\Bigl|_{u=0}\Bigr. \end{aligned} $$ -$$ - -On the other hand, since $\partial g/\partial \tau =2Ric$ and +On the other hand, since $\partial g/\partial \tau =2\mathit{Ric}$ and since $[\widetilde X,\widetilde Y]=0$, we have -$$ - $$ \begin{aligned} 2\frac{d}{d\tau}(\sqrt{\tau}\langle Y,X\rangle_{g(T-\tau)} ) & = & \frac{1}{\sqrt{\tau}}\langle Y,X\rangle + 2\sqrt{\tau}\langle\nabla_XY,X\rangle -+2\sqrt{\tau}\langle Y,\nabla_XX\rangle \\ & & +4\sqrt{\tau}Ric(Y,X) \\ ++2\sqrt{\tau}\langle Y,\nabla_XX\rangle \\ & & +4\sqrt{\tau}\mathit{Ric}(Y,X) \\ & = & \frac{1}{\sqrt{\tau}}\langle Y,X\rangle + 2\sqrt{\tau}\langle(\nabla_Y\widetilde X)|_{u=0},X\rangle -+2\sqrt{\tau}\langle Y,\nabla_XX\rangle \\ & & +4\sqrt{\tau}Ric(Y,X) ++2\sqrt{\tau}\langle Y,\nabla_XX\rangle \\ & & +4\sqrt{\tau}\mathit{Ric}(Y,X) \end{aligned} $$ -$$ - Using this we obtain -$$ $$ \begin{aligned} @@ -195,15 +204,13 @@ $$ Y,X\rangle\right]-\frac{1}{\sqrt{\tau}}\langle Y,X\rangle\Bigr. \\ & & +\sqrt{\tau}\bigl(\langle\nabla R,Y\rangle - - 2\langle Y,\nabla_{X}X\rangle - 4Ric(X,Y)\bigr)\Bigr)d\tau \nonumber \\ + - 2\langle Y,\nabla_{X}X\rangle - 4\mathit{Ric}(X,Y)\bigr)\Bigr)d\tau \nonumber \\ & = & 2\sqrt{\tau}\langle Y,X\rangle |_{\tau_{1}}^{\tau_{2}} \nonumber \\ & & + \int_{\tau_{1}}^{\tau_{2}} \sqrt{\tau}\langle Y,\bigl(\nabla R -- \frac{1}{\tau}X - 2\nabla_{X}X - 4Ric(X,\cdot)^*\bigr)\rangle d\tau. +- \frac{1}{\tau}X - 2\nabla_{X}X - 4\mathit{Ric}(X,\cdot)^*\bigr)\rangle d\tau. \end{aligned} $$ -$$ - Now we drop the assumption that the generalized Ricci flow is an ordinary Ricci flow. Still we can partition the interval $[\tau_1,\tau_2]$ into finitely many sub-intervals with the property @@ -211,68 +218,79 @@ that the restriction of $\gamma_0$ to each of the sub-intervals is contained in a patch of space-time on which the generalized Ricci flow is isomorphic to an ordinary Ricci flow. The above argument then applies to each of the sub-intervals. Adding up -Equation~() over these sub-intervals shows that the +Equation (6.4) over these sub-intervals shows that the same equation for the first variation of length for the entire family $\gamma_u$ holds. We consider a variation $\gamma(\tau,u)$ with fixed endpoints, so -that $Y(\tau_{1})=Y(\tau_{2})= 0.$ Thus, the condition that +that $Y(\tau_{1})=Y(\tau_{2})= 0.$ Thus, the condition that $\gamma$ be a critical path for the ${\mathcal L}$-length is that the integral expression vanish for all variations $Y$ satisfying -$Y(\tau_1)=Y(\tau_2)=0$. Equation~() holds for all +$Y(\tau_1)=Y(\tau_2)=0$. Equation (6.4) holds for all such $Y$ if and only if $\gamma$ satisfies -Equation~(). +Equation (6.3). + **Remark.** In the Euler-Lagrange equation, $\nabla R$ is the horizontal gradient, and the equation is an equation of horizontal vector fields along $\gamma$. + + **Definition.** A curve $\gamma$, parameterized by backward time, that is a critical -point of the ${\mathcal L}$-length is called an $\mathcal{L}$-*geodesic*$-geodesic|ii}. -Equation~() is the {\em ${\mathcal L}$-geodesic -equation}$-geodesic equation|ii}. +point of the ${\mathcal L}$-length is called an $\mathcal{L}$-*geodesic*. +Equation (6.3) is the *${\mathcal L}$-geodesic +equation*. + Written with respect to the variable $s=\sqrt{\tau}$ the $\mathcal{L}$-geodesic equation becomes $$ - -\nabla_{A(s)}A(s) -2s^{2}\nabla R + 4sRic(A(s),\cdot)^*=0. +\nabla_{A(s)}A(s) -2s^{2}\nabla R + 4s\mathit{Ric}(A(s),\cdot)^*=0. $$ Notice that in this form the ODE is regular even at $s=0$. + **Lemma.** Let $\gamma\colon [0,\tau_2]\to {\mathcal M}$ be an ${\mathcal L}$-geodesic. -Then $lim_{\tau\rightarrow 0}\sqrt{\tau}X_\gamma(\tau)$ exists. The +Then $\mathit{lim}_{\tau\rightarrow 0}\sqrt{\tau}X_\gamma(\tau)$ exists. The ${\mathcal L}$-geodesic $\gamma$ is completely determined by this limit (and by $\tau_2$). + -**Proof.** Since the ODE in Equation~() is non-singular even at +**Proof.** Since the ODE in Equation (6.5) is non-singular even at zero, it follows that $A(s)$ is a smooth function of $s$ in a neighborhood of $s=0$. , The lemma follows easily by the change of variables formula, $A(\tau)=2\sqrt{\tau}X_\gamma(\tau)$. -**Definition.** An ${\mathcal L}$-geodesic is said to be *minimizing*$-geodesic!minimizing} if there is no + +**Definition.** An ${\mathcal L}$-geodesic is said to be *minimizing* if there is no curve parameterized by backward time with the same endpoints and with smaller ${\mathcal L}$-length. + -### The ${\mathcal L -$-Jacobi equation} +### The ${\mathcal L}$-Jacobi equation Consider a family $\gamma(\tau,u)$ of ${\mathcal L}$-geodesics parameterized by $u$ and defined on $[\tau_1,\tau_2]$ with $0\le \tau_1<\tau_2$. Let $Y(\tau)$ be the horizontal vector field along $\gamma$ defined by -$$Y(\tau)=\frac{\partial}{\partial u}\gamma(\tau,u)|_{u=0}.$$ -**Lemma.** $Y(\tau)$ satisfies the {\em ${\mathcal L}$-Jacobi -equation}$-Jacobi equation|ii}: +$$ +Y(\tau)=\frac{\partial}{\partial u}\gamma(\tau,u)|_{u=0}. +$$ + + +**Lemma.** $Y(\tau)$ satisfies the *${\mathcal L}$-Jacobi +equation*: $$ \nabla_X\nabla_XY+{\mathcal R}(Y,X)X-\frac{1}{2}\nabla_Y(\nabla -R)+\frac{1}{2\tau}\nabla_XY+2(\nabla_YRic)(X,\cdot)^*+2Ric(\nabla_XY,\cdot)^*=0. +R)+\frac{1}{2\tau}\nabla_XY+2(\nabla_Y\mathit{Ric})(X,\cdot)^*+2\mathit{Ric}(\nabla_XY,\cdot)^*=0. $$ + This is a second-order linear equation for $Y$. Supposing that $\tau_1>0$, there is a unique horizontal vector field $Y$ along $\gamma$ solving this @@ -280,81 +298,117 @@ equation vanishing at $\tau_1$ with a given first-order derivative along $\gamma$ at $\tau_1$. Similarly, there is a unique solution $Y$ to this equation vanishing at $\tau_2$ and with a given first-order derivative at $\tau_2$. + **Proof.** Given a family $\gamma(\tau,u)$ of ${\mathcal L}$-geodesics, then -from Lemma~ we have -$$ \nabla_{\widetilde X}\widetilde X = \frac{1}{2}\nabla R(\gamma) --\frac{1}{2\tau}\widetilde X - 2Ric(\widetilde X,\cdot)^*.$$ +from \entryref{75695d1b1b54} we have + +$$ +\nabla_{\widetilde X}\widetilde X = \frac{1}{2}\nabla R(\gamma) +-\frac{1}{2\tau}\widetilde X - 2\mathit{Ric}(\widetilde X,\cdot)^*. +$$ + Differentiating this equation in the $u$-direction along the curve $u=0$ yields -$$\nabla_Y + +$$ +\nabla_Y \nabla_{\widetilde X}\widetilde X|_{u=0}=\frac{1}{2}\nabla_Y(\nabla -R)-\frac{1}{2\tau}\nabla_Y(\widetilde X)|_{u=0} -2\nabla_Y(Ric(\widetilde X,\cdot))^*|_{u=0}.$$ Of course, we have -$$\nabla_Y(Ric(\widetilde X,\cdot)^*)|_{u=0}=(\nabla_YRic)(X,\cdot)^* -+Ric(\nabla_Y\widetilde X|_{u=0},\cdot)^*.$$ Plugging this in, +R)-\frac{1}{2\tau}\nabla_Y(\widetilde X)|_{u=0} -2\nabla_Y(\mathit{Ric}(\widetilde X,\cdot))^*|_{u=0}. +$$ + + Of course, we have + +$$ +\nabla_Y(\mathit{Ric}(\widetilde X,\cdot)^*)|_{u=0}=(\nabla_Y\mathit{Ric})(X,\cdot)^* ++\mathit{Ric}(\nabla_Y\widetilde X|_{u=0},\cdot)^*. +$$ + + Plugging this in, interchanging the orders of differentiation on the left-hand side, using $\nabla_{\widetilde Y}\widetilde X=\nabla_{\widetilde X}\widetilde Y$, and restricting to $u=0$ yields the equation given in the statement of the lemma. -This equation is a regular, second-order linear equation for all $\tau>0$, and +This equation is a regular, second-order linear equation for all $\tau>0$, and hence is determined by specifying the value and first derivative at any $\tau>0$. -Equation~() is obtained by applying $\nabla_Y$ to -Equation~() and exchanging orders of differentiation. -The result Equation~() is a second-order differential +Equation (6.6) is obtained by applying $\nabla_Y$ to +Equation (6.3) and exchanging orders of differentiation. +The result Equation (6.6) is a second-order differential equation for $Y$ that makes no reference to an extension of $\gamma(\tau)$ to an entire family of curves. + **Definition.** A field $Y(\tau)$ along an ${\mathcal L}$-geodesic is -called an {\em ${\mathcal L}$-Jacobi field}$-Jacobi field|ii} if it satisfies the ${\mathcal L}$-Jacobi -equation, Equation~(), and if it vanishes at $\tau_1$. +called an *${\mathcal L}$-Jacobi field* if it satisfies the ${\mathcal L}$-Jacobi +equation, Equation (6.6), and if it vanishes at $\tau_1$. For any horizontal vector field $Y$ along $\gamma$ we denote by -$Jac(Y)$ the expression on the left-hand -side of Equation~(). +$\mathit{Jac}(Y)$ the expression on the left-hand +side of Equation (6.6). + In fact, there is a similar result even for $\tau_1=0$. + **Lemma.** Let $\gamma$ be an ${\mathcal L}$-geodesic defined on $[0,\tau_2]$ and let $Y(\tau)$ be an ${\mathcal L}$-Jacobi field along $\gamma$. Then -$$lim_{\tau\rightarrow 0}\sqrt{\tau}\nabla_XY$$ + +$$ +\mathit{lim}_{\tau\rightarrow 0}\sqrt{\tau}\nabla_XY +$$ + exists. Furthermore, $Y(\tau)$ is completely determined by this limit. + **Proof.** We use the variable $s=\sqrt{\tau}$, and let $A(s)$ be the horizontal component of $d\gamma/ds$. Then differentiating the ${\mathcal L}$-geodesic equation written with respect to this variable we see -$$\nabla_A\nabla_AY=-{\mathcal R}(Y,A)A+2s^2\nabla_Y(\nabla R)-4s(\nabla_YRic)(A,\cdot) --4sRic(\nabla_AY,\cdot).$$ Hence, for each tangent vector $Z$, + +$$ +\nabla_A\nabla_AY=-{\mathcal R}(Y,A)A+2s^2\nabla_Y(\nabla R)-4s(\nabla_Y\mathit{Ric})(A,\cdot) +-4s\mathit{Ric}(\nabla_AY,\cdot). +$$ + + Hence, for each tangent vector $Z$, there is a unique solution to this equation with the two initial conditions $Y(0)=0$ and $\nabla_AY(0)=Z$. -On the other hand, from Equation~() we have +On the other hand, from Equation (6.1) we have $\nabla_X(Y)=\frac{1}{2\sqrt{\tau}}\nabla_A(Y)$, so that -$$\sqrt{\tau}\nabla_X(Y)=\frac{1}{2}\nabla_A(Y).$$ -### Second order variation of ${\mathcal -L -$} We shall need the relationship of the ${\mathcal L}$-Jacobi +$$ +\sqrt{\tau}\nabla_X(Y)=\frac{1}{2}\nabla_A(Y). +$$ + +### Second order variation of ${\mathcal L}$ + We shall need the relationship of the ${\mathcal L}$-Jacobi equation to the second-order variation of ${\mathcal L}$. This is given in the next proposition. -**Proposition.** -Suppose that $\gamma$ is a minimizing ${\mathcal L}$-geodesic. Then, + +**Proposition.** Suppose that $\gamma$ is a minimizing ${\mathcal L}$-geodesic. Then, for any vector field $Y$ along $\gamma$, vanishing at both endpoints, and any family $\gamma_u$ of curves parameterized by backward time with $\gamma_0=\gamma$ and with the $u$-derivative of the family at $u=0$ being the vector field $Y$ along $\gamma$, we have -$$\frac{d^2}{du^2}{\mathcal L}(\gamma_u)|_{u=0}=-\int_{\tau_1}^{\tau_2}2\sqrt{\tau} -\langle Jac(Y),Y\rangle d\tau.$$ This quantity vanishes if and + +$$ +\frac{d^2}{du^2}{\mathcal L}(\gamma_u)|_{u=0}=-\int_{\tau_1}^{\tau_2}2\sqrt{\tau} +\langle \mathit{Jac}(Y),Y\rangle d\tau. +$$ + + This quantity vanishes if and only if $Y$ is an ${\mathcal L}$-Jacobi field. + Let us begin the proof of this proposition with the essential computation. -**Lemma.** -Let $\gamma$ be an ${\mathcal L}$-geodesic defined on $ + +**Lemma.** Let $\gamma$ be an ${\mathcal L}$-geodesic defined on $ [\tau_1,\tau_2]$, and let $Y_1$ and $Y_2$ be horizontal vector fields along $\gamma$ vanishing at $\tau_1$. Suppose that $\gamma_{u_1,u_2}$ is any family of curves parameterized by backward @@ -366,7 +420,6 @@ horizontal projection of $\partial/\partial \tau$ under this same map, so that the restrictions of these three vector fields to the curve $\gamma_{0,0}=\gamma$ are $Y_1,Y_2$ and $X$ respectively. Then we have -$$ $$ \begin{aligned} @@ -375,15 +428,12 @@ u_1}\frac{\partial}{\partial u_2}{\mathcal L}(\gamma_{u_1,u_2})|_{u_1=u_2=0} & = & 2\sqrt{\tau_2} Y_1(\tau_2)\langle \widetilde Y_2(\tau_2,u_1,0),\widetilde X(\tau_2,u_1,0)\rangle|_{u_1=0}\\ -& & -\int_{\tau_1}^{\tau_2}2\sqrt{\tau}\langle Jac(Y_1),Y_2\rangle d\tau. +& & -\int_{\tau_1}^{\tau_2}2\sqrt{\tau}\langle \mathit{Jac}(Y_1),Y_2\rangle d\tau. \end{aligned} $$ + -$$ - -**Proof.** According to Equation~() we have - -$$ +**Proof.** According to Equation (6.4) we have $$ \begin{aligned} @@ -395,20 +445,17 @@ X(\tau,u_1,u_2),\widetilde Y_2(\tau,u_1,u_2)\rangle d\tau, \end{aligned} $$ -$$ where $EL(\widetilde X(\tau,u_1,u_2))$ is the Euler-Lagrange expression for -geodesics, i.e., the left-hand side of Equation~(). +geodesics, i.e., the left-hand side of Equation (6.3). Differentiating again yields: -$$ - $$ \begin{aligned} -\lefteqn{\frac{\partial }{\partial u_1}\frac{\partial }{\partial u_2}{\mathcal +\frac{\partial }{\partial u_1}\frac{\partial }{\partial u_2}{\mathcal L}(\gamma_{u_1,u_2}\bigl)|_{u_1=u_2=0}\big. = 2\sqrt{\tau_2}Y_1(\tau_2)\langle \widetilde -Y_2(\tau_2,u_1,0),\widetilde X(\tau_2,u_1,0)\rangle\bigl|_{u_1=0}\bigr. } \nonumber\\ +Y_2(\tau_2,u_1,0),\widetilde X(\tau_2,u_1,0)\rangle\bigl|_{u_1=0}\bigr. \nonumber\\ & & - \int_{\tau_1}^{\tau_2}2\sqrt{\tau}\left(\langle\nabla_{Y_1}EL(\widetilde X), Y_2\rangle +\langle EL(X),\nabla_{Y_1}\widetilde @@ -416,38 +463,43 @@ Y_2\rangle\right)(\tau,0,0)d\tau. \end{aligned} $$ -$$ - Since $\gamma_{0,0}=\gamma$ is a geodesic, the second term in the integrand vanishes, and since $[\widetilde X,\widetilde Y_1]=0$, we -have $\nabla_{Y_1}EL(\widetilde X(\tau,0,0))=Jac(Y_1)(\tau)$. +have $\nabla_{Y_1}EL(\widetilde X(\tau,0,0))=\mathit{Jac}(Y_1)(\tau)$. This proves the lemma. -**Remark.** -Let $\gamma(\tau,u)$ be a family of curves as above with + +**Remark.** Let $\gamma(\tau,u)$ be a family of curves as above with $\gamma(\tau,0),\ \tau_1\le \tau\le \bar \tau$, being an ${\mathcal -L}$-geodesic. It follows from Lemma~ and the remark +L}$-geodesic. It follows from \entryref{f48558c60ca2} and the remark after the introduction of the ${\mathcal L}$-Jacobi equation that the second-order variation of length at $u=0$ of this family is determined by the vector field $Y(\tau)=\partial \gamma/\partial u$ along $\gamma(\cdot,0)$ and by the second-order information about the curve $\gamma(\bar\tau,u)$ at $u=0$. + -**Corollary.** -Let $\gamma$ be an ${\mathcal L}$-geodesic and let $Y_1,Y_2$ be vector fields + +**Corollary.** Let $\gamma$ be an ${\mathcal L}$-geodesic and let $Y_1,Y_2$ be vector fields along $\gamma$ vanishing at $\tau_1$. Suppose $Y_1(\tau_2)=Y_2(\tau_2)=0$. Then the bilinear pairing -$$-\int_{\tau_1}^{\tau_2}2\sqrt{\tau}\langle -Jac(Y_1),Y_2\rangle d\tau$$ is a symmetric function of $Y_1$ + +$$ +-\int_{\tau_1}^{\tau_2}2\sqrt{\tau}\langle +\mathit{Jac}(Y_1),Y_2\rangle d\tau +$$ + + is a symmetric function of $Y_1$ and $Y_2$. + **Proof.** Given $Y_1$ and $Y_2$ along $\gamma$ we construct a two-parameter family of curves parameterized by backward time as follows. Let $\gamma(\tau,u_1)$ be the value at $u_1$ of the geodesic through $\gamma(\tau)$ with tangent vector $Y_1(\tau)$. This defines a family of curves parameterized by backward time, the family being -parameterized by $u_1$ sufficiently close to $0$. We extend +parameterized by $u_1$ sufficiently close to $0$. We extend $Y_1$ and $X$ to vector fields on this entire family by defining them to be $\partial/\partial u_1$ and the horizontal projection of $\partial/\partial \tau$, respectively. @@ -459,23 +511,33 @@ $Y_2(\tau,u_1)$ we define a two-parameter family of curves parameterized by backward time by setting $\gamma(\tau,u_1,u_2)$ equal to the value at $u_2$ of the geodesic through $\gamma(\tau,u_1)$ in the direction $Y_2(\tau,u_1)$. - We then extend $Y_1$, $Y_2$, and $X$ over this entire family by letting them be + We then extend $Y_1$, $Y_2$, and $X$ over this entire family by letting them be $\partial/\partial u_1$, $\partial/\partial u_2$, and the horizontal projection of $\partial/ \partial -\tau$, respectively. Applying Lemma~ and using the fact that +\tau$, respectively. Applying \entryref{f48558c60ca2} and using the fact that $Y_i(\bar\tau)=0$ we conclude that -$$\frac{\partial}{\partial u_1}\frac{\partial}{\partial u_2}{\mathcal L}(\gamma)|_{u_1=u_2=0}= -\int_{\tau_1}^{\tau_2}2\sqrt{\tau}\langle -Jac(Y_1),Y_2\rangle d\tau$$ and symmetrically that -$$\frac{\partial}{\partial u_2}\frac{\partial}{\partial u_1}{\mathcal L}(\gamma)|_{u_1=u_2=0}= -\int_{\tau_1}^{\tau_2}2\sqrt{\tau}\langle -Jac(Y_2),Y_1\rangle d\tau.$$ Since the second cross partials + +$$ +\frac{\partial}{\partial u_1}\frac{\partial}{\partial u_2}{\mathcal L}(\gamma)|_{u_1=u_2=0}= -\int_{\tau_1}^{\tau_2}2\sqrt{\tau}\langle +\mathit{Jac}(Y_1),Y_2\rangle d\tau +$$ + + and symmetrically that + +$$ +\frac{\partial}{\partial u_2}\frac{\partial}{\partial u_1}{\mathcal L}(\gamma)|_{u_1=u_2=0}= -\int_{\tau_1}^{\tau_2}2\sqrt{\tau}\langle +\mathit{Jac}(Y_2),Y_1\rangle d\tau. +$$ + + Since the second cross partials are equal, the corollary follows. -Now we are in a position to establish Proposition~. +Now we are in a position to establish \entryref{26b7f69e803e}. -**Proof.** (Of Proposition~) -From the equation in Lemma~, the equality of the second variation +**Proof.** (Of \entryref{26b7f69e803e}) +From the equation in \entryref{f48558c60ca2}, the equality of the second variation of ${\mathcal L}$-length at $u=0$ and the integral is immediate from the fact -that $Y(\tau_2)=0$. It follows immediately that, if $Y$ is an ${\mathcal +that $Y(\tau_2)=0$. It follows immediately that, if $Y$ is an ${\mathcal L}$-Jacobi field vanishing at $\tau_2$, then the second variation of the length vanishes at $u=0$. Conversely, suppose given a family $\gamma_u$ with $\gamma_0=\gamma$ with the property that the second variation of length @@ -485,25 +547,31 @@ integral also vanishes. Since $\gamma$ is a minimizing ${\mathcal L}$-geodesic, for any variation $W$, vanishing at the endpoints, the first variation of the length vanishes and the second variation of length is non-negative. That is to say, -$$-\int_{\tau_1}^{\tau_2}2\sqrt{\tau}\langle Jac(W),W\rangle d\tau\ge 0$$ + +$$ +-\int_{\tau_1}^{\tau_2}2\sqrt{\tau}\langle \mathit{Jac}(W),W\rangle d\tau\ge 0 +$$ + for all vector fields $W$ along $\gamma$ vanishing at the endpoints. Hence, the restriction to the space of vector fields along $\gamma$ vanishing at the endpoints of the bilinear form -$$B(Y_1,Y_2)=-\int_{\tau_1}^{\tau_2}2\sqrt{\tau}\langle -Jac(Y_1),(Y_2)d\tau,$$ which is symmetric by Corollary~, is + +$$ +B(Y_1,Y_2)=-\int_{\tau_1}^{\tau_2}2\sqrt{\tau}\langle +\mathit{Jac}(Y_1),(Y_2)d\tau, +$$ + + which is symmetric by \entryref{abd40a465b53}, is positive semi-definite. Since $B(Y,Y)=0$, it follows that $B(Y,\cdot)=0$; that -is to say, $Jac(Y)=0$. +is to say, $\mathit{Jac}(Y)=0$. -## The ${\mathcal L -$-exponential map and its first-order -properties}$-exponential -map|ii} +## The ${\mathcal L}$-exponential map and its first-order properties -We use ${\mathcal L}$-geodesics in order to define the ${\mathcal +We use ${\mathcal L}$-geodesics in order to define the ${\mathcal L}$-exponential map. -For Section~ we fix $\tau_1\ge 0$ and a point $x\in -{\mathcal M}$ with $**t**(x)=T-\tau_1$. We suppose that $T-\tau_1$ +For Section 6.2 we fix $\tau_1\ge 0$ and a point $x\in +{\mathcal M}$ with $\mathbf{t}(x)=T-\tau_1$. We suppose that $T-\tau_1$ is greater than the initial time of the generalized Ricci flow. Then, for every $Z \in T_{x}M_{T-\tau_1},$ there is a maximal $\mathcal{L}$-geodesic, denoted $\gamma_{Z}$, defined on some @@ -512,104 +580,127 @@ $\sqrt{\tau_1}X(\tau_1)=Z$. (In the case $\tau_1=0$, this equation is interpreted to mean $\lim_{\tau\rightarrow 0} \sqrt{\tau}X(\tau) = Z$.) -**Definition.** -We define *the domain of definition of* ${\mathcal L}exp_x$exp$|ii}, denoted ${\mathcal D}_x$, to be the + +**Definition.** We define *the domain of definition of* ${\mathcal L}\mathit{exp}_x$, denoted ${\mathcal D}_x$, to be the subset of $T_xM_{T-\tau_1}\times (\tau_1,\infty)$ consisting of all pairs $(Z,\tau)$ for which $\tau> \tau_1$ is in the maximal domain of definition of -$\gamma_Z$. Then we define $\mathcal{L}exp_{x}\colon {\mathcal D}_x\to -{\mathcal M}$ by setting $\mathcal{L}exp_{x}(Z,\tau) = \gamma_{Z}(\tau)$ -for all $(Z,\tau)\in {\mathcal D}_x$. (See Fig.~.) We +$\gamma_Z$. Then we define $\mathcal{L}\mathit{exp}_{x}\colon {\mathcal D}_x\to +{\mathcal M}$ by setting $\mathcal{L}\mathit{exp}_{x}(Z,\tau) = \gamma_{Z}(\tau)$ +for all $(Z,\tau)\in {\mathcal D}_x$. (See Fig. 6.1.) We define the map $\widetilde L\colon {\mathcal D}_x\to \Ar$ by $\widetilde L(Z,\tau)={\mathcal L}\left(\gamma_Z|_{[\tau_1,\tau]}\right)$. -Lastly, for any $\tau> \tau_1$ we denote by ${\mathcal L}exp_x^\tau$exp_x^\tau$|ii} the restriction of -${\mathcal L}exp_x$ to the slice -$${\mathcal -D}^\tau_x={\mathcal D}_x\cap \left(T_xM_{T-\tau_1}\times \{\tau\}\right),$$ -which is *the domain of definition of* ${\mathcal L}exp_x^\tau$. We -also denote by $\widetilde L^\tau$ the +Lastly, for any $\tau> \tau_1$ we denote by ${\mathcal L}\mathit{exp}_x^\tau$ the restriction of +${\mathcal L}\mathit{exp}_x$ to the slice + +$$ +{\mathcal +D}^\tau_x={\mathcal D}_x\cap \left(T_xM_{T-\tau_1}\times \{\tau\}\right), +$$ + +which is *the domain of definition of* ${\mathcal L}\mathit{exp}_x^\tau$. We +also denote by $\widetilde L^\tau$ the restriction of $\widetilde L$ to this slice. We will implicitly identify ${\mathcal D}_x^\tau$ with a subset of $T_xM_{T-\tau_1}$. + -**Lemma.** -${\mathcal D}_x$ is an open subset of $T_xM_{T-\tau_1}\times (\tau_1,\infty)$; + +**Lemma.** ${\mathcal D}_x$ is an open subset of $T_xM_{T-\tau_1}\times (\tau_1,\infty)$; its intersection with each line $\{Z\}\times (\tau_1,\infty)$ is a non-empty -interval whose closure contains $\tau_1$. Furthermore, ${\mathcal L}exp_x\colon {\mathcal D}_x\to {\mathcal M}$ is a smooth map, and $\widetilde +interval whose closure contains $\tau_1$. Furthermore, ${\mathcal L}\mathit{exp}_x\colon {\mathcal D}_x\to {\mathcal M}$ is a smooth map, and $\widetilde L$ is a smooth function. + -**Proof.** The tangent vector in space-time of the ${\mathcal L}$-geodesic +**Proof.** The tangent vector in space-time of the ${\mathcal L}$-geodesic $\gamma$ is the vector field $-\chi+X_\gamma(\tau)$ along $\gamma$, -where $X_\gamma(\tau)$ satisfies Equation~(). As +where $X_\gamma(\tau)$ satisfies Equation (6.3). As above, in the case $\tau_1=0$, it is convenient to replace the independent variable $\tau$ by $s=\sqrt{\tau}$, so that the ODE -becomes Equation~() which is regular at $0$. With this +becomes Equation (6.5) which is regular at $0$. With this change, the lemma then follows immediately by the usual results on existence, uniqueness and $C^\infty$-variation with parameters of ODE's. -### The differential of ${\mathcal L -exp$} -Now we compute the differential of ${\mathcal L}exp$. +### The differential of ${\mathcal L}\mathit{exp}$ -**Lemma.** -Let $Z\in {\mathcal D}_x^{\bar\tau}\subset T_xM_{T-\tau_1}$. The -differential of ${\mathcal L}exp^{\bar\tau}_x$exp$!differential of|ii} +Now we compute the differential of ${\mathcal L}\mathit{exp}$. + + +**Lemma.** Let $Z\in {\mathcal D}_x^{\bar\tau}\subset T_xM_{T-\tau_1}$. The +differential of ${\mathcal L}\mathit{exp}^{\bar\tau}_x$ at the point $Z$ is given as follows: For each $W\in T_x(M_{T-\tau_1})$ there is a unique ${\mathcal L}$-Jacobi -field$-Jacobi field} $Y_W(\tau)$ along +field $Y_W(\tau)$ along $\gamma_Z$ with the property that $\sqrt{\tau_1}Y_W(\tau_1)=0$ and $\sqrt{\tau_1}\nabla_X(Y_W)(\tau_1)=W$. We have -$$d_Z{\mathcal L}exp^{\bar\tau}_x(W)=Y_W(\bar\tau).$$ + +$$ +d_Z{\mathcal L}\mathit{exp}^{\bar\tau}_x(W)=Y_W(\bar\tau). +$$ + Again, in case $\tau_1=0$, both of the conditions on $Y_W$ are interpreted as the limits as $\tau\rightarrow 0$. + **Proof.** Let $Z(u)$ be a curve in ${\mathcal D}_x^{\bar\tau}$ with $Z(0)=Z$. Let $\gamma_u$ be the ${\mathcal L}$-geodesic starting at $x$ with $\sqrt{\tau_1}X_{\gamma_u}(\tau_1)=Z(u)$. Then, clearly, -$$d_Z{\mathcal L}exp^{\bar\tau}_x\left(\frac{dZ}{du}(0)\right) -=\frac{\partial}{\partial u}\left(\gamma_u(\bar\tau)\right)|_{u=0}.$$ On the + +$$ +d_Z{\mathcal L}\mathit{exp}^{\bar\tau}_x\left(\frac{dZ}{du}(0)\right) +=\frac{\partial}{\partial u}\left(\gamma_u(\bar\tau)\right)|_{u=0}. +$$ + + On the other hand, the vector field $Y(\tau)=\left(\partial\gamma_u(\tau)/\partial -u\right)|_{u=0} $ is an ${\mathcal L}$-Jacobi field along $\gamma_Z$. Thus, to +u\right)|_{u=0} $ is an ${\mathcal L}$-Jacobi field along $\gamma_Z$. Thus, to complete the proof in the case when $\tau_1>0$ we need only see that $\nabla_X\widetilde Y(\tau_1)=\nabla_Y\widetilde X(\tau_1)$. This is clear since, as we have already seen, $[\widetilde X,\widetilde Y]=0$. - When $\tau_1=0$, we complete the + When $\tau_1=0$, we complete the argument using the following claim. + **Claim.** If $\tau_1=0$, then -$$\frac{\partial}{\partial u}\left(lim_{\tau\rightarrow 0}\sqrt{\tau}X(\tau,u)\right)|_{u=0}= -lim_{\tau\rightarrow 0}\sqrt{\tau}\frac{d}{d\tau}Y(\tau).$$ -\begin{proof} This follows immediately by changing variables, +$$ +\frac{\partial}{\partial u}\left(\mathit{lim}_{\tau\rightarrow 0}\sqrt{\tau}X(\tau,u)\right)|_{u=0}= +\mathit{lim}_{\tau\rightarrow 0}\sqrt{\tau}\frac{d}{d\tau}Y(\tau). +$$ + + +**Proof.** This follows immediately by changing variables, replacing $\tau$ by $s=\sqrt{\tau}$. -This completes the proof of Lemma~. -\end{proof} +This completes the proof of \entryref{d2e61ac52277}. -### Positivity of the second variation at a minimizing -${\mathcal L -$-geodesic} +### Positivity of the second variation at a minimizing ${\mathcal L}$-geodesic If $\gamma$ is a minimizing ${\mathcal L}$-geodesic, then variations -of $\gamma$ fixing the endpoints give curves whose ${\mathcal +of $\gamma$ fixing the endpoints give curves whose ${\mathcal L}$-length is no less than that of $\gamma$. In fact, there is a second-order version of this inequality which we shall need later. -**Corollary.** -Let $Z\in T_xM_{T-\tau_1}$. Suppose that the associated ${\mathcal + +**Corollary.** Let $Z\in T_xM_{T-\tau_1}$. Suppose that the associated ${\mathcal L}$-geodesic $\gamma_Z$ minimizes ${\mathcal L}$-length between its endpoints, $x$ and $\gamma_Z(\bar\tau)$, and that $d_Z{\mathcal -L}exp^{\bar\tau}_x$ is an isomorphism. Then for any family +L}\mathit{exp}^{\bar\tau}_x$ is an isomorphism. Then for any family $\gamma_u$ of curves parameterized by backward time with $Y=(\partial \gamma/\partial u)|_{u=0}$ vanishing at both endpoints, we have -$$\frac{d^2}{du^2}{\mathcal L}(\gamma_u)|_{u=0}\ge 0,$$ + +$$ +\frac{d^2}{du^2}{\mathcal L}(\gamma_u)|_{u=0}\ge 0, +$$ + with equality if and only if $Y=0$. + -**Proof.** According to Proposition~ the second variation in the +**Proof.** According to \entryref{26b7f69e803e} the second variation in the $Y$-direction is non-negative and vanishes if and only if $Y$ is an ${\mathcal -L}$-Jacobi field. But since $d_Z{\mathcal L}exp_x^{\bar\tau}$ is a -diffeomorphism, by Lemma~ there are no non-zero ${\mathcal +L}$-Jacobi field. But since $d_Z{\mathcal L}\mathit{exp}_x^{\bar\tau}$ is a +diffeomorphism, by \entryref{d2e61ac52277} there are no non-zero ${\mathcal L}$-Jacobi fields vanishing at both endpoints of $\gamma_Z$. ### The gradient of $\widetilde L^\tau$ @@ -619,12 +710,16 @@ L^{\tau}$ is the map from ${\mathcal D}^{\tau}_x$ to $\Ar$ that assigns to each $Z$ the ${\mathcal L}$-length of $\gamma_Z|_{[\tau_1,\tau]}$. We compute its gradient. -**Lemma.** -Suppose that $Z\in {\mathcal D}^{\tau}_x$. Then for any $\widetilde + +**Lemma.** Suppose that $Z\in {\mathcal D}^{\tau}_x$. Then for any $\widetilde Y\in T_xM_{T-\tau_1}=T_Z({\mathcal D}^{\tau}_x)$ we have -$$ \langle \nabla\widetilde L^{\tau},\widetilde Y\rangle =2\sqrt{\tau}\langle -X(\tau),d_Z\left({\mathcal L}exp_x^{\tau}\right)(\widetilde -Y)\rangle.$$ + +$$ +\langle \nabla\widetilde L^{\tau},\widetilde Y\rangle =2\sqrt{\tau}\langle +X(\tau),d_Z\left({\mathcal L}\mathit{exp}_x^{\tau}\right)(\widetilde +Y)\rangle. +$$ + **Proof.** Since ${\mathcal D}^{\tau}_x$ is an open subset of $T_x(M_{T-\tau_1})$, it follows that for any $\widetilde Y\in T_x(M_{T-\tau_1})$ there is a @@ -633,38 +728,47 @@ one-parameter family $\gamma_u(\tau')=\gamma(\tau',u)$ of starting at $x$ with $\gamma(\cdot,0)=\gamma_Z$ and with $\frac{\partial}{\partial u}\left(\sqrt{\tau_1}X(\tau_1)\right)=\widetilde Y$. (Again, when $\tau_1=0$, this equation is interpreted to mean -$\frac{\partial}{\partial u}lim_{\tau'\rightarrow +$\frac{\partial}{\partial u}\mathit{lim}_{\tau'\rightarrow 0}(\sqrt{\tau'}X(\tau',u))=\widetilde Y$.) Let $Y(\tau')=\frac{\partial}{\partial u}(\gamma(\tau',u))|_{u=0}$ be the corresponding ${\mathcal L}$-Jacobi field along $\gamma_Z$. Since $\gamma(\tau_1,u)=x$ for all $u$, we have $Y(\tau_1)=0$. Since $\gamma(\cdot,u)$ is an ${\mathcal L}$-geodesic for all $u$, according to -Equation~(), and in the case $\tau_1=0$, using the fact that +Equation (6.4), and in the case $\tau_1=0$, using the fact that $\sqrt{\tau}X(\tau')$ approaches a finite limit as $\tau\rightarrow 0$, we have -$$\frac{d}{du}{\mathcal -L}(\gamma_u)|_{u=0}=2\sqrt{\tau}\langle X(\tau),Y(\tau)\rangle.$$ By -Lemma~ we have $Y(\tau)=d_Z{\mathcal L}exp_x^{\tau}(\tilde + +$$ +\frac{d}{du}{\mathcal +L}(\gamma_u)|_{u=0}=2\sqrt{\tau}\langle X(\tau),Y(\tau)\rangle. +$$ + + By +\entryref{d2e61ac52277} we have $Y(\tau)=d_Z{\mathcal L}\mathit{exp}_x^{\tau}(\tilde Y)$. Thus, -$$\langle\nabla\widetilde L^{\tau},\widetilde Y\rangle=\frac{d}{du}{\mathcal + +$$ +\langle\nabla\widetilde L^{\tau},\widetilde Y\rangle=\frac{d}{du}{\mathcal L}(\gamma_u)|_{u=0}=2\sqrt{\tau}\langle X(\tau),Y(\tau)\rangle=2\sqrt{\tau}\langle X(\tau),d_Z({\mathcal -L}exp^{\tau}_x)(\widetilde Y)\rangle.$$ +L}\mathit{exp}^{\tau}_x)(\widetilde Y)\rangle. +$$ ### Local diffeomorphism near the initial $\tau$ Now let us use the nature of the ${\mathcal L}$-Jacobi equation to -study ${\mathcal L}exp_x$ for $\tau>\tau_1$ but $\tau$ +study ${\mathcal L}\mathit{exp}_x$ for $\tau>\tau_1$ but $\tau$ sufficiently close to $\tau_1$. -**Lemma.** -For any $x$ in ${\mathcal M}$ with $**t**(x)=T-\tau_1$ and any $Z\in T_xM_{t-\tau_1}$, there is $\delta>0$ + +**Lemma.** For any $x$ in ${\mathcal M}$ with $\mathbf{t}(x)=T-\tau_1$ and any $Z\in T_xM_{t-\tau_1}$, there is $\delta>0$ such that for any $\tau$ with $\tau_1<\tau<\tau_1+\delta$ the map -${\mathcal L}exp_x^{\tau}$ is a local diffeomorphism from a +${\mathcal L}\mathit{exp}_x^{\tau}$ is a local diffeomorphism from a neighborhood of $Z$ in $T_xM_{T-\tau_1}$ to $M_{T-\tau}$. + **Proof.** Fix $x$ and $Z$ as in the statement of the lemma. To establish the result it -suffices to prove that there is $\delta>0$ such that $d_Z{\mathcal L}exp_x^{\tau}$ is an isomorphism for all $\tau_1<\tau<\tau_1+\delta$. By -Lemma~ it is enough to find a $\delta>0$ such that any ${\mathcal +suffices to prove that there is $\delta>0$ such that $d_Z{\mathcal L}\mathit{exp}_x^{\tau}$ is an isomorphism for all $\tau_1<\tau<\tau_1+\delta$. By +\entryref{d2e61ac52277} it is enough to find a $\delta>0$ such that any ${\mathcal L}$-Jacobi field $Y$ along $\gamma_Z$ with $\sqrt{\tau_1}\nabla_XY(\tau_1)\not= 0$ does not vanish on the interval $(\tau_1,\tau_1+\delta)$. @@ -675,47 +779,46 @@ L}$-Jacobi field $Y$ along Let us consider first the case when $\tau_1\not= 0$. Then for any such tangent vector $\nabla_XY(\tau_1)\not=0$. Since $Y(\tau_1)=0$, it follows that $Y(\tau)\not=0$ in some interval - $(\tau_1,\tau_1+\delta)$, where $\delta$ can depend on $Y$. Using + $(\tau_1,\tau_1+\delta)$, where $\delta$ can depend on $Y$. Using compactness of the unit sphere in the tangent space, we see that there is $\delta>0$ independent of $Y$ so that the above holds. In case when $\tau_1=0$, it is convenient to shift to the $s=\sqrt{\tau}$ parameterization. Then the geodesic equation and the ${\mathcal L}$-Jacobi equation are non-singular at the origin. Also, letting - $A=d\gamma_Z/ds$ we have $\nabla_AY=2lim_{\tau\rightarrow + $A=d\gamma_Z/ds$ we have $\nabla_AY=2\mathit{lim}_{\tau\rightarrow 0}\sqrt{\tau}\nabla_XY$. In these variables, the argument for $\tau_1=0$ is the same as the one above for $\tau_1>0$. -**Remark.** -When $\tau_1>0$ it is possible to consider the ${\mathcal L}exp^{\tau}_x$ defined for $0<\tau<\tau_1$. In this case, the curves + +**Remark.** When $\tau_1>0$ it is possible to consider the ${\mathcal L}\mathit{exp}^{\tau}_x$ defined for $0<\tau<\tau_1$. In this case, the curves are moving backward in $\tau$ and hence are moving forward with -respect to the time parameter $**t**$. Two comments are in order. +respect to the time parameter $\mathbf{t}$. Two comments are in order. First of all, for $\tau<\tau_1$, the gradient of ${\widetilde L}^{\tau}_x$ is $-2\sqrt{\tau}X(\tau)$. The reason for the sign reversal is that the length is given by the integral from $\tau$ to $\tau_1$ and hence its derivative with respect to $\tau$ is the negative of the integrand. The second thing to remark is that -Lemma~ is true for $\tau<\tau_1$ with $\tau$ +\entryref{4180719ab2c8} is true for $\tau<\tau_1$ with $\tau$ sufficiently close to $\tau_1$. + -## Minimizing ${\mathcal L -$-geodesics and the injectivity -domain} +## Minimizing ${\mathcal L}$-geodesics and the injectivity domain Now we discuss the analogue of the interior of the cut locus for the usual exponential map of a Riemannian manifold. For -Section~ we keep the assumption that $x\in -{\mathcal M}$ with $**t**(x)=T-\tau_1$ for some $\tau_1\ge 0$. +Section 6.3 we keep the assumption that $x\in +{\mathcal M}$ with $\mathbf{t}(x)=T-\tau_1$ for some $\tau_1\ge 0$. -**Definition.** - The *injectivity set* + +**Definition.** The *injectivity set* $\widetilde{\mathcal U}_x\subset {\mathcal D}_x\subset -\left(T_xM_{T-\tau_1}\times (\tau_1,\infty)\right)$_x$|ii} is the subset of all $(Z,\tau)\in {\mathcal D}_x$ with the following +\left(T_xM_{T-\tau_1}\times (\tau_1,\infty)\right)$ is the subset of all $(Z,\tau)\in {\mathcal D}_x$ with the following properties: -\begin{enumerate} -\item The map ${\mathcal L}exp_x^{\tau}$ is a local + +1. The map ${\mathcal L}\mathit{exp}_x^{\tau}$ is a local diffeomorphism near $Z$ from $T_x(M_{T-\tau_1})$ to $M_{T-\tau}$. -\item There is a neighborhood ${\mathcal Z}$ of $Z$ in ${\mathcal D}^{\tau}_x$ +1. There is a neighborhood ${\mathcal Z}$ of $Z$ in ${\mathcal D}^{\tau}_x$ such that for every $Z'\in {\mathcal Z}$ the ${\mathcal L}$-geodesic $\gamma_{Z'}|_{[\tau_1,\tau]}$ is the unique minimizing path parameterized by backward time for the ${\mathcal L}$-length. That @@ -723,14 +826,18 @@ is to say, the ${\mathcal L}$-length of $\gamma_{Z'}|_{[\tau_1,\tau]}$ is less than the ${\mathcal L}$-length of any other path parameterized by backward time between the same endpoints. -\end{enumerate} + For any $\tau>\tau_1$, we set $\widetilde {\mathcal -U}_x(\tau)\subset T_xM_{T-\tau_1}$_x(\tau)$|ii} equal to the slice of $\widetilde{\mathcal U}_x$ at +U}_x(\tau)\subset T_xM_{T-\tau_1}$ equal to the slice of $\widetilde{\mathcal U}_x$ at $\tau$, i.e., $\widetilde {\mathcal U}_x(\tau)$ is determined by the equation -$$\widetilde{\mathcal + +$$ +\widetilde{\mathcal U}_x(\tau)\times\{\tau\}=\widetilde{\mathcal -U}_x\cap\left(T_xM_{T-\tau_1}\times\{\tau\}\right).$$ +U}_x\cap\left(T_xM_{T-\tau_1}\times\{\tau\}\right). +$$ + It is clear from the definition that $\widetilde{\mathcal U}_x\subset {\mathcal D}_x$ is an open subset and hence @@ -739,73 +846,90 @@ $T_xM_{T-\tau_1}\times (\tau_1,\infty)$. Of course, this implies that $\widetilde {\mathcal U}_x(\tau)$ is an open subset of ${\mathcal D}_x^\tau$ for every $\tau>\tau_1$. -**Definition.** -We set ${\mathcal U}_x\subset {\mathcal M}$_x$|ii} equal to $ {\mathcal L}exp_x(\widetilde{\mathcal + +**Definition.** We set ${\mathcal U}_x\subset {\mathcal M}$ equal to $ {\mathcal L}\mathit{exp}_x(\widetilde{\mathcal U}_x)$. We call this subset of ${\mathcal M}$ the *injectivity domain (of $x$)*. For any $\tau>\tau_1$ we set ${\mathcal U}_x(\tau)={\mathcal U}_x\cap -M_{T-\tau}$_x(\tau)$|ii}. +M_{T-\tau}$. + By definition, for every point $q\in {\mathcal U}_x$ for any -$(Z,\tau)\in \widetilde{\mathcal U}_x$ with ${\mathcal L}exp_x(Z,\tau)=q$, the ${\mathcal L}$-geodesic -$\gamma_{Z}|_{[\tau_1,\tau]}$ is a minimizing +$(Z,\tau)\in \widetilde{\mathcal U}_x$ with ${\mathcal L}\mathit{exp}_x(Z,\tau)=q$, the ${\mathcal L}$-geodesic +$\gamma_{Z}|_{[\tau_1,\tau]}$ is a minimizing ${\mathcal L}$-geodesic to $q$. In particular, there is a minimizing ${\mathcal L}$-geodesic from $x$ to $q$. + **Definition.** The function $L_x\colon {\mathcal U}_x\to \Ar$ assigns to each $q$ in ${\mathcal U}_x$ the length of any minimizing ${\mathcal L}$-geodesic from $x$ to $q$. For any $\tau>\tau_1$, we denote by $L_x^\tau$ the restriction of $L_x$ to the $T-\tau$ time-slice of ${\mathcal U}_x$, i.e., the restriction of $L_x$ to ${\mathcal U}_x(\tau)$. + - This brings us to the +This brings us to the analogue of the fact that in Riemannian geometry the restriction to the interior of the cut locus of the exponential mapping is a diffeomorphism onto an open subset of the manifold. -**Proposition.** -The map -$${\mathcal L}exp_x\colon\widetilde {\mathcal -U}_x\to {\mathcal M}$$ is a diffeomorphism onto the open subset + +**Proposition.** The map + +$$ +{\mathcal L}\mathit{exp}_x\colon\widetilde {\mathcal +U}_x\to {\mathcal M} +$$ + + is a diffeomorphism onto the open subset ${\mathcal U}_x$ of ${\mathcal M}$. The function $L\colon {\mathcal U}_x\to \Ar$ that associates to each $q\in {\mathcal U}_x$ the length of the unique minimizing ${\mathcal L}$-geodesic from $x$ to $q$ is a smooth function and -$$L_x\circ {\mathcal L}exp_x|_{\widetilde{\mathcal U}_x} -=\widetilde L|_{\widetilde{\mathcal U}_x}.$$ -**Proof.** We consider the differential of ${\mathcal L}exp_x$ at any +$$ +L_x\circ {\mathcal L}\mathit{exp}_x|_{\widetilde{\mathcal U}_x} +=\widetilde L|_{\widetilde{\mathcal U}_x}. +$$ + + +**Proof.** We consider the differential of ${\mathcal L}\mathit{exp}_x$ at any $(Z,\tau)\in\widetilde {\mathcal U}_x$. By construction the restriction of this differential to $T_xM_{T-\tau_1}$ is a local isomorphism onto ${\mathcal H}T{\mathcal M}$ at the image point. On the other - hand, the differential of ${\mathcal L}exp_x$ in the $\tau$ - direction is $\gamma_Z'(\tau)$, whose `vertical' component is + hand, the differential of ${\mathcal L}\mathit{exp}_x$ in the $\tau$ + direction is $\gamma_Z'(\tau)$, whose 'vertical' component is $-\chi$. By the inverse function theorem this - shows that ${\mathcal L}exp_x$ is a local diffeomorphism at + shows that ${\mathcal L}\mathit{exp}_x$ is a local diffeomorphism at $(Z,\tau)$, and its image is an open subset of ${\mathcal M}$. The uniqueness in Condition 2, of the definition immediately implies -that the restriction of ${\mathcal L}exp_x$ to +that the restriction of ${\mathcal L}\mathit{exp}_x$ to $\widetilde{\mathcal U}_x$ is one-to-one, and hence that it is a global diffeomorphism onto its image ${\mathcal U}_x$. Since for every $(Z,\tau)\in\widetilde{\mathcal U}_x$ the ${\mathcal -L}$-geodesic $\gamma_Z|_{[\tau_1,\tau]}$ is ${\mathcal -L}$-minimizing, we see that $L_x\circ {\mathcal L}exp_x|_{\widetilde{\mathcal U}_x}=\widetilde L|_{\widetilde +L}$-geodesic $\gamma_Z|_{[\tau_1,\tau]}$ is ${\mathcal +L}$-minimizing, we see that $L_x\circ {\mathcal L}\mathit{exp}_x|_{\widetilde{\mathcal U}_x}=\widetilde L|_{\widetilde {\mathcal U}_x}$ and that $L_x\colon {\mathcal U}_x\to \Ar$ is a smooth function. -According to Lemma~ we have: +According to \entryref{abf9cbb10c68} we have: -**Corollary.** -At any $q\in {\mathcal U}_x(\tau)$ we have -$$\nabla L_x^{\tau}(q)=2\sqrt{\tau}X(\tau)$$ -where $X(\tau)$ is the horizontal component of $\gamma'(\tau)$, + +**Corollary.** At any $q\in {\mathcal U}_x(\tau)$ we have + +$$ +\nabla L_x^{\tau}(q)=2\sqrt{\tau}X(\tau) +$$ + +where $X(\tau)$ is the horizontal component of $\gamma'(\tau)$, where $\gamma$ is the unique minimizing ${\mathcal L}$-geodesic -connecting $x$ to $q$. (See Fig.~ in the +connecting $x$ to $q$. (See Fig. 0.2 in the Introduction.) + At the level of generality that we are working (arbitrary generalized Ricci flows) there is no result analogous to the fact in @@ -813,33 +937,32 @@ Riemannian geometry that the image under the exponential mapping of the interior of the cut locus is an open dense subset of the manifold. There is an analogue in the special case of Ricci flows on compact manifolds or on complete manifolds of bounded curvature. -These will be discussed in Section~. +These will be discussed in Section 7. -### Monotonicity of the $\widetilde{\mathcal U -_x(\tau)$ with respect to -$\tau$}_x(\tau)$!monotonicty} +### Monotonicity of the $\widetilde{\mathcal U}_x(\tau)$ with respect to $\tau$ Next, we have the analogue of the fact in Riemannian geometry that the cut locus is star-shaped. -**Proposition.** -Let $\bar\tau'>\bar\tau$. Then $\widetilde{\mathcal + +**Proposition.** Let $\bar\tau'>\bar\tau$. Then $\widetilde{\mathcal U}_x(\bar\tau')\subset \widetilde{\mathcal U}_x(\bar\tau)\subset T_xM_{T-\tau_1}$. + **Proof.** For $Z\in \widetilde{\mathcal U}_x(\bar\tau')$, we shall show that: (i) the ${\mathcal L}$-geodesic $\gamma_{Z'}|_{[\tau_1,\bar\tau]}$ is the unique minimizing ${\mathcal L}$-geodesic from $x$ to $\gamma_{Z}(\bar\tau)$, and (ii) the differential $d_Z{\mathcal -L}exp_x^{\bar\tau}$ is an isomorphism. Given these two +L}\mathit{exp}_x^{\bar\tau}$ is an isomorphism. Given these two conditions, it follows from the definition that $\widetilde{\mathcal U}_x(\bar\tau')$ is contained in $\widetilde{\mathcal U}_x(\bar\tau)$. - We show that the ${\mathcal L}$-geodesic -$\gamma_{Z}|_{[\tau_1,\bar\tau]}$ is the unique minimizing -${\mathcal L}$- geodesic to its endpoint. If there is an ${\mathcal + We show that the ${\mathcal L}$-geodesic +$\gamma_{Z}|_{[\tau_1,\bar\tau]}$ is the unique minimizing +${\mathcal L}$- geodesic to its endpoint. If there is an ${\mathcal L}$-geodesic $\gamma_1$, distinct from $\gamma_{Z}|_{[\tau_1,\bar\tau]}$, from $x$ to $\gamma_Z(\bar\tau)$ whose ${\mathcal L}$-length is at most that of @@ -849,24 +972,29 @@ backward time whose ${\mathcal L}$-length is at most that of $\gamma_{Z}$. Since this latter path is not smooth, its ${\mathcal L}$-length cannot be the minimum, which is a contradiction. - Now suppose that $d_Z{\mathcal L}exp_x^{\bar\tau}$ is not an isomorphism. + Now suppose that $d_Z{\mathcal L}\mathit{exp}_x^{\bar\tau}$ is not an isomorphism. The argument is similar to the one above, using a non-zero ${\mathcal L}$-Jacobi field vanishing at both endpoints rather than another geodesic. Let $\tau'_2$ be the first $\tau$ for which -$d_Z{\mathcal L}exp_x^\tau$ is not an isomorphism. -According to Lemma~, $\tau_1<\tau'_2\le \bar\tau$. Since -${\mathcal L}exp_x^{\tau'_2}$ is not a local diffeomorphism at -$(Z,\tau'_2)$, by Lemma~ there is a non-zero +$d_Z{\mathcal L}\mathit{exp}_x^\tau$ is not an isomorphism. +According to \entryref{4180719ab2c8}, $\tau_1<\tau'_2\le \bar\tau$. Since +${\mathcal L}\mathit{exp}_x^{\tau'_2}$ is not a local diffeomorphism at +$(Z,\tau'_2)$, by \entryref{d2e61ac52277} there is a non-zero ${\mathcal L}$-Jacobi field $Y$ along $\gamma_Z|_{[\tau_1,\tau'_2]}$ vanishing at both ends. Since $\gamma_Z|_{[\tau_1,\tau_2']}$ is -${\mathcal L}$-minimizing, according to Proposition~, +${\mathcal L}$-minimizing, according to \entryref{26b7f69e803e}, the second variation of the length of $\gamma_Z|_{\tau_1,\tau'_2]}$ in the $Y$-direction vanishes, in the sense that if $\gamma(u,\tau)$ is any one-parameter family of paths parameterized by backward time from $x$ to $\gamma_Z(\tau'_2)$ with $(\partial \gamma/\partial u)|_{u=0}=Y$ then -$$\frac{\partial^2{\mathcal L}(\gamma_u)}{\partial -u^2}\bigl|_{u=0}\bigr.=0.$$ Extend $Y$ to a horizontal vector field + +$$ +\frac{\partial^2{\mathcal L}(\gamma_u)}{\partial +u^2}\bigl|_{u=0}\bigr.=0. +$$ + + Extend $Y$ to a horizontal vector field $\widehat Y$ along $\gamma_Z$ by setting $\widehat Y(\tau)=0$ for all $\tau\in [\tau'_2,\bar\tau]$. Of course, the extended horizontal vector field $\widehat Y$ is not $C^2$ at $\tau'_2$ since $Y$, being @@ -877,77 +1005,90 @@ $\tau\le \tau'_2$ and has $\hat\gamma(u,\tau)=\gamma_Z(\tau)$ for all $\tau\in [\tau'_2,\bar\tau]$. Of course, the second-order variation of this extended family at $u=0$ agrees with the second-order variation of the original family at $u=0$, and hence -vanishes. But according to Proposition~ this means that +vanishes. But according to \entryref{26b7f69e803e} this means that $\widehat Y$ is an ${\mathcal L}$-Jacobi field, which is absurd since it is not a $C^2$-vector field. We shall also need a closely related result. -**Proposition.** -Let $\gamma$ be a minimizing ${\mathcal L}$-geodesic defined for + +**Proposition.** Let $\gamma$ be a minimizing ${\mathcal L}$-geodesic defined for $[\tau_1,\bar\tau]$. Fix $0\le\tau_1<\tau_2<\bar\tau$, and set $q_2=\gamma(\tau_2)$, and $Z_2=\sqrt{\tau_2}X_{\gamma}(\tau_2)$. -Then, the map ${\mathcal L}exp_{q_2}$ is diffeomorphism from +Then, the map ${\mathcal L}\mathit{exp}_{q_2}$ is diffeomorphism from a neighborhood of $\{Z_2\}\times (\tau_2,\bar\tau]$ in $T_{q}M_{T-\tau_2}\times (\tau_2,\infty)$ onto a neighborhood of the -image of $\gamma|_{(\tau_2,\bar\tau]}$. +image of $\gamma|_{(\tau_2,\bar\tau]}$. + -**Proof.** It suffices to show that the differential of ${\mathcal L}exp_{q_2}^\tau$ is an isomorphism for all $\tau\in +**Proof.** It suffices to show that the differential of ${\mathcal L}\mathit{exp}_{q_2}^\tau$ is an isomorphism for all $\tau\in (\tau_2,\bar\tau]$. If this is not the case, then there is a $\tau'\in (\tau_2,\bar\tau]$ and a non-zero ${\mathcal L}$-Jacobi field $Y$ along $\gamma_Z|_{[\tau_2,\tau']}$ vanishing at both ends. We extend $Y$ to a horizontal vector field $\widehat Y$ along all of -$\gamma_Z|_{[\tau_1,\tau']}$ by setting it equal to zero on +$\gamma_Z|_{[\tau_1,\tau']}$ by setting it equal to zero on $[\tau_1,\tau_2]$. Since $Y$ is an ${\mathcal L}$-Jacobi field, the second-order variation of ${\mathcal L}$-length in the direction of $Y$ is zero, and consequently the second-order variation of the length of $\gamma_Z|_{[\tau_1,\tau']}$ vanishes. Hence by -Proposition~ it must be the case that $\widehat Y$ is a +\entryref{26b7f69e803e} it must be the case that $\widehat Y$ is a ${\mathcal L}$-Jacobi field. This is impossible since $\widehat Y$ is not smooth at $\tau'$. We finish this section with a computation of the $\tau$-derivative of $L_x$. -**Lemma.** - Suppose that -$q\in {\mathcal U}_x$ with $**t**(q)=T-\bar\tau$ for some $\bar\tau>\tau_1$. + +**Lemma.** Suppose that +$q\in {\mathcal U}_x$ with $\mathbf{t}(q)=T-\bar\tau$ for some $\bar\tau>\tau_1$. Let $\gamma\colon [\tau_1,\bar\tau]\to {\mathcal M}$ be the unique minimizing ${\mathcal L}$-geodesic from $x$ to $q$. Then we have $$ - \frac{\partial L_x}{\partial \tau}(q)=2\sqrt{\bar\tau}R( q)-\sqrt{\bar\tau}\left(R( q)+|X(\bar\tau)|^2\right). $$ + **Proof.** By definition and the Fundamental Theorem of Calculus, we have -$$\frac{d}{d\tau}L_x(\gamma(\tau))= \sqrt{\tau}\left(R(\gamma(\tau))+|X(\tau)|^2\right).$$ + +$$ +\frac{d}{d\tau}L_x(\gamma(\tau))= \sqrt{\tau}\left(R(\gamma(\tau))+|X(\tau)|^2\right). +$$ + On the other hand since $\gamma'(\tau)=-\partial/\partial t+X(\tau)$ the chain rule implies -$$\frac{d}{d\tau}L_x(\gamma(\tau))=\langle \nabla + +$$ +\frac{d}{d\tau}L_x(\gamma(\tau))=\langle \nabla L_x,X(\tau)\rangle+\frac{\partial L_x}{\partial -\tau}(\gamma(\tau)),$$ so that -$$\frac{\partial L_x}{\partial \tau}(\gamma(\tau))=\sqrt{\tau}\left(R(\gamma(\tau))+|X(\tau)|^2\right)-\langle\nabla -L_x,X(\tau)\rangle.$$ Now using Corollary~, and rearranging +\tau}(\gamma(\tau)), +$$ + + so that + +$$ +\frac{\partial L_x}{\partial \tau}(\gamma(\tau))=\sqrt{\tau}\left(R(\gamma(\tau))+|X(\tau)|^2\right)-\langle\nabla +L_x,X(\tau)\rangle. +$$ + + Now using \entryref{033786a89c6d}, and rearranging the terms gives the result. -## Second-order differential inequalities for $\widetilde L^{\bar\tau -$ -and $L_x^{\bar\tau}$} +## Second-order differential inequalities for $\widetilde L^{\bar\tau}$ and $L_x^{\bar\tau}$ -Throughout Section~ we fix $x\in {\mathcal M}$ with $x\in +Throughout Section 6.4 we fix $x\in {\mathcal M}$ with $x\in M_{T-\tau_1}$. -### The second variation formula for $\widetilde L^{\bar\tau -$} +### The second variation formula for $\widetilde L^{\bar\tau}$ Our goal here is to compute the second variation of $\widetilde L^{\bar\tau}$ in the direction of a horizontal vector field $Y(\tau)$ along an ${\mathcal L}$-geodesic $\gamma$. Here is the main result of this subsection. -**Proposition.** Fix $0\le \tau_1<\bar \tau$. + +**Proposition.** Fix $0\le \tau_1<\bar \tau$. Let $\gamma$ be an ${\mathcal L}$-geodesic defined on $[\tau_1,\bar\tau]$ and let $\gamma_u=\widetilde\gamma(\tau,u)$ be a smooth family of curves parameterized by backward time with @@ -959,178 +1100,185 @@ These are horizontal vector fields along the image of $\widetilde Y$ and $\widetilde X$, respectively, to $\gamma$. We assume that $Y(\tau_1)=0$. Then -$$ - $$ \begin{aligned} -\lefteqn{\frac{d^2}{du^2}\left({\mathcal L}(\gamma_u)\right)|_{u=0} +\frac{d^2}{du^2}\left({\mathcal L}(\gamma_u)\right)|_{u=0} = 2\sqrt{\bar\tau}\langle \nabla_{Y(\tau)}\widetilde -Y(\bar\tau,u)|_{u=0},X(\bar\tau)\rangle } \\ -& & + \int_{\tau_1}^{\bar\tau}\sqrt{\tau}(Hess(R)(Y,Y) +Y(\bar\tau,u)|_{u=0},X(\bar\tau)\rangle \\ +& & + \int_{\tau_1}^{\bar\tau}\sqrt{\tau}(\mathit{Hess}(R)(Y,Y) +2\langle{\mathcal R}(Y,X)Y,X\rangle - - 4(\nabla_{Y}Ric)(X,Y) \\ - & & + 2(\nabla_{X}Ric)(Y,Y)+ 2\abs{\nabla_{X} Y}^{2})d\tau. + - 4(\nabla_{Y}\mathit{Ric})(X,Y) \\ + & & + 2(\nabla_{X}\mathit{Ric})(Y,Y)+ 2\abs{\nabla_{X} Y}^{2})d\tau. \end{aligned} $$ - -$$ + As we shall see, this is simply a rewriting of the equation in -Lemma~ in the special case when $u_1=u_2$. +\entryref{f48558c60ca2} in the special case when $u_1=u_2$. We begin the proof of this result with the following computation. + **Claim.** Let $\gamma(\tau)$ be a curve parameterized by backward time. Let $Y$ be a horizontal vector field along $\gamma$ and let $X$ be the horizontal component of $\partial\widetilde\gamma/\partial \tau$. Then -$$ - $$ \begin{aligned} \frac{\partial}{\partial\tau}\langle \nabla_{ X}Y, Y\rangle &= \langle \nabla_{ X} Y, \nabla_{ X} Y\rangle + \langle \nabla_{ X}\nabla_{ X} Y, Y\rangle -\\&\hspace{0.5cm} +2Ric(\nabla_XY,Y)) +(\nabla_XRic)(Y,Y)) +\\&\hspace{0.5cm} +2\mathit{Ric}(\nabla_XY,Y)) +(\nabla_X\mathit{Ric})(Y,Y)) \end{aligned} $$ - -$$ + **Proof.** We can break $\frac{\partial}{\partial\tau}\langle \nabla_{X}Y,Y\rangle $ into -two parts: the first assumes that the metric is constant and the second deals +two parts: the first assumes that the metric is constant and the second deals with the variation with $\tau$ of the metric. The first contribution is the usual formula -$$\frac{\partial}{\partial\tau}\langle \nabla_{X}Y,Y\rangle _{G(T-\tau_0)}= + +$$ +\frac{\partial}{\partial\tau}\langle \nabla_{X}Y,Y\rangle _{G(T-\tau_0)}= \langle \nabla_{X}Y,\nabla_{X}Y\rangle _{G(T-\tau_0)}+ \langle -\nabla_{X}\nabla_{X}Y,Y\rangle _{G(T-\tau_0)}.$$ This gives us the first two +\nabla_{X}\nabla_{X}Y,Y\rangle _{G(T-\tau_0)}. +$$ + + This gives us the first two terms of the right-hand side of the equation in the claim. We show that the last two terms in that equation come from differentiating the metric with respect to $\tau$. To do this recall that in local coordinates, writing the metric $G(T-\tau)$ as $g_{ij}$, we have -$$\langle \nabla_{X}Y,Y \rangle=g_{ij}\bigl(X^k\partial_k -Y^i+\Gamma_{kl}^iX^kY^l\bigr)Y^j.$$ There are two contributions coming from + +$$ +\langle \nabla_{X}Y,Y \rangle=g_{ij}\bigl(X^k\partial_k +Y^i+\Gamma_{kl}^iX^kY^l\bigr)Y^j. +$$ + + There are two contributions coming from differentiating the metric with respect to $\tau$. The first is when we differentiate $g_{ij}$. This leads to -$$2Ric_{ij}\bigl(X^k\partial_k -Y^i+\Gamma_{kl}^iX^kY^l\bigr)Y^j=2Ric(\nabla_XY,Y\rangle.$$ + +$$ +2\mathit{Ric}_{ij}\bigl(X^k\partial_k +Y^i+\Gamma_{kl}^iX^kY^l\bigr)Y^j=2\mathit{Ric}(\nabla_XY,Y\rangle. +$$ + The other contribution is from differentiating the Christoffel symbols. This yields -$$g_{ij}\frac{\partial\Gamma^{i}_{kl}}{\partial \tau}X^{k}Y^{l}Y^{j}.$$ + +$$ +g_{ij}\frac{\partial\Gamma^{i}_{kl}}{\partial \tau}X^{k}Y^{l}Y^{j}. +$$ + Differentiating the formula $\Gamma_{kl}^i=\frac{1}{2}g^{si}(\partial_kg_{sl}+\partial_lg_{sk}-\partial_s g_{kl})$ leads to -$$ $$ \begin{aligned} g_{ij}\frac{\partial -\Gamma_{kl}^i}{\partial \tau} & = & -2Ric_{ij}\Gamma_{kl}^i+g_{ij}g^{si}(\partial_kRic_{sl}+\partial_lRic_{sk} --\partial_sRic_{kl})\\ -& = & -2Ric_{ij}\Gamma_{kl}^i+ -\partial_kRic_{jl}+\partial_lRic_{jk} -\partial_jRic_{kl}. +\Gamma_{kl}^i}{\partial \tau} & = & -2\mathit{Ric}_{ij}\Gamma_{kl}^i+g_{ij}g^{si}(\partial_k\mathit{Ric}_{sl}+\partial_l\mathit{Ric}_{sk} +-\partial_s\mathit{Ric}_{kl})\\ +& = & -2\mathit{Ric}_{ij}\Gamma_{kl}^i+ +\partial_k\mathit{Ric}_{jl}+\partial_l\mathit{Ric}_{jk} -\partial_j\mathit{Ric}_{kl}. \end{aligned} $$ -$$ - Thus, we have -$$ - $$ \begin{aligned} g_{ij}\frac{\partial\Gamma^{i}_{kl}}{\partial \tau}X^{k}Y^{l}Y^{j} - & = & \bigl(-2Ric_{ij}\Gamma^i_{kl}+\partial_kRic_{jl})\bigr)X^kY^lY^j \\ -& = & (\nabla_XRic)(Y,Y) + & = & \bigl(-2\mathit{Ric}_{ij}\Gamma^i_{kl}+\partial_k\mathit{Ric}_{jl})\bigr)X^kY^lY^j \\ +& = & (\nabla_X\mathit{Ric})(Y,Y) \end{aligned} $$ -$$ - This completes the proof of the claim. Now we return to the proof of the second variational formula in -Proposition~. +\entryref{3add96057999}. + +**Proof.** According to \entryref{f48558c60ca2} we have -**Proof.** According to Lemma~ we have -$$\frac{d^2}{du^2}{\mathcal L}_{u=0}=2\sqrt{\bar +$$ +\frac{d^2}{du^2}{\mathcal L}_{u=0}=2\sqrt{\bar \tau}Y(\bar\tau)(\langle \widetilde Y(\bar\tau,u),\widetilde X(\bar\tau,u)\rangle)|_{u=0}-\int_{\tau_1}^{\tau_2}2\sqrt{\tau}\langle -Jac(Y),Y\rangle d\tau.$$ We plug in -Equation~ for $Jac(Y)$ and this makes the -integrand - +\mathit{Jac}(Y),Y\rangle d\tau. $$ + We plug in +Equation 6.6 for $\mathit{Jac}(Y)$ and this makes the +integrand + $$ \begin{aligned} \sqrt{\tau}\langle\nabla_Y(\nabla R),Y\rangle+2\sqrt{\tau}\langle{\mathcal R}(Y,X)Y,X\rangle -\bigl(2\sqrt{\tau}\langle\nabla_X\nabla_XY,Y\rangle+\frac{1}{\sqrt{\tau}}\langle\nabla_XY,Y\rangle\bigr)\\ --4\sqrt{\tau}(\nabla_YRic)(X,Y)-4\sqrt{\tau}Ric(\nabla_XY,Y) +-4\sqrt{\tau}(\nabla_Y\mathit{Ric})(X,Y)-4\sqrt{\tau}\mathit{Ric}(\nabla_XY,Y) \end{aligned} $$ -$$ - -The first term is $\sqrt{\tau}Hess(R)(Y,Y)$. Let us deal with +The first term is $\sqrt{\tau}\mathit{Hess}(R)(Y,Y)$. Let us deal with the third and fourth terms, which are grouped together within parentheses. According to the previous claim, we have -$$ - $$ \begin{aligned} \frac{\partial}{\partial \tau}\bigl(2\sqrt{\tau}\langle \nabla_XY,Y\rangle\bigr) & = & \frac{1}{\sqrt{\tau}}\langle\nabla_XY,Y\rangle+2\sqrt{\tau}\langle\nabla_X\nabla_XY,Y\rangle - +2\sqrt{\tau}\langle\nabla_XY,\nabla_XY\rangle \\ & & +4\sqrt{\tau}Ric(\nabla_XY,Y) - +2\sqrt{\tau} (\nabla_XRic)(Y,Y). + +2\sqrt{\tau}\langle\nabla_XY,\nabla_XY\rangle \\ & & +4\sqrt{\tau}\mathit{Ric}(\nabla_XY,Y) + +2\sqrt{\tau} (\nabla_X\mathit{Ric})(Y,Y). \end{aligned} $$ -$$ - This allows us to replace the two terms under consideration by -$$-\frac{\partial}{\partial t}\bigl(2\sqrt{\tau}\langle -\nabla_XY,Y\rangle\bigr)+2\sqrt{\tau}\langle\nabla_XY,\nabla_XY\rangle+4\sqrt{\tau}Ric(\nabla_XY,Y) +2\sqrt{\tau} (\nabla_XRic)(Y,Y).$$ -Integrating the total derivative out of the integrand and canceling -terms leaves the integrand as $$ +-\frac{\partial}{\partial t}\bigl(2\sqrt{\tau}\langle +\nabla_XY,Y\rangle\bigr)+2\sqrt{\tau}\langle\nabla_XY,\nabla_XY\rangle+4\sqrt{\tau}\mathit{Ric}(\nabla_XY,Y) +2\sqrt{\tau} (\nabla_X\mathit{Ric})(Y,Y). +$$ + +Integrating the total derivative out of the integrand and canceling +terms leaves the integrand as $$ \begin{aligned} -\sqrt{\tau}Hess(R)(Y,Y)+2\sqrt{\tau}\langle{\mathcal R}(Y,X)Y,X\rangle +\sqrt{\tau}\mathit{Hess}(R)(Y,Y)+2\sqrt{\tau}\langle{\mathcal R}(Y,X)Y,X\rangle +2\sqrt{\tau}|\nabla_XY|^2 \\ --4\sqrt{\tau}(\nabla_YRic)(X,Y)+2\sqrt{\tau} (\nabla_XRic)(Y,Y), +-4\sqrt{\tau}(\nabla_Y\mathit{Ric})(X,Y)+2\sqrt{\tau} (\nabla_X\mathit{Ric})(Y,Y), \end{aligned} $$ -$$ - and makes the boundary term (the one in front of the integral) equal to -$$2\sqrt{\bar\tau}\bigl(Y(\bar\tau)\langle + +$$ +2\sqrt{\bar\tau}\bigl(Y(\bar\tau)\langle \widetilde Y(\bar\tau,u),\widetilde X(\bar\tau,u)\rangle|_{u=0}-\langle \nabla_XY(\bar\tau),Y(\bar\tau)\rangle\bigr)=2\sqrt{\bar\tau}\langle -X(\bar\tau),\nabla_Y\widetilde Y(\bar\tau,u)|_{u=0}\rangle.$$ This +X(\bar\tau),\nabla_Y\widetilde Y(\bar\tau,u)|_{u=0}\rangle. +$$ + + This completes the proof of the proposition. -### Inequalities for the Hessian of $L_x^{\bar\tau -$} +### Inequalities for the Hessian of $L_x^{\bar\tau}$ Now we shall specialize the type of vector fields along $\gamma$. This will allow us to give an inequality for the Hessian of ${\mathcal L}$ involving the integral of the vector field along $\gamma$. These lead to inequalities for the Hessian of -$L_x^{\bar\tau}$$}. The main result -of this section is Proposition~ below. In the end we +$L_x^{\bar\tau}$. The main result +of this section is \entryref{88d4ef67440f} below. In the end we are interested in the case when the $\tau_1=0$. In this case the formulas simplify. The reason for working here in the full generality of all $\tau_1$ is in order to establish differential @@ -1145,6 +1293,7 @@ general $\tau_1$ is needed so that we can establish the differential inequalities for the barrier functions that yield the weak inequalities at non-smooth points. + **Definition.** Let $q\in {\mathcal U}_x(\bar\tau)$ and let $\gamma\colon [\tau_1,\bar\tau]\to {\mathcal M}$ be the unique minimizing ${\mathcal L}$-geodesic from $x$ to $q$. We say that a horizontal @@ -1152,83 +1301,84 @@ vector field $Y(\tau)$ along $\gamma$ is *adapted* if it solves the following ODE on $[\tau_1,\bar \tau]$: $$ - -\nabla_{X}Y(\tau) = -Ric(Y(\tau),\cdot)^* + +\nabla_{X}Y(\tau) = -\mathit{Ric}(Y(\tau),\cdot)^* + \frac{1}{2\sqrt{\tau}(\sqrt{\tau}-\sqrt{\tau_1})}Y(\tau). $$ + Direct computation shows the following: -**Lemma.** - Suppose that $Y(\tau)$ is an adapted vector field along $\gamma$. + +**Lemma.** Suppose that $Y(\tau)$ is an adapted vector field along $\gamma$. Then -$$ - $$ \begin{aligned} - \frac{d}{d\tau}\langle Y(\tau),Y(\tau)\rangle & = & -2Ric(Y(\tau),Y(\tau)) + 2\langle \nabla_{X}Y(\tau),Y(\tau)\rangle \\ +2\mathit{Ric}(Y(\tau),Y(\tau)) + 2\langle \nabla_{X}Y(\tau),Y(\tau)\rangle \\ & = & \frac{1}{\sqrt{\tau}(\sqrt{\tau}-\sqrt{\tau_1})}\langle Y(\tau),Y(\tau)\rangle\nonumber. \end{aligned} $$ -$$ It follows that -$$|Y(\tau)|^2 = C\frac{(\sqrt{\tau}-\sqrt{\tau_1})^2}{(\sqrt{\bar\tau}-\sqrt{\tau_1})^2},$$ + +$$ +|Y(\tau)|^2 = C\frac{(\sqrt{\tau}-\sqrt{\tau_1})^2}{(\sqrt{\bar\tau}-\sqrt{\tau_1})^2}, +$$ + where $C=|Y(\bar\tau)|^2$. + Now we come to the main result of this subsection, which is an extremely important inequality for the Hessian of $L_x^{\bar\tau}$. -**Proposition.** -Suppose that $q\in {\mathcal U}_x(\bar\tau)$, that $Z\in -\widetilde{\mathcal U}_x(\bar\tau)$ is the pre-image of $q$, and -that $\gamma_Z$ is the ${\mathcal L}$-geodesic to $q$ determined by -$Z$. Suppose that $Y(\tau)$ is an adapted vector field along + +**Proposition.** Suppose that $q\in {\mathcal U}_x(\bar\tau)$, that $Z\in +\widetilde{\mathcal U}_x(\bar\tau)$ is the pre-image of $q$, and +that $\gamma_Z$ is the ${\mathcal L}$-geodesic to $q$ determined by +$Z$. Suppose that $Y(\tau)$ is an adapted vector field along $\gamma_Z$. Then $$ - -Hess(L_x^{\bar\tau})(Y(\bar\tau),Y(\bar\tau)) \leq +\mathit{Hess}(L_x^{\bar\tau})(Y(\bar\tau),Y(\bar\tau)) \leq \left(\frac{|Y(\bar\tau)|^2}{\sqrt{\bar\tau}-\sqrt{\tau_1}}\right)- -2\sqrt{\bar\tau}Ric(Y(\bar\tau),Y(\bar\tau)) - +2\sqrt{\bar\tau}\mathit{Ric}(Y(\bar\tau),Y(\bar\tau)) - \int_{\tau_1}^{\bar\tau}\sqrt{\tau}H(X,Y)d\tau, $$ - where -$$ + where $$ \begin{aligned} \nonumber -H(X,Y) & =&-Hess(R)(Y,Y) - 2\langle {\mathcal R}(Y,X)Y,X\rangle -\\& &-4(\nabla_{X}Ric)(Y,Y) +4(\nabla_{Y}Ric)(Y,X) -\\& & - 2\frac{\partialRic}{\partial\tau}(Y,Y) + 2\abs{Ric(Y,\cdot)}^{2} - -\frac{1}{\tau}Ric(Y,Y),\nonumber. +H(X,Y) & =&-\mathit{Hess}(R)(Y,Y) - 2\langle {\mathcal R}(Y,X)Y,X\rangle +\\& &-4(\nabla_{X}\mathit{Ric})(Y,Y) +4(\nabla_{Y}\mathit{Ric})(Y,X) +\\& & - 2\frac{\partial\mathit{Ric}}{\partial\tau}(Y,Y) + 2\abs{\mathit{Ric}(Y,\cdot)}^{2} - +\frac{1}{\tau}\mathit{Ric}(Y,Y),\nonumber. \end{aligned} -$$ - $$ We -have equality in Equation~() if and only if the adapted vector +have equality in Equation (6.11) if and only if the adapted vector field $Y$ is also a ${\mathcal L}$-Jacobi field. + + **Remark.** In spite of the notation, $H(X,Y)$ is a purely quadratic function of the vector field $Y$ along $\gamma_Z$. + We begin the proof of this proposition with three elementary lemmas. The first is an immediate consequence of the definition of $\widetilde {\mathcal U}_x(\bar\tau)$. -**Lemma.** Suppose that $q\in {\mathcal U}_x(\bar\tau)$ and that $\gamma\colon + +**Lemma.** Suppose that $q\in {\mathcal U}_x(\bar\tau)$ and that $\gamma\colon [\tau_1,\bar\tau]\to{\mathcal M}$ is the minimizing ${\mathcal L}$-geodesic from $x$ to $q$. Then for every tangent vector $ Y(\bar\tau)\in T_qM_{T-\bar\tau}$ there is a one-parameter family of -${\mathcal L}$-geodesics $\tilde\gamma(\tau,u)$ defined on +${\mathcal L}$-geodesics $\tilde\gamma(\tau,u)$ defined on $[\tau_1,\bar\tau]$ with $\tilde\gamma(0,u)=x$ for all $u$, with $\tilde\gamma(\tau,0)=\gamma(\tau)$ and $\partial\tilde \gamma(\bar\tau,0)/\partial u=Y(\bar \tau)$. Also, for every $Z\in @@ -1237,28 +1387,34 @@ $\tilde\gamma(\tau,u)$ such that $\gamma(0,u)=x$ for all $u$, $\tilde\gamma(\tau,0)=\gamma(\tau)$ and such that, setting $Y(\tau)=\frac{\partial}{\partial u} \tilde\gamma_u(\tau)|_{u=0}$, we have -$$\nabla_{\sqrt{\tau_1}X(\tau_1)}Y(\tau_1)=Z.$$ -**Lemma.** -Let $\gamma$ be a minimizing ${\mathcal L}$-geodesic from $x$, and +$$ +\nabla_{\sqrt{\tau_1}X(\tau_1)}Y(\tau_1)=Z. +$$ + + + +**Lemma.** Let $\gamma$ be a minimizing ${\mathcal L}$-geodesic from $x$, and let $Y(\tau)$ be an ${\mathcal L}$-Jacobi field along $\gamma$. Then -$$2\sqrt{\bar\tau}\langle\nabla_XY(\bar\tau),Y(\bar\tau)\rangle = Hess(L_x^{\bar\tau})(Y(\bar\tau), -Y(\bar\tau)).$$ + +$$ +2\sqrt{\bar\tau}\langle\nabla_XY(\bar\tau),Y(\bar\tau)\rangle = \mathit{Hess}(L_x^{\bar\tau})(Y(\bar\tau), +Y(\bar\tau)). +$$ + **Proof.** Let $\gamma(\tau,u)$ be a one-parameter family of ${\mathcal L}$-geodesics emanating from $x$ with $\gamma(u,0)$ being the -${\mathcal L}$-geodesic in the statement of the lemma and with +${\mathcal L}$-geodesic in the statement of the lemma and with $\frac{\partial}{\partial u}\gamma(\tau,0)=Y(\tau)$. We have the extensions of $X(\tau)$ and $Y(\tau)$ to vector fields $\widetilde X(\tau,u)$ and $\widetilde Y(\tau,u)$ defined at $\gamma(\tau,u)$ for all $\tau$ and $u$. Of course, -$$ - $$ \begin{aligned} -\lefteqn{2\sqrt{\bar\tau}\langle \nabla_Y\widetilde -X(\bar\tau,u)|_{u=0},Y(\bar\tau)\rangle } \\ +2\sqrt{\bar\tau}\langle \nabla_Y\widetilde +X(\bar\tau,u)|_{u=0},Y(\bar\tau)\rangle \\ & = & Y(\langle 2\sqrt{\bar\tau}\widetilde X(\bar\tau,u),\widetilde Y(\bar\tau,u)\rangle)|_{u=0}-\langle 2\sqrt{\bar\tau}X(\bar\tau),\nabla_Y\widetilde @@ -1266,11 +1422,8 @@ Y(\bar\tau,u)|_{u=0}\rangle. \end{aligned} $$ -$$ Then by -Corollary~ we have - -$$ +\entryref{033786a89c6d} we have $$ \begin{aligned} @@ -1283,43 +1436,50 @@ Y(\bar\tau,u)|_{u=0}\rangle \\ & = Y(\bar\tau)(\widetilde Y(\bar\tau,u) L_x^{\bar\tau})|_{u=0}-\nabla_{Y(\bar\tau)}\widetilde -Y(\bar\tau,u)|_{u=0}(L_x^{\bar\tau}) \\ & = Hess(L_x^{\bar\tau})(Y(\bar\tau),Y(\bar\tau)). +Y(\bar\tau,u)|_{u=0}(L_x^{\bar\tau}) \\ & = \mathit{Hess}(L_x^{\bar\tau})(Y(\bar\tau),Y(\bar\tau)). \end{aligned} $$ -$$ - Now suppose that we have a horizontal vector field that is both adapted and ${\mathcal L}$-Jacobi. We get: -**Lemma.** -Suppose that $q\in {\mathcal U}_x(\bar\tau)$, that $Z\in \widetilde + +**Lemma.** Suppose that $q\in {\mathcal U}_x(\bar\tau)$, that $Z\in \widetilde {\mathcal U}_x(\bar\tau)$ is the pre-image of $q$, and that -$\gamma_Z$ is the ${\mathcal L}$-geodesic to $q$ determined by $Z$. +$\gamma_Z$ is the ${\mathcal L}$-geodesic to $q$ determined by $Z$. Suppose further that $Y(\tau)$ is a horizontal vector field along $\gamma$ that is both adapted and an ${\mathcal L}$-Jacobi field. -Then, we have -$$\frac{1}{2\sqrt{\bar\tau}(\sqrt{\bar\tau}-\sqrt{\tau_1})}|Y(\bar\tau)|^2 -=\frac{1}{2\sqrt{\bar\tau}}Hess(L_x^{\bar\tau})(Y(\bar\tau),Y(\bar\tau))+Ric(Y(\bar\tau),Y(\bar\tau)).$$ +Then, we have + +$$ +\frac{1}{2\sqrt{\bar\tau}(\sqrt{\bar\tau}-\sqrt{\tau_1})}|Y(\bar\tau)|^2 +=\frac{1}{2\sqrt{\bar\tau}}\mathit{Hess}(L_x^{\bar\tau})(Y(\bar\tau),Y(\bar\tau))+\mathit{Ric}(Y(\bar\tau),Y(\bar\tau)). +$$ + -**Proof.** From the definition of an adapted vector field $Y(\tau)$ we have +**Proof.** From the definition of an adapted vector field $Y(\tau)$ we have $$ -Ric(Y(\tau),Y(\tau)) + \langle +\mathit{Ric}(Y(\tau),Y(\tau)) + \langle \nabla_{X}Y(\tau),Y(\tau)\rangle = \frac{1}{2\sqrt{\tau}(\sqrt{\tau}-\sqrt{\tau_1})}\langle Y(\tau),Y(\tau)\rangle. $$ Since $Y(\tau)$ is an ${\mathcal L}$-Jacobi field, according to -Lemma~ we have -$$\langle\nabla_XY(\bar\tau),Y(\bar\tau)\rangle = -\frac{1}{2\sqrt{\bar\tau}}Hess(L_x^{\bar\tau})(Y(\bar\tau),Y(\bar\tau)).$$ Putting these -together gives the result. - -Now we are ready to begin the proof of Proposition~. +\entryref{44abf07e60ec} we have -**Proof.** Let $\tilde\gamma(\tau,u)$ be a family of curves with +$$ +\langle\nabla_XY(\bar\tau),Y(\bar\tau)\rangle = +\frac{1}{2\sqrt{\bar\tau}}\mathit{Hess}(L_x^{\bar\tau})(Y(\bar\tau),Y(\bar\tau)). +$$ + + Putting these +together gives the result. + +Now we are ready to begin the proof of \entryref{88d4ef67440f}. + +**Proof.** Let $\tilde\gamma(\tau,u)$ be a family of curves with $\gamma(\tau,0)=\gamma_Z$ and with $\frac{\partial}{\partial u}\gamma(\tau,u)=\widetilde Y(\tau,u)$. We denote by $Y$ the horizontal vector field which is the restriction of $\widetilde Y$ @@ -1329,141 +1489,128 @@ neighborhood of $0$ in the $u$-direction, we can assume that $q(u)\in {\mathcal U}_x(\bar\tau)$ for all $u$. Then ${\mathcal L}(\tilde\gamma_u)\ge L_x^{\bar\tau}(q(u))$. Of course, $L_x^{\bar\tau}(q(0))={\mathcal L}(\gamma_Z)$. This implies that -$$\frac{d}{d u}L_x^{\bar\tau}(q(u))\bigl|\bigr._{u=0}= -\frac{d}{du}{\mathcal L}(\gamma_u)\bigl|\bigr._{u=0},$$ + +$$ +\frac{d}{d u}L_x^{\bar\tau}(q(u))\bigl|\bigr._{u=0}= +\frac{d}{du}{\mathcal L}(\gamma_u)\bigl|\bigr._{u=0}, +$$ + and + $$ Y(\bar\tau)(\widetilde Y(\bar\tau,u)(L^{\bar\tau}_x))|_{u=0}=\frac{d^2}{du^2}L_x^{\bar\tau}(q(u))\bigl|\bigr._{u=0}\le -\frac{d^2}{du^2}{\mathcal L}(\gamma_u)\bigl|\bigr._{u=0}.$$ +\frac{d^2}{du^2}{\mathcal L}(\gamma_u)\bigl|\bigr._{u=0}. +$$ + Recall that $\nabla L_x^{\bar\tau}(q) = 2\sqrt{\bar\tau}X(\bar\tau)$, so that -$$\nabla_{Y(\bar\tau)}\widetilde Y(\bar\tau,u)|_{u=0}(L_x^{\bar\tau})= + +$$ +\nabla_{Y(\bar\tau)}\widetilde Y(\bar\tau,u)|_{u=0}(L_x^{\bar\tau})= \langle \nabla_{Y(\bar\tau)}\widetilde Y(\bar\tau,u)|_{u=0},\nabla L^{\bar\tau}\rangle = 2\sqrt{\bar\tau}\langle \nabla_{Y(\bar\tau)}\widetilde -Y(\bar\tau,u)|_{u=0},X(\bar\tau)\rangle .$$ Thus, by -Proposition~, and using the fact that -$Y(\tau_1)=0$, we have - +Y(\bar\tau,u)|_{u=0},X(\bar\tau)\rangle . $$ + Thus, by +\entryref{3add96057999}, and using the fact that +$Y(\tau_1)=0$, we have + $$ \begin{aligned} -\lefteqn{Hess(L^{\bar\tau})(Y(\bar\tau),Y(\bar\tau)) = +\mathit{Hess}(L^{\bar\tau})(Y(\bar\tau),Y(\bar\tau)) = Y(\bar\tau)\left(\widetilde Y(\bar\tau,u)(L_x^{\bar\tau})\right)|_{u=0}- -\nabla_{Y(\bar\tau)}\widetilde Y(\bar\tau,u)|_{u=0}(L_x^{\bar\tau})} \\ +\nabla_{Y(\bar\tau)}\widetilde Y(\bar\tau,u)|_{u=0}(L_x^{\bar\tau}) \\ & \le & \frac{d^2}{du^2}{\mathcal L}(\gamma_u)-2\sqrt{\bar\tau}\langle \nabla_{Y(\bar\tau)}\widetilde Y(\bar\tau,u)|_{u=0},X(\bar\tau)\rangle \\ & = & \int_{\tau_1}^{\bar\tau} -\sqrt{\tau}\bigl(Hess(R)(Y,Y) + -2\langle {\mathcal R}(Y,X)Y,X\rangle - 4(\nabla_{Y}Ric)(X,Y)\bigr. \\ -& & \ \ \ \ \bigl.+ 2(\nabla_{X}Ric)(Y,Y) + +\sqrt{\tau}\bigl(\mathit{Hess}(R)(Y,Y) + +2\langle {\mathcal R}(Y,X)Y,X\rangle - 4(\nabla_{Y}\mathit{Ric})(X,Y)\bigr. \\ +& & \ \ \ \ \bigl.+ 2(\nabla_{X}\mathit{Ric})(Y,Y) + 2\abs{\nabla_XY}^2\bigr)d\tau. \end{aligned} $$ -$$ - - Plugging in Equation~(), and using the fact that + Plugging in Equation (6.9), and using the fact that $|Y(\tau)|^2= |Y(\bar\tau)|^2\frac{(\sqrt{\tau}-\sqrt{\tau_1})^2}{(\sqrt{\bar\tau}-\sqrt{\tau_1})^2}$, gives -$$ - $$ \begin{aligned} -\lefteqn{Hess(L^{\bar\tau})(Y(\bar\tau),Y(\bar\tau)) } \\ & -\le & \int_{\tau_1}^{\bar\tau} \sqrt{\tau}\bigl(Hess(R)(Y,Y) + -2\langle {\mathcal R}(Y,X)Y,X\rangle - 4(\nabla_{Y}Ric)(X,Y)\bigr. \\ -& & \ \ \ \ \bigl. +2(\nabla_XRic)(Y,Y) + - 2\abs{Ric(Y,\cdot)}^{2}\bigr)d\tau \\ & & +\mathit{Hess}(L^{\bar\tau})(Y(\bar\tau),Y(\bar\tau)) \\ & +\le & \int_{\tau_1}^{\bar\tau} \sqrt{\tau}\bigl(\mathit{Hess}(R)(Y,Y) + +2\langle {\mathcal R}(Y,X)Y,X\rangle - 4(\nabla_{Y}\mathit{Ric})(X,Y)\bigr. \\ +& & \ \ \ \ \bigl. +2(\nabla_X\mathit{Ric})(Y,Y) + + 2\abs{\mathit{Ric}(Y,\cdot)}^{2}\bigr)d\tau \\ & & +\int_{\tau_1}^{\bar\tau}\left[ \frac{|Y(\bar\tau)|^2}{2\sqrt{\tau}(\sqrt{\bar\tau}-\sqrt{\tau_1})^2} -- \frac{2}{(\sqrt{\tau}-\sqrt{\tau_1})}Ric(Y,Y)\right] d\tau +- \frac{2}{(\sqrt{\tau}-\sqrt{\tau_1})}\mathit{Ric}(Y,Y)\right] d\tau \end{aligned} $$ -$$ - - Using the definition of $H(X,Y)$ given in the statement, Equation~(), + Using the definition of $H(X,Y)$ given in the statement, Equation (6.12), allows us to write -$$ - $$ \begin{aligned} -\lefteqn{Hess(L^{\bar\tau})(Y(\bar\tau),Y(\bar\tau))} \\ +\mathit{Hess}(L^{\bar\tau})(Y(\bar\tau),Y(\bar\tau)) \\ & \le & -\int_{\tau_1}^{\bar\tau}\sqrt{\tau}H(X,Y)d\tau \\ & & -+\int_{\tau_1}^{\bar\tau}\Bigl[\sqrt{\tau} \bigl(-2(\nabla_{X}Ric)(Y,Y) --2\frac{\partialRic}{\partial\tau}(Y,Y) +4|Ric(Y,\cdot)|^2 \bigr) \Bigr.\\ ++\int_{\tau_1}^{\bar\tau}\Bigl[\sqrt{\tau} \bigl(-2(\nabla_{X}\mathit{Ric})(Y,Y) +-2\frac{\partial\mathit{Ric}}{\partial\tau}(Y,Y) +4|\mathit{Ric}(Y,\cdot)|^2 \bigr) \Bigr.\\ & & \Bigl.+ \frac{|Y(\bar\tau)|^2}{2\sqrt{\tau}(\sqrt{\bar\tau}-\sqrt{\tau_1})^2} -\left(\frac{2}{(\sqrt{\tau}- -\sqrt{\tau_1})}+\frac{1}{\sqrt{\tau}}\right) Ric(Y,Y)\Bigr]d\tau, +\sqrt{\tau_1})}+\frac{1}{\sqrt{\tau}}\right) \mathit{Ric}(Y,Y)\Bigr]d\tau, \end{aligned} -$$ - $$ To simplify -further, we compute, using Equation~() - -$$ +further, we compute, using Equation (6.9) $$ \begin{aligned} -\frac{d}{d\tau}\bigl(Ric(Y(\tau),Y(\tau))\bigr) &= & \frac{\partialRic}{\partial{\tau}}(Y,Y) + - 2Ric(\nabla_{X}Y,Y) + (\nabla_{X}Ric)(Y,Y)\\ - & = & \frac{\partialRic}{\partial{\tau}}(Y,Y)+ (\nabla_{X}Ric)(Y,Y) \\ & & + -\frac{1}{\sqrt{\tau}(\sqrt{\tau}-\sqrt{\tau_1})}Ric(Y,Y) - -2|Ric(Y,\cdot)|^{2}. +\frac{d}{d\tau}\bigl(\mathit{Ric}(Y(\tau),Y(\tau))\bigr) &= & \frac{\partial\mathit{Ric}}{\partial{\tau}}(Y,Y) + + 2\mathit{Ric}(\nabla_{X}Y,Y) + (\nabla_{X}\mathit{Ric})(Y,Y)\\ + & = & \frac{\partial\mathit{Ric}}{\partial{\tau}}(Y,Y)+ (\nabla_{X}\mathit{Ric})(Y,Y) \\ & & + +\frac{1}{\sqrt{\tau}(\sqrt{\tau}-\sqrt{\tau_1})}\mathit{Ric}(Y,Y) - +2|\mathit{Ric}(Y,\cdot)|^{2}. \end{aligned} -$$ - $$ Consequently, we have -$$ - $$ \begin{aligned} -\frac{d\left(2\sqrt{\tau}Ric(Y(\tau),Y(\tau))\right)}{d\tau} & -= & 2\sqrt{\tau}\left(\frac{\partialRic}{\partial{\tau}}(Y,Y) -+ (\nabla_{X}Ric)(Y,Y)- 2|Ric(Y,\cdot)|^{2}\right)\\ - & & + \left(\frac{2}{(\sqrt{\tau}-\sqrt{\tau_1})} +\frac{1}{\sqrt{\tau}}\right)Ric(Y,Y) +\frac{d\left(2\sqrt{\tau}\mathit{Ric}(Y(\tau),Y(\tau))\right)}{d\tau} & += & 2\sqrt{\tau}\left(\frac{\partial\mathit{Ric}}{\partial{\tau}}(Y,Y) ++ (\nabla_{X}\mathit{Ric})(Y,Y)- 2|\mathit{Ric}(Y,\cdot)|^{2}\right)\\ + & & + \left(\frac{2}{(\sqrt{\tau}-\sqrt{\tau_1})} +\frac{1}{\sqrt{\tau}}\right)\mathit{Ric}(Y,Y) \end{aligned} $$ -$$ - Using this, and the fact that $Y(\tau_1)=0$, gives -$$ - $$ \begin{aligned} - -\lefteqn{Hess(L_x^{\bar\tau})(Y(\bar\tau),Y(\bar\tau)) \le} \\ +\mathit{Hess}(L_x^{\bar\tau})(Y(\bar\tau),Y(\bar\tau)) \le \\ & & -\int_{\tau_1}^{\bar\tau}\left(\sqrt{\tau}H(X,Y) --\frac{d}{d\tau}\left(2\sqrt{\tau}Ric(Y,Y)\right) +-\frac{d}{d\tau}\left(2\sqrt{\tau}\mathit{Ric}(Y,Y)\right) -\frac{|Y(\bar\tau)|^2}{2\sqrt{\tau}(\sqrt{\bar\tau}-\sqrt{\tau_1})^2}\right)d\tau \nonumber\\ & = & \frac{|Y(\bar\tau)|^2}{\sqrt{\bar\tau}-\sqrt{\tau_1}} --2\sqrt{\bar\tau}Ric(Y(\bar\tau),Y(\bar\tau))-\int_{\tau_1}^{\bar\tau}\sqrt{\tau}H(X,Y)d\tau. +-2\sqrt{\bar\tau}\mathit{Ric}(Y(\bar\tau),Y(\bar\tau))-\int_{\tau_1}^{\bar\tau}\sqrt{\tau}H(X,Y)d\tau. \nonumber \end{aligned} $$ -$$ - -This proves Inequality~(). Now we examine when equality +This proves Inequality (6.11). Now we examine when equality holds in this expression. Given an adapted vector field $Y(\tau)$ along $\gamma$, let $\mu(v)$ be a geodesic through $\gamma(\bar\tau,0)$ with tangent vector $Y(\bar\tau)$. Then there @@ -1471,13 +1618,20 @@ is a one-parameter family $\mu(\tau,v)$ of minimizing ${\mathcal L}$-geodesics with the property that $\mu(\bar\tau,v)=\mu(v)$. Let $\widetilde Y'(\tau,v)$ be $\partial\mu(\tau,v)/\partial v$. It is an ${\mathcal L}$-Jacobi field with $\widetilde -Y'(\bar\tau,0)=Y(\bar\tau)$. Since $L_x\circ {\mathcal L}exp_x=\widetilde L$, we see that -$$\frac{d^2}{d v^2}{\mathcal -L}(\mu_v)|_{v=0}=\frac{d^2}{du^2}L^{\bar \tau}_x(\mu(u))|_{u=0}.$$ -Hence, the assumption that we have equality in~() +Y'(\bar\tau,0)=Y(\bar\tau)$. Since $L_x\circ {\mathcal L}\mathit{exp}_x=\widetilde L$, we see that + +$$ +\frac{d^2}{d v^2}{\mathcal +L}(\mu_v)|_{v=0}=\frac{d^2}{du^2}L^{\bar \tau}_x(\mu(u))|_{u=0}. +$$ + +Hence, the assumption that we have equality in (6.11) implies that -$$\frac{d^2}{d v^2}{\mathcal -L}(\mu_v)|_{v=0}=\frac{d^2}{d u^2}{\mathcal L}(\tilde\gamma_u)|_{u=0}.$$ + +$$ +\frac{d^2}{d v^2}{\mathcal +L}(\mu_v)|_{v=0}=\frac{d^2}{d u^2}{\mathcal L}(\tilde\gamma_u)|_{u=0}. +$$ Now we extend this one-parameter family to a two-parameter family $\mu(\tau,u,v)$ so @@ -1486,41 +1640,65 @@ $\partial\mu(\tau,0,0)/\partial u=Y(\tau)$. Let $w$ be the variable $u-v$, and let $\widetilde W$ be the tangent vector in this coordinate direction, so that $\widetilde W=\widetilde Y-\widetilde Y'$. We denote by $W$ the restriction of $\widetilde W$ to -$\gamma_{0,0}=\gamma_Z$. By Remark~ the second +$\gamma_{0,0}=\gamma_Z$. By \entryref{2fef81872a01} the second partial derivative of the length of this family in the $u$-direction at $u=v=0$ agrees with the second derivative of the length of the original family $\widetilde \gamma$ in the $u$-direction. -**Claim.** $$\frac{\partial}{\partial v}\frac{\partial}{\partial w}{\mathcal L}(\mu)|_{u=v=0}= + +**Claim.** + +$$ +\frac{\partial}{\partial v}\frac{\partial}{\partial w}{\mathcal L}(\mu)|_{u=v=0}= \frac{\partial}{\partial w}\frac{\partial}{\partial v}{\mathcal -L}(\mu)|_{v=w=0}=0.$$ +L}(\mu)|_{v=w=0}=0. +$$ + -\begin{proof} -Of course, the second partial derivatives are equal. According to -Lemma~ we have -$$\frac{\partial}{\partial v}\frac{\partial}{\partial w}{\mathcal L}(\mu)|_{v=w=0} +**Proof.** Of course, the second partial derivatives are equal. According to +\entryref{f48558c60ca2} we have + +$$ +\frac{\partial}{\partial v}\frac{\partial}{\partial w}{\mathcal L}(\mu)|_{v=w=0} =2\sqrt{\bar\tau} \widetilde Y'(\bar\tau)\langle \widetilde W(\bar\tau),X(\bar\tau)\rangle-\int_{\tau_1}^{\bar\tau}2\sqrt{\tau}\langle -Jac(\widetilde Y'),W\rangle d\tau.$$ Since $W(\bar\tau)=0$ and +\mathit{Jac}(\widetilde Y'),W\rangle d\tau. +$$ + + Since $W(\bar\tau)=0$ and since $\nabla_{\widetilde Y'}(\widetilde W)=\nabla_W(\widetilde Y')$, we see that the boundary term in the above expression vanishes. The integral vanishes since $\widetilde Y'$ is an ${\mathcal L}$-Jacobi field. -If Inequality~() is an equality, then -$$\frac{\partial^2}{\partial v^2}{\mathcal L}(\mu)|_{u=v=0}=\frac{\partial^2}{\partial u^2}{\mathcal -L}(\mu)|_{u=v=0}.$$ We write $\partial/\partial u=\partial/\partial +If Inequality (6.11) is an equality, then + +$$ +\frac{\partial^2}{\partial v^2}{\mathcal L}(\mu)|_{u=v=0}=\frac{\partial^2}{\partial u^2}{\mathcal +L}(\mu)|_{u=v=0}. +$$ + + We write $\partial/\partial u=\partial/\partial v+\partial/\partial w$. Expanding out the right-hand side and canceling the common terms gives -$$0=\left(\frac{\partial}{\partial v}\frac{\partial}{\partial w}+ + +$$ +0=\left(\frac{\partial}{\partial v}\frac{\partial}{\partial w}+ \frac{\partial}{\partial w}\frac{\partial}{\partial v}+\frac{\partial^2}{\partial w^2}\right){\mathcal -L}(\mu)|_{u=v=0}.$$ The previous claim tells us that the first two +L}(\mu)|_{u=v=0}. +$$ + + The previous claim tells us that the first two terms on the right-hand side of this equation vanish, and hence we conclude -$$\frac{\partial^2}{\partial w^2}{\mathcal L}(\mu)|_{u=v=0}=0$$ + +$$ +\frac{\partial^2}{\partial w^2}{\mathcal L}(\mu)|_{u=v=0}=0 +$$ + Since $W$ vanishes at both endpoints this implies, according to -Proposition~, that $\widetilde W(\tau,0,0)=0$ for all +\entryref{26b7f69e803e}, that $\widetilde W(\tau,0,0)=0$ for all $\tau$, or in other words $Y(\tau)=\widetilde Y'(\tau,0,0)$ for all $\tau$. Of course by construction $\widetilde Y'(\tau,0,0)$ is an ${\mathcal L}$-Jacobi field. This shows that equality holds only if @@ -1531,27 +1709,25 @@ Conversely, if the adapted vector field $Y(\tau)$ is also an ${\mathcal L}$-Jacobi field, then inequality between the second variations at the beginning of the proof is an equality. In the rest of the argument we dealt only with equalities. Hence, in this case -Inequality~() is an equality. +Inequality (6.11) is an equality. -This shows that we have equality in~() if and only if +This shows that we have equality in (6.11) if and only if the adapted vector field $Y(\tau)$ is also an ${\mathcal L}$-Jacobi field. -\end{proof} -### Inequalities for $\triangle L_x^{\bar\tau -$} +### Inequalities for $\triangle L_x^{\bar\tau}$ The inequalities for the Hessian of $L_x^{\bar\tau}$ lead to -inequalities for $\triangle L_x^{\bar\tau}$$} which we establish in this section. Here is the +inequalities for $\triangle L_x^{\bar\tau}$ which we establish in this section. Here is the main result. -**Proposition.** Suppose that $q\in {\mathcal U}_x(\bar\tau)$, that $Z\in \widetilde + +**Proposition.** Suppose that $q\in {\mathcal U}_x(\bar\tau)$, that $Z\in \widetilde {\mathcal U}_x(\bar\tau)$ is the pre-image of $q$ and that $\gamma_Z$ is the ${\mathcal L}$-geodesic determined by $Z$. Then $$ - - \triangle L_x^{\bar\tau}(q) \leq +\triangle L_x^{\bar\tau}(q) \leq \frac{n}{\sqrt{\bar\tau}-\sqrt{\tau_1}} -2\sqrt{\bar\tau}R(q) - \frac{1}{(\sqrt{\bar\tau}-\sqrt{\tau_1})^2}{\mathcal K}_{\tau_1}^{\bar\tau}(\gamma_Z), @@ -1561,189 +1737,255 @@ where, for any path $\gamma$ parameterized by backward time on the interval $[\tau_1,\bar\tau]$ taking value $x$ at $\tau=\tau_1$ we define -$${\mathcal K}_{\tau_1}^{\bar\tau}(\gamma)= \int_{\tau_1}^{\bar\tau} -\sqrt{\tau}(\sqrt{\tau}-\sqrt{\tau_1})^2H(X)d\tau,$$_{\tau_1}^{\bar\tau}$} with +$$ +{\mathcal K}_{\tau_1}^{\bar\tau}(\gamma)= \int_{\tau_1}^{\bar\tau} +\sqrt{\tau}(\sqrt{\tau}-\sqrt{\tau_1})^2H(X)d\tau, $$ + with + +$$ H(X)= -\frac{\partial R}{\partial \tau} - \frac{1}{\tau}R - 2\langle -\nabla R,X\rangle + 2Ric(X,X), +\nabla R,X\rangle + 2\mathit{Ric}(X,X), $$ + where $X$ is the horizontal projection of $\gamma'(\tau)$. Furthermore, -Inequality~() is an equality if and only if for every +Inequality (6.14) is an equality if and only if for every $Y\in T_q(M_{T-\bar\tau})$ the unique adapted vector field $Y(\tau)$ along $\gamma$ satisfying $Y(\bar\tau)=Y$ is an ${\mathcal L}$-Jacobi field. In this case -$$Ric+\frac{1}{2\sqrt{\bar\tau}}Hess(L_x^{\bar\tau}) -=\frac{1}{2\sqrt{\bar\tau}(\sqrt{\bar\tau}-\sqrt{\tau_1})}G(T-\bar\tau).$$ + +$$ +\mathit{Ric}+\frac{1}{2\sqrt{\bar\tau}}\mathit{Hess}(L_x^{\bar\tau}) +=\frac{1}{2\sqrt{\bar\tau}(\sqrt{\bar\tau}-\sqrt{\tau_1})}G(T-\bar\tau). +$$ + **Proof.** Choose an orthonormal basis $\{Y_{\alpha}\}$ for $T_q(M_{T-\bar\tau})$. For each $\alpha$, extend $\{Y_{\alpha}\}$ to an adapted vector field along the $ \mathcal{L}$-geodesic $\gamma_Z$ by solving -$$\nabla_{X}Y_{\alpha} = \frac{1}{2\sqrt{\tau}(\sqrt{\tau}-\sqrt{\tau_1})}Y_{\alpha} --Ric(Y_{\alpha},\cdot)^*.$$ As in Equation~(), -we have $$ +\nabla_{X}Y_{\alpha} = \frac{1}{2\sqrt{\tau}(\sqrt{\tau}-\sqrt{\tau_1})}Y_{\alpha} +-\mathit{Ric}(Y_{\alpha},\cdot)^*. +$$ + + As in Equation (6.10), +we have $$ \begin{aligned} \frac{d}{d\tau}\langle Y_{\alpha},Y_{\beta}\rangle &= \langle \nabla_{X}Y_{\alpha},Y_{\beta}\rangle + \langle -\nabla_{X}Y_{\beta}, Y_{\alpha}\rangle + 2Ric(Y_{\alpha},Y_{\beta}) +\nabla_{X}Y_{\beta}, Y_{\alpha}\rangle + 2\mathit{Ric}(Y_{\alpha},Y_{\beta}) \\&= \frac{1}{\sqrt{\tau}(\sqrt{\tau}-\sqrt{\tau_1})}\langle Y_{\alpha},Y_{\beta}\rangle . \end{aligned} $$ + By integrating we get + +$$ +\langle Y_{\alpha},Y_{\beta}\rangle (\tau) = +\frac{(\sqrt{\tau}-\sqrt{\tau_1})^2}{(\sqrt{\bar\tau}-\sqrt{\tau_1})^2}\delta_{\alpha\beta}. $$ - By integrating we get $$\langle Y_{\alpha},Y_{\beta}\rangle (\tau) = -\frac{(\sqrt{\tau}-\sqrt{\tau_1})^2}{(\sqrt{\bar\tau}-\sqrt{\tau_1})^2}\delta_{\alpha\beta}.$$ To simplify the notation we set -$$I(\tau)=\frac{\sqrt{\bar\tau}-\sqrt{\tau_1}}{\sqrt{\tau}-\sqrt{\tau_1}}$$ + +$$ +I(\tau)=\frac{\sqrt{\bar\tau}-\sqrt{\tau_1}}{\sqrt{\tau}-\sqrt{\tau_1}} +$$ + and $W_\alpha(\tau)=I(\tau)Y_\alpha(\tau)$. Then $\{W_{\alpha}(\tau)\}_\alpha$ form an orthonormal basis at $\tau$. Consequently, summing -Inequality~() over $\alpha$ gives +Inequality (6.13) over $\alpha$ gives $$ - \triangle L_x^{\bar\tau}(q)\le \frac{n}{\sqrt{\bar\tau}-\sqrt{\tau_1}} -2\sqrt{\bar\tau}R(q)-\sum_\alpha\int_{\tau_1}^{\bar\tau}\sqrt{\tau}H(X,Y_\alpha)d\tau. $$ -To establish Inequality~() it remains to prove the +To establish Inequality (6.14) it remains to prove the following claim. + **Claim.** -$$\sum_\alpha -H(X,Y_\alpha)=\frac{(\sqrt{\tau}-\sqrt{\tau_1})^2}{(\sqrt{\bar\tau}-\sqrt{\tau_1})^2}H(X).$$ - -\begin{proof} -To prove the claim we sum Equation~() giving $$ +\sum_\alpha +H(X,Y_\alpha)=\frac{(\sqrt{\tau}-\sqrt{\tau_1})^2}{(\sqrt{\bar\tau}-\sqrt{\tau_1})^2}H(X). +$$ + + +**Proof.** To prove the claim we sum Equation (6.12) giving $$ \begin{aligned} I^2(\tau) \sum_{\alpha}H(X,Y_{\alpha}) &= & \sum_\alpha H(X,W_\alpha) \\ -& = & -\triangle R + 2Ric(X,X) - 4\langle \nabla R,X\rangle -+4\sum_{\alpha}(\nabla_{W_\alpha} Ric)(W_{\alpha},X) \\ & & --2\sum_{\alpha}Ric_{\tau}( W_{\alpha},W_{\alpha}) +2|Ric|^{2} -\frac{1}{\tau}R. +& = & -\triangle R + 2\mathit{Ric}(X,X) - 4\langle \nabla R,X\rangle ++4\sum_{\alpha}(\nabla_{W_\alpha} \mathit{Ric})(W_{\alpha},X) \\ & & +-2\sum_{\alpha}\mathit{Ric}_{\tau}( W_{\alpha},W_{\alpha}) +2|\mathit{Ric}|^{2} -\frac{1}{\tau}R. \end{aligned} $$ +Taking the trace of the second Bianchi identity, we get + +$$ +\sum_{\alpha}(\nabla_{W_\alpha} +\mathit{Ric})(W_{\alpha},X) =\frac{1}{2}\langle \nabla R,X\rangle . $$ -Taking the trace of the second Bianchi identity, we get -$$\sum_{\alpha}(\nabla_{W_\alpha} -Ric)(W_{\alpha},X) =\frac{1}{2}\langle \nabla R,X\rangle .$$ In addition -by~(), recalling that $\partial R/\partial\tau=-\partial R/\partial + In addition +by (3.5), recalling that $\partial R/\partial\tau=-\partial R/\partial t$, we have -$$\frac{\partial R}{\partial \tau} = -\triangle R - 2|Ric|^{2}.$$ -On the other hand, $$\frac{\partial R}{\partial\tau} = -\partial(g^{ij}R_{ij})/\partial{\tau}= -2|Ric|^{2} + \sum_{\alpha}\frac{\partialRic}{\partial\tau}(W_{\alpha}, W_{\alpha}),$$ and so -$\sum_{\alpha}\frac{\partialRic}{\partial\tau}(W_{\alpha},W_{\alpha}) = -\triangle R$. Putting + +$$ +\frac{\partial R}{\partial \tau} = -\triangle R - 2|\mathit{Ric}|^{2}. +$$ + +On the other hand, + +$$ +\frac{\partial R}{\partial\tau} = +\partial(g^{ij}R_{ij})/\partial{\tau}= -2|\mathit{Ric}|^{2} + \sum_{\alpha}\frac{\partial\mathit{Ric}}{\partial\tau}(W_{\alpha}, W_{\alpha}), +$$ + + and so +$\sum_{\alpha}\frac{\partial\mathit{Ric}}{\partial\tau}(W_{\alpha},W_{\alpha}) = -\triangle R$. Putting all this together gives -$$I^2(\tau)\sum_{\alpha}H(X,Y_{\alpha})= H(X).$$ -Clearly, Inequality~() follows immediately from -Inequality~() and the claim. The last statement of -Proposition~ follows directly from the last statement of -Proposition~ and Lemma~. This completes -the proof of Proposition~. -\end{proof} +$$ +I^2(\tau)\sum_{\alpha}H(X,Y_{\alpha})= H(X). +$$ + +Clearly, Inequality (6.14) follows immediately from +Inequality (6.16) and the claim. The last statement of +\entryref{882677eecdc1} follows directly from the last statement of +\entryref{88d4ef67440f} and \entryref{72949e6350d4}. This completes +the proof of \entryref{882677eecdc1}. ## Reduced length We introduce the reduced length function both on the tangent space and on space-time. The reason that the reduced length $l_x$ is easier to work with is that it is scale invariant when $\tau_1=0$. -Throughout Section~ we fix $x\in{\mathcal M}$ with -$**t**(x)=T-\tau_1$. We shall always suppose that $T-\tau_1$ is +Throughout Section 6.5 we fix $x\in{\mathcal M}$ with +$\mathbf{t}(x)=T-\tau_1$. We shall always suppose that $T-\tau_1$ is greater than the initial time of the generalized Ricci flow. ### The reduced length function $l_x$ on space-time + **Definition.** We define - the $\mathcal{L}$-*reduced length*$-length!reduced} (from $x$) -$$l_x\colon {\mathcal U}_x\to \Ar$$ -by setting $$l_x(q)= \frac{L_x(q)}{2\sqrt{\tau}},$$ where $\tau=T-**t**(q)$. + the $\mathcal{L}$-*reduced length* (from $x$) + +$$ +l_x\colon {\mathcal U}_x\to \Ar +$$ + +by setting + +$$ +l_x(q)= \frac{L_x(q)}{2\sqrt{\tau}}, +$$ + + where $\tau=T-\mathbf{t}(q)$. We denote by $l_x^\tau$ the restriction of $l_x$ to the slice ${\mathcal U}_x(\tau)$. + In order to understand the differential inequalities that $l_x$ satisfies, we first need to introduce a quantity closely related to the function ${\mathcal K}^{\bar\tau}_{\tau_1}$ defined in -Proposition~. +\entryref{882677eecdc1}. + **Definition.** For any ${\mathcal L}$-geodesic $\gamma$ parameterized by $[\tau_1,\bar\tau]$ we define -$$K_{\tau_1}^{\bar\tau}(\gamma)= -\int_{\tau_1}^{\bar\tau}\tau^{3/2}H(X)d\tau.$$^{\bar\tau}$|ii} + +$$ +K_{\tau_1}^{\bar\tau}(\gamma)= +\int_{\tau_1}^{\bar\tau}\tau^{3/2}H(X)d\tau. +$$ + In the special case when $\tau_1=0$ we denote this integral by -$K^{\bar\tau}(\gamma)$$|ii}. +$K^{\bar\tau}(\gamma)$. + The following is immediate from the definitions. -**Lemma.** -For any ${\mathcal L}$-geodesic $\gamma$ defined on $[0,\bar\tau]$ both + +**Lemma.** For any ${\mathcal L}$-geodesic $\gamma$ defined on $[0,\bar\tau]$ both $K^{\bar\tau}_{\tau_1}(\gamma)$ and ${\mathcal K}^{\bar\tau}_{\tau_1}(\gamma)$ are continuous in $\tau_1$ and at $\tau_1=0$ they take the same value. Also, -$$\left(\frac{\tau_1}{\bar\tau}\right)^{3/2}\left(R(\gamma(\tau_1))+|X(\tau_1)|^2\right)$$ + +$$ +\left(\frac{\tau_1}{\bar\tau}\right)^{3/2}\left(R(\gamma(\tau_1))+|X(\tau_1)|^2\right) +$$ + is continuous for all $\tau_1> 0$ and has limit $0$ as $\tau_1\rightarrow 0$. Here, as always, $X(\tau_1)$ is the horizontal component of $\gamma'$ at $\tau=\tau_1$. + -**Lemma.** Let $q\in {\mathcal + +**Lemma.** Let $q\in {\mathcal U}_x(\bar\tau)$, let $Z\in \widetilde {\mathcal U}_x$ be the pre-image of $q$ and let $\gamma_Z$ be the ${\mathcal L}$-geodesic determined by $Z$. Then we have $$ - \bar\tau^{-\frac{3}{2}}K^{\bar\tau}_{\tau_1}(\gamma_Z)=\frac{ l_x(q)}{\bar\tau}-(R(q) +|X(\bar\tau)|^2)+\left(\frac{\tau_1}{\bar\tau}\right)^{3/2}\left(R(x)+|X(\tau_1)|^2\right). $$ -^{\bar\tau}$|(} In the case when $\tau_1=0$, the last term on the right-hand side of -Equation~() vanishes. +Equation (6.17) vanishes. + **Proof.** Using the ${\mathcal L}$-geodesic equation and the definition of $H$ we have -$$ - $$ \begin{aligned} -\lefteqn{\frac{d}{d\tau}(R(\gamma_Z(\tau))+ \abs{X(\tau)}^{2})} \\ & +\frac{d}{d\tau}(R(\gamma_Z(\tau))+ \abs{X(\tau)}^{2}) \\ & = & \frac{\partial R}{\partial\tau}(\gamma_Z(\tau)) + \langle \nabla R(\gamma_Z(\tau)), X(\tau)\rangle + 2\langle \nabla_{X}X(\tau),X(\tau)\rangle \\ -& & + 2Ric(X(\tau),X(\tau)) +& & + 2\mathit{Ric}(X(\tau),X(\tau)) \\&= & \frac{\partial R}{\partial\tau}(\gamma_Z(\tau)) + 2X(\tau)(R) - \frac{1}{\tau}\abs{X(\tau)}^{2} - -2Ric(X(\tau),X(\tau)) \\&= &-H(X(\tau)) +2\mathit{Ric}(X(\tau),X(\tau)) \\&= &-H(X(\tau)) -\frac{1}{\tau}(R(\gamma_Z(\tau)+ \abs{X(\tau)}^{2}). \end{aligned} $$ -$$ - Thus -$$\frac{d}{d\tau}(\tau^{\frac{3}{2}}(R(\gamma_Z(\tau)+\abs{X(\tau)}^{2}) )= + +$$ +\frac{d}{d\tau}(\tau^{\frac{3}{2}}(R(\gamma_Z(\tau)+\abs{X(\tau)}^{2}) )= \frac{1}{2}\sqrt{\tau}(R(\gamma_Z(\tau)+\abs{X(\tau)}^{2}) - -\tau^{\frac{3}{2}}H(X(\tau)).$$ Integration from $\tau_1$ to $\bar\tau$ gives -$$\bar\tau^{3/2}\left(R(q))+|X(\bar\tau)|^2\right)-\tau_1^{3/2}(R(x)+|X(\tau_1)|^2) -=\frac{L_x^{\bar\tau}(q)}{2}-K_{\tau_1}^{\bar\tau}(\gamma_Z),$$ -which is equivalent to Equation~(). In the case when +\tau^{\frac{3}{2}}H(X(\tau)). +$$ + + Integration from $\tau_1$ to $\bar\tau$ gives + +$$ +\bar\tau^{3/2}\left(R(q))+|X(\bar\tau)|^2\right)-\tau_1^{3/2}(R(x)+|X(\tau_1)|^2) +=\frac{L_x^{\bar\tau}(q)}{2}-K_{\tau_1}^{\bar\tau}(\gamma_Z), +$$ + +which is equivalent to Equation (6.17). In the case when $\tau_1=0$, the last term on the right-hand side vanishes since -$$lim_{\tau\rightarrow 0}\tau^{3/2}|X(\tau)|^2=0.$$ + +$$ +\mathit{lim}_{\tau\rightarrow 0}\tau^{3/2}|X(\tau)|^2=0. +$$ Now we come to the most general of the differential inequalities for $l_x$ that will be so important in what follows. Whenever the @@ -1752,14 +1994,12 @@ $\left(\frac{\tau_1}{\bar\tau}\right)^{3/2}\left(R(x)+|X(\tau_1)|^2\right)$ appears in a formula, it is interpreted to be zero in the case when $\tau_1=0$. -**Lemma.** -For any $q\in{\mathcal U}_x(\bar\tau)$, let $Z\in + +**Lemma.** For any $q\in{\mathcal U}_x(\bar\tau)$, let $Z\in \widetilde{\mathcal U}_x(\bar\tau)$ be the pre-image of $q$ and let $\gamma_Z$ be the ${\mathcal L}$-geodesic determined by $Z$. Then we have -$$ - $$ \begin{aligned} \frac{\partial l_x}{\partial \tau}(q) & = & @@ -1775,36 +2015,36 @@ L^{\bar\tau}_x(q)\le \frac{n}{2\sqrt{\bar\tau}(\sqrt{\bar\tau}-\sqrt{\tau_1})} K}_{\tau_1}^{\bar\tau}(\gamma_Z)}{2\sqrt{\bar\tau}(\sqrt{\bar\tau}-\sqrt{\tau_1})^{2}}. \end{aligned} $$ + + +**Proof.** It follows immediately from Equation (6.8) that $$ -_{\tau_1}^{\bar\tau}$} +\frac{\partial l_x}{\partial \tau}=R-\frac{1}{2}(R+|X|^2)-\frac{l_x}{2\tau}. +$$ -**Proof.** It follows immediately from Equation~() that -$$\frac{\partial l_x}{\partial \tau}=R-\frac{1}{2}(R+|X|^2)-\frac{l_x}{2\tau}.$$ -Using Equation~() this gives the first equality stated +Using Equation (6.17) this gives the first equality stated in the lemma. - It follows immediately from Corollary~ that + It follows immediately from \entryref{033786a89c6d} that $\nabla l_x^\tau=X(\tau)$ and hence $|\nabla -l_x^\tau|^2=|X(\tau)|^2$. From this and Equation~() the +l_x^\tau|^2=|X(\tau)|^2$. From this and Equation (6.17) the second equation follows. The last inequality is immediate from -Proposition~. +\entryref{882677eecdc1}. When $\tau_1=0$, the last terms on the right-hand sides of the first two equations vanish, since the last term on the right-hand side of -Equation~() vanishes in this case. +Equation (6.17) vanishes in this case. When $\tau_1=0$, which is the case of main interest, all these formulas simplify and we get: -**Theorem.** -Suppose that $x\in M_T$ so that $\tau_1=0$. For any $q\in{\mathcal + +**Theorem.** Suppose that $x\in M_T$ so that $\tau_1=0$. For any $q\in{\mathcal U}_x(\bar\tau)$, let $Z\in \widetilde{\mathcal U}_x(\bar\tau)$ be the pre-image of $q$ and let $\gamma_Z$ be the ${\mathcal L}$-geodesic determined by $Z$. As usual, let $X(\tau)$ be the horizontal projection of $\gamma_Z'(\tau)$. Then we have -$$ - $$ \begin{aligned} \frac{\partial l_x}{\partial \tau}(q) & = & @@ -1818,72 +2058,117 @@ R(q)-\frac{l_x(q)}{\bar\tau}+\frac{K^{\bar\tau}(\gamma_Z)}{2\bar\tau^{3/2}} \bar\tau}-R(q)-\frac{K^{\bar\tau}(\gamma_Z)}{2\bar\tau^{3/2}}. \end{aligned} $$ - -$$ + **Proof.** This is immediate from the formulas in the previous lemma. Now let us reformulate the differential inequalities in -Theorem~ in a way that will be useful later. +\entryref{4917e81d7684} in a way that will be useful later. -**Corollary.** -Suppose that $x\in M_T$ so that $\tau_1=0$. Then for $q\in{\mathcal + +**Corollary.** Suppose that $x\in M_T$ so that $\tau_1=0$. Then for $q\in{\mathcal U}^{\bar\tau}_x$ we have -$$\frac{\partial l_x}{\partial \tau}(q)+\triangle l_x^{\bar\tau}(q)\le \frac{(n/2)-l_x^{\bar\tau}(q)}{\bar\tau}.$$ -$$\frac{\partial l_x}{\partial \tau}(q)-\triangle l_x^{\bar\tau}(q)+|\nabla l_x^{\bar\tau}(q)|^2-R(q)+\frac{n}{2\bar\tau}\ge 0 .$$ -$$2\triangle l_x^{\bar\tau}(q)-|\nabla l_x^{\bar\tau}(q)|^2+R(q)+\frac{l_x^{\bar\tau}(q)-n}{\bar\tau} -\le 0.$$ -In fact, setting $$\delta=\frac{n}{2 +$$ +\frac{\partial l_x}{\partial \tau}(q)+\triangle l_x^{\bar\tau}(q)\le \frac{(n/2)-l_x^{\bar\tau}(q)}{\bar\tau}. +$$ + +$$ +\frac{\partial l_x}{\partial \tau}(q)-\triangle l_x^{\bar\tau}(q)+|\nabla l_x^{\bar\tau}(q)|^2-R(q)+\frac{n}{2\bar\tau}\ge 0 . +$$ + +$$ +2\triangle l_x^{\bar\tau}(q)-|\nabla l_x^{\bar\tau}(q)|^2+R(q)+\frac{l_x^{\bar\tau}(q)-n}{\bar\tau} +\le 0. +$$ + +In fact, setting + +$$ +\delta=\frac{n}{2 \bar\tau}-R(q)-\frac{K^{\bar\tau}(\gamma_Z)}{2\bar\tau^{3/2}}-\triangle -l_x^{\bar\tau}(q),$$ then $\delta\ge 0$ and -$$\frac{\partial l_x}{\partial \tau}(q)-\triangle l_x^{\bar\tau}(q) -+|\nabla l_x^{\bar\tau}(q)|^2-R(q)+\frac{n}{2\bar\tau}=\delta$$ -$$2\triangle l_x^{\bar\tau}(q)-|\nabla l_x^{\bar\tau}(q)|^2+R(q) -+\frac{l_x^{\bar\tau}(q)-n}{\bar\tau}=-2\delta.$$ +l_x^{\bar\tau}(q), +$$ -^{\bar\tau}$|)} + then $\delta\ge 0$ and + +$$ +\frac{\partial l_x}{\partial \tau}(q)-\triangle l_x^{\bar\tau}(q) ++|\nabla l_x^{\bar\tau}(q)|^2-R(q)+\frac{n}{2\bar\tau}=\delta +$$ + +$$ +2\triangle l_x^{\bar\tau}(q)-|\nabla l_x^{\bar\tau}(q)|^2+R(q) ++\frac{l_x^{\bar\tau}(q)-n}{\bar\tau}=-2\delta. +$$ + ### The tangential version $\widetilde l$ of the reduced length function For any path $\gamma\colon [\tau_1,\bar\tau]\to ({\mathcal M},G)$ parameterized by backward time we define -$$l(\gamma)=\frac{1}{2\sqrt{\bar\tau}}{\mathcal L}(\gamma).$$ + +$$ +l(\gamma)=\frac{1}{2\sqrt{\bar\tau}}{\mathcal L}(\gamma). +$$ + This leads immediately to a reduced length on $\widetilde {\mathcal U}_x$. + **Definition.** We define $\widetilde l\colon \widetilde {\mathcal U}_x\to -\Ar$ by -$$\widetilde l(Z,\tau)=\frac{\widetilde L(Z,\tau)}{2\sqrt{\tau}}=l(\gamma_Z|_{[\tau_1,\bar\tau]}).$$ +\Ar$ by + +$$ +\widetilde l(Z,\tau)=\frac{\widetilde L(Z,\tau)}{2\sqrt{\tau}}=l(\gamma_Z|_{[\tau_1,\bar\tau]}). +$$ + At first glance it may appear that the computations of the gradient and $\tau$-derivatives for $l_x$ pass immediately to those for $\widetilde l$. For the spatial derivative this is correct, but for the $\tau$-derivative it is not true. As the computation below shows, the $\tau$-derivatives of $\widetilde l$ and $l_x$ do not -agree under the identification ${\mathcal L}exp_x$. The reason +agree under the identification ${\mathcal L}\mathit{exp}_x$. The reason is that this identification does not line up the $\tau$-vector field in the domain with $-\partial/\partial t$ in the range. So it is an entirely different computation with a different answer. + **Lemma.** -$$\frac{\partial \widetilde + +$$ +\frac{\partial \widetilde l(Z,\tau)}{\partial \tau}=\frac{1}{2}\left(R(\gamma_Z(\tau))+X(\tau)|^2\right)-\frac{\widetilde -l(Z,\tau)}{2\tau}.$$ +l(Z,\tau)}{2\tau}. +$$ + **Proof.** By the Fundamental Theorem of Calculus -$$\frac{\partial}{\partial\tau}\widetilde L(Z,\tau)=\sqrt{\tau}\left(R(\gamma_z(\tau))+|X(\tau)|^2\right).$$ + +$$ +\frac{\partial}{\partial\tau}\widetilde L(Z,\tau)=\sqrt{\tau}\left(R(\gamma_z(\tau))+|X(\tau)|^2\right). +$$ + Thus, -$$\frac{\partial}{\partial\tau}\widetilde l(Z,\tau)=\frac{1}{2}\left(R(\gamma_z(\tau))+|X(\tau)|^2\right) --\frac{\widetilde l(Z,\tau)}{2\tau}.$$ -**Corollary.** Suppose that $x\in M_T$ so that +$$ +\frac{\partial}{\partial\tau}\widetilde l(Z,\tau)=\frac{1}{2}\left(R(\gamma_z(\tau))+|X(\tau)|^2\right) +-\frac{\widetilde l(Z,\tau)}{2\tau}. +$$ + + +**Corollary.** Suppose that $x\in M_T$ so that $\tau_1=0$. Then -$$\frac{\partial}{\partial\tau}\widetilde -l(Z,\tau)=-\frac{K^\tau(\gamma_Z)}{2\tau^{\frac{3}{2}}}.$$ -**Proof.** This is immediate from Lemma~ and Lemma~ (after the +$$ +\frac{\partial}{\partial\tau}\widetilde +l(Z,\tau)=-\frac{K^\tau(\gamma_Z)}{2\tau^{\frac{3}{2}}}. +$$ + + +**Proof.** This is immediate from \entryref{5e142c14df9d} and \entryref{7a68e2160e8a} (after the latter is rewritten using $\widetilde L$ instead of $L_x$). ## Local Lipschitz estimates for $l_x$ @@ -1893,45 +2178,51 @@ Lipschitz properties of $l_x$, or equivalently $L_x$. Of course, these are the analogues of the fact that in Riemannian geometry the distance function from a point is Lipschitz. The proof of the Lipschitz property given here is based on the exposition in -. In Section~, we fix $x\in +[Ye]. In Section 6.6, we fix $x\in M_{T-\tau_1}\subset {\mathcal M}$. ### Statement and corollaries + **Definition.** Let $({\mathcal M},G)$ be a generalized Ricci flow and let $x\in M_{T-\tau_1}\subset {\mathcal M}$. The reduced length function $l_x$ is defined on the subset of ${\mathcal M}$ consisting of all points $y\in {\mathcal M}$ for which there is a minimizing ${\mathcal L}$-geodesic from $x$ to $y$. The value $l_x(y)$ is the quotient of ${\mathcal L}$-length of any such minimizing ${\mathcal L}$-geodesic divided by $2\sqrt{\tau}$. + Here is the main result of this subsection. -**Proposition.** -Let $({\mathcal M},G)$ be a generalized Ricci flow and let $x\in + +**Proposition.** Let $({\mathcal M},G)$ be a generalized Ricci flow and let $x\in M_{T-\tau_1}\subset {\mathcal M}$. Let $\epsilon>0$ be given and let $A\subset -{\mathcal M}\cap **t**^{-1}(-\infty,T-\tau_1+\epsilon)$. Suppose that there -is a subset $F\subset {\mathcal M}$ on which $|Ric|$ and $|\nabla R|$ are +{\mathcal M}\cap \mathbf{t}^{-1}(-\infty,T-\tau_1+\epsilon)$. Suppose that there +is a subset $F\subset {\mathcal M}$ on which $|\mathit{Ric}|$ and $|\nabla R|$ are bounded and a neighborhood $\nu(A)$ of $A$ contained in $F$ with the property that for every point $z\in \nu(A)$ there is a minimizing ${\mathcal -L}$-geodesic from $x$ to $z$ contained in $F$. Then $l_x$ is defined on all of +L}$-geodesic from $x$ to $z$ contained in $F$. Then $l_x$ is defined on all of $\nu(A)$. Furthermore, there is a smaller neighborhood $\nu_0(A)\subset \nu(A)$ of $A$ on which $l_x$ is a locally Lipschitz function with respect to the Riemannian metric, denoted $\widehat G$, on ${\mathcal M}$ which is defined as the orthogonal sum of the Riemannian metric $G$ on ${\mathcal H}T{\mathcal M}$ and the metric $dt^2$ on the tangent line spanned by $\chi$. + -**Corollary.** With $A$ and $\nu_0(A)$ as in Proposition~, the -restriction of $l_x$ to $\nu_0(A)\cap M_{T-\bar\tau}$ is a locally + +**Corollary.** With $A$ and $\nu_0(A)$ as in \entryref{db4ced24b05f}, the +restriction of $l_x$ to $\nu_0(A)\cap M_{T-\bar\tau}$ is a locally Lipschitz function with respect to the metric $G_{T-\bar\tau}$. + -### The proof of Proposition~\protect{} +### The proof of Proposition \entryref{db4ced24b05f} -Proposition~ follows from a much more precise, though more +\entryref{db4ced24b05f} follows from a much more precise, though more complicated to state, result. In order to state this more technical result we introduce the following definition. -**Definition.** Let $y\in {\mathcal M}$ with $**t**(y)=t$ and suppose that for + +**Definition.** Let $y\in {\mathcal M}$ with $\mathbf{t}(y)=t$ and suppose that for some $\epsilon>0$ there is an embedding $\iota\colon B(y,t,r)\times (t-\epsilon,t+\epsilon)\to {\mathcal M}$ that is compatible with time and the vector field. Then we denote by $\widetilde @@ -1939,19 +2230,20 @@ P(y,r,\epsilon)\subset {\mathcal M}$ the image of $\iota$. Whenever we introduce $\widetilde P(y,r,\epsilon)\subset {\mathcal M}$ implicitly we are asserting that such an embedding exists. -For $A\subset {\mathcal M}$, if $\widetilde +For $A\subset {\mathcal M}$, if $\widetilde P(a,\epsilon,\epsilon)\subset {\mathcal M}$ exists for every $a\in A$, then we denote by $\nu_\epsilon(A)$ the union over all $a\in A$ of $\widetilde P(a,\epsilon,\epsilon)$. + Now we are ready for the more precise, technical result. -**Proposition.** -Given constants $\epsilon>0$, $\bar\tau_0<\infty$, $l_0<\infty$, + +**Proposition.** Given constants $\epsilon>0$, $\bar\tau_0<\infty$, $l_0<\infty$, and $C_0<\infty$, there are constants $ C<\infty$ and -$0<\delta<\epsilon$ depending only the given constants such that +$0<\delta<\epsilon$ depending only the given constants such that the following holds. Let $({\mathcal M},G)$ be a generalized Ricci -flow and let $x\in{\mathcal M}$ be a point with $**t**(x)=T-\tau_1$. Let $y\in {\mathcal M}$ be a point with $**t**(y)=t=T-\bar\tau$ where $\tau_1+\epsilon\le \bar\tau\le +flow and let $x\in{\mathcal M}$ be a point with $\mathbf{t}(x)=T-\tau_1$. Let $y\in {\mathcal M}$ be a point with $\mathbf{t}(y)=t=T-\bar\tau$ where $\tau_1+\epsilon\le \bar\tau\le \bar\tau_0$. Suppose that there is a minimizing ${\mathcal L}$-geodesic $\gamma$ from $x$ to $y$ with $l(\gamma)\le l_0$. Suppose that the ball $B(y,t,\epsilon)$ has compact closure in @@ -1960,82 +2252,107 @@ M}$ exists and that the sectional curvatures of the restriction of $G$ to this submanifold are bounded by $C_0$. Lastly, suppose that for every point of the form $z\in \widetilde P(y,\delta,\delta)$ there is a minimizing ${\mathcal L}$-geodesic from $x$ to $z$ with -$|Ric|$ and $|\nabla R|$ bounded by $C_0$ along this geodesic. +$|\mathit{Ric}|$ and $|\nabla R|$ bounded by $C_0$ along this geodesic. Then for all $(b,t')\in B(y,t,\delta)\times(t-\delta,t+\delta)$ we have -$$|l_x(y)-l_x(\iota(b,t'))|\le C\sqrt{d_{t}(y,b)^2+|t-t'|^2}.$$ -Before proving Proposition~, let us show how it implies -Proposition~. +$$ +|l_x(y)-l_x(\iota(b,t'))|\le C\sqrt{d_{t}(y,b)^2+|t-t'|^2}. +$$ + + +Before proving \entryref{96ced8c484df}, let us show how it implies +\entryref{db4ced24b05f}. -**Proof.** (that Proposition~ implies -Proposition~) Suppose given $\epsilon>0$, $A$, $\nu(A)$ and $F$ as in -the statement of Proposition~. For each $y\in A$ there is +**Proof.** (that \entryref{96ced8c484df} implies +\entryref{db4ced24b05f}) Suppose given $\epsilon>0$, $A$, $\nu(A)$ and $F$ as in +the statement of \entryref{db4ced24b05f}. For each $y\in A$ there is $0<\epsilon'<\epsilon$ and a neighborhood $\nu'(y)$ with (i) the closure $\overline{\nu}'(y)$ of $\nu'(y)$ being a compact subset of $\nu(A)$ and (ii) for each $z\in \overline{\nu}'(y)$ the parabolic neighborhood $\widetilde P(z,\epsilon',\epsilon')$ exists and has compact closure in $\nu(A)$. It -follows that for every $z\in \overline{\nu}'(A)$, $Rm_G$ is bounded on +follows that for every $z\in \overline{\nu}'(A)$, $\mathit{Rm}_G$ is bounded on $\widetilde P(z,\epsilon',\epsilon')$ and every point of $\widetilde P(z,\epsilon',\epsilon')$ is connected to $x$ by a minimizing ${\mathcal -L}$-geodesic in $F$. Thus, Proposition~, with $\epsilon$ replaced +L}$-geodesic in $F$. Thus, \entryref{96ced8c484df}, with $\epsilon$ replaced by $\epsilon'$, applies to $z$. In particular, $l_x$ is continuous at $z$, and hence is continuous on all of $\overline{\nu}'(y)$. Thus, $l_x$ is bounded on $\overline{\nu}'(y)$. Since we have uniform bounds for the curvature on -$\widetilde P(z,\epsilon',\epsilon')$ according to Proposition~ +$\widetilde P(z,\epsilon',\epsilon')$ according to \entryref{96ced8c484df} there are constants $C<\infty$ and $0<\delta<\epsilon'$ such that for any $z\in \overline{\nu}'(y)$ and any $z'\in \widetilde P(z,\delta,\delta)$, we have -$$|l_x(z)-l_x(z')|\le C|z-z'|_{G(**t**(z)+dt^2}.$$ + +$$ +|l_x(z)-l_x(z')|\le C|z-z'|_{G(\mathbf{t}(z)+dt^2}. +$$ + Since we have a uniform bound for the curvature on $\widetilde P(z,\epsilon',\epsilon')$ independent of $z\in \nu'(y)$, the metrics $\widehat -G=G+dt^2$ and $G(**t**(z))+dt^2$ are uniformly comparable on all of +G=G+dt^2$ and $G(\mathbf{t}(z))+dt^2$ are uniformly comparable on all of $\widetilde P(z,\delta,\delta)$. It follows that there is a constant $C'<\infty$ such that for all $z\in \nu'(y)$ and all $z'\in \widetilde P(z,\delta,\delta)$ we have -$$|l_x(z)-l_x(z')|\le C'|Z-z'|_{\widehat G}.$$ + +$$ +|l_x(z)-l_x(z')|\le C'|Z-z'|_{\widehat G}. +$$ We set $\nu_0(A)=\cup_{y\in A}\nu'(y)$. This is an open neighborhood of $A$ contained in $\nu(A)$ on which $l_x$ is locally Lipschitz with respect to the metric $\widehat G$. -Now we turn to the proof of Proposition~. We begin with +Now we turn to the proof of \entryref{96ced8c484df}. We begin with several preliminary results. -**Lemma.** - Suppose that + +**Lemma.** Suppose that $\gamma$ is an ${\mathcal L}$-geodesic defined on $[\tau_1,\bar\tau]$, and suppose that for all $\tau\in [\tau_1,\bar\tau]$ we have $|\nabla R(\gamma(\tau))|\le C_0$ and -$|Ric(\gamma(\tau))|\le C_0$. Then -$$max_{\tau}\left(\sqrt{\tau}|X_\gamma(\tau)|\right)\le C_1min_{\tau}\left(\sqrt{\tau}|X_\gamma(\tau)|\right)+\frac{(C_1-1)}{2}\sqrt{\bar\tau},$$ +$|\mathit{Ric}(\gamma(\tau))|\le C_0$. Then + +$$ +\mathit{max}_{\tau}\left(\sqrt{\tau}|X_\gamma(\tau)|\right)\le C_1\mathit{min}_{\tau}\left(\sqrt{\tau}|X_\gamma(\tau)|\right)+\frac{(C_1-1)}{2}\sqrt{\bar\tau}, +$$ + where $C_1=e^{2C_0\bar\tau}$. + **Proof.** The geodesic equation in terms of the variable $s$, -Equation~(), gives - -$$ +Equation (6.5), gives $$ \begin{aligned} \frac{d|\gamma'(s)|^2}{ds} & = & 2\langle -\nabla_{\gamma'(s)}\gamma'(s),\gamma'(s)\rangle+4sRic(\gamma'(s),\gamma'(s))\nonumber \\ -& = & 4s^2\langle\nabla R,\gamma'(s)\rangle -4sRic(\gamma'(s),\gamma'(s)). +\nabla_{\gamma'(s)}\gamma'(s),\gamma'(s)\rangle+4s\mathit{Ric}(\gamma'(s),\gamma'(s))\nonumber \\ +& = & 4s^2\langle\nabla R,\gamma'(s)\rangle -4s\mathit{Ric}(\gamma'(s),\gamma'(s)). \end{aligned} $$ +Thus,, by our assumption on $|\nabla R|$ and $|\mathit{Ric}|$ along +$\gamma$, we have + +$$ +\left|\frac{d|\gamma'(s)|^2}{ds}\right| \le +4C_0s^2|\gamma'(s)|+4C_0s|\gamma'(s)|^2. $$ -Thus,, by our assumption on $|\nabla R|$ and $|Ric|$ along -$\gamma$, we have -$$\left|\frac{d|\gamma'(s)|^2}{ds}\right| \le -4C_0s^2|\gamma'(s)|+4C_0s|\gamma'(s)|^2.$$ It follows that -$$\left|\frac{d|\gamma'(s)|}{ds}\right|\le 2C_0s^2+2C_0s|\gamma'(s)|\le 2C_0\bar\tau+2 -C_0\sqrt{\bar\tau}|\gamma'(s)|,$$ and hence that -$$-2C_0\sqrt{\bar\tau}ds\le \frac{d|\gamma'(s)|}{\sqrt{\bar\tau}+|\gamma'(s)|} -\le 2C_0\sqrt{\bar\tau}ds.$$ Suppose that $s_00$, and $l_0<\infty$, there + +**Corollary.** Given $\bar\tau_0<\infty$, $C_0<\infty$, $\epsilon>0$, and $l_0<\infty$, there is a constant $C_2$ depending only on $C_0$, $l_0$, $\epsilon$ and $\bar\tau_0$ such that the following holds. Let $\gamma$ be an ${\mathcal L}$-geodesic defined on $[\tau_1,\bar\tau]$ with $\tau_1+\epsilon\le\bar\tau\le \bar\tau_0$ -and with $|\nabla R(\gamma(\tau))|\le C_0$ and $|Ric(\gamma(\tau))|\le +and with $|\nabla R(\gamma(\tau))|\le C_0$ and $|\mathit{Ric}(\gamma(\tau))|\le C_0$ for all $\tau\in [\tau_1,\bar\tau]$. Suppose also that $ l(\gamma)\le l_0$. Then, we have -$$max_{\tau}\left(\sqrt{\tau}|X_\gamma(\tau)|\right)\le C_2.$$ + +$$ +\mathit{max}_{\tau}\left(\sqrt{\tau}|X_\gamma(\tau)|\right)\le C_2. +$$ + **Proof.** From the definition ${\mathcal L}(\gamma)=\int_{\sqrt{\tau_1}}^{\sqrt{\bar\tau}} -(2s^2R+\frac{1}{2}|\gamma'(s)|^2)ds$. Because of the bound on $|Ric|$ (which implies that $|R|\le 3C_0$) we have -$$\frac{1}{2}\int_{\sqrt{\tau_1}}^{\sqrt{\bar\tau}}|\gamma'(s)|^2ds\le {\mathcal L}(\gamma)+ -2C_0\bar\tau^{3/2}.$$ Thus, -$$(\sqrt{\bar \tau}-\sqrt{\tau_1})min(|\gamma'(s)|^2)\le 2{\mathcal L}(\gamma)+ -4C_0\bar\tau^{3/2}.$$ The bounds $\tau_1+\epsilon\le \bar\tau\le\bar\tau_0$, -then imply that $min|\gamma'(s)|^2\le C''$ for some $C''$ depending on +(2s^2R+\frac{1}{2}|\gamma'(s)|^2)ds$. Because of the bound on $|\mathit{Ric}|$ (which implies that $|R|\le 3C_0$) we have + +$$ +\frac{1}{2}\int_{\sqrt{\tau_1}}^{\sqrt{\bar\tau}}|\gamma'(s)|^2ds\le {\mathcal L}(\gamma)+ +2C_0\bar\tau^{3/2}. +$$ + + Thus, + +$$ +(\sqrt{\bar \tau}-\sqrt{\tau_1})\mathit{min}(|\gamma'(s)|^2)\le 2{\mathcal L}(\gamma)+ +4C_0\bar\tau^{3/2}. +$$ + + The bounds $\tau_1+\epsilon\le \bar\tau\le\bar\tau_0$, +then imply that $\mathit{min}|\gamma'(s)|^2\le C''$ for some $C''$ depending on $C_0,l_0,\epsilon,$ and $\bar\tau_0$. Since $\sqrt{\tau}X_\gamma(\tau)=\frac{1}{2}\gamma'(s)$, we have -$$min_{\tau}\left(\sqrt{\tau}|X_\gamma(\tau)|\right)\le C'$$ for some constant $C'$ depending + +$$ +\mathit{min}_{\tau}\left(\sqrt{\tau}|X_\gamma(\tau)|\right)\le C' +$$ + + for some constant $C'$ depending only on $C_0$, $l_0$, $\epsilon$ and $\bar\tau_0$. The result is now immediate -from Lemma~. +from \entryref{333be2e58780}. -Now we are ready to show that, for $z$ sufficiently close to $y$, +Now we are ready to show that, for $z$ sufficiently close to $y$, the reduced length $l_x(z)$ is bounded above by a constant depending on the curvature bounds, on $l_x(y)$, and on the distance in space-time from $z$ to $y$. -**Lemma.** -Given constants $\epsilon>0$, $\bar\tau_0<\infty$, $C_0<\infty$, + +**Lemma.** Given constants $\epsilon>0$, $\bar\tau_0<\infty$, $C_0<\infty$, and $l_0<\infty$, there are $C_3<\infty$ and $0<\delta_2\le -\epsilon/4$ depending only on the given constants such that the -following holds. Let $y\in {\mathcal M}$ be a point with $**t**(y)=t_0=T-\bar\tau$ where $\tau_1+\epsilon\le \bar\tau\le +\epsilon/4$ depending only on the given constants such that the +following holds. Let $y\in {\mathcal M}$ be a point with $\mathbf{t}(y)=t_0=T-\bar\tau$ where $\tau_1+\epsilon\le \bar\tau\le \bar\tau_0$. Suppose that there is a minimizing ${\mathcal L}$-geodesic $\gamma$ from $x$ to $y$ with $l_x(\gamma)\le l_0$. -Suppose that $|\nabla R|$ and $|Ric|$ are bounded by $C_0$ +Suppose that $|\nabla R|$ and $|\mathit{Ric}|$ are bounded by $C_0$ along $\gamma$. Suppose also that the ball $B(y,t_0,\epsilon)$ has compact closure in $M_{t_0}$ and that there is an embedding -$$\iota\colon B(y,t_0,\epsilon)\times (t_0-\epsilon,t_0+\epsilon)\stackrel + +$$ +\iota\colon B(y,t_0,\epsilon)\times (t_0-\epsilon,t_0+\epsilon)\stackrel {\cong}{\longrightarrow} \tilde P(y,\epsilon,\epsilon)\subset -{\mathcal M}$$ compatible with time and the vector field so that +{\mathcal M} +$$ + + compatible with time and the vector field so that the sectional curvatures of the restriction of $G$ to the image of this embedding are bounded by $C_0$. Then for any point $b\in B(y,t_0,\delta_2)$ and for any $t'\in (t_0-\delta_2,t_0+\delta_2)$ there is a curve $\gamma_1$ from $x$ to the point $z=\iota(b,t')$, parameterized by backward time, such that -$$l(\gamma_1)\le l(\gamma)+C_3\sqrt{d_{t_0}(y,b)^2+|t_0-t'|^2}.$$ + +$$ +l(\gamma_1)\le l(\gamma)+C_3\sqrt{d_{t_0}(y,b)^2+|t_0-t'|^2}. +$$ + **Proof.** Let $C_2$ be the constant depending on $C_0$, $l_0$, $\epsilon$, and -$\bar\tau_0$ from Corollary~, and set -$$C'=\frac{\sqrt{2}}{\sqrt{\epsilon}}C_2.$$ +$\bar\tau_0$ from \entryref{a7c2ebee07a3}, and set + +$$ +C'=\frac{\sqrt{2}}{\sqrt{\epsilon}}C_2. +$$ + Since $\bar\tau\ge \epsilon$, it follows that $\bar\tau-\epsilon/2\ge \epsilon/2$, so that by -Corollary~ we have $|X_\gamma(\tau)|\le C'$ for all +\entryref{a7c2ebee07a3} we have $|X_\gamma(\tau)|\le C'$ for all $\tau\in [\bar\tau-\epsilon/2,\bar\tau]$. Set $0<\delta_0$ sufficiently small (how small depends only on $C_0$) such that for all $(z,t)\in \tilde P(y,\epsilon,\delta_0)$ we have -$$\frac{1}{2}g(z,t)\le g(z,t_0)\le 2g(z,t),$$ and define -$$\delta_2=min\left(\frac{\epsilon}{8},\frac{\epsilon}{8C'},\frac{\delta_0}{4}\right).$$ + +$$ +\frac{1}{2}g(z,t)\le g(z,t_0)\le 2g(z,t), +$$ + + and define + +$$ +\delta_2=\mathit{min}\left(\frac{\epsilon}{8},\frac{\epsilon}{8C'},\frac{\delta_0}{4}\right). +$$ + Let $b\in B(y,t_0,\delta_2)$ and $t'\in (t_0-\delta_2,t_0+\delta_2)$ be given. Set $\alpha=\sqrt{d_{t_0}(y,b)^2+|t_0-t'|^2}$, set $t_1=t_0-2\alpha$, @@ -2124,38 +2480,51 @@ $\alpha<\sqrt{2}\delta_2<\epsilon/4$, so that the norm of the Ricci curvature is bounded by $C'$ on $\iota(B(y,t_0,\epsilon)\times(t_1,t_0+2\alpha))$. -**Claim.** -$\gamma(\tau_1)\in \widetilde P(y,\epsilon,\epsilon)$ and writing + +**Claim.** $\gamma(\tau_1)\in \widetilde P(y,\epsilon,\epsilon)$ and writing $\gamma(\tau_1)=\iota(c,t_1)$ we have $d_{t_0}(c,b)\le (4C'+1)\alpha$. + -\begin{proof} -Since $|X_\gamma(\tau)|\le C'$ for all $\tau\in -[\bar\tau-2\alpha,\bar\tau]$, and $\delta_2\le \delta_0/4$, it +**Proof.** Since $|X_\gamma(\tau)|\le C'$ for all $\tau\in +[\bar\tau-2\alpha,\bar\tau]$, and $\delta_2\le \delta_0/4$, it follows that $2\alpha\le \delta_0$ and hence that -$|X_\gamma(\tau)|_{g(t_0)}\le 2C'$ for all $\tau\in +$|X_\gamma(\tau)|_{g(t_0)}\le 2C'$ for all $\tau\in [\bar\tau-2\alpha,\bar\tau]$. Since $\gamma(\bar\tau)=y$, this implies that -$$d_{t_0}(y,c)\le 4C'\alpha.$$ + +$$ +d_{t_0}(y,c)\le 4C'\alpha. +$$ + The claim then follows from the triangle inequality. Now let $\overline\mu\colon [\bar\tau-2\alpha,T-t']\to B(y,t_0,\epsilon)$ be a shortest $g(t_0)$-geodesic from $c$ to $b$, parameterized at constant $g(t_0)$-speed, and let $\mu$ be the path parameterized by backward time defined by -$$\mu(\tau)=\iota(\overline\mu(\tau),T-\tau)$$ + +$$ +\mu(\tau)=\iota(\overline\mu(\tau),T-\tau) +$$ + for all $\tau\in [\bar\tau-2\alpha,T-t']$. Then the concatenation $\gamma_1=\gamma|_{[\tau_1,\bar\tau-2\alpha]}*\mu$ is a path parameterized by backward time from $x$ to $\iota(b,t')$. + **Claim.** There is a constant $C'_1$ depending only on $C_0$, $C'$, and $\bar\tau_0$ such that -$$l(\gamma_1)\le l(\gamma|_{[\tau_1,\bar\tau-2\alpha]})+C'_1\alpha$$ + +$$ +l(\gamma_1)\le l(\gamma|_{[\tau_1,\bar\tau-2\alpha]})+C'_1\alpha +$$ + **Proof.** First notice that since $\bar\tau=T-t_0$ and $|t'-t_0|\le \alpha$ we have $(T-t')-(\bar\tau-2\alpha)=2\alpha+(t'-t_0)\ge \alpha$. According to -Claim~ this implies that the $g(t_0)$-speed of $\mu$ is at most -$(4C'+1)$, and hence that $|X_{\mu}(\tau)|_{g(T-\tau)}\le 8C'+2$ for all +\entryref{409f236af149} this implies that the $g(t_0)$-speed of $\mu$ is at most +$(4C'+1)$, and hence that $|X_{\mu}(\tau)|_{g(T-\tau)}\le 8C'+2$ for all $\tau\in [\bar\tau-2\alpha,T-t']$. Consequently, $R+|X_\mu|^2$ is bounded above along $\mu$ by a constant $\widetilde C$ depending only on $C'$ and $C_0$. This implies that ${\mathcal L}(\mu)\le \widetilde C\alpha\sqrt{T-t'}$. Of course, @@ -2164,45 +2533,59 @@ of the claim. On the other hand, since $R\ge -3C_0$ in $P(y,\epsilon,\epsilon)$ and $|X|^2\ge 0$, we see that -$${\mathcal L}(\gamma|_{[\tau_1,\bar\tau-2\alpha]})\le {\mathcal L}(\gamma) -+ 6C_0\alpha\sqrt{\bar\tau_0}.$$ Together with the previous claim -this establishes Lemma~. -\end{proof} + +$$ +{\mathcal L}(\gamma|_{[\tau_1,\bar\tau-2\alpha]})\le {\mathcal L}(\gamma) ++ 6C_0\alpha\sqrt{\bar\tau_0}. +$$ + + Together with the previous claim +this establishes \entryref{7d130fba9693}. This is a one-sided inequality which says that the nearby values of -$l_x$ are bounded above in terms of $l_x(y)$, the curvature bounds, +$l_x$ are bounded above in terms of $l_x(y)$, the curvature bounds, and the distance in space-time from $y$. In order to complete the -proof of Proposition~ we must establish inequalities in +proof of \entryref{96ced8c484df} we must establish inequalities in the opposite direction. This requires reversing the roles of the points. -**Proof.** (of Proposition~) +**Proof.** (of \entryref{96ced8c484df}) Let $\delta_2$ and $C_3$ be the constants given by -Lemma~ associated to $\epsilon/2$, $\bar\tau_0$, $C_0$, +\entryref{7d130fba9693} associated to $\epsilon/2$, $\bar\tau_0$, $C_0$, and $l_0$. We shall choose $C\ge C_3$ and $\delta\le \delta_2$ so -that by Lemma~ we will automatically have -$$l_x(\iota(b,t'))\le l_x(y)+C_3\sqrt{d_{t_0}(y,b)^2+|t_0-t'|^2} -\le l_x(y)+C\sqrt{d_{t_0}(y,b)^2+|t_0-t'|^2}$$ for all +that by \entryref{7d130fba9693} we will automatically have + +$$ +l_x(\iota(b,t'))\le l_x(y)+C_3\sqrt{d_{t_0}(y,b)^2+|t_0-t'|^2} +\le l_x(y)+C\sqrt{d_{t_0}(y,b)^2+|t_0-t'|^2} +$$ + + for all $\iota(b,t')\in P(y,\delta,\delta)$. It remains to choose $C$ and $\delta$ so that -$$l_x(y)\le l_x(\iota(b,t'))+C\sqrt{d_{t_0}(y,b)^2+|t_0-t'|^2}.$$ + +$$ +l_x(y)\le l_x(\iota(b,t'))+C\sqrt{d_{t_0}(y,b)^2+|t_0-t'|^2}. +$$ + Let $\delta'_2$ and $ C'_3$ be the constants given by -Lemma~ for the following set of input constants: +\entryref{7d130fba9693} for the following set of input constants: $C_0'=C_0$, $\bar\tau_0$ replaced by $\bar\tau_0'=\bar\tau_0+\epsilon/2$, and $l_0$ replaced by $l_0'=l_0+\sqrt{2}C_3\delta_2$, and $\epsilon$ replaced by -$\epsilon'=\epsilon/4$. Then set $C=max(2C_3',C_3)$. +$\epsilon'=\epsilon/4$. Then set $C=\mathit{max}(2C_3',C_3)$. Let $z=\iota(b,t')\in \widetilde P(y,\delta,\delta)$. + **Claim.** For $\delta$ sufficiently small (how small depending on $\delta_2$ and $\delta'_2$) we have $B(z,t',\epsilon/4)\subset B(y,t_0,\epsilon)$. + -\begin{proof} -Since $|t_0-t'|<\delta\le \delta_2$, and by construction -$\delta_2<\delta_0$, it follows that for any $c\in +**Proof.** Since $|t_0-t'|<\delta\le \delta_2$, and by construction +$\delta_2<\delta_0$, it follows that for any $c\in B(y,t_0,\epsilon)$ we have $d_{t'}(b,c)\le 2d_{t_0}(b,c)$. Since $d_{t_0}(y,b)<\delta\le \epsilon/4$, the result is immediate from the triangle inequality. @@ -2210,30 +2593,34 @@ the triangle inequality. By the above and the fact that $ \delta\le\epsilon/4$, the sectional curvatures on $\widetilde P(z,\epsilon/4,\epsilon/4)$ are bounded by -$C_0$. By Lemma~ there is a curve parameterized by +$C_0$. By \entryref{7d130fba9693} there is a curve parameterized by backward time from $x$ to $z$ whose $l$-length is at most $l_0'$. Thus the $l$-length any minimizing ${\mathcal L}$-geodesic from $x$ to $z$ is at most $l_0'$. By assumption we have a minimizing -${\mathcal L}$-geodesic with the property that $|Ric|$ +${\mathcal L}$-geodesic with the property that $|\mathit{Ric}|$ and$|\nabla R|$ are bounded by $C_0$ along the ${\mathcal L}$-geodesic. Of course, $t_0-\delta **Claim.** $y\in \widetilde P(z,\delta_2',\delta'_2)$. + **Proof.** By construction $|t'-t_0|<\delta\le \delta_2'$. Also, $d_{t_0}(y,b)<\delta\le \delta'_2/2$. Since $d_{t_0}\le 2d_{t'}$, we have $d_{t'}(y,b)< \delta'_2$ the claim is then immediate. It follows immediately that -$$ $$ \begin{aligned} @@ -2244,56 +2631,55 @@ l_x(z)+C\sqrt{d_{t_0}(b,y)^2+|t_0-t'|^2}. \end{aligned} $$ -$$ - -This completes the proof of Proposition~. -\end{proof} +This completes the proof of \entryref{96ced8c484df}. -**Corollary.** -Let $({\mathcal M},G)$ be a generalized Ricci flow and let $x\in {\mathcal M}$ -with $**t**(x)=T-\tau_1$. Let + +**Corollary.** Let $({\mathcal M},G)$ be a generalized Ricci flow and let $x\in {\mathcal M}$ +with $\mathbf{t}(x)=T-\tau_1$. Let $A\subset {\mathcal -M}\cap **t**^{-1}(-\infty,T-\tau_1)$ be a subset whose intersection with each -time-slice $M_t$ is measurable. Suppose that there is a subset $F\subset -{\mathcal M}$ such that $|\nabla R|$ and $|Ric|$ are bounded on $F$ and +M}\cap \mathbf{t}^{-1}(-\infty,T-\tau_1)$ be a subset whose intersection with each +time-slice $M_t$ is measurable. Suppose that there is a subset $F\subset +{\mathcal M}$ such that $|\nabla R|$ and $|\mathit{Ric}|$ are bounded on $F$ and such that every minimizing ${\mathcal L}$ geodesic from $x$ to any point in a neighborhood, $\nu(A)$, of $A$ is contained in $F$. Then for each $\tau\in (\tau_1,\bar\tau]$ the intersection of $A$ with ${\mathcal U}_x(\tau)$ is an open subset of full measure in $A\cap M_{T-\tau}$. + **Proof.** Since ${\mathcal U}_x(\tau)$ is an open subset of $M_{T-\tau}$, the complement of $\nu(A)\cap{\mathcal U}_x(\tau)$ in $\nu(A)\cap M_{T-\tau}$ is a closed subset of $\nu(A)\cap M_{T-\tau}$. Since there is a minimizing ${\mathcal L}$-geodesic to every point of $\nu(A)\cap M_{T-\tau}$, the ${\mathcal -L}$-exponential map ${\mathcal L}exp^\tau_x$ is onto $\nu(A)\cap +L}$-exponential map ${\mathcal L}\mathit{exp}^\tau_x$ is onto $\nu(A)\cap M_{T-\tau}$. -**Claim.** The complement of $\nu(A)\cap {\mathcal U}_{x}(\tau)$ + +**Claim.** The complement of $\nu(A)\cap {\mathcal U}_{x}(\tau)$ in $\nu(A)$ is contained in the union of two sets: The first is the set of points $z$ where there is more than one minimizing ${\mathcal L}$-geodesic from $x$ ending at $z$ and if $Z$ is the initial condition for any minimizing -${\mathcal L}$-geodesic to $z$ then the differential of ${\mathcal L}exp_x^\tau$ at any $Z$ is an isomorphism. The second is the intersection of -the set of critical values of ${\mathcal L}exp^\tau$ with $\nu(A)\cap +${\mathcal L}$-geodesic to $z$ then the differential of ${\mathcal L}\mathit{exp}_x^\tau$ at any $Z$ is an isomorphism. The second is the intersection of +the set of critical values of ${\mathcal L}\mathit{exp}^\tau$ with $\nu(A)\cap M_{T-\tau}$. + -\begin{proof} -Suppose that $q\in \nu(A)\cap M_{T-\tau}$ is not contained in ${\mathcal U}_x$. -Let $\gamma_Z$ be a minimal ${\mathcal L}$-geodesic from $x$ to $q$. If the -differential of ${\mathcal L}exp_x$ is not an isomorphism at $Z$, then +**Proof.** Suppose that $q\in \nu(A)\cap M_{T-\tau}$ is not contained in ${\mathcal U}_x$. +Let $\gamma_Z$ be a minimal ${\mathcal L}$-geodesic from $x$ to $q$. If the +differential of ${\mathcal L}\mathit{exp}_x$ is not an isomorphism at $Z$, then $q$ is contained in the second set given in the claim. Thus, we can assume that -the differential of ${\mathcal L}exp_x$ at $Z$, and hence ${\mathcal -L}exp_x$ identifies a neighborhood $\widetilde V$ of $Z$ in ${\mathcal -H}T_z{\mathcal M}$ with a neighborhood $V\subset \nu(A)$ of $q$ in +the differential of ${\mathcal L}\mathit{exp}_x$ at $Z$, and hence ${\mathcal +L}\mathit{exp}_x$ identifies a neighborhood $\widetilde V$ of $Z$ in ${\mathcal +H}T_z{\mathcal M}$ with a neighborhood $V\subset \nu(A)$ of $q$ in $M_{T-\tau}$. Suppose that there is no neighborhood $\widetilde V'\subset \widetilde V$ of $Z$ so that the ${\mathcal L}$-geodesics are unique minimal ${\mathcal L}$-geodesics to their endpoints in $M_{T_\tau}$. Then there is a sequence of minimizing ${\mathcal L}$-geodesics $\gamma_n$ whose endpoints converge to $q$, but so that no $\gamma_n$ has initial condition contained in $\widetilde V'$. By hypothesis all of these geodesics are contained in $F$ and -hence $|Ric|$ and $|\nabla R|$ are uniformly bounded on these geodesics. -Also, by the continuity of ${\mathcal L}$, the ${\mathcal L}$-lengths of +hence $|\mathit{Ric}|$ and $|\nabla R|$ are uniformly bounded on these geodesics. +Also, by the continuity of ${\mathcal L}$, the ${\mathcal L}$-lengths of $\gamma_n$ are uniformly bounded as $n$ tends to infinity. By -Corollary~ we see that the initial conditions +\entryref{a7c2ebee07a3} we see that the initial conditions $Z_n=\sqrt{\tau_1}X_{\gamma_n}(\tau_1)$ (meaning the limit as $\tau\rightarrow 0$ of these quantities in the case when $\tau_1=0$) are of uniformly bounded norm. Hence, passing to a subsequence we can arrange that the $Z_n$ converge to @@ -2307,62 +2693,75 @@ throughout some neighborhood $\widetilde V'$ of $Z$ the ${\mathcal L}$-geodesics are unique minimizing ${\mathcal L}$-geodesics and completing the proof of the claim. - According to the next claim, the first subset given in Claim~ is contained in the + According to the next claim, the first subset given in \entryref{a673ebd63624} is contained in the set of points of $\nu(A)\cap M_{T-\tau}$ where $L_x^\tau$ is non-differentiable. Since $L_x^\tau$ is a locally Lipschitz function on $\nu(A)$, this subset is of measure zero in $\nu(A)$; see Rademacher's Theorem -on p. 81 of . The second set is of measure zero by Sard's +on p. 81 of [EvansGariepy]. The second set is of measure zero by Sard's theorem. This proves, modulo the next claim, that ${\mathcal U}_x(\tau)\cap A$ is full measure in $A\cap M_{T-\tau}$. + **Claim.** Let $z\in M_{T-\tau}$. Suppose that there is a neighborhood of $z$ in $M_{T-\tau}$ with the property that every point of the neighborhood is the endpoint of a minimizing ${\mathcal L}$-geodesic from $x$, so that $L^\tau_x$ is defined on this neighborhood of $z$. Suppose that there are two distinct, minimizing ${\mathcal L}$-geodesics $\gamma_{Z_1}$ and $\gamma_{Z_2}$ from $x$ -ending at $z$ with the property that the differential of ${\mathcal L}exp^{\tau}$ is an isomorphism at both $Z_1$ and $Z_2$. Then the function +ending at $z$ with the property that the differential of ${\mathcal L}\mathit{exp}^{\tau}$ is an isomorphism at both $Z_1$ and $Z_2$. Then the function $L^\tau_x$ is non-differentiable at $z$. + **Proof.** Suppose that $\gamma_{Z_0}|_{[0,\tau]}$ is an ${\mathcal L}$-minimal ${\mathcal L}$-geodesic -and that $d_{Z_0}{\mathcal L}exp_x^\tau$ is an isomorphism. Then use -${\mathcal L}exp_x^{\tau}$ to identify a neighborhood of $Z_0\in T_xM$ -with a neighborhood of $z$ in $M_{T-\tau}$, and push the function +and that $d_{Z_0}{\mathcal L}\mathit{exp}_x^\tau$ is an isomorphism. Then use +${\mathcal L}\mathit{exp}_x^{\tau}$ to identify a neighborhood of $Z_0\in T_xM$ +with a neighborhood of $z$ in $M_{T-\tau}$, and push the function $\widetilde{\mathcal L}^\tau_x$ on this neighborhood of $Z_0$ down to a function $L_{Z_0}$ on a neighborhood in $M_{T-\tau}$ of $z$. According to -Lemma~ the resulting function $L_{Z_0}$ is smooth and its gradient +\entryref{abf9cbb10c68} the resulting function $L_{Z_0}$ is smooth and its gradient at $z$ is $2\sqrt{\tau}X(\tau)$. Now suppose that there is a second ${\mathcal L}$-minimizing ${\mathcal L}$-geodesic to $z$ with initial condition -$Z_1\not=Z_0$ and with $d_{Z_1}{\mathcal L}exp_x^\tau$ being an +$Z_1\not=Z_0$ and with $d_{Z_1}{\mathcal L}\mathit{exp}_x^\tau$ being an isomorphism. Then near $z$ the function $L_x^{\tau}$ is less than or equal to the minimum of two smooth functions $L_{Z_0}$ and $L_{Z_1}$. We have -$L_{Z_0}(z)=L_{Z_1}(z)=L_x^\tau(z)$, and furthermore, $L_{Z_0}$ and $L_{Z_1}$ +$L_{Z_0}(z)=L_{Z_1}(z)=L_x^\tau(z)$, and furthermore, $L_{Z_0}$ and $L_{Z_1}$ have distinct gradients at $z$. It follows that $L_x^{\tau}$ is not smooth at $z$. -This completes the proof of the Corollary~. -\end{proof} +This completes the proof of the \entryref{7aad50e49af2}. ## Reduced volume -Here, we assume that $x\in M_T\subset {\mathcal M}$, so that +Here, we assume that $x\in M_T\subset {\mathcal M}$, so that $\tau_1=0$ in this subsection. -**Definition.** -Let $A\subset {\mathcal U}_x(\tau)$ be a measurable subset of + +**Definition.** Let $A\subset {\mathcal U}_x(\tau)$ be a measurable subset of $M_{T-\tau}$. The $\mathcal{L}$-*reduced volume of $A$ from $x$* (or the *reduced volume* for short) is defined to be -$$\widetilde V_x(A) = \int_{A} \tau^{-\frac{n}{2}}exp(-l_x(q))dq$$ where $dq$ is the volume + +$$ +\widetilde V_x(A) = \int_{A} \tau^{-\frac{n}{2}}\mathit{exp}(-l_x(q))dq +$$ + + where $dq$ is the volume element of the metric $G(T-\tau)$. + -**Lemma.** -Let $A\subset {\mathcal U}_x(\tau)$ be a measurable subset. Define + +**Lemma.** Let $A\subset {\mathcal U}_x(\tau)$ be a measurable subset. Define $\widetilde A\subset \widetilde{\mathcal U}_x(\tau)$ to be the -pre-image under ${\mathcal L}exp_x^{\tau}$ of $A$. Then -$$\widetilde V_x(A)=\int_{\widetilde A}\tau^{-\frac{n}{2}}exp(-\tilde l(Z, \tau)){\mathcal J}(Z,\tau)dZ,$$ where $dZ$ is the usual +pre-image under ${\mathcal L}\mathit{exp}_x^{\tau}$ of $A$. Then + +$$ +\widetilde V_x(A)=\int_{\widetilde A}\tau^{-\frac{n}{2}}\mathit{exp}(-\tilde l(Z, \tau)){\mathcal J}(Z,\tau)dZ, +$$ + + where $dZ$ is the usual Euclidean volume element and ${\mathcal J}(Z,\tau)$ is the Jacobian determinant -of ${\mathcal L}exp_x^{\tau}$ at $Z\in T_xM_T$. +of ${\mathcal L}\mathit{exp}_x^{\tau}$ at $Z\in T_xM_T$. + **Proof.** This is simply the change of variables formula for integration. @@ -2375,15 +2774,15 @@ limit as $\tau\rightarrow 0$ requires a rescaling argument. Fix $Q>0$. We rescale to form $(Q{\mathcal M},QG)$ and then we shift the time by $T-QT$ so that the time-slice $M_T$ in the original flow is the $T$ time-slice of the new flow. We call the result -$({\mathcal M}',G')$. Recall that $\tau=T-**t**$ is the parameter +$({\mathcal M}',G')$. Recall that $\tau=T-\mathbf{t}$ is the parameter for ${\mathcal L}$-geodesics in $({\mathcal M},G)$ The corresponding -parameter in the rescaled flow $({\mathcal M}',G')$ is $\tau'=T-**t'**=Q\tau$. We denote by ${\mathcal L}'exp_x$ the ${\mathcal +parameter in the rescaled flow $({\mathcal M}',G')$ is $\tau'=T-\mathbf{t'}=Q\tau$. We denote by ${\mathcal L}'\mathit{exp}_x$ the ${\mathcal L}$-exponential map from $x$ in $({\mathcal M}',G')$, and by $l_x'$ the reduced length function for this Ricci flow. The associated function on the tangent space is denoted $\tilde l'$. -**Lemma.** -Let $({\mathcal M},G)$ be a generalized Ricci flow and let $x\in + +**Lemma.** Let $({\mathcal M},G)$ be a generalized Ricci flow and let $x\in M_T\subset {\mathcal M}$. Fix $Q>0$ and let $({\mathcal M}',G')$ be the $Q$ scaling and shifting of $({\mathcal M},G)$ as described in the previous paragraph. Let $\iota\colon{\mathcal M}\to {\mathcal @@ -2391,15 +2790,22 @@ M}'$ be the identity map. Suppose that $\gamma\colon [0,\bar\tau]\to {\mathcal M}$ is a path parameterized by backward time with $\gamma(0)=x$. Let $\beta\colon [0,Q\bar\tau]\to Q{\mathcal M}$ be defined by -$$\beta(\tau')=\iota(\gamma(\tau'/Q)).$$ + +$$ +\beta(\tau')=\iota(\gamma(\tau'/Q)). +$$ + Then $\beta(0)=x$ and $\beta$ is parameterized by backward time in $({\mathcal M}',G')$, and ${\mathcal L}(\beta)=\sqrt{Q}{\mathcal L}(\gamma)$. Furthermore, $\beta$ is an ${\mathcal L}$-geodesic if -and only if $\gamma$ is. In this case, if $Z=lim_{\tau\rightarrow 0}\sqrt{\tau}X_\gamma(\tau)$ then -$\sqrt{Q^{-1}}Z=lim_{\tau'\rightarrow +and only if $\gamma$ is. In this case, if $Z=\mathit{lim}_{\tau\rightarrow 0}\sqrt{\tau}X_\gamma(\tau)$ then +$\sqrt{Q^{-1}}Z=\mathit{lim}_{\tau'\rightarrow 0}\sqrt{\tau'}X_\beta(\tau')$ + + **Remark.** Notice that $|Z|_G^2=|\sqrt{Q^{-1}}Z|^2_{G'}$. + **Proof.** It is clear that $\beta(0)=x$ and that $\beta$ is parameterized by backward time in $({\mathcal M}',G')$. Because of the scaling of @@ -2407,91 +2813,140 @@ space and time by $Q$, we have $R_{G'}=R_G/Q$ and $X_\beta(\tau')=d\iota(X_\gamma(\tau))/Q$, and hence $|X_\beta(\tau')|^2_{G'}=\frac{1}{Q}|X_\gamma(\tau)|^2_{G}$. A direct change of variables in the integral then shows that -$${\mathcal L}(\beta)=\sqrt{Q}{\mathcal L}(\gamma).$$ + +$$ +{\mathcal L}(\beta)=\sqrt{Q}{\mathcal L}(\gamma). +$$ + It follows that $\beta$ is an ${\mathcal L}$-geodesic if and only if $\gamma$ is. The last statement follows directly. Immediately from the definitions we see the following: -**Corollary.** -With notation as above, and with the substitution $\tau'=Q\tau$, for + +**Corollary.** With notation as above, and with the substitution $\tau'=Q\tau$, for any $Z\in {\mathcal H}T_x{\mathcal M}$ and any $\tau>0$ we have -$${\mathcal -L}'exp_x(\sqrt{Q^{-1}}Z,\tau')=\iota({\mathcal L}exp_x(Z,\tau))$$ and -$$ \tilde l'(\sqrt{Q^{-1}}Z,\tau')=\tilde l(Z,\tau),$$ + +$$ +{\mathcal +L}'\mathit{exp}_x(\sqrt{Q^{-1}}Z,\tau')=\iota({\mathcal L}\mathit{exp}_x(Z,\tau)) +$$ + + and + +$$ +\tilde l'(\sqrt{Q^{-1}}Z,\tau')=\tilde l(Z,\tau), +$$ + whenever these are defined. + ### The integrand in the reduced volume integral Now we turn our attention to the integrand (over $\widetilde {\mathcal U}_x(\tau)$) in the reduced volume integral. Namely, set -$$f(\tau)=\tau^{-n/2}e^{-\tilde l(Z,\tau)}{\mathcal J}(Z,\tau),$$ -where ${\mathcal J}(Z,\tau)$(Z,\tau)$\ii} is the Jacobian -determinant of ${\mathcal L}exp_x^\tau$ at the point $Z\in \widetilde + +$$ +f(\tau)=\tau^{-n/2}e^{-\tilde l(Z,\tau)}{\mathcal J}(Z,\tau), +$$ + +where ${\mathcal J}(Z,\tau)$ is the Jacobian +determinant of ${\mathcal L}\mathit{exp}_x^\tau$ at the point $Z\in \widetilde U_x(\tau)\subset T_xM_T$. We wish to see that this quantity is invariant under the rescaling. -**Lemma.** -With the notation as above let ${\mathcal J}'(Z,\tau')$ denote the -Jacobian determinant of ${\mathcal L}'exp_x$. Then, with the + +**Lemma.** With the notation as above let ${\mathcal J}'(Z,\tau')$ denote the +Jacobian determinant of ${\mathcal L}'\mathit{exp}_x$. Then, with the substitution $\tau'=Q\tau$, we have -$$(\tau')^{-n/2}e^{-\tilde l'(\sqrt{Q^{-1}}Z,\tau')}{\mathcal J}'(\sqrt{Q^{-1}}Z,\tau')= -\tau^{-n/2}e^{-\tilde l(Z,\tau)}{\mathcal J}(Z,\tau).$$ -**Proof.** It follows from the first equation in Corollary~ that -$$J(\iota){\mathcal J}(Z,\tau)=J(\sqrt{Q^{-1}}){\mathcal -J}'(\sqrt{Q^{-1}}Z,\tau'),$$ where $J(\iota)$ is the Jacobian -determinant of $\iota$ at ${\mathcal L}exp_x(Z,\tau)$ and +$$ +(\tau')^{-n/2}e^{-\tilde l'(\sqrt{Q^{-1}}Z,\tau')}{\mathcal J}'(\sqrt{Q^{-1}}Z,\tau')= +\tau^{-n/2}e^{-\tilde l(Z,\tau)}{\mathcal J}(Z,\tau). +$$ + + +**Proof.** It follows from the first equation in \entryref{541f71387f84} that + +$$ +J(\iota){\mathcal J}(Z,\tau)=J(\sqrt{Q^{-1}}){\mathcal +J}'(\sqrt{Q^{-1}}Z,\tau'), +$$ + + where $J(\iota)$ is the Jacobian +determinant of $\iota$ at ${\mathcal L}\mathit{exp}_x(Z,\tau)$ and $J(\sqrt{Q^{-1}})$ is the Jacobian determinant of multiplication by $\sqrt{Q^{-1}}$ as a map from $T_xM_T$ to itself, where the domain has the metric $G$ and the range has metric $G'=QG$. Clearly, with these conventions, we have $J(\iota)=Q^{n/2}$ and $J(\sqrt{Q^{-1}})=1$. Hence, we conclude -$$Q^{n/2}{\mathcal J}(Z,\tau)={\mathcal J}'(\sqrt{Q^{-1}}Z,\tau').$$ + +$$ +Q^{n/2}{\mathcal J}(Z,\tau)={\mathcal J}'(\sqrt{Q^{-1}}Z,\tau'). +$$ Letting $\gamma$ be the ${\mathcal L}$-geodesic in $({\mathcal M},G)$ with initial condition $Z$ and $\beta$ the ${\mathcal L}$-geodesic in $({\mathcal M}',G')$ with initial condition -$\sqrt{Q^{-1}}Z$, by Lemma~ we have -$\gamma(\tau)=\beta(\tau')$. From Corollary~ and the -definition of the reduced length, we get $$\tilde -l'(\sqrt{Q^{-1}}Z,\tau')=\tilde l(\gamma,\tau).$$ Plugging these in +$\sqrt{Q^{-1}}Z$, by \entryref{ad48b42bb0a0} we have +$\gamma(\tau)=\beta(\tau')$. From \entryref{541f71387f84} and the +definition of the reduced length, we get + +$$ +\tilde +l'(\sqrt{Q^{-1}}Z,\tau')=\tilde l(\gamma,\tau). +$$ + + Plugging these in gives the result. Let us evaluate $f(\tau)$ in the case of $\Ar^n$ with the Ricci flow being the constant family of Euclidean metrics. -\begin{exam} -Let the Ricci flow be the constant family of standard metrics on + +**Example.** Let the Ricci flow be the constant family of standard metrics on $\Ar^n$. Fix $x=(p,T)\in \Ar ^n\times (-\infty,\infty)$. Then -$${\mathcal L}exp_x(Z,\tau)=(p+2\sqrt{\tau}Z,T-\tau).$$ -In particular, the Jacobian determinant of ${\mathcal L}exp_{(x,T)}^\tau$ is constant and equal to $2^n\tau^{n/2}$. The + +$$ +{\mathcal L}\mathit{exp}_x(Z,\tau)=(p+2\sqrt{\tau}Z,T-\tau). +$$ + +In particular, the Jacobian determinant of ${\mathcal L}\mathit{exp}_{(x,T)}^\tau$ is constant and equal to $2^n\tau^{n/2}$. The $\widetilde l$-length of the ${\mathcal L}$-geodesic $\gamma_Z(\tau)=(p+2\sqrt{\tau}Z, T-\tau),\ 0\le \tau\le \bar\tau$, is $|Z|^2$. -\end{exam} + Putting these computations together gives the following. -**Claim.** -In the case of the constant flow on Euclidean space we have -$$f(\tau)=\tau^{-n/2}e^{-\tilde l(Z,\tau)}{\mathcal J}(Z,\tau)=2^ne^{-|Z|^2}.$$ + +**Claim.** In the case of the constant flow on Euclidean space we have + +$$ +f(\tau)=\tau^{-n/2}e^{-\tilde l(Z,\tau)}{\mathcal J}(Z,\tau)=2^ne^{-|Z|^2}. +$$ + This computation has consequences for all Ricci flows. -**Proposition.** -Let $({\mathcal M},G)$ be a generalized Ricci flow and let $x\in M_T\subset + +**Proposition.** Let $({\mathcal M},G)$ be a generalized Ricci flow and let $x\in M_T\subset {\mathcal M}$. Then, for any $A<\infty$, there is $\delta>0$ such that the map -${\mathcal L}exp_x$ is defined on $B(0,A)\times (0,\delta)$, where +${\mathcal L}\mathit{exp}_x$ is defined on $B(0,A)\times (0,\delta)$, where $B(0,A)$ is the ball of radius $A$ centered at the origin in $T_xM_T$. -Moveover, ${\mathcal L}exp_x$ defines a diffeomorphism of $B(0,A)\times +Moveover, ${\mathcal L}\mathit{exp}_x$ defines a diffeomorphism of $B(0,A)\times (0,\delta)$ onto an open subset of ${\mathcal M}$. Furthermore, -$$lim_{\tau\rightarrow 0}\tau^{-n/2}e^{-\tilde l(Z,\tau)}{\mathcal J}(Z,\tau)=2^ne^{-|Z|^2},$$ + +$$ +\mathit{lim}_{\tau\rightarrow 0}\tau^{-n/2}e^{-\tilde l(Z,\tau)}{\mathcal J}(Z,\tau)=2^ne^{-|Z|^2}, +$$ + where the convergence is uniform on each compact subset of $T_xM_T$. + **Proof.** First notice that since $T$ is greater than the initial time of ${\mathcal M}$, there is $\epsilon>0$, and an embedding $\rho\colon B(x,T,\epsilon)\times -[T-\epsilon,T]\to {\mathcal M}$ compatible with time and the vector field. By +[T-\epsilon,T]\to {\mathcal M}$ compatible with time and the vector field. By taking $\epsilon>0$ smaller if necessary, we can assume that the image of $\rho$ has compact closure in ${\mathcal M}$. By compactness every higher partial derivative (both spatial and temporal) of the metric is bounded on the @@ -2502,13 +2957,18 @@ $k\rightarrow\infty$, and set $Q_k=\tau_k^{-1}$. We let $({\mathcal M}_k,G_k)$ be the $Q_k$-rescaling and shifting of $({\mathcal M},G)$ as described at the beginning of this section. The rescaled version of $\rho$ is an embedding -$$\rho_k\colon B_{G_k}(x,T,\sqrt{Q_k}\epsilon)\times [T-Q_k\epsilon,T]\to -{\mathcal M}_k$$ compatible with the time function $**t**_k$ and the vector + +$$ +\rho_k\colon B_{G_k}(x,T,\sqrt{Q_k}\epsilon)\times [T-Q_k\epsilon,T]\to +{\mathcal M}_k +$$ + + compatible with the time function $\mathbf{t}_k$ and the vector field. Furthermore, uniformly on the image of $\rho_k$, every higher partial derivative of the metric is bounded by a constant that goes to zero with $k$. Thus, the generalized Ricci flows $({\mathcal M}_k,G_k)$ based at $x$ converge geometrically to the constant family of Euclidean metrics on $\Ar^n$. Since the -ODE given in Equation~() is regular even at $0$, this implies that +ODE given in Equation (6.5) is regular even at $0$, this implies that the ${\mathcal L}$-exponential maps for these flows converge uniformly on the balls of finite radius centered at the origin of the tangent spaces at $x$ to the ${\mathcal L}$-exponential map of $\Ar^n$ at the origin. Of course, if @@ -2521,57 +2981,73 @@ $\sqrt{Q_k^{-1}}Z\in B_{G_k}(0,A)$. that for any $A<\infty$, for all $k$ sufficiently large, the ${\mathcal L}$-geodesics are defined on $B_{G_k}(0,A)\times (0,1]$ and the image is contained in the image of $\rho_k$. Rescaling shows -that for any $A<\infty$ there is $k$ for which the ${\mathcal +that for any $A<\infty$ there is $k$ for which the ${\mathcal L}$-exponential map is defined on $B_G(0,A)\times (0,\tau_k]$ and has image contained in $\rho$. Let $Z\in B_Q(0,A)\subset T_xM_T$, and let $\gamma$ be the -${\mathcal L}$-geodesic with $lim_{\tau\rightarrow +${\mathcal L}$-geodesic with $\mathit{lim}_{\tau\rightarrow 0}\sqrt{\tau}X_\gamma(\tau)=Z$. Let $\gamma_k$ be the corresponding -${\mathcal L}$-geodesic in $({\mathcal M}_k,G_k)$. Then $lim_{\tau\rightarrow +${\mathcal L}$-geodesic in $({\mathcal M}_k,G_k)$. Then $\mathit{lim}_{\tau\rightarrow 0}\sqrt{\tau}X_{\gamma_k}(\tau)=\sqrt{\tau_k}Z=Z_k$. Of course, $|Z_k|^2_{G_k}=|Z|^2_G$, meaning that $Z_k$ is contained in the ball $B_{G_k}(0,A)\subset T_xM_T$ for all $k$. Hence, by passing to a subsequence we can assume that, in the geometric limit, the $\sqrt{\tau_k}Z$ converge to a tangent vector $Z'$ in the ball of radius $A$ centered at the origin in the tangent space to Euclidean -space. Of course $|Z'|^2=|Z|_G^2$. By Claim~, this +space. Of course $|Z'|^2=|Z|_G^2$. By \entryref{9d4d83df4874}, this means that we have -$$lim_{k\rightarrow \infty}1^{-n/2}e^{-\tilde l_k(\sqrt{Q_k^{-1}}Z,1)}{\mathcal -J}_k(\sqrt{Q_k^{-1}}Z,1)=2^ne^{-|Z|^2},$$ where ${\mathcal J}_k$ is + +$$ +\mathit{lim}_{k\rightarrow \infty}1^{-n/2}e^{-\tilde l_k(\sqrt{Q_k^{-1}}Z,1)}{\mathcal +J}_k(\sqrt{Q_k^{-1}}Z,1)=2^ne^{-|Z|^2}, +$$ + + where ${\mathcal J}_k$ is the Jacobian determinant of the ${\mathcal L}$-exponential map for $({\mathcal M}_k,G_k)$. Of course, since -$\tau_k=Q_k^{-1}$, by Lemma~ we have -$$1^{-n/2}e^{-\tilde l_k(\sqrt{Q_k^{-1}}Z,1)}{\mathcal +$\tau_k=Q_k^{-1}$, by \entryref{76c5f88ad2b4} we have + +$$ +1^{-n/2}e^{-\tilde l_k(\sqrt{Q_k^{-1}}Z,1)}{\mathcal J}_k(\sqrt{Q_k^{-1}}Z,1)=\tau_k^{-n/2}e^{-\tilde -l(Z,\tau_k)}{\mathcal J}(Z,\tau_k).$$ This establishes the limiting +l(Z,\tau_k)}{\mathcal J}(Z,\tau_k). +$$ + + This establishes the limiting result. Since the geometric limits are uniform on balls of finite radius centered at the origin in the tangent space, the above limit also is uniform over each of these balls. -**Corollary.** -Let $({\mathcal M},G)$ be a generalized Ricci flow whose sectional + +**Corollary.** Let $({\mathcal M},G)$ be a generalized Ricci flow whose sectional curvatures are bounded. For any $x\in M_T$ and any $R<\infty$ for all $\tau>0$ sufficiently small, the ball of radius $R$ centered at the origin in $T_xM_T$ is contained in $\widetilde {\mathcal U}_x(\tau)$. + **Proof.** According to the last result, given $R<\infty$, for all $\delta>0$ sufficiently small the ball of radius $R$ centered at the origin in $T_xM_T$ is contained in -${\mathcal D}_x^\delta$, in the domain of definition of ${\mathcal L}exp_x^\delta$ as given in Definition~, and ${\mathcal L}exp_x$ is a diffeomorphism on this subset. We shall show that if $\delta>0$ is +${\mathcal D}_x^\delta$, in the domain of definition of ${\mathcal L}\mathit{exp}_x^\delta$ as given in \entryref{df842ef2ab9f}, and ${\mathcal L}\mathit{exp}_x$ is a diffeomorphism on this subset. We shall show that if $\delta>0$ is sufficiently small, then the resulting ${\mathcal L}$-geodesic $\gamma$ is the unique minimizing ${\mathcal L}$-geodesic. If not then there must be another, distinct ${\mathcal L}$-geodesic to this point whose ${\mathcal L}$-length is -no greater than that of $\gamma$. According to Lemma~ there is a +no greater than that of $\gamma$. According to \entryref{333be2e58780} there is a constant $C_1$ depending on the curvature bound and on $\delta$ such that if $Z$ is an initial condition for an ${\mathcal L}$-geodesic then for all $\tau\in (0,\delta)$ we have -$$C_1^{-1}\left(|Z|-\frac{(C_1-1)}{2}\sqrt{\delta}\right)\le \sqrt{\tau}|X(\tau)|\le -C_1|Z|+\frac{(C_1-1)}{2}\sqrt{\delta}.$$ From the formula given in -Lemma~ for $C_1$, it follows that, fixing the bound of + +$$ +C_1^{-1}\left(|Z|-\frac{(C_1-1)}{2}\sqrt{\delta}\right)\le \sqrt{\tau}|X(\tau)|\le +C_1|Z|+\frac{(C_1-1)}{2}\sqrt{\delta}. +$$ + + From the formula given in +\entryref{333be2e58780} for $C_1$, it follows that, fixing the bound of the curvature and its derivatives, $C_1\rightarrow 1$ as $\delta\rightarrow 0$. Thus, with a given curvature bound, for $\delta$ sufficiently small, @@ -2582,8 +3058,8 @@ value of the integral of $\sqrt{\tau}R(\gamma(\tau))$ is at most $2C_0\delta^{3/2}/3$ where $C_0$ is an upper bound for the absolute value of the scalar curvature. -Given $R<\infty$, choose $\delta>0$ sufficiently small such that ${\mathcal -L}exp_x$ is a diffeomorphism on the ball of radius $9R$ centered at the +Given $R<\infty$, choose $\delta>0$ sufficiently small such that ${\mathcal +L}\mathit{exp}_x$ is a diffeomorphism on the ball of radius $9R$ centered at the origin and such that the following estimate holds: The ${\mathcal L}$-length of an ${\mathcal L}$-geodesic defined on $[0,\delta]$ with initial condition $Z$ is between $\sqrt{\delta}|Z|^2$ and $3\sqrt{\delta}|Z|^2$. To ensure the latter @@ -2600,58 +3076,75 @@ $|Z|$ with $|Z| +**Theorem.** Fix $x\in M_T\subset {\mathcal M}$. Let $A\subset{\mathcal +U}_x\subset {\mathcal M}$ be an open subset. We suppose that for any $0<\tau\le \bar\tau$ and any $y\in A_\tau=A\cap M_{T-\tau}$ the minimizing ${\mathcal L}$-geodesic from $x$ to $y$ contained in -$A\cup \{x\}$. Then $\widetilde V_x(A_\tau)$ is a non-increasing +$A\cup \{x\}$. Then $\widetilde V_x(A_\tau)$ is a non-increasing function of $\tau$ for all $0<\tau\le \bar\tau$. + **Proof.** Fix $\tau_0\in (0,\bar\tau]$. To prove the theorem we shall show that for any $0<\tau<\tau_0$ we have $\widetilde V_x(A_\tau)\ge \widetilde V_x(A_{\tau_0})$. Let $\widetilde A_{\tau_0}\subset \widetilde {\mathcal U}_x(\tau_0)$ be the pre-image under ${\mathcal -L}exp_x^{\tau_0}$ of $A_{\tau_0}$. For each $0<\tau\le \tau_0$ +L}\mathit{exp}_x^{\tau_0}$ of $A_{\tau_0}$. For each $0<\tau\le \tau_0$ we set -$$A_{\tau,\tau_0}={\mathcal L}exp_x^\tau(\widetilde -A_{\tau_0})\subset M_{T-\tau}.$$ It follows from the assumption on + +$$ +A_{\tau,\tau_0}={\mathcal L}\mathit{exp}_x^\tau(\widetilde +A_{\tau_0})\subset M_{T-\tau}. +$$ + + It follows from the assumption on $A$ that $A_{\tau,\tau_0}\subset A_\tau$, so that $\widetilde V_x(A_{\tau,\tau_0})\le \widetilde V_x(A_\tau)$. Thus, it suffices to show that for all $0<\tau\le \tau_0$ we have -$$\widetilde V_x(A_{\tau,\tau_0})\ge \widetilde V_x(\tau_0).$$ + +$$ +\widetilde V_x(A_{\tau,\tau_0})\ge \widetilde V_x(\tau_0). +$$ Since -$$ \widetilde V_x(A_{\tau,\tau_0})=\int_{\widetilde A_{\tau_0}} \tau^{-\frac{n}{2}} -exp(-\tilde l(Z,\tau)){\mathcal J}(Z,\tau)dZ,$$ the theorem + +$$ +\widetilde V_x(A_{\tau,\tau_0})=\int_{\widetilde A_{\tau_0}} \tau^{-\frac{n}{2}} +\mathit{exp}(-\tilde l(Z,\tau)){\mathcal J}(Z,\tau)dZ, +$$ + + the theorem follows from: -**Proposition.** For each $Z\in \widetilde + +**Proposition.** For each $Z\in \widetilde {\mathcal U}_x(\bar\tau)\subset T_xM_T$ the function -$$f(Z,\tau)=\tau^{-\frac{n}{2}}e^{-\tilde l(Z,\tau)}{\mathcal J}(Z,\tau)$$ + +$$ +f(Z,\tau)=\tau^{-\frac{n}{2}}e^{-\tilde l(Z,\tau)}{\mathcal J}(Z,\tau) +$$ + is a non-increasing function of $\tau$ on the interval -$(0,\bar\tau]$ with $lim_{\tau\rightarrow +$(0,\bar\tau]$ with $\mathit{lim}_{\tau\rightarrow 0}f(Z,\tau)=2^ne^{-|Z|^2}$, the limit being uniform on any compact subset of $T_xM_T$. + -\begin{proof} - First, we analyze the Jacobian -${\mathcal J}(Z,\tau)$. We know that $\mathcal{L}exp_x^{\tau}$ +**Proof.** First, we analyze the Jacobian +${\mathcal J}(Z,\tau)$. We know that $\mathcal{L}\mathit{exp}_x^{\tau}$ is smooth in a neighborhood of $Z$. Choose a basis $\{\partial_\alpha\}$ for $T_xM_T$ such that $\partial_\alpha$ -pushes forward under the differential at $Z$ of ${\mathcal L}exp_x^\tau$ to an orthonormal basis $\{Y_\alpha\}$ for +pushes forward under the differential at $Z$ of ${\mathcal L}\mathit{exp}_x^\tau$ to an orthonormal basis $\{Y_\alpha\}$ for $M_{T-\tau}$ at $\gamma_Z(\tau)$. Notice that, letting $\tau'$ range from $0$ to $\tau$ and taking the push-forward of the -$\partial_\alpha$ under the differential at $Z$ of ${\mathcal L}exp_x^{\tau'}$ produces a basis of ${\mathcal L}$-Jacobi fields +$\partial_\alpha$ under the differential at $Z$ of ${\mathcal L}\mathit{exp}_x^{\tau'}$ produces a basis of ${\mathcal L}$-Jacobi fields $\{Y_\alpha(\tau')\}$ along $\gamma_Z$. With this understood, we have: -$$ - $$ \begin{aligned} -\frac{\partial}{\partial \tau}ln\mathcal{J}|_\tau & = & -\frac{d}{d\tau}ln(\sqrt{det(\langle +\frac{\partial}{\partial \tau}\mathit{ln}\mathcal{J}|_\tau & = & +\frac{d}{d\tau}\mathit{ln}(\sqrt{\mathit{det}(\langle Y_{\alpha},Y_{\beta}\rangle )})\\ & = & @@ -2660,34 +3153,27 @@ Y_{\alpha},Y_{\beta}\rangle \end{aligned} $$ -$$ - -By Lemma~ and by Proposition~ (recall that +By \entryref{44abf07e60ec} and by \entryref{88d4ef67440f} (recall that $\tau_1=0$) we have -$$ - $$ \begin{aligned} \nonumber \frac{1}{2}\frac{d}{d\tau}|Y_\alpha(\tau)|^2 & = & -\frac{1}{2\sqrt{\tau}}Hess(L)(Y_\alpha,Y_\alpha) +Ric(Y_\alpha,Y_\alpha) \\ & \le & +\frac{1}{2\sqrt{\tau}}\mathit{Hess}(L)(Y_\alpha,Y_\alpha) +\mathit{Ric}(Y_\alpha,Y_\alpha) \\ & \le & \frac{1}{2\tau}-\frac{1}{2\sqrt{\tau}}\int_0^{\tau}\sqrt{\tau'}H(X,\tilde Y_\alpha(\tau'))d\tau', \end{aligned} $$ -$$ where $\tilde Y_\alpha(\tau')$ is the adapted vector field along $\gamma$ with $\tilde Y(\tau)=Y_\alpha(\tau)$. Summing over $\alpha$ as in the -proof of Proposition~ and Claim~ yields - -$$ +proof of \entryref{882677eecdc1} and \entryref{11e3d00eb60d} yields $$ \begin{aligned} -\frac{\partial}{\partial\tau}ln\mathcal{J}(Z,\tau)|_{\tau} & +\frac{\partial}{\partial\tau}\mathit{ln}\mathcal{J}(Z,\tau)|_{\tau} & \le & \frac{n}{2\tau} - \frac{1}{2\sqrt{\tau}}\sum_{\alpha}\int_{0}^{\tau}\sqrt{\tau'}H(X,\tilde Y_{\alpha(\tau')})d\tau' \nonumber @@ -2695,44 +3181,49 @@ Y_{\alpha(\tau')})d\tau' \nonumber \end{aligned} $$ -$$ - On $\widetilde {\mathcal U}_x(\tau)$ the expression $\tau^{-\frac{n}{2}}e^{-\widetilde l(Z,\tau)}{\mathcal J}(Z,\tau)$ is positive, and so we have -$$\frac{\partial}{\partial\tau}ln\left(\tau^{-\frac{n}{2}}e^{-\widetilde + +$$ +\frac{\partial}{\partial\tau}\mathit{ln}\left(\tau^{-\frac{n}{2}}e^{-\widetilde l(Z,\tau)}{\mathcal J}(Z,\tau)\right)\le \left(-\frac{n}{2\tau}-\frac{d\widetilde l}{d -\tau}+\frac{n}{2\tau}-\frac{1}{2}\tau^{-\frac{3}{2}}K^\tau(\gamma_Z)\right).$$ +\tau}+\frac{n}{2\tau}-\frac{1}{2}\tau^{-\frac{3}{2}}K^\tau(\gamma_Z)\right). +$$ -Corollary~ says that the right-hand side of the +\entryref{451b4cc590c1} says that the right-hand side of the previous inequality is zero. Hence, we conclude $$ - \frac{d}{d\tau}\left(\tau^{-\frac{n}{2}}e^{-\tilde l(X,\tau)}{\mathcal J}(X,\tau)\right)\le 0. $$ This proves the inequality given in the statement of the proposition. The limit statement as $\tau\rightarrow 0$ is contained -in Proposition~. +in \entryref{4e6330b2aa46}. As we have already seen, this proposition implies - Theorem~, and hence the proof of this theorem is complete. -\end{proof} + \entryref{6a52537bef2f}, and hence the proof of this theorem is complete. Notice that we have established the following: -**Corollary.** -For any measurable subset $A\subset {\mathcal U}_x(\tau)$ the -reduced volume $\widetilde V_x(A)$ is at most $(4\pi)^{n/2}$. + +**Corollary.** For any measurable subset $A\subset {\mathcal U}_x(\tau)$ the +reduced volume $\widetilde V_x(A)$ is at most $(4\pi)^{n/2}$. + **Proof.** Let $\widetilde A\subset \widetilde {\mathcal U}_x(\tau)$ be the pre-image of $A$. We have seen that -$$\widetilde V_x(A)=\int_{A}\tau^{-n/2}e^{-l(q,\tau)}dq=\int_{\widetilde -A}\tau^{-n/2}e^{-\tilde l(Z,\tau)}{\mathcal J}(Z,\tau)dz.$$ By -Theorem~ we see that $\tau^{-n/2}e^{-\tilde + +$$ +\widetilde V_x(A)=\int_{A}\tau^{-n/2}e^{-l(q,\tau)}dq=\int_{\widetilde +A}\tau^{-n/2}e^{-\tilde l(Z,\tau)}{\mathcal J}(Z,\tau)dz. +$$ + + By +\entryref{6a52537bef2f} we see that $\tau^{-n/2}e^{-\tilde l(Z,\tau)}{\mathcal J}(Z,\tau)$ is a non-increasing function of $\tau$ whose limit as $\tau\rightarrow 0$ is the restriction of $2^ne^{-|Z|^2}$ to $\widetilde A$. The result is immediate from diff --git a/projects/poincare-conjecture/.astrolabe/docs/07-newcomp2.mdx b/projects/poincare-conjecture/.astrolabe/docs-src/08-complete-ricci-flows-of-bounded-curvature.mdx similarity index 60% rename from projects/poincare-conjecture/.astrolabe/docs/07-newcomp2.mdx rename to projects/poincare-conjecture/.astrolabe/docs-src/08-complete-ricci-flows-of-bounded-curvature.mdx index ab7cb378..b8fc004b 100644 --- a/projects/poincare-conjecture/.astrolabe/docs/07-newcomp2.mdx +++ b/projects/poincare-conjecture/.astrolabe/docs-src/08-complete-ricci-flows-of-bounded-curvature.mdx @@ -1,11 +1,13 @@ -# Complete Ricci flows of bounded curvature + - In this chapter we establish strong results for ${\mathcal L}exp_x$ in the case of ordinary Ricci flow on complete $n$-manifolds +# Chapter 7 -- Complete Ricci flows of bounded curvature + + In this chapter we establish strong results for ${\mathcal L}\mathit{exp}_x$ in the case of ordinary Ricci flow on complete $n$-manifolds with appropriate curvature bounds. In particular, for these flows we show that there is a minimizing ${\mathcal L}$-geodesic to every point. This means that $l_x$ is everywhere defined. We extend the differential inequalities for $l_x$ established in -Section~ at the `smooth points' to weak +Section 6.4 at the 'smooth points' to weak inequalities (i.e., inequalities in the distributional sense) valid on the whole manifold. Using this we prove an upper bound for the minimum of $l_x^\tau$. @@ -13,33 +15,33 @@ minimum of $l_x^\tau$. Let us begin with a definition that captures the necessary curvature bound for these results. -**Definition.** Let $(M,g(t)),\ a\le t\le b$, be a Ricci flow. We say that the flow + +**Definition.** Let $(M,g(t)),\ a\le t\le b$, be a Ricci flow. We say that the flow is *complete of bounded curvature* if for each $t\in [a,b]$ the Riemannian manifold $(M,g(t))$ is complete and if there is -$C<\infty$ such that $|Rm|(p,t)\le C$ for all $p\in M$ and all +$C<\infty$ such that $|\mathit{Rm}|(p,t)\le C$ for all $p\in M$ and all $t\in [a,b]$. Let $I$ be an interval and let $(M,g(t)),\ t\in I$, be a Ricci flow. Then we say that the flow is *complete with curvature locally bounded in time* if for each compact subinterval $J\subset I$ the restriction of the flow to $(M,g(t)),\ t\in J$, is complete of bounded curvature. + -## The functions $L_{x -$ and $l_{x}$} +## The functions $L_{x}$ and $l_{x}$ -Throughout Chapter~ we have a Ricci flow $(M,g(t)),\ +Throughout Chapter 7 we have a Ricci flow $(M,g(t)),\ 0\le t\le T<\infty$, and we set $\tau=T-t$. All the results of the last chapter apply in this context, but in fact in this context there are much stronger results, which we develop here. -### Existence of ${\mathcal L -$-geodesics} +### Existence of ${\mathcal L}$-geodesics We assume here that $(M,g(t)),\ 0\le t\le T<\infty$, is a Ricci flow which is complete of bounded curvature. In Shi's Theorem -(Theorem~) we take $K$ equal to the bound of the norm of +(\entryref{ed4b72caf9f9}) we take $K$ equal to the bound of the norm of the Riemannian curvature on $M\times [0,T]$, we take $\alpha=1$, and -we take $t_0=T$. It follows from Theorem~ that there is a +we take $t_0=T$. It follows from \entryref{ed4b72caf9f9} that there is a constant $C(K,T)$ such that $|\nabla R(x,t)|\le C/t^{1/2}$. Thus, for any $\epsilon>0$ we have a uniform bound for $|\nabla R|$ on $M\times [\epsilon,T]$. Also, because of the uniform bound for the @@ -47,37 +49,53 @@ Riemann curvature and the fact that $T<\infty$, there is a constant $C$, depending on the curvature bound and $T$ such that $$ - C^{-1}g(x,t)\le g(x,0)\le Cg(x,t) $$ for all $(x,t)\in M\times [0,T]$. -**Lemma.** Assume that $M$ is connected. + +**Lemma.** Assume that $M$ is connected. Given $p_1,p_2\in M$ and $0\le\tau_1<\tau_2\le T$, there is a minimizing ${\mathcal L}$-geodesic: $\gamma\colon[\tau_{1},\tau_{2}]\to M\times [0,T]$ connecting $(p_1,\tau_1)$ to $(p_2,\tau_2)$. + **Proof.** For any curve $\gamma$ parameterized by backward time, we set $\bar \gamma$ equal to the path in $M$ that is the image under projection of $\gamma$. We set $A(s)=\bar\gamma'(s)$. Define -$$c((p_1,\tau_1),(p_2,\tau_{2})) = inf + +$$ +c((p_1,\tau_1),(p_2,\tau_{2})) = \mathit{inf} \{\mathcal{L}(\gamma)|\gamma\colon [\tau_{1},\tau_{2}] \rightarrow M\times [0,T], \bar\gamma(\tau_{1}) = p_1, \bar\gamma(\tau_{2})= -p_2\}.$$ From Equation~() we see that the infimum exists +p_2\}. +$$ + + From Equation (6.2) we see that the infimum exists since, by assumption, the curvature is uniformly bounded (below). Furthermore, for a minimizing sequence $\gamma_{i}$, we have $\int_{s_1}^{s_{2}}\abs{A_i(s)}^{2}ds \leq C_0$, for some constant $C_0$, where $s_i=\sqrt{\tau_i}$ for $i=1,2$. It follows from this -and the inequality in Equation~() that there is a +and the inequality in Equation (7.1) that there is a constant $C_1<\infty$ such that for all $i$ we have -$$\int_{s_{1}}^{s_{2}} \abs{ -A_{i}}^{2}_{g(0)}d\tau \leq C_1.$$ Therefore the sequence $\{\gamma_{i}\}$ is + +$$ +\int_{s_{1}}^{s_{2}} \abs{ +A_{i}}^{2}_{g(0)}d\tau \leq C_1. +$$ + + Therefore the sequence $\{\gamma_{i}\}$ is uniformly continuous with respect to the metric $g(0)$; by Cauchy-Schwarz we have -$$\abs{\bar\gamma_{i}(s) - \bar\gamma_{i}(s')}_{g(0)} \leq -\int_{s'}^{s} \abs{A_{i}}_{g(0)}ds \leq \sqrt{C_1}\sqrt{s-s'}.$$ By + +$$ +\abs{\bar\gamma_{i}(s) - \bar\gamma_{i}(s')}_{g(0)} \leq +\int_{s'}^{s} \abs{A_{i}}_{g(0)}ds \leq \sqrt{C_1}\sqrt{s-s'}. +$$ + + By the uniform continuity, we see that a subsequence of the $\gamma_i$ converges uniformly pointwise to a continuous curve $\gamma$ parameterized by $s$, the square root backward time. By passing to a @@ -87,8 +105,8 @@ continuous limit $\gamma$. That is to say, after passing to a subsequence, the $\gamma_i$ converge uniformly and weakly in $H^{2,1}$ to a continuous curve $\gamma$. Let $A(s)$ be the $L^2$-derivative of $\gamma$. Weak convergence in $H^{2,1}$ implies -that $\int_{s'}^s|A(s)|^2ds\le lim_{i\rightarrow\infty}\int_{s'}^s|A_i(s)|^2ds$, so that -${\mathcal L}(\gamma)\le lim_{i\rightarrow\infty}{\mathcal +that $\int_{s'}^s|A(s)|^2ds\le \mathit{lim}_{i\rightarrow\infty}\int_{s'}^s|A_i(s)|^2ds$, so that +${\mathcal L}(\gamma)\le \mathit{lim}_{i\rightarrow\infty}{\mathcal L}(\gamma_i)$. This means that $\gamma$ minimizes the ${\mathcal L}$-length. Being a minimizer of ${\mathcal L}$-length, $\gamma$ satisfies the Euler-Lagrange equation and is smooth by the @@ -99,25 +117,32 @@ $(p_2,\tau_2)$. Let us now show that it is always possible to uniquely extend ${\mathcal L}$-geodesics up to time $T$. + **Lemma.** For any $0\le \tau_1<\tau_2 **Proof.** We work with the parameter $s=\sqrt{\tau}$. According to -Equation~(), we have -$$\nabla_{\gamma'(s)}\gamma'(s)=2s^2\nabla -R-4sRic(\gamma'(s),\cdot).$$ This is an everywhere +Equation (6.5), we have + +$$ +\nabla_{\gamma'(s)}\gamma'(s)=2s^2\nabla +R-4s\mathit{Ric}(\gamma'(s),\cdot). +$$ + + This is an everywhere non-singular ODE. Since the manifolds $(M,g(t))$ are complete and their metrics are uniformly related as in -Inequality~(), to show that the solution is defined +Inequality (7.1), to show that the solution is defined on the entire interval $s\in [0,\sqrt{T})$ we need only show that there is a uniform bound to the length, or equivalently the energy of $\gamma$ of any compact subinterval of $[0,T)$ on which it is defined. Fix $\epsilon>0$. It follows immediately from -Lemma~, and the fact that the quantities $R$, $|\nabla -R|$ and $|Rm|$ are bounded on $M\times [\epsilon,T]$, that -there is a bound on $max|\gamma'(s)|$ in terms of +\entryref{333be2e58780}, and the fact that the quantities $R$, $|\nabla +R|$ and $|\mathit{Rm}|$ are bounded on $M\times [\epsilon,T]$, that +there is a bound on $\mathit{max}|\gamma'(s)|$ in terms of $|\gamma'(\tau_1)|$, for all $s\in [0,\sqrt{T-\epsilon}]$ for which $\gamma$ is defined. Since $(M,g(0))$ is complete, this, together with a standard extension result for second-order ODE's, implies @@ -128,13 +153,13 @@ to the entire interval $[0,T-\epsilon]$. Since this is true for every $\epsilon>0$, this completes the proof. Let $p\in M$ and set $x=(p,T)\in M\times [0,T]$. Recall that from -Definition~ for every $\tau>0$, the injectivity set +\entryref{90fe97584bdd} for every $\tau>0$, the injectivity set $\widetilde{\mathcal U}_{x}(\tau)\subset T_pM$ consists of all $Z\in T_pM$ for which (i) the ${\mathcal L}$-geodesic $\gamma_Z|_{[0,\tau]}$ is the unique minimizing ${\mathcal L}$-geodesic from $x$ to its endpoint, (ii) the differential of -${\mathcal L}exp_{x}^\tau$ is an isomorphism at $Z$, and (iii) -for all $Z'$ sufficiently close to +${\mathcal L}\mathit{exp}_{x}^\tau$ is an isomorphism at $Z$, and (iii) +for all $Z'$ sufficiently close to $Z$ the ${\mathcal L}$-geodesic $\gamma_{Z'}|_{[0,\tau]}$ is the unique minimizing ${\mathcal L}$-geodesic to its endpoint. The image of $\widetilde{\mathcal U}_{x}(\tau)$ is denoted ${\mathcal U}_{x}(\tau)\subset @@ -144,13 +169,14 @@ The existence of minimizing ${\mathcal L}$-geodesics from $x$ to every point of $M\times (0,T)$ means that the functions $L_{x}$ and $l_{x}$ are defined on all of $M\times (0,T)$. This leads to: + **Definition.** Suppose that $(M,g(t)),\ 0\le t\le T<\infty$, is a Ricci -flow, complete of bounded curvature. We define the function +flow, complete of bounded curvature. We define the function $L_{x}\colon M\times [0,T)\to \Ar$ by assigning to each $(q,t)$ the length of any ${\mathcal L}$-minimizing ${\mathcal L}$-geodesic from $x$ to $y=(q,t)\in M\times [0,T)$. Clearly, the restriction of this function to ${\mathcal U}_{x}$ agrees with the smooth function -$L_{x}$ given in Definition~. We define +$L_{x}$ given in \entryref{440e7d27aee8}. We define $L_{x}^\tau\colon M\to \Ar$ to be the restriction of $L_x$ to $M\times \{T-\tau\}$. Of course, the restriction of $L_{x}^\tau$ to ${\mathcal @@ -158,47 +184,50 @@ U}_{x}(\tau)$ agrees with the smooth function $L_{x}^\tau$ defined in the last chapter. We define $l_{x}\colon M\times [0,T)\to \Ar$ by $l_{x}(y)=L_{x}(y)/2\sqrt{\tau}$, where, as always $\tau=T-t$, and we define $l_{x}^\tau(q)=l_{x}(q,T-\tau)$. + -### Results about $l_{x -$ and ${\mathcal -U}_{x}(\tau)$} +### Results about $l_{x}$ and ${\mathcal U}_{x}(\tau)$ Now we come to our main result about the nature of ${\mathcal -U}_{x}(\tau)$_x(\tau)$} and the function $l_{x}$ +U}_{x}(\tau)$ and the function $l_{x}$ in the context of Ricci flows which are complete and of bounded curvature. -**Proposition.** -Let $(M,g(t)),\ 0\le t\le T<\infty$, be a Ricci flow that is + +**Proposition.** Let $(M,g(t)),\ 0\le t\le T<\infty$, be a Ricci flow that is complete and of bounded curvature. Let $p\in M$, let $x=(p,T)\in M\times [0,T]$, and let $\tau\in (0,T)$. -\begin{enumerate} -\item[(1)] The functions $L_{x}$ and $l_{x}$ are locally Lipschitz functions + +- **(1)** The functions $L_{x}$ and $l_{x}$ are locally Lipschitz functions on $M\times (0,T)$. -\item[(2)] ${\mathcal L}exp_{x}^\tau$ +- **(2)** ${\mathcal L}\mathit{exp}_{x}^\tau$ is a diffeomorphism from $\widetilde{\mathcal U}_{x}(\tau)$ onto -an open subset ${\mathcal U}_{x}(\tau)$_x(\tau)$} of $M$. -\item[(3)] The complement of ${\mathcal U}_{x}(\tau)$ in $M$ is a closed -subset of $M$ of zero Lebesgue measure. \item[(4)] - For every +an open subset ${\mathcal U}_{x}(\tau)$ of $M$. +- **(3)** The complement of ${\mathcal U}_{x}(\tau)$ in $M$ is a closed +subset of $M$ of zero Lebesgue measure. +- **(4)** For every $\tau<\tau' + +**Proof.** By Shi's Theorem (\entryref{ed4b72caf9f9}) the curvature bound on $M\times [0,T]$ implies that for each $\epsilon>0$ there is a bound for $|\nabla R|$ on $M\times (\epsilon,T]$. Thus, -Proposition~ shows that $L_{x}$ is a +\entryref{96ced8c484df} shows that $L_{x}$ is a locally Lipschitz function on $M\times (\epsilon,T)$. Since this is true for every $\epsilon>0$, $L_x$ is a locally Lipschitz function on $M\times (0,T)$. Of course, the same is true for $l_{x}$. The -second statement is contained in Proposition~, and the -last one is contained in Proposition~. It remains to +second statement is contained in \entryref{17254b50b27a}, and the +last one is contained in \entryref{8b9b3403dbc1}. It remains to prove the third statement, namely that the complement of ${\mathcal U}_{x}(\tau)$ is closed nowhere dense. This follows immediately from -Corollary~ since $|Ric|$ and $|\nabla R|$ are +\entryref{7aad50e49af2} since $|\mathit{Ric}|$ and $|\nabla R|$ are bounded on $F=M\times [T-\tau,T]$. + **Corollary.** The function $l_x$ is a continuous function on $M\times (0,T)$ and is smooth on the complement of a closed subset ${\mathcal C}$ that has the property that its intersection with each $M\times \{t\}$ is @@ -211,72 +240,87 @@ the restriction of $|\nabla l_x^\tau|$ to $V\setminus\left(V\cap {\mathcal C}\right)$ is a bounded smooth function. Similarly, $\partial l_x/\partial t$ is an essentially bounded smooth vector field on $M\times (0,T)$. + -## A bound for $min -\, l^\tau_x$ +## A bound for $\mathit{min} l^\tau_x$ We continue to assume that we have a Ricci flow $(M,g(t)),\ 0\le t\le T<\infty$, complete and of bounded curvature and a point $x=(p,T)\in M\times [0,T]$. Our purpose here is to extend the first differential inequality given in - Corollary~ to a differential + \entryref{967c0f8bb774} to a differential inequality in the weak or distributional sense for $l_{x}$ valid on -all of $M\times (0,T)$. We then use this to establish that $min_{q\in M} l^\tau_{x}(q)\le n/2$ for all $0<\tau **Definition.** Let $P$ be a smooth manifold and let $f\colon P\to \Ar$ be a continuous function. An *upper barrier* for $f$ at $p\in P$ is a smooth function $\varphi$ defined on a neighborhood of $p$ in $P$, say $U$, satisfying $\varphi(p)=f(p)$ and $\varphi(u)\ge f(u)$ for all $u\in U$, see -Fig.~. +Fig. 7.1. + -**Proposition.** -Let $(M,g(t)),\ 0\le t\le T<\infty$, be an $n$-dimensional Ricci -flow, complete of bounded curvature. Fix a point $x=(p,T)\in + +**Proposition.** Let $(M,g(t)),\ 0\le t\le T<\infty$, be an $n$-dimensional Ricci +flow, complete of bounded curvature. Fix a point $x=(p,T)\in M\times [0,T]$, and for any $(q,t)\in M\times [0,T]$, set $\tau=T-t$. Then for any $(q,t)$, with $00$ there is a neighborhood $U$ of $(q,t)$ in $M\times [0,T]$ and an upper barrier $\varphi$ for $l_x$ at this point defined on $U$ satisfying -$$\frac{\partial \varphi}{\partial \tau}(q,\tau)+\triangle \varphi(q,\tau) -\le \frac{(n/2)-l_x(q,\tau)}{\tau}+\epsilon.$$ +$$ +\frac{\partial \varphi}{\partial \tau}(q,\tau)+\triangle \varphi(q,\tau) +\le \frac{(n/2)-l_x(q,\tau)}{\tau}+\epsilon. +$$ + + + **Remark.** The operator $\triangle$ in the above statement is the horizontal Laplacian, i.e., the Laplacian of the restriction of the indicated function to the slice $M\times\{t=T-\tau\}$ as defined using the metric $g(T-\tau)$ on this slice. + **Proof.** If $(q,T-\tau)\in {\mathcal U}_{x}$, then $l_x$ is smooth near $(q,T-\tau)$, and the result is immediate from the first inequality in -Corollary~. +\entryref{967c0f8bb774}. Now consider a general point $(q,t=T-\tau)$ with $0 +**Theorem.** Suppose that $(M,g(t)),\ 0\le t\le T<\infty$, is an $n$-dimensional Ricci flow, complete of bounded curvature. Then for any $x=(p,T)\in M\times [0,T]$ and for every $0<\tau -**Proof.** We set $l_min(\tau)=inf_{q\in +**Proof.** We set $l_\mathit{min}(\tau)=\mathit{inf}_{q\in M}l_x(q,\tau)$. (We are not excluding the possibility that this infimum is $-\infty$.) To prove this corollary we first need to establish the following claim. -**Claim.** -For every + +**Claim.** For every $\tau\in (0,T)$ the function $l_x(\cdot,\tau)$ achieves its minimum. Furthermore, for every compact interval $I\subset (0,T)$ the subset of $(q,\tau)\in M\times I$ - for which $l_x(q,\tau)=l_min(\tau)$ is a compact set. + for which $l_x(q,\tau)=l_\mathit{min}(\tau)$ is a compact set. + - First, let us assume this claim and use it to prove the -theorem. We set $l_min(\tau)=min_{q\in M}l_x(q,\tau)$. +First, let us assume this claim and use it to prove the +theorem. We set $l_\mathit{min}(\tau)=\mathit{min}_{q\in M}l_x(q,\tau)$. (This minimum exists by the first statement in the claim.) From the compactness result in the claim, it follows (see for example -Proposition~) that $l_min(\tau)$ is a +\entryref{37d7d059a526}) that $l_\mathit{min}(\tau)$ is a continuous function of $\tau$. Suppose that $l_x(\cdot,\tau)$ achieves its minimum at $q$. Then by the previous result, for any $\epsilon>0$ there is an upper barrier $\varphi$ for $l_x$ at $(q,\tau)$ defined on an open subset $U$ of $(q,\tau)\in M\times (0,T)$ and satisfying -$$\frac{d\varphi}{d\tau}(q,\tau)+\triangle \varphi(q,\tau)\le \frac{(n/2)-l_x(q,t)}{\tau}+\epsilon.$$ -Since $l_x(q,\tau)=l_min(\tau)$, it follows that + +$$ +\frac{d\varphi}{d\tau}(q,\tau)+\triangle \varphi(q,\tau)\le \frac{(n/2)-l_x(q,t)}{\tau}+\epsilon. +$$ + +Since $l_x(q,\tau)=l_\mathit{min}(\tau)$, it follows that $\varphi(q',\tau)\ge \varphi(q,\tau)$ for all $(q',\tau)\in U\cap M_{T-\tau}$. This means that $\triangle \varphi(q,\tau)\ge 0$, and we conclude that -$$\frac{d\varphi}{d\tau}(q,\tau)\le \frac{(n/2)-l_min(\tau)}{\tau}+\epsilon.$$ + +$$ +\frac{d\varphi}{d\tau}(q,\tau)\le \frac{(n/2)-l_\mathit{min}(\tau)}{\tau}+\epsilon. +$$ + Since $\varphi$ is an upper barrier for $l_x$ at $(q,\tau)$ it follows immediately that -$$limsup_{\tau'\rightarrow + +$$ +\mathit{limsup}_{\tau'\rightarrow \tau^+}\frac{l_x(q,\tau')-l_x(q,\tau)}{\tau'-\tau}\le -\frac{(n/2)-l_x(q,\tau)}{\tau}+\epsilon.$$ Since this is true for every +\frac{(n/2)-l_x(q,\tau)}{\tau}+\epsilon. +$$ + + Since this is true for every $\epsilon>0$, we see that -$$limsup_{\tau'\rightarrow + +$$ +\mathit{limsup}_{\tau'\rightarrow \tau^+}\frac{l_x(q,\tau')-l_x(q,\tau)}{\tau'-\tau}\le -\frac{(n/2)-l_x(q,\tau)}{\tau}.$$ Since $l_min(\tau)=l_x(q,\tau)$, the same inequality holds for the forward -difference quotient of $l_min$ at $\tau$. That is to say, we +\frac{(n/2)-l_x(q,\tau)}{\tau}. +$$ + + Since $l_\mathit{min}(\tau)=l_x(q,\tau)$, the same inequality holds for the forward +difference quotient of $l_\mathit{min}$ at $\tau$. That is to say, we have -$$limsup_{\tau'\rightarrow \tau^+}\frac{l_min(\tau')-l_min(\tau)}{\tau'-\tau}\le \frac{(n/2)-l_min(\tau)}{\tau}.$$ -The preceding equation implies that if $l_min(\tau)\le n/2$ -then $l_min(\tau')\le n/2$ for every $\tau'\ge \tau$. On the -other hand $lim_{\tau\rightarrow 0}l_min(\tau)=0$. Then + +$$ +\mathit{limsup}_{\tau'\rightarrow \tau^+}\frac{l_\mathit{min}(\tau')-l_\mathit{min}(\tau)}{\tau'-\tau}\le \frac{(n/2)-l_\mathit{min}(\tau)}{\tau}. +$$ + +The preceding equation implies that if $l_\mathit{min}(\tau)\le n/2$ +then $l_\mathit{min}(\tau')\le n/2$ for every $\tau'\ge \tau$. On the +other hand $\mathit{lim}_{\tau\rightarrow 0}l_\mathit{min}(\tau)=0$. Then reason for this is that the path $\tau'\mapsto (P,T-\tau')$ for $\tau'\in [0,\tau]$ has ${\mathcal L}$-length $ O(\tau^{3/2})$ as -$\tau\rightarrow 0$. It follows that $l_min(\tau) +**Corollary.** Suppose that $({\mathcal M},G)$ is a generalized $n$-dimensional Ricci flow and +that $x\in {\mathcal M}$ is given and set $t_0=\mathbf{t}(x)$. We suppose that +there is an open subset $U\subset \mathbf{t}^{-1}(-\infty,t_0)$ with the following properties: -\begin{enumerate} -\item For every $y\in U$ there is a minimizing ${\mathcal L}$-geodesic from $x$ + +1. For every $y\in U$ there is a minimizing ${\mathcal L}$-geodesic from $x$ to $y$. -\item There are $r>0$ and $\Delta t>0$ such that the backward parabolic neighborhood -$P(x,t_0,r,-\Delta t)$ of $x$ exists in ${\mathcal M}$ and has the property -that $P\cap **t**^{-1}(-\infty,t_0)$ is contained in $U$. -\item For each compact interval (including the case of degenerate intervals +1. There are $r>0$ and $\Delta t>0$ such that the backward parabolic neighborhood +$P(x,t_0,r,-\Delta t)$ of $x$ exists in ${\mathcal M}$ and has the property +that $P\cap \mathbf{t}^{-1}(-\infty,t_0)$ is contained in $U$. +1. For each compact interval (including the case of degenerate intervals consisting of a single point) $I\subset (-\infty,t_0)$ the subset of points -$y\in **t**^{-1}(I)\cap U$ for which ${\mathcal L}(y)=inf_{z\in **t**^{-1}(**t**(y))\cap U}{\mathcal L}(z)$ is compact and non-empty. -\end{enumerate} +$y\in \mathbf{t}^{-1}(I)\cap U$ for which ${\mathcal L}(y)=\mathit{inf}_{z\in \mathbf{t}^{-1}(\mathbf{t}(y))\cap U}{\mathcal L}(z)$ is compact and non-empty. + Then for every $t -### Extension of the other inequalities in - Corollary~\protect{} +### Extension of the other inequalities in Corollary \entryref{967c0f8bb774} -The material in this subsection is adapted from . It +The material in this subsection is adapted from [Ye]. It captures (in a weaker way) the fact that, in the case of geodesics on a Riemannian manifold, the interior of the cut locus in $T_xM$ is star-shaped from the origin. -**Theorem.** -Let $(M,g(t)),\ 0\le t\le T<\infty$, be a Ricci flow, complete and + +**Theorem.** Let $(M,g(t)),\ 0\le t\le T<\infty$, be a Ricci flow, complete and of bounded curvature, and let $x=(p,T)\in M\times[0,T]$. The last -two inequalities in Corollary~, namely -$$\frac{\partial l_x}{\partial \tau}+|\nabla l^\tau_x|^2-R+\frac{n}{2\tau} -- \triangle l^\tau_x\ge 0$$ -$$-|\nabla l^\tau_x|^2+R+\frac{l^\tau_x-n}{\tau}+2\triangle l^\tau_x\le 0$$ +two inequalities in \entryref{967c0f8bb774}, namely + +$$ +\frac{\partial l_x}{\partial \tau}+|\nabla l^\tau_x|^2-R+\frac{n}{2\tau} +- \triangle l^\tau_x\ge 0 +$$ + +$$ +-|\nabla l^\tau_x|^2+R+\frac{l^\tau_x-n}{\tau}+2\triangle l^\tau_x\le 0 +$$ + hold in the weak or distributional sense on all of $M\times\{\tau\}$ for all $\tau>0$. This means that for any $\tau>0$ and for any non-negative, compactly supported, smooth function $\phi(q)$ on $M$ we have the following two inequalities: -$$\int_{M\times \{\tau\}}\Bigl[ \phi\cdot\left(\frac{\partial l_x}{\partial \tau}+|\nabla + +$$ +\int_{M\times \{\tau\}}\Bigl[ \phi\cdot\left(\frac{\partial l_x}{\partial \tau}+|\nabla l_x^\tau|^2-R+\frac{n}{2\tau}\right) - l_x^\tau\triangle \phi\Bigr]\ -dvol(g(t))\ge 0$$ -$$\int_{M\times \{\tau\}}\Big[ \phi\cdot\left(-|\nabla +d\mathit{vol}(g(t))\ge 0 +$$ + +$$ +\int_{M\times \{\tau\}}\Big[ \phi\cdot\left(-|\nabla l_x^\tau|^2+R+\frac{l_x^\tau-n}{\tau}\right)+2l^\tau_x\triangle -\phi\Bigr]\ dvol(g(t))\le 0.$$ Furthermore, equality holds in +\phi\Bigr]\ d\mathit{vol}(g(t))\le 0. +$$ + + Furthermore, equality holds in either of these weak inequalities for all functions $\phi$ as above and all $\tau$ if and only if it holds in both. In that case $l_x$ is a smooth function on space-time and the equalities hold in the usual smooth sense. + + **Remark.** The terms in these inequalities are interpreted in the following way: First of all, $\nabla l^\tau_x$ and $\triangle l^\tau_x$ are computed using only the spatial derivatives (i.e., they are @@ -462,7 +563,7 @@ locally bounded on all of $M\times\{\tau\}$ in the sense that for any $q\in M$ there is a neighborhood $V$ of $q$ such that the restriction of $|\nabla l_x^\tau|^2$ to $V\cap {\mathcal U}_x(\tau)$ is bounded. This means that $\partial l_x/\partial t$ and $|\nabla -l_x^\tau|^2$ are elements of $L_loc^\infty(M)$ and hence can +l_x^\tau|^2$ are elements of $L_\mathit{loc}^\infty(M)$ and hence can be integrated against any smooth function with compact support. In particular, they are distributions. @@ -470,15 +571,27 @@ particular, they are distributions. $\nabla l^\tau_x$ is a smooth, locally bounded vector field on an open subset of full measure, for any compactly supported test function $\phi$, integration by parts yields -$$\int\triangle \phi\cdot l^\tau_x\ dvol(g(t))=-\int \langle \nabla \phi,\nabla -l_x^\tau\rangle\ dvol(g(t)).$$ Thus, formulas in -Theorem~ can also be taken to mean: -$$\int_{M\times \{\tau\}} \Bigl[\phi\cdot(\frac{\partial l_x}{\partial \tau} + +$$ +\int\triangle \phi\cdot l^\tau_x\ d\mathit{vol}(g(t))=-\int \langle \nabla \phi,\nabla +l_x^\tau\rangle\ d\mathit{vol}(g(t)). +$$ + + Thus, formulas in +\entryref{7d5d94efe4e9} can also be taken to mean: + +$$ +\int_{M\times \{\tau\}} \Bigl[\phi\cdot(\frac{\partial l_x}{\partial \tau} +|\nabla l^\tau_x|^2-R+\frac{n}{2\tau})+ \langle \nabla -l_x^\tau,\nabla \phi \rangle\Bigr] dvol(g(t))\ge 0$$ -$$\int_{M\times\{\tau\}} \Bigl[\phi\cdot(-|\nabla +l_x^\tau,\nabla \phi \rangle\Bigr] d\mathit{vol}(g(t))\ge 0 +$$ + +$$ +\int_{M\times\{\tau\}} \Bigl[\phi\cdot(-|\nabla l^\tau_x|^2+R+\frac{l^\tau_x-n}{\tau})-2\langle\nabla -l^\tau_x,\nabla\phi \rangle\Bigr] dvol(g(t))\le 0.$$ +l^\tau_x,\nabla\phi \rangle\Bigr] d\mathit{vol}(g(t))\le 0. +$$ + The rest of this subsection is devoted to the proof of these inequalities. We fix $(M,g(t)),\ 0\le t\le T<\infty$, as in the @@ -489,34 +602,39 @@ We also fix $\tau$, and we denote by $L^\tau$ and $l^\tau$ the restrictions of $L$ and $l$ to the slice $M\times \{T-\tau\}$. We begin with a lemma. -**Lemma.** -There is a continuous function $C\colon M\times (0,T)\to \Ar$ such + +**Lemma.** There is a continuous function $C\colon M\times (0,T)\to \Ar$ such that for each point $(q,t)\in M\times (0,T)$, setting $\tau=T-t$, the following holds. There is an upper barrier $\varphi_{(q,t)}$ for $L_x^\tau$ at the point $q$ defined on a neighborhood $U_{(q,t)}$ of $q$ in $M$ satisfying $|\nabla \varphi_{(q,t)}(q)|\le C(q,t)$ and -$$Hess(\varphi)(v,v)\le C(q,t)|v|^2$$ + +$$ +\mathit{Hess}(\varphi)(v,v)\le C(q,t)|v|^2 +$$ + for all tangent vectors $v\in T_qM$. + -**Proof.** By Proposition~, $L$ is a locally Lipschitz +**Proof.** By \entryref{16fbe263a7c8}, $L$ is a locally Lipschitz function on $M\times (0,T)$, and in particular is continuous. The bound $C(q,t)$ will depend only on the bounds on curvature and its first two derivatives and on the function $L(q,t)$. Fix $(q,t)$ and let $\gamma$ be a minimizing ${\mathcal L}$-geodesic from $x$ to $(q,t)$. (The existence of such a minimizing geodesic is established -in Lemma~.) Fix $\tau_1>0$, with $\tau_1<(T-t)/2$, let +in \entryref{cd062526b91a}.) Fix $\tau_1>0$, with $\tau_1<(T-t)/2$, let $t_1=T-\tau_1$, and let $q_1=\gamma(\tau_1)$. Consider $\varphi_{(q,t)}={\mathcal L}(\gamma_{[0,\tau_1]})+ -L^\tau_{(q_1,t_1)}$. This is an upper barrier for $L_x^\tau$ at $q$ -defined in some neighborhood $V\subset M$ of $q$. Clearly, $\nabla -\varphi_{(q,t)}=\nabla L^\tau_{(q_1,t_1)}$ and $Hess(\varphi_{(q,t)}=Hess( L^\tau_{(q_1,t_1)})$. +L^\tau_{(q_1,t_1)}$. This is an upper barrier for $L_x^\tau$ at $q$ +defined in some neighborhood $V\subset M$ of $q$. Clearly, $\nabla +\varphi_{(q,t)}=\nabla L^\tau_{(q_1,t_1)}$ and $\mathit{Hess}(\varphi_{(q,t)}=\mathit{Hess}( L^\tau_{(q_1,t_1)})$. -According to Corollary~ we have $\nabla +According to \entryref{033786a89c6d} we have $\nabla L^\tau_{(q_1,t_1)}(q)=2\sqrt{\tau}X_{\gamma}(\tau)$. On the other -hand, by Corollary~ there is a bound on +hand, by \entryref{a7c2ebee07a3} there is a bound on $\sqrt{\tau}|X_{\gamma}(\tau)|$ depending only on the bounds on curvature and its first derivatives, on $\tau$ and $\tau_1$ and on -$l_x(q,\tau)$. Of course, by Shi's theorem (Theorem~) +$l_x(q,\tau)$. Of course, by Shi's theorem (\entryref{dbe4fabd0076}) for every $\epsilon>0$ the norms of the first derivatives of curvature on $M\times [\epsilon,T]$ are bounded in terms of $\epsilon$ and the bounds on curvature. @@ -524,7 +642,7 @@ $\epsilon$ and the bounds on curvature. proves that $|\nabla \varphi_{(q,t)}(q)|$ is bounded by a continuous function $C(q,t)$ defined on all of $M\times (0,T)$. - Now consider Inequality~() for $\gamma$ at $\bar\tau=\tau$. + Now consider Inequality (6.11) for $\gamma$ at $\bar\tau=\tau$. It is clear that the first two terms on the right-hand side are bounded by $C|Y(\tau)|^2$, where $C$ depends on the curvature bound and on $T-t$. We consider @@ -534,97 +652,145 @@ function $C(q,t)$ defined on all of $M\times (0,T)$. second derivatives along $\gamma_1$ and on $T-t$. We consider $\tau'\in [\tau_1,\tau]$. Of course, $\sqrt{\tau'}|X(\tau')|$ is bounded on this interval. Also, - $$|Y(\tau')|=\left(\frac{\sqrt{\tau'}-\sqrt{\tau_1}}{\sqrt{\tau} - -\sqrt{\tau_1}}\right)|Y(\tau)|\le \frac{\sqrt{\tau'}}{\sqrt{\tau}}|Y(\tau)|.$$ + +$$ +|Y(\tau')|=\left(\frac{\sqrt{\tau'}-\sqrt{\tau_1}}{\sqrt{\tau} + -\sqrt{\tau_1}}\right)|Y(\tau)|\le \frac{\sqrt{\tau'}}{\sqrt{\tau}}|Y(\tau)|. +$$ + Hence $|Y(\tau')|/\sqrt{\tau'}$ and $|Y(\tau')||X(\tau')|$ are bounded in terms of $T-t$, $|Y(\tau)|$, and the bound on $\sqrt{\tau'}|X(\tau')|$ along the ${\mathcal L}$-geodesic. From -this it follows immediately from Equation~() that $H(X,Y)$ -is bounded along the ${\mathcal L}$-geodesic by $C|Y(\bar\tau)|^2$ +this it follows immediately from Equation (6.12) that $H(X,Y)$ +is bounded along the ${\mathcal L}$-geodesic by $C|Y(\bar\tau)|^2$ where the constant $C$ depends on $T-t$ and the bounds on curvature and its first two derivatives. Of course, if $(q,t)\in {\mathcal U}_{x}(\tau)$, then this argument shows that the Hessian of $L^\tau_{x}$ is bounded near $(q,t)$. -At this point in the proof of Theorem~ we wish to employ +At this point in the proof of \entryref{7d5d94efe4e9} we wish to employ arguments using convexity. To carry these out we find it convenient to work with a Euclidean metric and usual convexity rather than the given metric $g(t)$ and convexity measured using $g(t)$-geodesics. In order to switch to a Euclidean metric we must find one that well approximates $g(t)$. The following is straightforward to prove. + **Claim.** For each point $(q,t)\in M\times(0,T)$ there is an open metric ball $B_{(q,t)}$ centered at $q$ in $(M,g(t))$ which is the diffeomorphic image of a ball $\widetilde B\subset T_qM$ under the exponential map for $g(t)$ centered at $q$ such that the following hold: -\begin{enumerate} -\item $B_{(q,t)}\subset U_{(q,t)}$ so that the upper barrier $\varphi_{(q,t)}$ -from Lemma~ is defined on all of $B_{(q,t)}$. -\item The constants $C(z,t)$ of Lemma~ satisfy + +1. $B_{(q,t)}\subset U_{(q,t)}$ so that the upper barrier $\varphi_{(q,t)}$ +from \entryref{5e0538ed62cf} is defined on all of $B_{(q,t)}$. +1. The constants $C(z,t)$ of \entryref{5e0538ed62cf} satisfy $C(z,t)\le 2C(q,t)$ for all $z\in B_{(q,t)}$. -\item The push-forward, $h$, under the exponential mapping of the Euclidean -metric on $T_qM$ satisfies $$h/2\le g\le 2h.$$ -\item The Christoffel symbols $\Gamma_{ij}^k$ for the +1. The push-forward, $h$, under the exponential mapping of the Euclidean +metric on $T_qM$ satisfies + +$$ +h/2\le g\le 2h. +$$ + +1. The Christoffel symbols $\Gamma_{ij}^k$ for the metric $g(t)$ written using the Gaussian normal coordinates (the image under the exponential mapping of orthonormal linear coordinates on $T_qM$) are bounded in absolute value by $1/(8n^3C(q,t))$ where $n$ is the dimension of $M$. -\end{enumerate} + Instead of working in the given metric $g(t)$ on $B_{(q,t)}$ we shall use the Euclidean metric $h$ as in the above claim. For any -function $f$ on $B_{(q,t)}$ we denote by $Hess(f)$ the Hessian -of $f$ with respect to the metric $g(t)$ and by $Hess^h(f)$ +function $f$ on $B_{(q,t)}$ we denote by $\mathit{Hess}(f)$ the Hessian +of $f$ with respect to the metric $g(t)$ and by $\mathit{Hess}^h(f)$ the Hessian of $f$ with respect to the metric $h$. By -Formula~(), for any $z\in B_{(q,t)}$ and any $v\in +Formula (1.2), for any $z\in B_{(q,t)}$ and any $v\in T_zM$, we have -$$Hess(\varphi_{(z,t)})(v,v)=Hess^h(\varphi_{(z,t)})(v,v) + +$$ +\mathit{Hess}(\varphi_{(z,t)})(v,v)=\mathit{Hess}^h(\varphi_{(z,t)})(v,v) -\sum_{i,j,k}v^iv^j\Gamma_{ij}^k\frac{\partial -\varphi_{(z,t)}}{\partial x^k}.$$ Thus, it follows from the above +\varphi_{(z,t)}}{\partial x^k}. +$$ + + Thus, it follows from the above assumptions on the $\Gamma_{ij}^k$ and the bound on $|\nabla \varphi_{(z,t)}|$ that for all $z\in B_{(q,t)}$ we have $$ - -\left|Hess(\varphi_{(z,t)})(v,v)-Hess^h(\varphi_{(z,t)})(v,v)\right|\le +\left|\mathit{Hess}(\varphi_{(z,t)})(v,v)-\mathit{Hess}^h(\varphi_{(z,t)})(v,v)\right|\le \frac{1}{4}|v|^2_h, $$ + and hence for every $z\in B_{(q,t)}$ we have -$$Hess^h(\varphi_{(z,t)})(v,v)\le 2C(q,t)|v|^2_g+\frac{|v|_h^2}{4}\le -\left(4C(q,t)+\frac{1}{4}\right)|v|_h^2.$$ + +$$ +\mathit{Hess}^h(\varphi_{(z,t)})(v,v)\le 2C(q,t)|v|^2_g+\frac{|v|_h^2}{4}\le +\left(4C(q,t)+\frac{1}{4}\right)|v|_h^2. +$$ This means: + **Claim.** For each $(q,t)\in M\times(0,T)$ there is a smooth function -$$\psi_{(q,t)}\colon B_{(q,t)}\to \Ar$$ + +$$ +\psi_{(q,t)}\colon B_{(q,t)}\to \Ar +$$ + with the property that at each $z\in B_{(q,t)}$ there is an upper barrier $b_{(z,t)}$ for $L^\tau+\psi_{(q,t)}$ at $z$ with -$$Hess^h(b_{(z,t)})(v,v)\le -3|v|_h^2/2$$ + +$$ +\mathit{Hess}^h(b_{(z,t)})(v,v)\le -3|v|_h^2/2 +$$ + for all $v\in T_zM$. + **Proof.** Set -$$\psi_{(q,t)}=-(2C(q,t)+1)d^2_h(q,\cdot).$$ -Then for any $z\in B_{(q,t)}$ the function -$b_{(z,t)}=\varphi_{(z,t)}+\psi_{(q,t)}$ is an upper barrier for + +$$ +\psi_{(q,t)}=-(2C(q,t)+1)d^2_h(q,\cdot). +$$ + +Then for any $z\in B_{(q,t)}$ the function +$b_{(z,t)}=\varphi_{(z,t)}+\psi_{(q,t)}$ is an upper barrier for $L^\tau+\psi_{(q,t)}$ at $z$. Clearly, for all $v\in T_zM$ we have -$$Hess^h(b_{(z,t)})(v,v)=Hess^h(\varphi_{(z,t)})(v,v)+Hess^h(\psi_{(q,t)})(v,v)\le -3|v|_h^2/2.$$ + +$$ +\mathit{Hess}^h(b_{(z,t)})(v,v)=\mathit{Hess}^h(\varphi_{(z,t)})(v,v)+\mathit{Hess}^h(\psi_{(q,t)})(v,v)\le -3|v|_h^2/2. +$$ This implies that if $\alpha\colon [a,b]\to B_{(q,t)}$ is any Euclidean straight-line segment in $B_{(q,t)}$ parameterized by Euclidean arc length and if $z= \alpha(s)$ for some $s\in (a,b)$, then -$$(b_{(z,t)}\circ\alpha)''(s)\le -3/2.$$ +$$ +(b_{(z,t)}\circ\alpha)''(s)\le -3/2. +$$ + + **Claim.** Suppose that $\beta\colon [-a,a]\to \Ar$ is a continuous function and that at each $s\in (-a,a)$ there is an upper barrier $\hat b_s$ -for $\beta$ at $s$ with $\hat b_s''\le -3/2$. Then -$$\frac{\beta(a)+\beta(-a)}{2}\le \beta(0)-\frac{3}{4}a^2.$$ +for $\beta$ at $s$ with $\hat b_s''\le -3/2$. Then + +$$ +\frac{\beta(a)+\beta(-a)}{2}\le \beta(0)-\frac{3}{4}a^2. +$$ + **Proof.** Fix $c<3/4$ and define a continuous function -$$A(s)=\frac{(\beta(-s)+\beta(s))}{2}+cs^2-\beta(0)$$ for $s\in [0,a]$. + +$$ +A(s)=\frac{(\beta(-s)+\beta(s))}{2}+cs^2-\beta(0) +$$ + + for $s\in [0,a]$. Clearly, $A(0)=0$. Also, using the upper barrier at $0$ we see that for $s>0$ sufficiently small $A(s)<0$. For any $s\in (0,a)$ there is an upper barrier $c_s=(\hat b_{s}+\hat b_{-s})/2+cs^2-\beta(0)$ for @@ -637,60 +803,82 @@ this is true for every $c<3/4$, the result follows. Now applying this to Euclidean intervals in $B_{(q,t)}$ we conclude: -**Corollary.** -For any $(q,t)\in M\times (0,T)$, the function -$$\beta_{(q,t)}=L^\tau+\psi_{(q,t)}\colon B_{(q,t)}\to\Ar$$ is + +**Corollary.** For any $(q,t)\in M\times (0,T)$, the function + +$$ +\beta_{(q,t)}=L^\tau+\psi_{(q,t)}\colon B_{(q,t)}\to\Ar +$$ + + is uniformly strictly convex with respect to $h$. In fact, let $\alpha\colon [a,b]\to B_{(q,t)}$ be a Euclidean geodesic arc. Let $y,z$ be the endpoints of $\alpha$, let $w$ be its midpoint, and let $|\alpha|$ denote the length of this arc (all defined using the Euclidean metric). We have -$$\beta_{(q,t)}(w)\ge \frac{\left(\beta_{(q,t)}(y)+\beta_{(q,t)}(z)\right)}{2}+\frac{3}{16}|\alpha|^2.$$ -What follows is a simple interpolation result (see ). +$$ +\beta_{(q,t)}(w)\ge \frac{\left(\beta_{(q,t)}(y)+\beta_{(q,t)}(z)\right)}{2}+\frac{3}{16}|\alpha|^2. +$$ + + +What follows is a simple interpolation result (see [GreeneWu]). For each $q\in M$ we let $B_{(q,t)}'\subset B_{(q,t)}$ be a smaller ball centered at $q$, so that $B'_{(q,t)}$ has compact closure in $B_{(q,t)}$. + **Claim.** Fix $(q,t)\in M\times(0,T)$, and let $\beta_{(q,t)}\colon B_{(q,t)}\to \Ar$ be as above. Let $S\subset M$ be the singular locus of $L^\tau$, i.e., $S=M\setminus {\mathcal U}_{x}(\tau)$. Set $S_{(q,t)}=B_{(q,t)}\cap S$. Of course, $\beta_{(q,t)}$ is smooth on $B_{(q,t)}\setminus S_{(q,t)}$. Then there is a sequence of smooth functions $\{f_k\colon B'_{(q,t)}\to \Ar\}_{k=1}^\infty$ with the following properties: -\begin{enumerate} -\item As $k\rightarrow \infty$ the functions $f_k$ converge uniformly to $\beta_{(q,t)}$ on + +1. As $k\rightarrow \infty$ the functions $f_k$ converge uniformly to $\beta_{(q,t)}$ on $B_{(q,t)}'$. -\item For any $\epsilon>0$ sufficiently small, let $\nu_\epsilon(S_{(q,t)})$ be the +1. For any $\epsilon>0$ sufficiently small, let $\nu_\epsilon(S_{(q,t)})$ be the $\epsilon$-neighborhood (with respect to the Euclidean metric) in $B_{(q,t)}$ of $S_{(q,t)}\cap B_{(q,t)}$. Then, as $k\rightarrow \infty$ the restrictions of $f_k$ to $B_{(q,t)}'\setminus \left(B_{(q,t)}'\cap \nu_\epsilon(S_{(q,t)})\right)$ converge uniformly in the $C^\infty$-topology to the restriction of $\beta_{(q,t)}$ to this subset. -\item For each $k$, and for any $z\in B_{(q,t)}'$ and any $v\in T_zM$ we have -$$Hess(f_k)(v,v)\le -|v|_{g(t)}^2/2.$$ +1. For each $k$, and for any $z\in B_{(q,t)}'$ and any $v\in T_zM$ we have + +$$ +\mathit{Hess}(f_k)(v,v)\le -|v|_{g(t)}^2/2. +$$ + That is to say, $f_k$ is strictly convex with respect to the metric $g(t)$. -\end{enumerate} + **Proof.** Fix $\epsilon>0$ sufficiently small so that for any $z\in B_{(q,t)}'$ the Euclidean $\epsilon$-ball centered at $z$ is contained in $B_{(q,t)}$. Let $B_0$ be the ball of radius $\epsilon$ centered at the origin in $\Ar^n$ and let $\xi\colon B_0\to \Ar$ be a non-negative $C^\infty$-function with compact support and with -$\int_{B_0}\xi dvol_h=1$. We define -$$\beta_{(q,t)}^\epsilon(z)=\int_{B_0}\xi(y)\beta_{(q,t)}(z+y)dy,$$ +$\int_{B_0}\xi d\mathit{vol}_h=1$. We define + +$$ +\beta_{(q,t)}^\epsilon(z)=\int_{B_0}\xi(y)\beta_{(q,t)}(z+y)dy, +$$ + for all $z\in B_{(q,t)}'$. It is clear that for each $\epsilon>0$ sufficiently small, the function $\beta_{(q,t)}^\epsilon\colon B_{(q,t)}'\to \Ar$ is $C^\infty$ and that as $\epsilon\rightarrow 0$ the $\beta_{(q,t)}^\epsilon$ converge uniformly on $B_{(q,t)}'$ to $\beta_{(q,t)}$. It is also clear that for every $\epsilon>0$ -sufficiently small, the conclusion of Corollary~ holds +sufficiently small, the conclusion of \entryref{528f2ee171eb} holds for $\beta_{(q,t)}^\epsilon$ and for each Euclidean straight-line -segment $\alpha$ in $B_{(q,t)}'$. This implies that $Hess^h(\beta_{(q,t)}^\epsilon)(v,v)\le -3|v|_h^2/2$, and hence that -by Inequality~() that -$$Hess(\beta_{(q,t)}^\epsilon)(v,v)\le -|v|^2_h=-|v|^2_{g(t)}/2.$$ +segment $\alpha$ in $B_{(q,t)}'$. This implies that $\mathit{Hess}^h(\beta_{(q,t)}^\epsilon)(v,v)\le -3|v|_h^2/2$, and hence that +by Inequality (7.2) that + +$$ +\mathit{Hess}(\beta_{(q,t)}^\epsilon)(v,v)\le -|v|^2_h=-|v|^2_{g(t)}/2. +$$ + This means that $\beta_{(q,t)}^\epsilon$ is convex with respect to $g(t)$. Now take a sequence $\epsilon_k\rightarrow 0$ and let $f_k=\beta_{(q,t)}^{\epsilon_k}$. @@ -699,44 +887,61 @@ Lastly, it is a standard fact that $f_k$ converge uniformly in the $C^\infty$-topology to $\beta_{(q,t)}$ on any subset of $B_{(q,t)}'$ whose closure is disjoint from $S_{(q,t)}$. + **Definition.** For any continuous function $\psi$ defined on $B_{(q,t)}'\setminus \left(S_{(q,t)}\cap B_{(q,t)}'\right)$ we define -$$\int_{(B_{(q,t)}')^*}\psi dvol(g(t))=lim_{\epsilon\rightarrow + +$$ +\int_{(B_{(q,t)}')^*}\psi d\mathit{vol}(g(t))=\mathit{lim}_{\epsilon\rightarrow 0}\int_{B_{(q,t)}'\setminus \nu_\epsilon(S_{(q,t)})\cap -B_{(q,t)}'}\psi dvol(g(t)).$$ +B_{(q,t)}'}\psi d\mathit{vol}(g(t)). +$$ + We now have: -**Claim.** -Let $\phi\colon B_{(q,t)}'\to \Ar$ be a non-negative, smooth + +**Claim.** Let $\phi\colon B_{(q,t)}'\to \Ar$ be a non-negative, smooth function with compact support. Then -$$\int_{B_{(q,t)}'}\beta_{(q,t)}\triangle \phi dvol(g(t))\le \int_{(B_{(q,t)}')^*}\phi -\triangle \beta_{(q,t)} dvol(g(t)).$$ +$$ +\int_{B_{(q,t)}'}\beta_{(q,t)}\triangle \phi d\mathit{vol}(g(t))\le \int_{(B_{(q,t)}')^*}\phi +\triangle \beta_{(q,t)} d\mathit{vol}(g(t)). +$$ + + + **Remark.** Here $\triangle$ denotes the Laplacian with respect to the metric $g(t)$. + **Proof.** Since $f_k\rightarrow \beta_{(q,t)}$ uniformly on $B_{(q,t)}'$ we have -$$\int_{B_{(q,t)}'}\beta_{(q,t)}\triangle \phi dvol(g(t))=lim_{k\rightarrow -\infty}\int_{B_{(q,t)}'}f_k\triangle \phi dvol(g(t)).$$ Since + +$$ +\int_{B_{(q,t)}'}\beta_{(q,t)}\triangle \phi d\mathit{vol}(g(t))=\mathit{lim}_{k\rightarrow +\infty}\int_{B_{(q,t)}'}f_k\triangle \phi d\mathit{vol}(g(t)). +$$ + + Since $f_k$ is strictly convex with respect to the metric $g(t)$, $\triangle f_k\le 0$ on all of $B_{(q,t)}'$. Since $\phi\ge 0$, for every $\epsilon$ and $k$ we have -$$\int_{\nu_\epsilon(S_{(q,t)})\cap B_{(q,t)}'}\phi\triangle f_k dvol(g(t))\le -0.$$ Hence, for every $k$ and for every $\epsilon$ we have +$$ +\int_{\nu_\epsilon(S_{(q,t)})\cap B_{(q,t)}'}\phi\triangle f_k d\mathit{vol}(g(t))\le +0. $$ + Hence, for every $k$ and for every $\epsilon$ we have + $$ \begin{aligned} -\int_{B_{(q,t)}'}f_k\triangle \phi dvol(g(t)) & = & -\int_{B_{(q,t)}'}\phi\triangle f_k dvol(g(t)) \\ +\int_{B_{(q,t)}'}f_k\triangle \phi d\mathit{vol}(g(t)) & = & +\int_{B_{(q,t)}'}\phi\triangle f_k d\mathit{vol}(g(t)) \\ & \le & \int_{B_{(q,t)}'\setminus \left(B_{(q,t)}'\cap -\nu_\epsilon(S_{(q,t)})\right)}\phi\triangle f_k dvol(g(t)). +\nu_\epsilon(S_{(q,t)})\right)}\phi\triangle f_k d\mathit{vol}(g(t)). \end{aligned} -$$ - $$ Taking the limit as $k\rightarrow\infty$, @@ -745,25 +950,39 @@ $B_{(q,t)}'$ and that restricted to $B_{(q,t)}'\setminus(B_{(q,t)}'\cap \nu_\epsilon (S_{(q,t)}))$ the $f_k$ converge uniformly in the $C^\infty$-topology to $\beta_{(q,t)}$ yields -$$\int_{B_{(q,t)}'}\beta_{(q,t)}\triangle \phi dvol(g(t)) + +$$ +\int_{B_{(q,t)}'}\beta_{(q,t)}\triangle \phi d\mathit{vol}(g(t)) \le \int_{B_{(q,t)}'\setminus \left(B_{(q,t)}'\cap -\nu_\epsilon(S_{(q,t)})\right)}\phi\triangle \beta_q dvol(g(t)).$$ Now taking the limit as $\epsilon\rightarrow 0$ +\nu_\epsilon(S_{(q,t)})\right)}\phi\triangle \beta_q d\mathit{vol}(g(t)). +$$ + + Now taking the limit as $\epsilon\rightarrow 0$ establishes the claim. -**Corollary.** -Let $\phi\colon B_{(q,t)}'\to \Ar$ be a non-negative, smooth + +**Corollary.** Let $\phi\colon B_{(q,t)}'\to \Ar$ be a non-negative, smooth function with compact support. Then -$$\int_{B_{(q,t)}'}l^\tau\triangle \phi dvol(g(t))\le \int_{(B_{(q,t)}')^*}\phi -\triangle l^\tau dvol(g(t)).$$ + +$$ +\int_{B_{(q,t)}'}l^\tau\triangle \phi d\mathit{vol}(g(t))\le \int_{(B_{(q,t)}')^*}\phi +\triangle l^\tau d\mathit{vol}(g(t)). +$$ + **Proof.** Recall that $\beta_{(q,t)}=L^\tau+\psi_{(q,t)}$ and that $\psi_{(q,t)}$ is a $C^\infty$-function. Hence, -$$\int_{B_{(q,t)}'}\psi_{(q,t)}\triangle \phi dvol(g(t))= \int_{(B_{(q,t)}')^*}\phi -\triangle \psi_{(q,t)} dvol(g(t)).$$ Subtracting this equality + +$$ +\int_{B_{(q,t)}'}\psi_{(q,t)}\triangle \phi d\mathit{vol}(g(t))= \int_{(B_{(q,t)}')^*}\phi +\triangle \psi_{(q,t)} d\mathit{vol}(g(t)). +$$ + + Subtracting this equality from the inequality in the previous claim and dividing by $2\sqrt{\tau}$ gives the result. -Now we turn to the proof proper of Theorem~. +Now we turn to the proof proper of \entryref{7d5d94efe4e9}. **Proof.** Let $\phi\colon M\to \Ar$ be a non-negative, smooth function of compact support. Cover $M$ by open subsets of the form $B_{(q,t)}'$ @@ -775,144 +994,198 @@ the result for each $\phi_i$. This allows us to assume (and we shall assume) that $\phi$ is supported in $B_{(q,t)}'$ for some $q\in M$. Since $l_x^\tau$ is a locally Lipschitz function, the restriction of -$|\nabla l_x^\tau|^2$ to $B_{(q,t)}'$ is an $L_loc^\infty$-function. Similarly, $\partial l_x/\partial \tau$ is an -$L_loc^\infty$-function. Hence - -$$ +$|\nabla l_x^\tau|^2$ to $B_{(q,t)}'$ is an $L_\mathit{loc}^\infty$-function. Similarly, $\partial l_x/\partial \tau$ is an +$L_\mathit{loc}^\infty$-function. Hence $$ \begin{aligned} -\lefteqn{\int_{B_{(q,t)}'} +\int_{B_{(q,t)}'} \phi\cdot\left(\frac{\partial l_x}{\partial \tau}+|\nabla -l_x^\tau|^2-R+\frac{n}{2\tau}\right) dvol(g(t))} \\ & = & +l_x^\tau|^2-R+\frac{n}{2\tau}\right) d\mathit{vol}(g(t)) \\ & = & \int_{(B_{(q,t)}')^*}\phi\left(\frac{\partial l_x}{\partial -\tau}+|\nabla l_x^\tau|^2-R+\frac{n}{2\tau}\right) dvol(g(t)). +\tau}+|\nabla l_x^\tau|^2-R+\frac{n}{2\tau}\right) d\mathit{vol}(g(t)). \end{aligned} $$ -$$ On the other hand, by -Corollary~ we have -$$\int_{B_{(q,t)}'}l_x^\tau\triangle \phi dvol(g(t))\le \int_{(B_{(q,t)}')^*}\phi\triangle l_x^\tau -dvol(g(t)).$$ Putting these together we see +\entryref{490290f4c322} we have +$$ +\int_{B_{(q,t)}'}l_x^\tau\triangle \phi d\mathit{vol}(g(t))\le \int_{(B_{(q,t)}')^*}\phi\triangle l_x^\tau +d\mathit{vol}(g(t)). $$ + Putting these together we see + $$ \begin{aligned} -\lefteqn{\int_{B_{(q,t)}'}\phi\left(\frac{\partial l_x}{\partial +\int_{B_{(q,t)}'}\phi\left(\frac{\partial l_x}{\partial \tau}+|\nabla -l_x^\tau|^2-R+\frac{n}{2\tau}\right) -l_x^\tau\triangle \phi dvol(g(t))} \\ +l_x^\tau|^2-R+\frac{n}{2\tau}\right) -l_x^\tau\triangle \phi d\mathit{vol}(g(t)) \\ & \ge & \int_{(B_{(q,t)}')^*}\phi\left(\frac{\partial l_x}{\partial \tau}+|\nabla l_x^\tau|^2-R+\frac{n}{2\tau} -\triangle -l_x^\tau\right) dvol(g(t)). +l_x^\tau\right) d\mathit{vol}(g(t)). \end{aligned} $$ -$$ - It follows immediately from the second inequality in -Corollary~ that, since $\phi\ge 0$ and +\entryref{967c0f8bb774} that, since $\phi\ge 0$ and $\left(B'_{(q,t)}\right)^*\subset {\mathcal U}_x(\tau)$, we have -$$\int_{(B_{(q,t)}')^*}\phi\left(\frac{\partial l_x}{\partial \tau}+|\nabla l_x^\tau|^2-R+ -\frac{n}{2\tau}-\triangle l_x^\tau\right) dvol(g(t))\ge 0.$$ + +$$ +\int_{(B_{(q,t)}')^*}\phi\left(\frac{\partial l_x}{\partial \tau}+|\nabla l_x^\tau|^2-R+ +\frac{n}{2\tau}-\triangle l_x^\tau\right) d\mathit{vol}(g(t))\ge 0. +$$ + This proves the first inequality in the statement of the theorem. The second inequality in the statement of the theorem is proved in exactly the same way using the third inequality in -Corollary~. +\entryref{967c0f8bb774}. Now let us consider the distributions -$$D_1=\frac{\partial l_x}{\partial \tau}+|\nabla l_x^\tau|^2-R+\frac{n}{2\tau}- \triangle -l_x^\tau $$ and -$$D_2=-|\nabla + +$$ +D_1=\frac{\partial l_x}{\partial \tau}+|\nabla l_x^\tau|^2-R+\frac{n}{2\tau}- \triangle +l_x^\tau +$$ + + and + +$$ +D_2=-|\nabla l_x^\tau|^2+R+\frac{l_x^\tau-n}{\tau}+2\triangle l_x^\tau -$$ on $M\times\{\tau\}$. According to Corollary~ the following equality +$$ + + on $M\times\{\tau\}$. According to \entryref{967c0f8bb774} the following equality holdes on ${\mathcal U}_x(\tau)$: -$$2\frac{\partial l_x}{\partial \tau}+|\nabla l_x^\tau|^2-R+\frac{l_x^\tau}{\tau}=0.$$ -By Proposition~ the open set ${\mathcal U}_x(\tau)$ + +$$ +2\frac{\partial l_x}{\partial \tau}+|\nabla l_x^\tau|^2-R+\frac{l_x^\tau}{\tau}=0. +$$ + +By \entryref{16fbe263a7c8} the open set ${\mathcal U}_x(\tau)$ has full measure in $M$ and $|\nabla l_x^\tau|^2$ and $\partial l_x/\partial\tau$ are locally essentially bounded. Thus, this equality is an equality of locally essentially bounded, measurable -functions, i.e., elements of $L_loc^\infty(M)$, and hence is +functions, i.e., elements of $L_\mathit{loc}^\infty(M)$, and hence is an equality of distributions on $M$. Subtracting $2D_1$ from this equality yields $D_2$. Thus, -$$D_2=-2D_1,$$ + +$$ +D_2=-2D_1, +$$ + as distributions on $M$. This shows that $D_2$ vanishes as a distribution if and only if $D_1$ does. But if $D_2=0$ as a distribution for some $\tau$, then by elliptic regularity $l_x^\tau$ -is smooth on $M\times\{\tau\}$ and the equality is the na\"ive one +is smooth on $M\times\{\tau\}$ and the equality is the naïve one for smooth functions. Thus, if $D_2=0$ for all $\tau$, then $l_x^\tau$ and $\partial l/\partial \tau$ are $C^\infty$ functions on each slice $M\times\{\tau\}$ and both $D_1$ and $D_2$ hold in the -na\"ive sense on each slice $M\times \{\tau\}$. It follows from a +naïve sense on each slice $M\times \{\tau\}$. It follows from a standard bootstrap argument that in this case $l_x^\tau$ is smooth on all of space-time. ## Reduced volume We have established that for a Ricci flow $(M,g(t)),\ 0\le t\le T$, -and a point $x=(p,T)\in M\times [0,T]$ the reduced length function +and a point $x=(p,T)\in M\times [0,T]$ the reduced length function $l_x$ is defined on all of $M\times (0,T)$. This allows us to defined the reduced volume of $M\times \{\tau\}$ for any $\tau\in (0,T)$ Recall that the *reduced volume* of $M$ is defined to be -$$\widetilde V_x(M,\tau) = \int_{M} \tau^{-\frac{n}{2}}exp(-l_x(q,\tau))dq.$$ + +$$ +\widetilde V_x(M,\tau) = \int_{M} \tau^{-\frac{n}{2}}exp(-l_x(q,\tau))dq. +$$ + This function is defined for $0<\tau +**Lemma.** If $(M,g(t))$ is flat Euclidean $n$-space (independent of $t$), then for any $x\in \Ar^n\times (-\infty,\infty)$ we have -$$\widetilde V_x(M,\tau)=(4\pi)^{n/2}$$ for all $\tau>0$. + +$$ +\widetilde V_x(M,\tau)=(4\pi)^{n/2} +$$ + + for all $\tau>0$. + **Proof.** By symmetry we can assume that $x=(0,T)\in \Ar^n\times [0,T]$, where $0\in \Ar^n$ is the origin. We have already seen that the ${\mathcal L}$-geodesics in flat space are the usual geodesics when parameterized by $s=\sqrt{\tau}$. Thus, for any $X\in \Ar^n=T_0\Ar^n$ $\gamma_X(\tau)=2\sqrt{\tau}X$, and hence ${\mathcal -L}exp(X,\bar\tau)=2\sqrt{\bar\tau}X$. This means that for any +L}\mathit{exp}(X,\bar\tau)=2\sqrt{\bar\tau}X$. This means that for any $\tau>0$ and any $X\in T_0\Ar^n$ we have ${\mathcal U}(\tau)=T_pM$, and ${\mathcal J}(X,\tau)=2^n\tau^{n/2}$. Also, $L_x(X,\tau)=2\sqrt{\tau}|X|^2$, so that $l_x(X,\tau)=|X|^2$. Thus, for any $\tau>0$ - $$\widetilde - V_x(\Ar^n,\tau)=\int_{\Ar^n}\tau^{-n/2}e^{-|X|^2}2^n\tau^{n/2}dX=(4\pi)^{n/2}.$$ + +$$ +\widetilde + V_x(\Ar^n,\tau)=\int_{\Ar^n}\tau^{-n/2}e^{-|X|^2}2^n\tau^{n/2}dX=(4\pi)^{n/2}. +$$ In the case when $M$ is non-compact, it is not clear *a priori* that the integral defining the reduced volume is finite in general. In fact, as the next proposition shows, it is always finite and -indeed, it is bounded above by the integral for $\Ar^n$. +indeed, it is bounded above by the integral for $\Ar^n$. -**Theorem.** Let $(M,g(t)),\ 0\le t\le T$, be a Ricci flow + +**Theorem.** Let $(M,g(t)),\ 0\le t\le T$, be a Ricci flow of bounded curvature with the property that for each $t\in [0,T]$ the Riemannian manifold $(M,g(t))$ is complete. Fix a point $x=(p,T)\in M\times [0,T]$. For every $0<\tau + +**Proof.** By \entryref{16fbe263a7c8} ${\mathcal U}_x(\tau)$ is an open +subset of full measure in $M$. Hence, + +$$ +\widetilde +V_x(M,\tau)=\int_{{\mathcal U}_x(\tau)}\tau^{-\frac{n}{2}}\mathit{exp}(-l_x(q,\tau))dq. +$$ + Take linear orthonormal coordinates $(z^1,\ldots,z^n)$ on $T_pM$. It -follows from the previous equality and Lemma~ that -$$\widetilde V_x(M,\tau)=\int_{\widetilde {\mathcal U}_{x}(\tau)}f(Z,\tau)dz^1\cdots dz^n,$$ +follows from the previous equality and \entryref{c3d4eaed00fd} that + +$$ +\widetilde V_x(M,\tau)=\int_{\widetilde {\mathcal U}_{x}(\tau)}f(Z,\tau)dz^1\cdots dz^n, +$$ + where $f(Z,\tau)=\tau^{-\frac{n}{2}}e^{-\widetilde -l(Z,\tau)}{\mathcal J}(Z,\tau)$. By Proposition~ for +l(Z,\tau)}{\mathcal J}(Z,\tau)$. By \entryref{4dc40f37cd6d} for each $Z$ the integrand, $f(Z,\tau)$, is a non-increasing function of $\tau$ and the function converges uniformly on compact sets as $\tau\rightarrow 0$ to $2^ne^{-|Z|^2}$. This implies that $f(Z,\tau)\le 2^ne^{-|Z|^2}$ for all $\tau>0$, and hence that -$$\int_{\widetilde{\mathcal U}_x(\tau)}f(Z,\tau)dz^1\ldots dz^n$$ + +$$ +\int_{\widetilde{\mathcal U}_x(\tau)}f(Z,\tau)dz^1\ldots dz^n +$$ + converges absolutely for each $\tau>0$, and the integral has value at most $(4\pi)^{n/2}$. -Fix $0<\tau_00$ sufficiently small + each $A<\infty$ for all $\tau>0$ sufficiently small $\widetilde {\mathcal U}_x(\tau)$ contains the ball of radius $A$ centered at the origin - in $T_pM$. Since the curvature is bounded, this is exactly - the content of Corollary~. + in $T_pM$. Since the curvature is bounded, this is exactly + the content of \entryref{f1123507a2ea}. -### Converse to Lemma~\protect{} +### Converse to Lemma \entryref{4a4c8aae3a8d} -In Lemma~ we showed that for the trivial flow on flat +In \entryref{4a4c8aae3a8d} we showed that for the trivial flow on flat Euclidean $n$-space and for any point $x\in \Ar^n\times \{T\}$ the reduced volume $\widetilde V_x(\Ar^n,\tau)$ is independent of -$\tau>0$ and is equal to $(4\pi)^{n/2}$. In this subsection we use +$\tau>0$ and is equal to $(4\pi)^{n/2}$. In this subsection we use the monotonicity results of the last subsection to establish the -converse to Lemma~, namely to show that if $(M,g(t)),0\le +converse to \entryref{4a4c8aae3a8d}, namely to show that if $(M,g(t)),0\le t\le T$, is a Ricci flow complete with bounded curvature and if $\widetilde V_x(M,\bar\tau)=(4\pi)^{n/2}$ for some $\bar\tau>0$ and some $x\in M\times\{T\}$, then the flow on the interval $[T-\bar\tau,T]$ is the trivial flow on flat Euclidean $n$-space. -**Proposition.** Suppose that $(M,g(\tau)),\ 0\le \tau\le T$, + +**Proposition.** Suppose that $(M,g(\tau)),\ 0\le \tau\le T$, is a solution to the backward Ricci flow equation, complete of bounded curvature. Let $x=(p,T)\in M\times \{T\}$, and suppose that $0<\bar\tau **Proof.** If $\widetilde V_x(M,\bar\tau)=(4\pi)^{n/2}$, then by -Lemma~, $\widetilde V_x(M,\tau)$ is constant on the +\entryref{4a4c8aae3a8d}, $\widetilde V_x(M,\tau)$ is constant on the interval $(0,\bar\tau]$. Hence, it follows from the proof of -Theorem~ that the closure of $\widetilde{\mathcal U}(\tau)$ +\entryref{ae6f1c6d00d9} that the closure of $\widetilde{\mathcal U}(\tau)$ is all of $T_pM$ for all $\tau\in (0,\bar\tau]$ and that $f(Z,\tau)=e^{-|Z|^2}2^n$ for all $Z\in T_pM$ and all $\tau\le \bar\tau$. In particular, -$$\frac{\partial ln(f(Z,\tau))}{\partial \tau}=0.$$ + +$$ +\frac{\partial \mathit{ln}(f(Z,\tau))}{\partial \tau}=0. +$$ + From the proof of - Proposition~ this means that -Inequality~() is an equality and consequently, so is -Inequality~(). Thus, by Proposition~ + \entryref{4dc40f37cd6d} this means that +Inequality (6.20) is an equality and consequently, so is +Inequality (6.19). Thus, by \entryref{88d4ef67440f} (with $\tau_1=0$) each of the vector fields $Y_\alpha(\tau)=\tilde Y_\alpha(\tau)$ is both a Jacobi field and adapted. By -Proposition~ we then have -$$Ric+Hess(l^\tau_x)=\frac{g}{2\tau}.$$ +\entryref{882677eecdc1} we then have + +$$ +\mathit{Ric}+\mathit{Hess}(l^\tau_x)=\frac{g}{2\tau}. +$$ + In particular, $l_x$ is smooth. Let $\varphi_\tau\colon M\to M,\ -0<\tau\le \bar\tau$,\ be the one-parameter family of diffeomorphisms +0<\tau\le \bar\tau$, be the one-parameter family of diffeomorphisms obtained by solving -$$\frac{d\varphi_\tau}{d\tau}=\nabla -l_x(\cdot,\tau)\ \ \ \ and \ \ \ \varphi_{\bar\tau}=Id.$$ We now consider -$$h(\tau)=\frac{\bar\tau}{\tau}\varphi_\tau^*g(\tau).$$ -We compute + +$$ +\frac{d\varphi_\tau}{d\tau}=\nabla +l_x(\cdot,\tau)\ \ \ \ \mathit{and} \ \ \ \varphi_{\bar\tau}=\mathit{Id}. +$$ + + We now consider + $$ +h(\tau)=\frac{\bar\tau}{\tau}\varphi_\tau^*g(\tau). +$$ + +We compute $$ \begin{aligned} @@ -979,24 +1270,27 @@ $$ -\frac{\bar\tau}{\tau^2}\varphi_\tau^*g(\tau) +\frac{\bar\tau}{\tau}\varphi_\tau^*{\mathcal L}_{\frac{d\varphi_\tau}{d\tau}}(g(\tau)) -+\frac{\bar\tau}{\tau}\varphi_t^*2Ric(g(\tau)) \\ ++\frac{\bar\tau}{\tau}\varphi_t^*2\mathit{Ric}(g(\tau)) \\ & = & -\frac{\bar\tau}{\tau^2}\varphi_\tau^*g(\tau) + -\frac{\bar\tau}{\tau}\varphi_\tau^*2Hess(l_x^\tau)+\frac{\bar\tau}{\tau} -\varphi_\tau^*\left(\frac{1}{\tau}g(\tau)-2Hess(l_x^\tau)\right)=0. +\frac{\bar\tau}{\tau}\varphi_\tau^*2\mathit{Hess}(l_x^\tau)+\frac{\bar\tau}{\tau} +\varphi_\tau^*\left(\frac{1}{\tau}g(\tau)-2\mathit{Hess}(l_x^\tau)\right)=0. \end{aligned} $$ -$$ That is to say the family -of metrics $h(\tau)$ is constant in $\tau$: for all $\tau\in +of metrics $h(\tau)$ is constant in $\tau$: for all $\tau\in (0,\bar\tau]$ we have $h(\tau)=h(\bar\tau)=g(\bar\tau)$. It then follows that -$$g(\tau)=\frac{\tau}{\bar\tau}(\varphi_\tau^{-1})^*g(\bar\tau),$$ + +$$ +g(\tau)=\frac{\tau}{\bar\tau}(\varphi_\tau^{-1})^*g(\bar\tau), +$$ + which means that the entire flow in the interval $(0,\bar\tau]$ differs by diffeomorphism and scaling from $g(\bar\tau)$. Suppose that $g(\bar\tau)$ is not flat, i.e., suppose that there is some -$(x,\bar\tau)$ with $|Rm(x,\bar\tau)|=K>0$. Then from the flow -equation we see that $|Rm(\varphi^{-1}_\tau(x),\tau)|=K\bar\tau^2/\tau^2$, and these +$(x,\bar\tau)$ with $|\mathit{Rm}(x,\bar\tau)|=K>0$. Then from the flow +equation we see that $|\mathit{Rm}(\varphi^{-1}_\tau(x),\tau)|=K\bar\tau^2/\tau^2$, and these curvatures are not bounded as $\tau\rightarrow 0$. This is a contradiction. We conclude that $g(\bar\tau)$ is flat, and hence, again by the flow equation so are all the $g(\tau)$ for $0<\tau\le diff --git a/projects/poincare-conjecture/.astrolabe/docs/08-noncoll.mdx b/projects/poincare-conjecture/.astrolabe/docs-src/09-non-collapsed-results.mdx similarity index 51% rename from projects/poincare-conjecture/.astrolabe/docs/08-noncoll.mdx rename to projects/poincare-conjecture/.astrolabe/docs-src/09-non-collapsed-results.mdx index 7a4717f9..3e62d034 100644 --- a/projects/poincare-conjecture/.astrolabe/docs/08-noncoll.mdx +++ b/projects/poincare-conjecture/.astrolabe/docs-src/09-non-collapsed-results.mdx @@ -1,4 +1,6 @@ -# Non-collapsed results + + +# Chapter 8 -- Non-collapsed results In this chapter we apply the results for the reduced length function and reduced volume established in the last two sections to prove @@ -14,31 +16,39 @@ initial metrics. The main result of this chapter is a $\kappa$-non-collapsed result. -**Theorem.** - Fix positive constants $\bar\tau_0<\infty$, $l_0<\infty$, and $V>0$. + +**Theorem.** Fix positive constants $\bar\tau_0<\infty$, $l_0<\infty$, and $V>0$. Then there is $\kappa>0$ depending on $\bar\tau_0$, $V$, and $l_0$ and the dimension $n$ such that the following holds. Let $({\mathcal M},G)$ be a generalized $n$-dimensional Ricci flow, and let $0<\tau_0\le \bar\tau_0$. Let -$x\in {\mathcal M}$ be fixed. Set $T=**t**(x)$. Suppose that +$x\in {\mathcal M}$ be fixed. Set $T=\mathbf{t}(x)$. Suppose that $0 + +See Fig. 8.1. In this section we denote by $g(\tau), \ 0\le \tau\le r^2$, the family of metrics on $B(x,T,r)$ induced from pulling back $G$ under @@ -46,107 +56,161 @@ the embedding $B(x,T,r)\times [T-r^2,T]\to {\mathcal M}$. Of course, this family of metrics satisfies the backward Ricci flow equation. **Proof.** Clearly from the definition of the reduced volume, we have -$$\widetilde V_x(W(\tau_0))\ge \tau_0^{-n/2}Ve^{-l_0}\ge -\bar\tau_0^{-n/2}Ve^{-l_0}.$$ By the -monotonicity result (Theorem~) it follows that for any + +$$ +\widetilde V_x(W(\tau_0))\ge \tau_0^{-n/2}Ve^{-l_0}\ge +\bar\tau_0^{-n/2}Ve^{-l_0}. +$$ + + By the +monotonicity result (\entryref{6a52537bef2f}) it follows that for any $\tau\le \tau_0$, and in particular for any $\tau\le r^2$, we have $$ - \widetilde V_x(W(\tau))\ge +\widetilde V_x(W(\tau))\ge \bar\tau_0^{-n/2}Ve^{-l_0}. $$ -Let $\varepsilon=\sqrt[n]{Vol(B(x,T,r))}/r$, so that $Vol\,B(x,T,r)=\varepsilon^nr^n$. The basic result we need to +Let $\varepsilon=\sqrt[n]{\mathit{Vol}(B(x,T,r))}/r$, so that $\mathit{Vol} B(x,T,r)=\varepsilon^nr^n$. The basic result we need to establish in order to prove this theorem is the following: -**Proposition.** There is a positive constant + +**Proposition.** There is a positive constant $\varepsilon_0\le 1/4n(n-1)$ depending on $\bar\tau_0$ and $l_0$ such that if $\varepsilon\le \varepsilon_0$ then, setting $\tau_1=\varepsilon r^2$, we have $\widetilde V_x(W(\tau_1)) < 3\varepsilon^{\frac{n}{2}}$. + Given this proposition, it follows immediately that either $\varepsilon>\varepsilon_0$ or -$$\varepsilon\ge \left(\frac{\widetilde V_x(W(\tau_1))}{3}\right)^{2/n}\ge -\frac{1}{3^{2/n}\bar\tau_0}V^{2/n}e^{-2l_0/n}.$$ Since + +$$ +\varepsilon\ge \left(\frac{\widetilde V_x(W(\tau_1))}{3}\right)^{2/n}\ge +\frac{1}{3^{2/n}\bar\tau_0}V^{2/n}e^{-2l_0/n}. +$$ + + Since $\kappa=\varepsilon^n$, this proves the theorem. -\begin{proof} We divide $\widetilde W$ +**Proof.** We divide $\widetilde W$ into -$$\widetilde W_sm=\widetilde W\cap \left\{Z\in -T_xM_T\bigl|\bigr. |Z|\le \frac{1}{8}\varepsilon^{-1/2}\right\}$$ + +$$ +\widetilde W_\mathit{sm}=\widetilde W\cap \left\{Z\in +T_xM_T\bigl|\bigr. |Z|\le \frac{1}{8}\varepsilon^{-1/2}\right\} +$$ + and -$$\widetilde W_lg=\widetilde W\setminus \widetilde W_sm,$$ + +$$ +\widetilde W_\mathit{lg}=\widetilde W\setminus \widetilde W_\mathit{sm}, +$$ + (see Fig. 8.2). -We set $W_sm(\tau_1)={\mathcal L}exp_x^{\tau_1}(\widetilde W_sm)$ and $W_lg(\tau_1)={\mathcal L}exp_x^{\tau_1}(\widetilde W_lg).$ Clearly, since $W(\tau_1)$ is the union of $W_sm(\tau_1)$ and $W_lg(\tau_1)$ and since these subsets are +We set $W_\mathit{sm}(\tau_1)={\mathcal L}\mathit{exp}_x^{\tau_1}(\widetilde W_\mathit{sm})$ and $W_\mathit{lg}(\tau_1)={\mathcal L}\mathit{exp}_x^{\tau_1}(\widetilde W_\mathit{lg}).$ Clearly, since $W(\tau_1)$ is the union of $W_\mathit{sm}(\tau_1)$ and $W_\mathit{lg}(\tau_1)$ and since these subsets are disjoint measurable subsets, we have -$$\widetilde V_x(W(\tau_1))=\widetilde V_x(W_sm(\tau_1))+\widetilde V_x(W_lg(\tau_1)).$$ We shall show that there is $\varepsilon_0$ such that either -$\varepsilon>\varepsilon_0$ or $\widetilde V_x(W_sm(\tau_1))\le -2\varepsilon^{n/2}$ and $\widetilde V_x(W_lg(\tau_1))\le -\varepsilon^{n/2}$. This will establish Proposition~ and hence -Theorem~. -### Upper bound for $\widetilde V_x(W_sm -(\tau_1))$ +$$ +\widetilde V_x(W(\tau_1))=\widetilde V_x(W_\mathit{sm}(\tau_1))+\widetilde V_x(W_\mathit{lg}(\tau_1)). +$$ + + We shall show that there is $\varepsilon_0$ such that either +$\varepsilon>\varepsilon_0$ or $\widetilde V_x(W_\mathit{sm}(\tau_1))\le +2\varepsilon^{n/2}$ and $\widetilde V_x(W_\mathit{lg}(\tau_1))\le +\varepsilon^{n/2}$. This will establish \entryref{12d93b430400} and hence +\entryref{3defa9b2209c}. + +### Upper bound for $\widetilde V_x(W_\mathit{sm}(\tau_1))$ The idea here is that ${\mathcal L}$-geodesics with initial vector -in $\widetilde W_sm$ remain in the parabolic neighborhood +in $\widetilde W_\mathit{sm}$ remain in the parabolic neighborhood $P=B(x,T,r)\times [T-r^2,T]$ for $\tau\in [0,r^2]$. Once we know this it is easy to see that their ${\mathcal L}$-lengths are bounded from below. Then if the volume of $B(x,T,r)$ was arbitrarily small, -the reduced volume of $W_sm(\tau_1)$ would be arbitrarily +the reduced volume of $W_\mathit{sm}(\tau_1)$ would be arbitrarily small. -**Lemma.** -Setting $\tau_1=\varepsilon r^2$, there is a constant + +**Lemma.** Setting $\tau_1=\varepsilon r^2$, there is a constant $\varepsilon_0>0$ depending on $\bar\tau_0$ such that, if $\varepsilon\le \varepsilon_0$, we have -$$\int_{\widetilde -W_sm}\tau_1^{-n/2}e^{-\tilde l(Z,\tau_1)}{\mathcal -J}(Z,\tau_1)dZ\le 2\varepsilon^{\frac{n}{2}}.$$ -Of course, we have -$$\widetilde V_x(W_sm(\tau_1))=\int_{W_sm(\tau_1)}\tau_1^{-n/2}e^{-l(q,\tau_1)}dvol_{g(\tau_1)}=\int_{\widetilde W_sm}\tau_1^{-n/2}e^{-\tilde -l(Z,\tau_1)}{\mathcal J}(Z,\tau_1)dZ,$$ so that it will follow +$$ +\int_{\widetilde +W_\mathit{sm}}\tau_1^{-n/2}e^{-\tilde l(Z,\tau_1)}{\mathcal +J}(Z,\tau_1)dZ\le 2\varepsilon^{\frac{n}{2}}. +$$ + + +Of course, we have + +$$ +\widetilde V_x(W_\mathit{sm}(\tau_1))=\int_{W_\mathit{sm}(\tau_1)}\tau_1^{-n/2}e^{-l(q,\tau_1)}d\mathit{vol}_{g(\tau_1)}=\int_{\widetilde W_\mathit{sm}}\tau_1^{-n/2}e^{-\tilde +l(Z,\tau_1)}{\mathcal J}(Z,\tau_1)dZ, +$$ + + so that it will follow immediately from the lemma that: -**Corollary.** There is a constant $\varepsilon_0>0$ depending on $\bar\tau_0$ + +**Corollary.** There is a constant $\varepsilon_0>0$ depending on $\bar\tau_0$ such that, if $\varepsilon\le \varepsilon_0$, we have -$$\widetilde V_x(W_sm(\tau_1))\le 2\varepsilon^{\frac{n}{2}}.$$ -\begin{proof}{\em (Of Lemma~)} -In order to establish Lemma~ we need two preliminary +$$ +\widetilde V_x(W_\mathit{sm}(\tau_1))\le 2\varepsilon^{\frac{n}{2}}. +$$ + + +**Proof.** *(Of \entryref{26e1a6292fef})* +In order to establish \entryref{26e1a6292fef} we need two preliminary estimates: -**Claim.** -There is a universal positive constant $\varepsilon'_0$ such that, if + +**Claim.** There is a universal positive constant $\varepsilon'_0$ such that, if $\varepsilon\le \varepsilon'_0$, then there is a constant $C_1<\infty$ depending only on the dimension $n$ such that the following hold for all $y\in B(x,T,r/2)$, and for all $t\in[T-\tau_1,T]$: -\begin{enumerate} -\item[(1)] $$|\nabla R(y,t)|\le \frac{C_1}{r^3}$$ -\item[(2)] $$(1-C_1\varepsilon)\le \frac{g(y,t)}{g(y,T)}\le (1+C_1\varepsilon).$$ -\end{enumerate} -\begin{proof} -Recall that by hypothesis $|Rm(y,t)|\le 1/r^2$ on +- **(1)** + +$$ +|\nabla R(y,t)|\le \frac{C_1}{r^3} +$$ + +- **(2)** + +$$ +(1-C_1\varepsilon)\le \frac{g(y,t)}{g(y,T)}\le (1+C_1\varepsilon). +$$ + + +**Proof.** Recall that by hypothesis $|\mathit{Rm}(y,t)|\le 1/r^2$ on $B(x,T,r)\times [T-r^2,T]$. Rescale the flow by multiplying the metric and time by $r^{-2}$ resulting in a ball $\widetilde B$ of radius one and a flow defined for a time interval of length one with -$|Rm|\le 1$ on the entire parabolic neighborhood -$B(x,T,1)\times [T-1,T]$. Then according to Theorem~ there +$|\mathit{Rm}|\le 1$ on the entire parabolic neighborhood +$B(x,T,1)\times [T-1,T]$. Then according to \entryref{ed4b72caf9f9} there is a universal constant $C_1$ such that $|\nabla R(y,t)|\le C_1$ for all $(y,t)\in B(x,T,1/2)\times [T-1/2,T]$. Rescaling back by $r^2$ to the original flow, we see that on this flow $|\nabla R(y,t)|\le C_1/r^3$ for all $(y,t)\in B(x,T,r/2)\times [T-r^2/2,T]$. Taking $\varepsilon'_0\le 1/2$ gives the first item in the claim. - Since $|Ric|\le (n-1)/r^2$ for all + Since $|\mathit{Ric}|\le (n-1)/r^2$ for all $(y,t)\in B\times[T-r^2,T]$ it follows by integrating that -$$e^{-2(n-1)(T-t)/r^2}\le \frac{g(x,t)}{g(x,T)}\le e^{2(n-1)(T-t)/r^2}.$$ + +$$ +e^{-2(n-1)(T-t)/r^2}\le \frac{g(x,t)}{g(x,T)}\le e^{2(n-1)(T-t)/r^2}. +$$ + Thus, for $t\in [T-\tau_1,T]$ we have -$$e^{-2(n-1)\varepsilon}\le \frac{g(x,t)}{g(x,T)}\le e^{2(n-1)\varepsilon}.$$ + +$$ +e^{-2(n-1)\varepsilon}\le \frac{g(x,t)}{g(x,T)}\le e^{2(n-1)\varepsilon}. +$$ + From this the second item in the claim is immediate. At this point we view the ${\mathcal L}$-geodesics as paths @@ -155,70 +219,107 @@ the path in space-time is given by the composition of the path $ (\gamma(\tau),T-\tau)$ in $B(x,T,r)\times [T-r^2,T]$ followed by the given inclusion of this product into ${\mathcal M}$. -The next step in the proof is to show that for any $Z\in \widetilde -W_sm$ the ${\mathcal L}$-geodesic $\gamma_Z$ (the one having -$lim_{\tau\rightarrow 0}\sqrt\tau X_{\gamma_Z}(\tau)=Z$) +The next step in the proof is to show that for any $Z\in \widetilde +W_\mathit{sm}$ the ${\mathcal L}$-geodesic $\gamma_Z$ (the one having +$\mathit{lim}_{\tau\rightarrow 0}\sqrt\tau X_{\gamma_Z}(\tau)=Z$) remains in $B(x,T,r/2)$ up to time $\tau_1$. Because of this, as we shall see, these paths contribute a small amount to the reduced volume since $B(x,T,r/2)$ has small volume. We set $X(\tau)=X_{\gamma_Z}(\tau)$ -**Claim.** There is a + +**Claim.** There is a positive constant $\varepsilon_0\le 1/4n(n-1)$ depending on $\bar\tau_0$, such that the following holds. Suppose that - $\varepsilon\le \varepsilon_0$ and $\tau_1' \leq + $\varepsilon\le \varepsilon_0$ and $\tau_1' \leq \tau_1=\varepsilon r^2$. Let $Z\in T_xM_T$ and let $\gamma_Z$ be the -associated ${\mathcal L}$-geodesic from $x$. Suppose that +associated ${\mathcal L}$-geodesic from $x$. Suppose that $\gamma_Z(\tau) \in B(x,T,r/2)$ for all $\tau < \tau_1'$. Then for all $\tau<\tau_1'$ we have -$$\abs{|\sqrt{\tau}X(\tau)|_{g(T)} - |Z|} \leq 2\varepsilon (1+|Z|).$$ + +$$ +\abs{|\sqrt{\tau}X(\tau)|_{g(T)} - |Z|} \leq 2\varepsilon (1+|Z|). +$$ + **Proof.** First we make sure that $\varepsilon_0$ is less than or equal to the universal constant $\varepsilon_0'$ of the last claim. For all -$(y,t)\in B(x,T,r)\times [T-r^2,T]$ we have $|Rm(y,t)|\le +$(y,t)\in B(x,T,r)\times [T-r^2,T]$ we have $|\mathit{Rm}(y,t)|\le r^{-2}$ and $|\nabla R(y,t)|\le C_1/r^3$ for some universal constant $C_1$. Of course, $r^2\le \bar\tau$. Thus, at the expense of replacing $C_1$ by a larger constant, we can (and shall) assume that -$C_1/r^3>(n-1)r^{-2}\ge |Ric(y,t)|$ for all $(y,t)\in +$C_1/r^3>(n-1)r^{-2}\ge |\mathit{Ric}(y,t)|$ for all $(y,t)\in B(x,T,r)\times [T-r^2,T]$. Thus, we can take the constant $C_0$ in -the hypothesis of Lemma~ to be $C_1/r^3$. We take the +the hypothesis of \entryref{333be2e58780} to be $C_1/r^3$. We take the constant $\bar\tau$ in the hypothesis of that lemma to be $\varepsilon r^2$. Then, we have that -$$max_{0\le \tau\le \tau_1'}\sqrt{\tau}|X(\tau)|\le + +$$ +\mathit{max}_{0\le \tau\le \tau_1'}\sqrt{\tau}|X(\tau)|\le e^{2C_1\varepsilon^2}|Z|+\frac{e^{2C_1\varepsilon^2}-1}{2}\sqrt{\varepsilon} -r,$$ and -$$|Z|\le e^{2C_1\varepsilon^2}min_{0\le \tau\le -\tau_1'}\sqrt{\tau}|X(\tau)|+\frac{e^{2C_1\varepsilon^2}-1}{2}\sqrt{\varepsilon}r.$$ +r, +$$ + + and + +$$ +|Z|\le e^{2C_1\varepsilon^2}\mathit{min}_{0\le \tau\le +\tau_1'}\sqrt{\tau}|X(\tau)|+\frac{e^{2C_1\varepsilon^2}-1}{2}\sqrt{\varepsilon}r. +$$ + By choosing $\varepsilon_0>0$ sufficiently small (as determined by the universal constant $C_1$ and by $\bar\tau_0$), we have -$$max_{0\le \tau\le \tau_1'}\sqrt{\tau}|X(\tau)|_{g(T-\tau)}\le -(1+\frac{\varepsilon}{2})|Z|+\frac{\varepsilon}{2},$$ and -$$|Z|\le (1+\frac{\varepsilon}{2})min_{0\le \tau\le -\tau_1'}\sqrt{\tau}|X(\tau)|_{g(T-\tau)}+\frac{\varepsilon}{2}.$$ It + +$$ +\mathit{max}_{0\le \tau\le \tau_1'}\sqrt{\tau}|X(\tau)|_{g(T-\tau)}\le +(1+\frac{\varepsilon}{2})|Z|+\frac{\varepsilon}{2}, +$$ + + and + +$$ +|Z|\le (1+\frac{\varepsilon}{2})\mathit{min}_{0\le \tau\le +\tau_1'}\sqrt{\tau}|X(\tau)|_{g(T-\tau)}+\frac{\varepsilon}{2}. +$$ + + It is now immediate that -$$\abs{|\sqrt{\tau}X(\tau)|_{g(T-\tau)} - |Z|} \leq \varepsilon (1+|Z|).$$ + +$$ +\abs{|\sqrt{\tau}X(\tau)|_{g(T-\tau)} - |Z|} \leq \varepsilon (1+|Z|). +$$ Again choosing $\varepsilon_0$ sufficiently small the result now -follows from the second inequality in Claim~ +follows from the second inequality in \entryref{bf8f356cfd7a} Now we are ready to establish that the ${\mathcal L}$-geodesics -whose initial conditions are elements of $\widetilde W_sm$ do -not leave $B(x,T,r/2)\times [T-r^2,T]$ for any $\tau\le\tau_1$. +whose initial conditions are elements of $\widetilde W_\mathit{sm}$ do +not leave $B(x,T,r/2)\times [T-r^2,T]$ for any $\tau\le\tau_1$. -**Claim.** -Suppose $\varepsilon_0\le 1/4n(n-1)$ is the constant from the last + +**Claim.** Suppose $\varepsilon_0\le 1/4n(n-1)$ is the constant from the last claim. Set $\tau_1=\varepsilon r^2$, and suppose that $\varepsilon\le \varepsilon_0$. Lastly, assume that $|Z|\le \frac{1}{8\sqrt{\varepsilon}}$. Then $\gamma_Z(\tau) \in B(x,T,r/2)$ for all $\tau\le \tau_1$. + **Proof.** Since $\varepsilon\le \varepsilon_0\le 1/4n(n-1)\le 1/8$, by the last claim we have -$$|\sqrt{\tau}X(\tau)|_{g(T)}\le (1+2\varepsilon )|Z|+2\varepsilon\le -\frac{5}{4}|Z|+\frac{3}{32\sqrt\varepsilon},$$ provided that + +$$ +|\sqrt{\tau}X(\tau)|_{g(T)}\le (1+2\varepsilon )|Z|+2\varepsilon\le +\frac{5}{4}|Z|+\frac{3}{32\sqrt\varepsilon}, +$$ + + provided that $\gamma|_{[0,\tau)}$ is contained in $B(x,T,r/2)\times [T-\tau,T]$. Since $|Z|\le (8\sqrt\varepsilon)^{-1}$ we conclude that -$$|\sqrt{\tau}X(\tau)|_{g(T)}\le \frac{1}{4\sqrt\varepsilon},$$ + +$$ +|\sqrt{\tau}X(\tau)|_{g(T)}\le \frac{1}{4\sqrt\varepsilon}, +$$ + as long as $\gamma([0,\tau))$ is contained in $B(x,T,r/2)\times [T-\tau,T]$. @@ -226,105 +327,132 @@ as long as $\gamma([0,\tau))$ is contained in $B(x,T,r/2)\times $\tau'<\tau_1=\varepsilon r^2$ for which $\gamma_Z$ exits $B(x,T,r/2)\times [T-r^2,T]$. We take $\tau'$ to be the first such time. Then we have + $$ |\gamma_Z(\tau')- x|_{g(T)} \leq \int_{0}^{\tau'}|X(\tau)|_{g(T)}d\tau \le \frac{1}{4\varepsilon^{\frac{1}{2}}}\int_{0}^{\tau'}\frac{d\tau}{\sqrt{\tau}} =\frac{1}{2\varepsilon^{\frac{1}{2}}}\sqrt{\tau'} < r/2. $$ -This contradiction implies that $\gamma_Z(\tau) \in B(x,T,r/2)$ for + +This contradiction implies that $\gamma_Z(\tau) \in B(x,T,r/2)$ for all $\tau< \tau_1=\varepsilon r^2$. Now we assume that $\varepsilon_0>0$ depending on $\bar\tau_0$ is as above and that $\varepsilon\le \varepsilon_0$, and we shall estimate + +$$ +\widetilde V_x(W_\mathit{sm}(\tau_1)) = \int_{W_\mathit{sm}(\tau_1)} +(\tau_1)^{-\frac{n}{2}}e^{- l(q,\tau_1)}d\mathit{vol}_{g(\tau_1)}. $$ -\widetilde V_x(W_sm(\tau_1)) = \int_{W_sm(\tau_1)} -(\tau_1)^{-\frac{n}{2}}e^{- l(q,\tau_1)}dvol_{g(\tau_1)}.$$ In -order to do this we estimate $ l_x(q,\tau_1)$ on $ W_sm(\tau_1)$. By hypothesis $|Rm|\le 1/r^2$ on -$B(x,T,r/2)\times [0,\tau_1]$ and by Lemma~ every + + In +order to do this we estimate $ l_x(q,\tau_1)$ on $ W_\mathit{sm}(\tau_1)$. By hypothesis $|\mathit{Rm}|\le 1/r^2$ on +$B(x,T,r/2)\times [0,\tau_1]$ and by \entryref{0a1eac346e21} every ${\mathcal L}$-geodesic $\gamma_Z$, defined on $[0,\tau_1]$, with initial conditions $Z$ satisfying $|Z|\leq \frac{1}{8}\varepsilon^{-\frac{1}{2}}$ remains in $B(x,T,r/2)$. Thus, for such $\gamma_Z$ we have $R(\gamma_Z(\tau))\ge --n(n-1)/r^2$. Thus, for any $q\in W_sm(\tau_1)$ we have -$$ L_x(q,\tau_1) = +-n(n-1)/r^2$. Thus, for any $q\in W_\mathit{sm}(\tau_1)$ we have + +$$ +L_x(q,\tau_1) = \int_{0}^{\tau_1}\sqrt{\tau}(R+|X(\tau)|^{2})d \tau \geq -\frac{2n(n-1)}{3r^{2}}(\tau_1)^{\frac{3}{2}}= --\frac{2n(n-1)}{3}\varepsilon^{\frac{3}{2}}r,$$ and hence -$$l+x(q,\tau_1)=\frac{L_x(q,\tau_1)}{2\sqrt{\tau_1}}\ge -\frac{n(n-1)}{3}\varepsilon.$$ Since $W_sm(\tau)\subset B(x,T,r/2)\subset B(x,T,r)$, we have: +-\frac{2n(n-1)}{3}\varepsilon^{\frac{3}{2}}r, +$$ + and hence + +$$ +l+x(q,\tau_1)=\frac{L_x(q,\tau_1)}{2\sqrt{\tau_1}}\ge -\frac{n(n-1)}{3}\varepsilon. $$ + Since $W_\mathit{sm}(\tau)\subset B(x,T,r/2)\subset B(x,T,r)$, we have: + $$ \begin{aligned} - - \widetilde V_x(W_sm(\tau_1)) & \leq & -\varepsilon^{-\frac{n}{2}}r^{-n}e^{n(n-1)\varepsilon/3}Vol_{g(T-\tau_1)}\, W_sm(\tau) \\ -& \le & \varepsilon^{-\frac{n}{2}}r^{-n}e^{n(n-1)\varepsilon/3}Vol_{g(T-\tau_1)}\,B(x,T,r).\nonumber +\widetilde V_x(W_\mathit{sm}(\tau_1)) & \leq & +\varepsilon^{-\frac{n}{2}}r^{-n}e^{n(n-1)\varepsilon/3}\mathit{Vol}_{g(T-\tau_1)} W_\mathit{sm}(\tau) \\ +& \le & \varepsilon^{-\frac{n}{2}}r^{-n}e^{n(n-1)\varepsilon/3}\mathit{Vol}_{g(T-\tau_1)} B(x,T,r).\nonumber \end{aligned} $$ -$$ - -**Claim.** -There is a universal constant $\varepsilon_0>0$ such that if + +**Claim.** There is a universal constant $\varepsilon_0>0$ such that if $\varepsilon\le \varepsilon_0$, for any open subset $U$ of $B(x,T,r)$, and for any $0\le \tau_1\le \tau_0$, we have -$$0.9\le Vol_{g(T)}U/Vol_{g(T-\tau_1)}U\le 1.1.$$ -**Proof.** This is immediate from the second item in Claim~. +$$ +0.9\le \mathit{Vol}_{g(T)}U/\mathit{Vol}_{g(T-\tau_1)}U\le 1.1. +$$ + + +**Proof.** This is immediate from the second item in \entryref{bf8f356cfd7a}. Now assume that $\varepsilon_0$ also satisfies this claim. Plugging -this into Equation~(), and using the fact that +this into Equation (8.2), and using the fact that $\varepsilon\le \varepsilon_0\le 1/4n(n-1)$, , so that $n(n-1)\varepsilon/3\le 1/12$, and the fact that from the definition -we have $Vol_{g(T)}\,B(x,T,r)=\varepsilon^nr^n$, gives -$$\widetilde V_x(W_sm(\tau_1))\leq \varepsilon^{-\frac{n}{2}}r^{-n}e^{n(n-1)\varepsilon/3} -(1.1)Vol_{g(T)}B(x,T,r)\le(1.1) -\varepsilon^{\frac{n}{2}}e^{\frac{1}{12}}.$$ Thus, -$$\widetilde V_x(W_sm(\tau_1))\le -2\varepsilon^{\frac{n}{2}}.$$ +we have $\mathit{Vol}_{g(T)} B(x,T,r)=\varepsilon^nr^n$, gives -This completes the proof of Lemma~. -\end{proof} +$$ +\widetilde V_x(W_\mathit{sm}(\tau_1))\leq \varepsilon^{-\frac{n}{2}}r^{-n}e^{n(n-1)\varepsilon/3} +(1.1)\mathit{Vol}_{g(T)}B(x,T,r)\le(1.1) +\varepsilon^{\frac{n}{2}}e^{\frac{1}{12}}. +$$ + + Thus, + +$$ +\widetilde V_x(W_\mathit{sm}(\tau_1))\le +2\varepsilon^{\frac{n}{2}}. +$$ -### Upper bound for $\widetilde V_x( W_lg -(\tau_1))$ +This completes the proof of \entryref{26e1a6292fef}. + +### Upper bound for $\widetilde V_x( W_\mathit{lg}(\tau_1))$ Here the basic point is to approximate the reduced volume integrand by the heat kernel, which drops off exponentially fast as we go away from the origin. -Recall that $Vol\,B(x,T,r)=\varepsilon^nr^n$ and +Recall that $\mathit{Vol} B(x,T,r)=\varepsilon^nr^n$ and $\tau_1=\varepsilon r^2$. -**Lemma.** -There is a universal positive constant $\varepsilon_0>0$ such that + +**Lemma.** There is a universal positive constant $\varepsilon_0>0$ such that if $\varepsilon\le \varepsilon_0$, we have + $$ -\widetilde V_x(W_lg(\tau_1)) \le \int_{{\widetilde +\widetilde V_x(W_\mathit{lg}(\tau_1)) \le \int_{{\widetilde U}(\tau_1)\cap \{Z\bigl|\bigr. |Z|\geq \frac{1}{8}\varepsilon^{-\frac{1}{2}}\}}(\tau_1)^{-\frac{n}{2}} e^{-\tilde l(q,\tau_1)}\mathcal{J}(Z,\tau_1)dZ\le -\varepsilon^{\frac{n}{2}}.$$ +\varepsilon^{\frac{n}{2}}. +$$ + -**Proof.** By the monotonicity result (Proposition~), we see +**Proof.** By the monotonicity result (\entryref{4dc40f37cd6d}), we see that the restriction of the function $\tau_1^{-\frac{n}{2}}e^{-\tilde l(Z,\tau_1)}{\mathcal J}(Z,\tau_1)$ to ${\widetilde U}(\tau_1)$ is less than or equal to the restriction of the function $2^ne^{-|Z|^2}$ to the same subset. This means that -$$\widetilde V_x(W_lg(\tau_1))\le \int_{{\widetilde U}(\tau_1)\setminus + +$$ +\widetilde V_x(W_\mathit{lg}(\tau_1))\le \int_{{\widetilde U}(\tau_1)\setminus \widetilde U(\tau_1)\cap B(0,\frac{1}{8}\varepsilon^{-1/2})}2^n e^{-|Z|^2}dZ\le \int_{T_p{\mathcal M}_T\setminus -B(0,\frac{1}{8}\varepsilon^{-1/2})}2^ne^{-|Z|^2}dZ.$$ So it suffices +B(0,\frac{1}{8}\varepsilon^{-1/2})}2^ne^{-|Z|^2}dZ. +$$ + + So it suffices to estimate this latter integral. -Fix some $a>0$ and let $I(a)=\int_{B(0,a)}2^ne^{-|Z|^2}dZ$. Let +Fix some $a>0$ and let $I(a)=\int_{B(0,a)}2^ne^{-|Z|^2}dZ$. Let $R(a/\sqrt{n})$ be the $n$-cube centered at the origin with side lengths $2a/\sqrt{n}$. Then $R(a/\sqrt{n})\subset B(0,a)$, so that -$$ - $$ \begin{aligned} I(a) & \ge & \int_{R(a/\sqrt{n})}2^ne^{-|Z|^2}dZ \\ @@ -336,26 +464,36 @@ I(a) & \ge & \int_{R(a/\sqrt{n})}2^ne^{-|Z|^2}dZ \\ \end{aligned} $$ +Now + +$$ +\int_0^{2\pi}\int_0^{a/\sqrt{n}}4e^{-r^2}rdrd\theta=4\pi(1-e^{-\frac{a^2}{n}}). $$ -Now -$$\int_0^{2\pi}\int_0^{a/\sqrt{n}}4e^{-r^2}rdrd\theta=4\pi(1-e^{-\frac{a^2}{n}}).$$ Applying this with $a=(8\sqrt\varepsilon)^{-1}$ we have -$$\widetilde V_x(W_lg(\tau_1))\le \int_{\Ar^n}2^ne^{-|Z|^2}dZ-I(1/8\sqrt\varepsilon) -\le(4\pi)^{n/2}\left(1-\left(1-e^{-1/(64n\varepsilon)}\right)^{n/2}\right).$$ + +$$ +\widetilde V_x(W_\mathit{lg}(\tau_1))\le \int_{\Ar^n}2^ne^{-|Z|^2}dZ-I(1/8\sqrt\varepsilon) +\le(4\pi)^{n/2}\left(1-\left(1-e^{-1/(64n\varepsilon)}\right)^{n/2}\right). +$$ + Thus, -$$\widetilde V_x(W_lg(\tau_1))\le -(4\pi)^{n/2}\frac{n}{2}e^{-1/(64n\varepsilon)}.$$ There is $\varepsilon_0>0$ such that the expression on the + +$$ +\widetilde V_x(W_\mathit{lg}(\tau_1))\le +(4\pi)^{n/2}\frac{n}{2}e^{-1/(64n\varepsilon)}. +$$ + + There is $\varepsilon_0>0$ such that the expression on the right-hand side is less than $\varepsilon^{n/2}$ if $\varepsilon\le -\varepsilon_0$. This completes the proof of Lemma~. +\varepsilon_0$. This completes the proof of \entryref{4b41e7ff8317}. -Putting Lemmas~ and~ together, establishes -Proposition~. \end{proof} +Putting Lemmas \entryref{26e1a6292fef} and \entryref{4b41e7ff8317} together, establishes +\entryref{12d93b430400}. -As we have already remarked, Proposition~ immediately -implies Theorem~. This completes the proof of -Theorem~. -\end{proof} +As we have already remarked, \entryref{12d93b430400} immediately +implies \entryref{3defa9b2209c}. This completes the proof of +\entryref{3defa9b2209c}. ## Application to compact Ricci flows @@ -367,24 +505,26 @@ follows for we shall prove a more delicate result in the context of Ricci flows with surgery. Still, this result is much simpler and serves as a paradigm of what will come. + **Theorem.** Fix positive constants $\omega>0$ and $T_0<\infty$. Then there is $\kappa>0$ depending only on these constants such that the following holds. Let $(M,g(t)),\ 0\le t0$ with $r^2\le t_0$ and any $(p,t_0)\in M\times -\{t_0\}$, if $|Rm(q,t)|\le r^{-2}$ on $B(p,t_0,r)\times -[t_0-r^2,t_0]$ then $Vol\,B(p,t_0,r)\ge \kappa r^3$. +flow with $M$ compact. Suppose that $|\mathit{Rm}(p,0)|\le 1$ and also +that $\mathit{Vol} B(p,0,1)\ge \omega$ for all $p\in M$. Then for any +$t_0\le T$, any $r>0$ with $r^2\le t_0$ and any $(p,t_0)\in M\times +\{t_0\}$, if $|\mathit{Rm}(q,t)|\le r^{-2}$ on $B(p,t_0,r)\times +[t_0-r^2,t_0]$ then $\mathit{Vol} B(p,t_0,r)\ge \kappa r^3$. + **Proof.** Fix any $x=(p,t_0)\in M\times [0,T]$. First, we claim that we can suppose that $t_0\ge 1$. For if not, then rescale the flow by $Q=1/t_0$. This does not affect the curvature inequality at time zero. Furthermore, there is $\omega'>0$ depending only on $\omega$ such that for any ball $B$ at time zero and of radius one in the -rescaled flow we have $Vol\,B\ge \omega'$. The reason for the +rescaled flow we have $\mathit{Vol} B\ge \omega'$. The reason for the latter fact is the following: By the Bishop-Gromov inequality -(Theorem~) there is $\omega'>0$ depending only on -$\omega$ such that for any $q\in M$ and any $r\le 1$ we have $Vol\,B(q,0,r)\ge \omega'r^3$. Of course, the rescaling increases +(\entryref{1a1b2c4fdc9b}) there is $\omega'>0$ depending only on +$\omega$ such that for any $q\in M$ and any $r\le 1$ we have $\mathit{Vol} B(q,0,r)\ge \omega'r^3$. Of course, the rescaling increases $T$, but simply restrict to the rescaled flow on $[0,1]$. Next, we claim that we can assume that $r\le \sqrt{t_0}/2$. If $r$ @@ -392,25 +532,32 @@ does not satisfy this inequality, then we replace $r$ with $r'=\sqrt{t_0}/2$. Of course, the curvature inequalities hold for $r'$ if they hold for $r$. Suppose that we have established the result for $r'$. Then -$$Vol\,B(p,T,r)\ge Vol\,B(p,T,r')\ge \kappa(r')^3\ge -\kappa\left(\frac{r}{2}\right)^3=\frac{\kappa}{8}r^3.$$ + +$$ +\mathit{Vol} B(p,T,r)\ge \mathit{Vol} B(p,T,r')\ge \kappa(r')^3\ge +\kappa\left(\frac{r}{2}\right)^3=\frac{\kappa}{8}r^3. +$$ From now on we assume that $t_0\ge 1$ and $r\le\sqrt{t_0}/2$. -According to Proposition~ for any $(p,t)\in M\times -[0,2^{-4}]$ we have $|Rm(p,t)|\le 2$ and $Vol\,B(p,t,r)\ge \kappa_0 r^3$ for all $r\le 1$. +According to \entryref{c1330c5a8be3} for any $(p,t)\in M\times +[0,2^{-4}]$ we have $|\mathit{Rm}(p,t)|\le 2$ and $\mathit{Vol} B(p,t,r)\ge \kappa_0 r^3$ for all $r\le 1$. -Once we know that $|Rm|$ is universally bounded on $M\times +Once we know that $|\mathit{Rm}|$ is universally bounded on $M\times [0,2^{-4}]$ it follows that there is a universal constant $C_1$ such that $C_1^{-1}g(q,0)\le g(q,t) \le C_1g(q,0)$ for all $q\in M$ and all $t\in [0,2^{-4}]$. This means that there is a universal constant $C<\infty $ such that the following holds. For any points $q_0,q\in M$ with $d_0(q_0,q)\le 1$ let $\gamma_{q_0,q}$ be the path in $M\times [2^{-5},2^{-4}]$ given by -$$\gamma_{q_0,q}(\tau)=(A_{q_0,q}(\tau),2^{-4}-\tau),\ 0\le \tau\le 2^{-5},$$ + +$$ +\gamma_{q_0,q}(\tau)=(A_{q_0,q}(\tau),2^{-4}-\tau),\ 0\le \tau\le 2^{-5}, +$$ + where $A_{q_0,q}$ is a shortest $g(0)$-geodesic from $q_0$ to $q$. Then ${\mathcal L}(\gamma_{q_0,q})\le C$. -By Theorem~ there is a point $q_0\in M$ and an ${\mathcal +By \entryref{aebc3ff2bd3b} there is a point $q_0\in M$ and an ${\mathcal L}$-geodesic $\gamma_0$ from $x=(p,t_0)$ to $(q_0,2^{-4})$ with $l(\gamma_0)\le 3/2$. Since $t_0\ge 1$, this means that there is a universal constant $C'<\infty$ such that for each point $q\in @@ -418,8 +565,8 @@ B(q_0,0,1)$ the path which is the composite of $\gamma_0$ followed by $\gamma_{q_0,q}$ has $\tilde l$-length at most $C'$. Setting $\tau_0=t_0-2^{-5}$, this implies that $l_x(q,\tau_0)\le C'$ for every $q\in B(q_0,0,1)$. This ball has volume at least $\kappa_0$. -By Proposition~, the open subset ${\mathcal +By \entryref{16fbe263a7c8}, the open subset ${\mathcal U}_x(\tau_0)$ is of full measure in $M\times\{2^{-5}\}$. Hence, $W(\tau_0)=\left(B(q_0,0,1)\times \{2^{-5}\}\right)\cap {\mathcal U}_x(\tau_0)$ also has volume at least $\kappa_0$. Since $r^2\le -t_0/4<\tau_0$, Theorem~ now gives the result. (See Fig.~.) +t_0/4<\tau_0$, \entryref{3defa9b2209c} now gives the result. (See Fig. 8.3.) diff --git a/projects/poincare-conjecture/.astrolabe/docs/09-temp2kappa.mdx b/projects/poincare-conjecture/.astrolabe/docs-src/10-kappa-non-collapsed-ancient-solutions.mdx similarity index 63% rename from projects/poincare-conjecture/.astrolabe/docs/09-temp2kappa.mdx rename to projects/poincare-conjecture/.astrolabe/docs-src/10-kappa-non-collapsed-ancient-solutions.mdx index 64558618..2fa9ac36 100644 --- a/projects/poincare-conjecture/.astrolabe/docs/09-temp2kappa.mdx +++ b/projects/poincare-conjecture/.astrolabe/docs-src/10-kappa-non-collapsed-ancient-solutions.mdx @@ -1,4 +1,6 @@ -# $\kappa$-non-collapsed ancient solutions + + +# Chapter 9 -- $\kappa$-non-collapsed ancient solutions In this chapter we discuss the qualitative properties of $\kappa$-non-collapsed, ancient solutions. One of the most important is the @@ -20,29 +22,33 @@ with each $(M,g(t))$ being a complete manifold of bounded non-negative curvature. The first remark to make is that the appropriate notion of non-negative curvature is that the Riemann curvature operator -$$Rm\colon \wedge^2TM\to \wedge^2TM,$$ + +$$ +\mathit{Rm}\colon \wedge^2TM\to \wedge^2TM, +$$ + which is a symmetric operator, is non-negative. In general, this implies, but is stronger than, the condition that the sectional curvatures are all non-negative. In case the dimension of $M$ is at most three, every element of $\wedge^2TM$ is represented by a $2$-plane (with area form) and hence the Riemann curvature operator is non-negative if and only if all the sectional curvatures are -non-negative. In the case of non-negative curvature operator, +non-negative. In the case of non-negative curvature operator, bounded curvature operator is equivalent to bounded scalar curvature. It follows immediately from the Ricci flow equation that since the $(M,g(t))$ -have non-negative Ricci curvature, the metric is non-increasing in time in the +have non-negative Ricci curvature, the metric is non-increasing in time in the sense that for any point $p\in M$ and any $v\in T_pM$ the function $|v|^2_{g(t)}$ is a non-increasing function of $t$. -There are stronger results under the assumption of bounded, non-negative +There are stronger results under the assumption of bounded, non-negative curvature operator. These are consequences of the Harnack inequality (see -). As was established in Corollary~, since +[Hamiltonharnack]). As was established in \entryref{36dd2f6fa9c2}, since the flow exists for $t\in (-\infty,0]$ and since the curvature operator is non-negative and bounded for each $(q,t)\in M\times (-\infty,0]$, it follows that $\partial R(q,t)/\partial t\ge 0$ for all $q$ and $t$. That is to say, for -each $q\in M$ the scalar curvature $R(q,t)$ is a non-decreasing function of +each $q\in M$ the scalar curvature $R(q,t)$ is a non-decreasing function of $t$. ### Definition @@ -50,18 +56,19 @@ $t$. Now we turn to the definition of what it means for a Ricci flow to be $\kappa$-non-collapsed. -**Definition.** -Fix $\kappa>0$. Let $(M,g(t)),\ a0$. We say that $(M,g(t))$ is {\em -$\kappa$-non-collapsed on scales at -most $r_0$} if the following holds for any $(p,t)\in M\times (a,b]$ -and any $0 +**Definition.** Fix $\kappa>0$. Let $(M,g(t)),\ a0$. We say that $(M,g(t))$ is *$\kappa$-non-collapsed on scales at +most $r_0$* if the following holds for any $(p,t)\in M\times (a,b]$ +and any $0 + **Definition.** An *ancient solution* is a Ricci flow $(M,g(t))$ defined for $-\infty 0$, an ancient solution is *$\kappa$-non-collapsed* if it is $\kappa$-non-collapsed on all scales. We also use the terminology $\kappa$-*solution* for a $\kappa$-non-collapsed, ancient solution. + Notice that a $\kappa$-solution is a $\kappa'$-solution for any $0<\kappa'\le \kappa$. @@ -76,24 +84,27 @@ $0<\kappa'\le \kappa$. ### Examples Here are some examples of $\kappa$-solutions: -\begin{exam} -Let $(S^{2},g_{0})$ be the standard round $2$-sphere of scalar + + +**Example.** Let $(S^{2},g_{0})$ be the standard round $2$-sphere of scalar curvature $1$ (and hence Ricci tensor $g_0/2$). Set -$g(t)=(1-t)g_{0}$. Then $\partial g(t)/\partial t=-2Ric(g(t))$, $-\infty According to a result of Hamilton which we shall prove below -(Corollary~): +(\entryref{da767c71996b}): + **Theorem.** Every orientable, $2$-dimensional $\kappa$-solution is a rescaling of the previous example, i.e., is a family of shrinking round $2$-spheres. + -\begin{exam} -Let $(S^n,g_0)$ be the standard round $n$-sphere of scalar curvature + +**Example.** Let $(S^n,g_0)$ be the standard round $n$-sphere of scalar curvature $n/2$. Set $g(t)=(1-t)g_0$. This is a $\kappa$-solution for any $\kappa$ which is at most the volume of the ball of radius one in the unit $n$-sphere. If $\Gamma$ is a finite subgroup of the @@ -102,34 +113,43 @@ $S^n/\Gamma$ inherits an induced family of metrics $\bar g(t)$ satisfying the Ricci flow equation. The result is a $\kappa$-solution for any $\kappa$ at most $1/|\Gamma|$ times the volume of the ball of radius one in the unit sphere. -\end{exam} + -\begin{exam} -Consider the product $S^{2}\times \mathbb{R} $, with the metric $ + +**Example.** Consider the product $S^{2}\times \mathbb{R} $, with the metric $ g(t) =(1-t)g_{0}+ds^{2}.$ This is a $\kappa$-solution for any $\kappa$ at most the volume of a ball of radius one in the product of the unit $2$-sphere with $\Ar$. -\end{exam} + -\begin{exam} -The quotient $S^{2}\times \mathbb{R}/\langle\iota\rangle$, where the + +**Example.** The quotient $S^{2}\times \mathbb{R}/\langle\iota\rangle$, where the involution $\iota$ is the product of the antipodal map on $S^2$ with -$s\mapsto -s$ on the $\Ar$ factor, is an orientable +$s\mapsto -s$ on the $\Ar$ factor, is an orientable $\kappa$-solution for some $\kappa>0$. -\end{exam} + -\begin{exam} -Consider the metric product $(S^{2},g_0) \times (S^{1}_{R},ds^2)$ + +**Example.** Consider the metric product $(S^{2},g_0) \times (S^{1}_{R},ds^2)$ where $(S^1_R,ds^2)$ is the circle of radius $R$. We define $g(t)=(1-t)g_0+ds^2$. This is an ancient solution to the Ricci flow. But it is not $\kappa$-non-collapsed for any $\kappa>0$. The reason is that -$$|Rm(p,t)|=\frac{1}{1-t},$$ and -$$\frac{Vol_{g(t)}\,B(p,\sqrt{1-t})}{(1-t)^{3/2}} -\le \frac{Vol_{g(t)}(S^{2}\times S^{1}_{R})}{(1-t)^{3/2}} -=\frac{2\pi R(1-t)4\pi}{(1-t)^{3/2}}=\frac{8\pi^2R}{\sqrt{1-t}}.$$ + +$$ +|\mathit{Rm}(p,t)|=\frac{1}{1-t}, +$$ + + and + +$$ +\frac{\mathit{Vol}_{g(t)} B(p,\sqrt{1-t})}{(1-t)^{3/2}} +\le \frac{\mathit{Vol}_{g(t)}(S^{2}\times S^{1}_{R})}{(1-t)^{3/2}} +=\frac{2\pi R(1-t)4\pi}{(1-t)^{3/2}}=\frac{8\pi^2R}{\sqrt{1-t}}. +$$ + Thus, as $t\rightarrow-\infty$ this ratio goes to zero. -\end{exam} + ### A consequence of Hamilton's Harnack inequality @@ -139,49 +159,68 @@ inequality which is a consequence of Hamilton's Harnack inequality for Ricci flows with non-negative curvature operator. -**Proposition.** -Let $(M,g(t)),\ -\tau_0\le t\le 0$, be an $n$-dimensional Ricci flow + +**Proposition.** Let $(M,g(t)),\ -\tau_0\le t\le 0$, be an $n$-dimensional Ricci flow such that for each $t\in [-\tau_0,0]$ the Riemannian manifold $(M,g(t))$ is complete with non-negative, bounded curvature operator. Let $\tau=-t$. Fix a point $p\in M$ and let $x=(p,0)\in M\times [-\tau_0,0]$. Then for any $0 + +**Proof.** Recall that from Definition 6.15 we have + +$$ +H(X)=-\frac{\partial R}{\partial +\tau}-\frac{R}{\tau}-2\langle \nabla R,X\rangle+2\mathit{Ric}(X,X). +$$ -**Proof.** Recall that from Definition~ we have -$$H(X)=-\frac{\partial R}{\partial -\tau}-\frac{R}{\tau}-2\langle \nabla R,X\rangle+2Ric(X,X).$$ -Using Hamilton's Harnack's inequality (Theorem~) with +Using Hamilton's Harnack's inequality (\entryref{aa9d836895d8}) with $\chi=-X$, we have $$ -\frac{\partial R}{\partial \tau}-\frac{R}{\tau_0-\tau}-2\langle -\nabla R,X\rangle+2Ric(X,X)\ge 0. +\nabla R,X\rangle+2\mathit{Ric}(X,X)\ge 0. $$ Together these imply - $$H(X)\ge \left(\frac{1}{\tau-\tau_0}-\frac{1}{\tau}\right)R - =\frac{\tau_0}{\tau(\tau-\tau_0)}R.$$ + +$$ +H(X)\ge \left(\frac{1}{\tau-\tau_0}-\frac{1}{\tau}\right)R + =\frac{\tau_0}{\tau(\tau-\tau_0)}R. +$$ + Restricting to $\tau\le (1-c)\tau_0$ gives -$$H(X)\ge -\frac{c^{-1}}{\tau}R.$$ + +$$ +H(X)\ge -\frac{c^{-1}}{\tau}R. +$$ + Take a minimal ${\mathcal L}$-geodesic from $x$ to $(q,\bar\tau)$, we have $$ - K^{\bar\tau}(\gamma)=\int_0^{\bar\tau}\tau^{3/2}H(X)d\tau\ge -c^{-1}\int_0^{\bar\tau}\sqrt{\tau}Rd\tau\ge -2c^{-1}\sqrt{\bar\tau}l_x(q,\bar\tau). $$ - Together with -the second equality in Theorem~, this gives -$$ + Together with +the second equality in \entryref{4917e81d7684}, this gives $$ \begin{aligned} @@ -192,38 +231,58 @@ K^{\bar\tau}(\gamma)\\ & \le -4\bar\tau R(q,\bar\tau) \end{aligned} $$ -$$ - Dividing through by $4\bar\tau$, and replacing $\bar\tau$ with $\tau$ yields the first inequality in the statement of the proposition: -$$|\nabla l_x(q,\tau)|^2+R(q,\tau)\le \frac{(1+2c^{-1})l_x(q,\tau)}{\tau}$$ + +$$ +|\nabla l_x(q,\tau)|^2+R(q,\tau)\le \frac{(1+2c^{-1})l_x(q,\tau)}{\tau} +$$ + for all $0<\tau\le (1-c)\tau_0.$ This is an equation of smooth functions on the open dense subset ${\mathcal U}(\bar \tau)$ but it -extends as an equation of $L^\infty_loc$-functions on all of +extends as an equation of $L^\infty_\mathit{loc}$-functions on all of $M$. As to the second inequality in the statement, by the first equation -in Theorem~ we have -$$\frac{\partial l_x(q,\tau)}{\partial \tau} -=R(q,\tau)-\frac{l_x(q,\tau)}{\tau}+\frac{1}{2\tau^{3/2}}K^{\tau}(\gamma).$$ -The estimate on $K^\tau$ in Equation~() then gives -$$R(q,\tau)-\frac{(1+c^{-1})l_x(q,\tau)}{\tau}\le -\frac{\partial l_x(q,\tau)}{\partial \tau}.$$ This establishes the +in \entryref{4917e81d7684} we have + +$$ +\frac{\partial l_x(q,\tau)}{\partial \tau} +=R(q,\tau)-\frac{l_x(q,\tau)}{\tau}+\frac{1}{2\tau^{3/2}}K^{\tau}(\gamma). +$$ + +The estimate on $K^\tau$ in Equation (9.1) then gives + +$$ +R(q,\tau)-\frac{(1+c^{-1})l_x(q,\tau)}{\tau}\le +\frac{\partial l_x(q,\tau)}{\partial \tau}. +$$ + + This establishes the second inequality. -**Corollary.** -Let $(M,g(t)),\ -\infty< t\le 0$, be a Ricci flow on a complete, + +**Corollary.** Let $(M,g(t)),\ -\infty< t\le 0$, be a Ricci flow on a complete, $n$-dimensional manifold with bounded, non-negative curvature operator. Fix a point $p\in M$ and let $x=(p,0)\in M\times (-\infty,0]$. Then for any $\tau>0$ we have -$$|\bigtriangledown l_x(q,\tau))|^{2} +R(q,\tau) \leq \frac{3l_x(q,\tau)}{\tau},$$ -$$-\frac{2l_x(q,\tau)}{\tau}\le \frac{\partial l_x(q,\tau)}{\partial \tau} -\le \frac{l_x(q,\tau)}{\tau}.$$ where these inequalities are valid + +$$ +|\bigtriangledown l_x(q,\tau))|^{2} +R(q,\tau) \leq \frac{3l_x(q,\tau)}{\tau}, +$$ + +$$ +-\frac{2l_x(q,\tau)}{\tau}\le \frac{\partial l_x(q,\tau)}{\partial \tau} +\le \frac{l_x(q,\tau)}{\tau}. +$$ + + where these inequalities are valid in the sense of smooth functions on the open subset of full measure of $M\times \{\tau\}$ on which $l_x$ is a smooth function, and are -valid as inequalities of $L^\infty_loc$-functions on all of +valid as inequalities of $L^\infty_\mathit{loc}$-functions on all of $M\times \{\tau\}$. + **Proof.** Fix $\tau$ and take a sequence of $\tau_0\rightarrow \infty$, allowing us to take $c\rightarrow 1$, and apply the previous @@ -234,11 +293,13 @@ bound for $\partial l_x/\partial \tau$ in the second inequality. consider the path that is the concatenation of a minimal ${\mathcal L}$-geodesic $\gamma$ from $x$ to $(q,\tau)$ followed by the path $\mu(\tau')=(q,\tau')$ for $\tau'\ge \tau$. Then -$$l_x(\gamma*\mu|_{[\tau,\tau_1]})=\frac{1}{2\sqrt{\tau_1}}\left({\mathcal L}(\gamma) -+\int_{\tau}^{\tau_1} \sqrt{\tau'}R(q,\tau')d\tau'\right).$$ -Differentiating at $\tau_1=\tau$ gives $$ +l_x(\gamma*\mu|_{[\tau,\tau_1]})=\frac{1}{2\sqrt{\tau_1}}\left({\mathcal L}(\gamma) ++\int_{\tau}^{\tau_1} \sqrt{\tau'}R(q,\tau')d\tau'\right). +$$ + +Differentiating at $\tau_1=\tau$ gives $$ \begin{aligned} @@ -250,35 +311,44 @@ l_x(\gamma*\mu)}{\partial\tau}\Bigl|_{\tau_1=\tau}\Bigr. & = & \end{aligned} $$ +By the first inequality in this statement, we have + +$$ +-\frac{l_x(q,\tau)}{2\tau}+\frac{R(q,\tau)}{2}\le + \frac{l_x(q,\tau)}{\tau}. $$ -By the first inequality in this statement, we have -$$-\frac{l_x(q,\tau)}{2\tau}+\frac{R(q,\tau)}{2}\le - \frac{l_x(q,\tau)}{\tau}.$$ Since $l_x(q,\tau')\le \tilde l(\gamma*\mu|_{[\tau,\tau']})$ + Since $l_x(q,\tau')\le \tilde l(\gamma*\mu|_{[\tau,\tau']})$ for all $\tau'\ge \tau$, this establishes the claimed upper bound for $\partial l_x/\partial \tau$. -## The asymptotic gradient shrinking soliton for -$\kappa$-solutions +## The asymptotic gradient shrinking soliton for $\kappa$-solutions We fix $\kappa>0$ and we consider an $n$-dimensional $\kappa$-solution $(M,g(t)),\ -\infty 0$ there is a point $q(\tau)\in +By \entryref{aebc3ff2bd3b} for every $\tau>0$ there is a point $q(\tau)\in M$ at which the function $l_x(\cdot,^\tau)$ achieves its minimum, and furthermore, we have -$$l_x(q(\tau),\tau) \leq \frac{n}{2}.$$ + +$$ +l_x(q(\tau),\tau) \leq \frac{n}{2}. +$$ For $\bar\tau>0,$ define -$$g_{\bar\tau}(t) = \frac{1}{\bar\tau}g(\bar\tau t),\ \ -\infty +**Theorem.** Let $(M,g(t)),\ -\infty -See Fig.~. +See Fig. 9.1. + **Remark.** We are not claiming that the gradient shrinking soliton is a $\kappa$-solution (or more precisely an extension forward in time of a time-shifted version of a $\kappa$-solution) because we are not claiming that the time-slices have bounded curvature operator. Indeed, we do not know if this is true in general. We shall -establish below (see Corollary~ and -Corollary~) that in the case $n=2,3$, the gradient +establish below (see \entryref{da767c71996b} and +\entryref{351e4ac14e4e}) that in the case $n=2,3$, the gradient shrinking soliton does indeed have time-slices of bounded curvature, and hence is an extension of a $\kappa$-solution. We are also not claiming at this point that the limiting flow is a gradient @@ -323,12 +394,13 @@ with the property that $|t|\varphi_t^*g_\infty(-1)=g_\infty(t)$ and with the property that the $\varphi_t$ are generated by the gradient vector field of a function. We shall also establish this result in dimensions $2$ and $3$ later in this chapter. + -We will divide the proof of Theorem~ into steps. +We will divide the proof of \entryref{671402eb454b} into steps. First, we will show that the reduced length and norm of the -curvature $|Rm|$ are bounded throughout the sequence in some +curvature $|\mathit{Rm}|$ are bounded throughout the sequence in some way. Then using the $\kappa$-non-collapsed assumption, by the -compactness theorem (Theorem~), we conclude that a +compactness theorem (\entryref{a0212d0eefcf}), we conclude that a subsequence of the sequence of flows converges geometrically to a limiting flow. Then, using the fact that the limit of the reduced volumes, denoted $\widetilde @@ -336,7 +408,7 @@ V_{\infty}(M_\infty\times\{t\})$, is constant we show that the limit flow is a gradient shrinking soliton. Finally we argue that the limit is non-flat. The proof occupies the rest of -Section~. +Section 9.2. ### Bounding the reduced length and the curvature @@ -346,88 +418,116 @@ hypothesis, all the Ricci flows $(M_k,g_k(t))$ are $\kappa$-non-collapsed on all scales. Next, we have the effect on reduced volume. -**Claim.** For each $k\ge 1$ denote by $x_k\in M_k$ the point $(p,0)\in + +**Claim.** For each $k\ge 1$ denote by $x_k\in M_k$ the point $(p,0)\in M_k$. Let $\widetilde V_{x_k}(\tau)=\widetilde V_{x_k}(M_k\times\{\tau\})$ denote the reduced volume function for the Ricci flow $(M_k,g_k(t))$ from the point $x_k$, and let $\widetilde V_x(\tau)$ denote the reduced volume of $M\times\{\tau\}$ for the Ricci flow $(M,g(t))$ from the point $x$. Then -$$\widetilde V_{x_k}(\tau) = \widetilde V_x (\bar \tau_{k}\tau).$$ + +$$ +\widetilde V_{x_k}(\tau) = \widetilde V_x (\bar \tau_{k}\tau). +$$ + **Proof.** This is a special case of the reparameterization equation for -reduced volume (Lemma~). +reduced volume (\entryref{76c5f88ad2b4}). -By Theorem~ the reduced volume function $\widetilde -V_x(\tau)$ is a non-increasing function of $\tau$ with $lim_{\tau\rightarrow 0}\widetilde V_x(\tau)=(4\pi)^{\frac{n}{2}}$. +By \entryref{ae6f1c6d00d9} the reduced volume function $\widetilde +V_x(\tau)$ is a non-increasing function of $\tau$ with $\mathit{lim}_{\tau\rightarrow 0}\widetilde V_x(\tau)=(4\pi)^{\frac{n}{2}}$. Since the integrand for $\widetilde V_x(\tau)$ is everywhere positive, it is clear that $\widetilde V_x(\tau)>0$ for all $\tau$. -Hence, $lim_{\tau\rightarrow\infty}\widetilde V_x(\tau)$ -exists. By Corollary~ either this limit as $\tau$ goes +Hence, $\mathit{lim}_{\tau\rightarrow\infty}\widetilde V_x(\tau)$ +exists. By \entryref{613fb3994956} either this limit as $\tau$ goes to infinity is less than $(4\pi)^{n/2}$ or the flow is the constant flow on flat Euclidean space. The latter is ruled out by our assumption that the manifolds are non-flat. It follows immediately -from this and Claim~ that: +from this and \entryref{dbf6a3c0bae7} that: -**Corollary.** -There is a non-negative constant $V_\infty<(4\pi)^{n/2}$ such that + +**Corollary.** There is a non-negative constant $V_\infty<(4\pi)^{n/2}$ such that for all $\tau\in(0,\infty)$, we have $$ - -lim_{k\rightarrow\infty}\widetilde V_{x_k}(\tau)=V_\infty. +\mathit{lim}_{k\rightarrow\infty}\widetilde V_{x_k}(\tau)=V_\infty. $$ + Now let us turn to the length functions $l_{x_k}$. -**Claim.** -For any $\tau>0$ we have -$$l_{x_k}(q_k,\tau)\le \frac{n}{2\tau^2}+\frac{n\tau}{2}.$$ + +**Claim.** For any $\tau>0$ we have + +$$ +l_{x_k}(q_k,\tau)\le \frac{n}{2\tau^2}+\frac{n\tau}{2}. +$$ + **Proof.** By the choice of $q_k$ we have $l_{x_k}(q_k,\tau_k)\le \frac{n}{2}$. - By the scale invariance of $l$ (Corollary~) we have + By the scale invariance of $l$ (\entryref{541f71387f84}) we have $l_{x_k}(q_k,-1)\le n/2$ for all $k$. Fix $0<\tau<1$. Integrating the inequality -$$\frac{-2l_x(q_k,\tau)}{\tau}\le \frac{\partial l_{x_k}(q_k,\tau)}{\partial \tau}$$ - from $\tau$ to $1$ yields -$$l_{x_k}(q_k,\tau)\le \frac{n}{2\tau^2}.$$ + +$$ +\frac{-2l_x(q_k,\tau)}{\tau}\le \frac{\partial l_{x_k}(q_k,\tau)}{\partial \tau} +$$ + + from $\tau$ to $1$ yields + +$$ +l_{x_k}(q_k,\tau)\le \frac{n}{2\tau^2}. +$$ + If $\tau>1$, then - integrating the second inequality in the second displayed line of Corollary~ gives + integrating the second inequality in the second displayed line of \entryref{8b8e3a0f3e36} gives $l_{x_k}(q_k,\tau)\le \frac{n\tau}{2}$. -**Corollary.** -There is a positive continuous function $C_1(\tau)$ defined for + +**Corollary.** There is a positive continuous function $C_1(\tau)$ defined for $\tau>0$ such that for any $q\in M_k$ we have: -$$l_{x_k}(q,\tau)\le -\left(\sqrt{\frac{3}{\tau}}d_{g_k(-\tau)}(q_k,q)+C_1(\tau)\right)^2,$$ -$$|\nabla l_{x_k}(q,\tau)|\le -\frac{3}{\tau}d_{g_k(-\tau)}(q_k,q)+\sqrt{\frac{3}{\tau}}C_1(\tau).$$ -**Proof.** By Corollary~, for any $q\in M_k$ we have $|\nabla +$$ +l_{x_k}(q,\tau)\le +\left(\sqrt{\frac{3}{\tau}}d_{g_k(-\tau)}(q_k,q)+C_1(\tau)\right)^2, +$$ + +$$ +|\nabla l_{x_k}(q,\tau)|\le +\frac{3}{\tau}d_{g_k(-\tau)}(q_k,q)+\sqrt{\frac{3}{\tau}}C_1(\tau). +$$ + + +**Proof.** By \entryref{8b8e3a0f3e36}, for any $q\in M_k$ we have $|\nabla l_{x_k}(q,\tau)|^2\le 3l_{x_k}(q,\tau)/\tau$. Since $l_{x_k}(q_k,\tau)\le \frac {n}{2\tau_0^2}+\frac{n\tau}{2}$, integrating yields -$$l_{x_k}(q,\tau)\le -\left(\sqrt{\frac{3}{\tau}}d_{g_k(-\tau)}(q_k,q)+C_1(\tau)\right)^2,$$ + +$$ +l_{x_k}(q,\tau)\le +\left(\sqrt{\frac{3}{\tau}}d_{g_k(-\tau)}(q_k,q)+C_1(\tau)\right)^2, +$$ + with $C_1(\tau)$ being $\sqrt{(n/2\tau^2)+(n\tau/2)}$. The second -statement follows from this and Proposition~. +statement follows from this and \entryref{9387b1b04824}. -It follows immediately from Corollary~ that for each +It follows immediately from \entryref{611a73900deb} that for each $A<\infty$ and $\tau_0>0$, the functions $l_{x_k}$ are uniformly bounded (by a bound that is independent of $k$ but depends on $\tau_0$ and $A$) on the balls $B(q_k,-\tau_0,A)$. Once we know that the $l_{x_k}$ are uniformly bounded on $B(q_k,-\tau_0,A)$, it -follows from Corollary~ that $R_{g_k}$ are also +follows from \entryref{8b8e3a0f3e36} that $R_{g_k}$ are also uniformly bounded on the $B(q_k,-\tau_0,A)$. Invoking -Corollary~, +\entryref{36dd2f6fa9c2}, we see that for any $A<\infty$ the scalar curvatures of the metrics $g_k$ are uniformly bounded on $B_{g_k}(q_k,-\tau_0,A)\times (-\infty, -\tau_0]$. Since the metrics have non-negative curvature operator, -this implies that the eigenvalues of this operator are uniformly +this implies that the eigenvalues of this operator are uniformly bounded on these regions. Since we are assuming that the original Ricci flows are $\kappa$-non-collapsed on all -scales, it follows from Theorem~ that after passing +scales, it follows from \entryref{a0212d0eefcf} that after passing to a subsequence there is a geometric limit $(M_\infty,g_\infty(t),(q_\infty,-1)),\ -\infty **Corollary.** After passing to a subsequence of the $\tau_k$ there is a smooth limiting flow of the $(M_k,g_k(t),(q_k,-1)),-\infty -**Proof.** Since the flows in the sequence are all +**Proof.** Since the flows in the sequence are all $\kappa$-non-collapsed on all scales and have non-negative curvature operator, the limiting flow is $\kappa$-non-collapsed on all scales and has non-negative curvature operator. By the consequence of -Hamilton's Harnack inequality (Corollary~), we have +Hamilton's Harnack inequality (\entryref{36dd2f6fa9c2}), we have $\partial R/\partial t\ge 0$ for the original $\kappa$-solution. This condition also passes to the limit. @@ -482,7 +589,7 @@ covariant derivative and the Laplacian in the limiting metric $g_\infty$. Now let us consider the functions $l_{x_k}$. According to -Corollary~, for any $A<\infty$ and any $0<\tau_0 +**Corollary.** For any $\tau>0$ and any $q$ we have + +$$ +|\nabla l_\infty(q,\tau)|\le +\frac{3}{\tau}d_{g_\infty}(-\tau)(q_\infty,q)+\sqrt{\frac{3}{\tau}}C_1(\tau), +$$ -**Corollary.** -For any $\tau>0$ and any $q$ we have -$$|\nabla l_\infty(q,\tau)|\le -\frac{3}{\tau}d_{g_\infty}(-\tau)(q_\infty,q)+\sqrt{\frac{3}{\tau}}C_1(\tau),$$ where $C_1(\tau)$ is the continuous function from -Corollary~. +\entryref{611a73900deb}. + -**Proof.** This is immediate from Corollary~ and Fatou's lemma. +**Proof.** This is immediate from \entryref{611a73900deb} and Fatou's lemma. + **Remark.** **N.B.** We are not claiming that $l_\infty$ is the reduced length function from a point of $M_\infty\times (-\infty,0)$. + ### Differential inequalities for $l_\infty$ The next step is to establish differential equalities for $l_\infty$ related to, but stronger than, those that we established in -Chapter~ for $l_x$. Here is a crucial result. +Chapter 7 for $l_x$. Here is a crucial result. -**Proposition.** The function $l_\infty$ is a smooth function + +**Proposition.** The function $l_\infty$ is a smooth function on $M\times (-\infty,0)$ and satisfies the following two differential equalities: -$$ +$$ \frac{\partial l_\infty}{\partial \tau}+|\nabla l_\infty|^2-R+\frac{n}{2\tau}-\triangle l_\infty=0 $$ @@ -534,15 +649,15 @@ $$ 2\triangle l_\infty-|\nabla l_\infty|^2+R+\frac{l_\infty-n}{\tau}=0. $$ + -The proof of this result is contained in Sections~ -through~ +The proof of this result is contained in Sections 9.2.4 +through 9.2.6 -### Preliminary results toward the proof of -Proposition~\protect{} +### Preliminary results toward the proof of Proposition \entryref{2e9e6c89f40c} -In this subsection we shall prove that the left-hand side of -Equation~() is a distribution and is $\ge 0$ in the +In this subsection we shall prove that the left-hand side of +Equation (9.4) is a distribution and is $\ge 0$ in the distributional sense. We shall also show that this distribution extends to a continuous linear functional on compactly supported functions in $W^{1,2}$. @@ -550,15 +665,21 @@ functions in $W^{1,2}$. The first step in the proof of this result is the following, somewhat delicate lemma. -**Lemma.** -For any $t\in (-\infty,0)$ we have -$$lim_{k\rightarrow\infty}|\nabla^{h_k} l_k|_{h_k}^2dvol(h_k) -=|\nabla l_\infty|_{g_\infty}^2 dvol(g_\infty)$$ in the sense + +**Lemma.** For any $t\in (-\infty,0)$ we have + +$$ +\mathit{lim}_{k\rightarrow\infty}|\nabla^{h_k} l_k|_{h_k}^2d\mathit{vol}(h_k) +=|\nabla l_\infty|_{g_\infty}^2 d\mathit{vol}(g_\infty) +$$ + + in the sense of distributions on $M_\infty\times \{t\}$. + **Proof.** It suffices to fix $0<\tau_0<|t|$. The inequality in one direction ($\ge$) is a general -result. Here is the argument. Since the $|\nabla^{g_k} l_{x_k}|_{g_k}$ are +result. Here is the argument. Since the $|\nabla^{g_k} l_{x_k}|_{g_k}$ are uniformly essentially bounded on every $B(x_k,-\tau_0,A)\times [-T,-\tau_0]$, the $|\nabla^{h_k}l_k|_{h_k}$ are uniformly essentially bounded on $B(x_\infty,-\tau_0,A)\times [-T,-\tau_0]$. (Of course, @@ -566,57 +687,67 @@ $\nabla^{h_k}l_k=dl_k=\nabla l_k$.) Since the $h_k$ converge uniformly on compact sets to $g_\infty$, it is clear that $$ - lim_{k\rightarrow\infty}\left(|\nabla^{h_k}l_k|^2_{h_k}dvol(h_k)-|\nabla l_k|_{g_\infty}^2dvol(g_\infty)\right)=0 +\mathit{lim}_{k\rightarrow\infty}\left(|\nabla^{h_k}l_k|^2_{h_k}d\mathit{vol}(h_k)-|\nabla l_k|_{g_\infty}^2d\mathit{vol}(g_\infty)\right)=0 $$ + in the sense of distributions on $M\times \{t\}$. Since the $ l_k$ converge uniformly on compact subsets to $l_\infty$, it follows immediately from Fatou's lemma that -$$lim_{k\rightarrow \infty}|\nabla -l_k|^2_{g_\infty}dvol(g_\infty)\ge|\nabla -l_\infty|^2_{g_\infty}dvol (g_\infty)$$ in the sense of + +$$ +\mathit{lim}_{k\rightarrow \infty}|\nabla +l_k|^2_{g_\infty}d\mathit{vol}(g_\infty)\ge|\nabla +l_\infty|^2_{g_\infty}d\mathit{vol} (g_\infty) +$$ + + in the sense of distributions on $M_\infty\times \{t\}$. Thus, we have the following inequality of distributions: -$$lim_{k\rightarrow\infty}|\nabla^{h_k}l_k|^2_{h_k}dvol(h_k)\ge -|\nabla l_\infty|^2_{g_\infty}dvol (g_\infty).$$ + +$$ +\mathit{lim}_{k\rightarrow\infty}|\nabla^{h_k}l_k|^2_{h_k}d\mathit{vol}(h_k)\ge +|\nabla l_\infty|^2_{g_\infty}d\mathit{vol} (g_\infty). +$$ We need to establish the opposite -inequality which is not a general result, but rather relies on the +inequality which is not a general result, but rather relies on the bounds on $\triangle^{g_k}l_{x_k}$ (or equivalently on $\triangle^{h_k}l_k$) given in second inequality in -Theorem~. We must show that for each $t\le -\tau_0$ and +\entryref{7d5d94efe4e9}. We must show that for each $t\le -\tau_0$ and for any $\varphi$, a non-negative, smooth function with compact support in $M_\infty\times \{t\}$, we have -$$lim_{k\rightarrow\infty}\int_{M\times\{t\}}\varphi -\left(|\nabla^{h_k} l_k|_{h_k}^2dvol(h_k)-|\nabla -l_\infty|_{g_\infty}^2dvol(g_\infty)\right)\le 0.$$ First, + +$$ +\mathit{lim}_{k\rightarrow\infty}\int_{M\times\{t\}}\varphi +\left(|\nabla^{h_k} l_k|_{h_k}^2d\mathit{vol}(h_k)-|\nabla +l_\infty|_{g_\infty}^2d\mathit{vol}(g_\infty)\right)\le 0. +$$ + + First, notice that since, on the support of $\varphi$, the metrics $h_k$ converge uniformly in the $C^\infty$-topology to $g_\infty$ and since $|\nabla^{h_k}l_k|_{h_k}^2$ and $|\nabla l_\infty|_{g_\infty}^2$ are essentially bounded on the support of $\varphi$, we have -$$ - $$ \begin{aligned} \nonumber -\lefteqn{lim_{k\rightarrow\infty}\int_{M\times\{t\}}\varphi -\left(|\nabla^{h_k}l_k|_{h_k}^2dvol(h_k)-|\nabla -l_\infty|_{g_\infty}^2dvol(g_\infty) \right)} & & \\ -& = & lim_{k\rightarrow\infty}\int_{M\times\{t\}}\varphi (| +\mathit{lim}_{k\rightarrow\infty}\int_{M\times\{t\}}\varphi +\left(|\nabla^{h_k}l_k|_{h_k}^2d\mathit{vol}(h_k)-|\nabla +l_\infty|_{g_\infty}^2d\mathit{vol}(g_\infty) \right) & & \\ +& = & \mathit{lim}_{k\rightarrow\infty}\int_{M\times\{t\}}\varphi (| \nabla^{h_k} l_k|_{h_k}^2-|\nabla -l_\infty|_{h_k}^2)dvol(h_k) \nonumber \\ -\nonumber& = & lim_{k\rightarrow\infty}\int_{M\times\{t\}}\langle\nabla^{h_k} -l_k-\nabla l_\infty),\varphi\nabla^{h_k} l_k\rangle_{h_k}dvol(h_k) +l_\infty|_{h_k}^2)d\mathit{vol}(h_k) \nonumber \\ +\nonumber& = & \mathit{lim}_{k\rightarrow\infty}\int_{M\times\{t\}}\langle\nabla^{h_k} +l_k-\nabla l_\infty),\varphi\nabla^{h_k} l_k\rangle_{h_k}d\mathit{vol}(h_k) \\ & & \hskip.55in+\int_{M\times\{t\}}\langle\nabla^{h_k}l_k-\nabla l_\infty), -\varphi\nabla l_\infty\rangle_{h_k}dvol(h_k). - +\varphi\nabla l_\infty\rangle_{h_k}d\mathit{vol}(h_k). \end{aligned} $$ -$$ We claim that, in the limit, the last term in this expression vanishes. Using the fact that the $h_k$ converge uniformly in the $C^\infty$-topology to $g_\infty$ on the support of @@ -624,73 +755,78 @@ $\varphi$, and $|\nabla l_\infty|$ is bounded on this support we can rewrite the last term as $$ - -lim_{k\rightarrow\infty}\int_{M\times\{t\}}\langle\nabla( -l_k-l_\infty),\varphi\nabla l_\infty\rangle_{g_\infty}dvol( +\mathit{lim}_{k\rightarrow\infty}\int_{M\times\{t\}}\langle\nabla( +l_k-l_\infty),\varphi\nabla l_\infty\rangle_{g_\infty}d\mathit{vol}( g_\infty). $$ + Since $ l_k-l_\infty$ goes to zero weakly -in $W^{1,2}$ on the support of $\varphi$ whereas $ l_\infty$ is an +in $W^{1,2}$ on the support of $\varphi$ whereas $ l_\infty$ is an element of $W^{1,2}$ of this compact set, we see that the expression -given in~() vanishes and hence that -$$lim_{k\rightarrow\infty}\int_{M\times\{t\}}\langle\nabla^{h_k}( -l_k-l_\infty),\varphi\nabla l_\infty\rangle_{h_k}dvol(h_k)=0.$$ +given in (9.8) vanishes and hence that + +$$ +\mathit{lim}_{k\rightarrow\infty}\int_{M\times\{t\}}\langle\nabla^{h_k}( +l_k-l_\infty),\varphi\nabla l_\infty\rangle_{h_k}d\mathit{vol}(h_k)=0. +$$ It remains to consider the first term in the last expression in -Equation~(). (This is where we shall need the differential +Equation (9.7). (This is where we shall need the differential inequality for the $\triangle^{g_k}l_{x_k}$.) Since the $l_k$ converge uniformly to $l_\infty$ on the support of $\varphi$, we can -choose positive constants $\epsilon_k$ tending to $0$ as $k$ tends +choose positive constants $\epsilon_k$ tending to $0$ as $k$ tends to $\infty$ so that $l_\infty- l_k+\epsilon_k>0$ on the support of $\varphi$. We can rewrite -$$ - $$ \begin{aligned} -\lefteqn{lim_{k\rightarrow\infty}\int_{M\times\{t\}}\langle\left(\nabla^{h_k} +\mathit{lim}_{k\rightarrow\infty}\int_{M\times\{t\}}\langle\left(\nabla^{h_k} l_k-\nabla l_\infty\right),\varphi\nabla^{h_k} l_k\rangle_{h_k} -dvol(h_k) = } \\ -& & lim_{k\rightarrow\infty}\int_{M\times\{t\}}\langle\nabla^{h_k}( -l_k-l_\infty-\epsilon_k),\varphi\nabla^{h_k}l_k\rangle_{h_k} dvol(h_k). +d\mathit{vol}(h_k) = \\ +& & \mathit{lim}_{k\rightarrow\infty}\int_{M\times\{t\}}\langle\nabla^{h_k}( +l_k-l_\infty-\epsilon_k),\varphi\nabla^{h_k}l_k\rangle_{h_k} d\mathit{vol}(h_k). \end{aligned} $$ -$$ + +**Claim.** -**Claim.** $$lim_{k\rightarrow\infty}\int_{M\times\{t\}}\langle\nabla^{h_k}(l_k-l_\infty-\epsilon_k), -\varphi\nabla^{h_k} l_k\rangle_{h_k} dvol(h_k)\le 0.$$ +$$ +\mathit{lim}_{k\rightarrow\infty}\int_{M\times\{t\}}\langle\nabla^{h_k}(l_k-l_\infty-\epsilon_k), +\varphi\nabla^{h_k} l_k\rangle_{h_k} d\mathit{vol}(h_k)\le 0. +$$ + -\begin{proof} -Since $\varphi$ is a compactly supported, non-negative smooth -function, it follows from Theorem~ that we have the +**Proof.** Since $\varphi$ is a compactly supported, non-negative smooth +function, it follows from \entryref{7d5d94efe4e9} that we have the following inequality of distributions: -$$\varphi\triangle^{h_k} l_k\le + +$$ +\varphi\triangle^{h_k} l_k\le \frac{\varphi}{2}\left(|\nabla^{h_k} l_k|_{h_k}^2-R_{h_k}-\frac{ -l_k-n}{\tau}\right).$$ (Here $R_{h_k}$ is the scalar curvature of +l_k-n}{\tau}\right). +$$ + + (Here $R_{h_k}$ is the scalar curvature of $h_k$.) That is to say, for any non-negative $C^\infty$-function $f$ we have -$$ - $$ \begin{aligned} -\lefteqn{\int_{M\times\{t\}}-\langle\nabla^{h_k} -l_k,\nabla^{h_k}(\varphi \cdot f)\rangle_{h_k} dvol(h_k) \le} +\int_{M\times\{t\}}-\langle\nabla^{h_k} +l_k,\nabla^{h_k}(\varphi \cdot f)\rangle_{h_k} d\mathit{vol}(h_k) \le \\ & & \int_{M\times\{t\}}\frac{\varphi f}{2}\left(|\nabla^{h_k} -l_k|_{h_k}^2-R_{h_k}-\frac{l_k-n}{\tau}\right)dvol(h_k). +l_k|_{h_k}^2-R_{h_k}-\frac{l_k-n}{\tau}\right)d\mathit{vol}(h_k). \end{aligned} $$ -$$ - We claim that the same inequality holds as long as $f$ is a -non-negative, locally Lipschitz function. The point is that given +non-negative, locally Lipschitz function. The point is that given such a function $f$, we can find a sequence of non-negative $C^\infty$-functions $f_k$ on the support of $\varphi$ (by say -mollifying $f$) that converge to $f$ strongly in the $W^{1,2}$-norm +mollifying $f$) that converge to $f$ strongly in the $W^{1,2}$-norm on the support of $\varphi$. The sought-after inequality holds for every $f_k$. Since both sides of the inequality are continuous in the $W^{1,2}$-norm of the function, the result holds for the limit @@ -699,47 +835,59 @@ function $f$ as well. Now we apply this with $f$ being the non-negative locally Lipschitz function $l_\infty-l_k+\epsilon_k$. We conclude that -$$ - $$ \begin{aligned} -\lefteqn{\int_{M\times\{t\}}\langle \nabla^{h_k}(\varphi( -l_k-l_\infty-\epsilon_k)),\nabla^{h_k} l_k\rangle_{h_k}dvol(h_k) \le} \\ +\int_{M\times\{t\}}\langle \nabla^{h_k}(\varphi( +l_k-l_\infty-\epsilon_k)),\nabla^{h_k} l_k\rangle_{h_k}d\mathit{vol}(h_k) \le \\ & & \int_{M\times\{t\}}\frac{\varphi(l_\infty- l_k+\epsilon_k)}{2}\left(|\nabla^{h_k} -l_k|_{h_k}^2-R_{h_k}-\frac{l_k-n}{\tau}\right)dvol(h_k). +l_k|_{h_k}^2-R_{h_k}-\frac{l_k-n}{\tau}\right)d\mathit{vol}(h_k). \end{aligned} $$ -$$ - Now taking the limit as $k\rightarrow\infty$, we see that the right-hand side of this inequality tends to zero since $(l_\infty- l_k+\epsilon_k)$ tends uniformly to zero on the support of $\varphi$ and $|\nabla^{h_k}l_k|_{h_k}^2$, $R_k$ and $l_k$ are all uniformly essentially bounded on the support of $\varphi$. Thus, the term -$$\int_{M\times\{t\}}\langle\nabla^{h_k}(\varphi( -l_k-l_\infty-\epsilon_k)),\nabla^{h_k} l_k\rangle_{h_k}dvol(h_k)$$ has a limsup $\le 0$ as $k$ tends to $\infty$. Now we + +$$ +\int_{M\times\{t\}}\langle\nabla^{h_k}(\varphi( +l_k-l_\infty-\epsilon_k)),\nabla^{h_k} l_k\rangle_{h_k}d\mathit{vol}(h_k) +$$ + + has a limsup $\le 0$ as $k$ tends to $\infty$. Now we expand -$$\nabla^{h_k}(\varphi(l_k-l_\infty-\epsilon_k))=\nabla^{h_k}(\varphi)(l_k-l_\infty-\epsilon_k)+ -\varphi\nabla^{h_k}(l_k-l_\infty-\epsilon_k).$$ The first term on + +$$ +\nabla^{h_k}(\varphi(l_k-l_\infty-\epsilon_k))=\nabla^{h_k}(\varphi)(l_k-l_\infty-\epsilon_k)+ +\varphi\nabla^{h_k}(l_k-l_\infty-\epsilon_k). +$$ + + The first term on the right-hand side converges to zero as $k\rightarrow\infty$ since $l_k-l_\infty-\epsilon_k$ tends uniformly to zero on the support of $\varphi$. This completes the proof of the claim. -We have now established the inequalities in both directions and -hence completed the proof of Lemma~. -\end{proof} +We have now established the inequalities in both directions and +hence completed the proof of \entryref{366e3fdadf6f}. + **Lemma.** Consider the distribution -$${\mathcal D}=\frac{\partial + +$$ +{\mathcal D}=\frac{\partial l_\infty}{\partial \tau}+|\nabla -l_\infty|^2-R+\frac{n}{2\tau}-\triangle l_\infty$$ on +l_\infty|^2-R+\frac{n}{2\tau}-\triangle l_\infty +$$ + + on $M_\infty\times (-\infty,0)$. Then ${\mathcal D}$ extends to a continuous linear functional on the space of compactly supported $W^{1,2}$-functions on $M_\infty\times (-\infty,0)$. Furthermore, if $\psi$ is a non-negative Lipschitz function on $M_\infty\times (-\infty,0)$ with compact support, then ${\mathcal D}(\psi)\le 0$. + **Proof.** Clearly, since the $l_k$ converge uniformly on compact subsets of $M_{\infty}\times (-\infty,0)$ to $l_\infty$ and the metrics $h_k$ @@ -747,11 +895,10 @@ converge smoothly to $g_\infty$, uniformly on compact sets, it follows that the $\triangle ^{h_k}l_k $ converge in the weak sense to $\triangle l_\infty$ and similarly, the $\partial l_k/\partial \tau$ converge in the weak sense to $\partial l_\infty/\partial -\tau$. Hence, by taking limits from Theorem~, using -Lemma~, we see that +\tau$. Hence, by taking limits from \entryref{7d5d94efe4e9}, using +\entryref{366e3fdadf6f}, we see that $$ - {\mathcal D}=\frac{\partial l_\infty}{\partial \tau}+|\nabla l_\infty|^2-R+\frac{n}{2\tau}-\triangle l_\infty\ge 0 $$ @@ -762,7 +909,7 @@ Since $R$ and $\frac{n}{2\tau}$ are $C^\infty$-functions, it is clear that the distributions given by these terms extend to continuous linear functionals on the space of compactly supported $W^{1,2}$-functions. Similarly, since $|\nabla l_\infty|^2$ is an -element of $L^\infty_loc$, it also extends to a continuous +element of $L^\infty_\mathit{loc}$, it also extends to a continuous linear functional on compactly supported $W^{1,2}$-functions. Since $|\partial l_\infty/\partial \tau|$ is an locally essentially bounded function, $\partial l_\infty/\partial \tau$ extends to a @@ -770,24 +917,35 @@ continuous functional on the space of compactly supported $W^{1,2}$ functions. Lastly, we consider $\triangle l_\infty$. As we have seen, the value of the associated distribution on $\varphi$ is given by -$$\int_{M\times (-\infty,,0)}-\langle \nabla \varphi,\nabla -l_\infty\rangle_{g_\infty} dvol(g_\infty)d\tau.$$ Since + +$$ +\int_{M\times (-\infty,,0)}-\langle \nabla \varphi,\nabla +l_\infty\rangle_{g_\infty} d\mathit{vol}(g_\infty)d\tau. +$$ + + Since $|\nabla l_\infty|$ is a locally essentially bounded function, this expression also extends to a continuous linear functional on compactly supported $W^{1,2}$-functions. Lastly, if $\psi$ is an element of $W^{1,2}$ with compact support and hence can be approximated in the $W^{1,2}$-norm by non-negative -smooth functions. The last statement is now immediate from -Equation~(). +smooth functions. The last statement is now immediate from +Equation (9.9). This leads immediately to: -**Corollary.** -The functional -$$\varphi\mapsto{\mathcal D}(e^{-l_\infty}\varphi)$$ is a distribution and + +**Corollary.** The functional + +$$ +\varphi\mapsto{\mathcal D}(e^{-l_\infty}\varphi) +$$ + + is a distribution and its value on any non-negative, compactly supported $C^\infty$-function $\varphi$ is $\ge 0$. + **Proof.** If $\varphi$ is a compactly supported non-negative $C^\infty$-function, then $e^{-l_\infty}\varphi$ is a compactly @@ -797,15 +955,19 @@ immediate consequence of the previous corollary. ### Extension to non-compactly supported functions The next step in this proof is to estimate the $l_{x_k}$ uniformly from below -in order to show that the integrals involved in the distributions in -Proposition~ are absolutely convergent so that they extend to -continuous functionals on a certain space of functions that includes +in order to show that the integrals involved in the distributions in +\entryref{2e9e6c89f40c} are absolutely convergent so that they extend to +continuous functionals on a certain space of functions that includes non-compactly supported functions. -**Lemma.** -There is a constant $c_1>0$ depending only on the dimension $n$ such + +**Lemma.** There is a constant $c_1>0$ depending only on the dimension $n$ such that for any $p,q\in M_k$ we have -$$l_{x_k}(p,\tau)\ge -l_{x_k}(q,\tau)-1+c_1\frac{d^2_{g(-\tau)}(p,q)}{\tau}.$$ + +$$ +l_{x_k}(p,\tau)\ge -l_{x_k}(q,\tau)-1+c_1\frac{d^2_{g(-\tau)}(p,q)}{\tau}. +$$ + **Proof.** Since both sides of this inequality and also Ricci flow are invariant if the metric and time are simultaneously rescaled, it @@ -818,11 +980,13 @@ $l_{x_k}(q,1)\le l_{x_k}(p,1)$. $\gamma_1$ and $\gamma_2$ be the minimizing ${\mathcal L}$-geodesics from $x$ to $(p,1)$ and $(q,1)$ respectively. We define a function $f\colon M_k\times M_k\times [0,\infty)\to \Ar$ by -$$f(a,b,\tau)=d_{g_k(-\tau)}(a,b).$$ -Since $\gamma_1(0)=\gamma_2(0)$ we have +$$ +f(a,b,\tau)=d_{g_k(-\tau)}(a,b). $$ +Since $\gamma_1(0)=\gamma_2(0)$ we have + $$ \begin{aligned} \nonumber @@ -836,25 +1000,29 @@ f_b,\gamma_2'(\tau)\rangle\bigr)d\tau, \end{aligned} $$ -$$ - -where $\nabla_af$ and $\nabla_bf$ refer respectively to the +where $\nabla_af$ and $\nabla_bf$ refer respectively to the gradient of $f$ with respect to the first copy of $M_k$ in the domain and the second copy of $M_k$ in the domain. Of course, $|\nabla f_a|=1$ and $|\nabla f_b|=1$. -By Corollary~, we have $\gamma_1'(\tau)=\nabla +By \entryref{033786a89c6d}, we have $\gamma_1'(\tau)=\nabla l_{x_k}(\gamma_1(\tau),\tau)$ and $\gamma_2'(\tau)=\nabla l_{x_k}(\gamma_2(\tau),\tau)$. Since $R\ge 0$ we have -$$l_{x_k}(\gamma_1(\tau),\tau)=\frac{1}{2\sqrt{\tau}}{\mathcal + +$$ +l_{x_k}(\gamma_1(\tau),\tau)=\frac{1}{2\sqrt{\tau}}{\mathcal L}_{x_k}(\gamma_1|_{[0,\tau]})\le \frac{1}{2\sqrt{\tau}}{\mathcal -L}_{x_k}(\gamma_1)=\frac{1}{\sqrt{\tau}}l_{x_k}(p,1).$$ +L}_{x_k}(\gamma_1)=\frac{1}{\sqrt{\tau}}l_{x_k}(p,1). +$$ + Symmetrically, we have -$$l_{x_k}(\gamma_2(\tau),\tau)\le \frac{1}{\sqrt{\tau}}l_{x_k}(q,1).$$ -From this inequality, Corollary~, and the fact that -$R\ge 0$, we have $$ +l_{x_k}(\gamma_2(\tau),\tau)\le \frac{1}{\sqrt{\tau}}l_{x_k}(q,1). +$$ + +From this inequality, \entryref{8b8e3a0f3e36}, and the fact that +$R\ge 0$, we have $$ \begin{aligned} @@ -869,11 +1037,10 @@ f_a(\gamma_1(\tau),\gamma_2(\tau),\tau),\gamma_1'(\tau)\rangle \end{aligned} $$ -$$ - Symmetrically, we have + $$ - \left|\langle +\left|\langle \nabla f_b(\gamma_1(\tau),\gamma_2(\tau),\tau),\gamma'_2(\tau)\rangle\right|\le \frac{\sqrt{3}}{\tau^{3/4}}\sqrt{l_{x_k}(q,1)}\le @@ -881,36 +1048,64 @@ f_b(\gamma_1(\tau),\gamma_2(\tau),\tau),\gamma'_2(\tau)\rangle\right|\le $$ It follows from -Corollary~ that for any $p$ -$$|\nabla (\sqrt{l_{x_k}(p,\tau)})|\le -\frac{\sqrt{3}}{2\sqrt{\tau}}.$$ Set +\entryref{8b8e3a0f3e36} that for any $p$ + +$$ +|\nabla (\sqrt{l_{x_k}(p,\tau)})|\le +\frac{\sqrt{3}}{2\sqrt{\tau}}. +$$ + + Set $r_0(\tau)=\tau^{3/4}(l_{x_k}(q,1)+1)^{-1/2}$. For any $p'\in B_{g_k}(\gamma_1(\tau),\tau,r_0(\tau))$ integrating gives -$$l_{x_k}^{1/2}(p',\tau)\le + +$$ +l_{x_k}^{1/2}(p',\tau)\le l_{x_k}^{1/2}(\gamma_1(\tau),\tau)+\frac{\sqrt{3}}{2\sqrt{\tau}}r_0(\tau)\le -\left(\tau^{-1/4}+\frac{\sqrt{3}}{2}\tau^{1/4}\right)\sqrt{l_{x_k}(p,1)+1},$$ +\left(\tau^{-1/4}+\frac{\sqrt{3}}{2}\tau^{1/4}\right)\sqrt{l_{x_k}(p,1)+1}, +$$ + where in the last inequality we have used the fact that $1\le l_{x_k}(q,1)+1\le l_{x_k}(p,1)+1$. Again using -Corollary~ we have -$$R(p',\tau)\le -\frac{3}{\tau}\left(\tau^{-1/4}+\frac{\sqrt{3}}{2}\tau^{1/4}\right)^2(l_{x_k}(p,1)+1).$$ +\entryref{8b8e3a0f3e36} we have + +$$ +R(p',\tau)\le +\frac{3}{\tau}\left(\tau^{-1/4}+\frac{\sqrt{3}}{2}\tau^{1/4}\right)^2(l_{x_k}(p,1)+1). +$$ + Now consider $q'\in B_{g_k(\tau)}(\gamma_2(\tau),\tau,r_0(\tau))$. Similarly to the above computations, we have -$$l^{1/2}_{x_k}(q',\tau)\le -l_{x_k}^{1/2}(q,1)+\frac{\sqrt{3}}{2\sqrt{\tau}}r_0(\tau),$$ so that -$$l_{x_k}^{1/2}(q',\tau)\le \left(\tau^{-1/4} -+\frac{\sqrt{3}}{2}\tau^{1/4}\right)\sqrt{l_{x_k}(q,1)+1},$$ and -$$|Ric(q',\tau)\le R(q',\tau)\le -\frac{3}{\tau}\left(\tau^{-1/4}+\frac{\sqrt{3}}{2}\tau^{1/4}\right)^2(l_{x_k}(q,1)+1).$$ -We set -$$K=\frac{3}{\tau}\left(\tau^{-1/4}+\frac{\sqrt{3}}{2}\tau^{1/4}\right)^2(l_{x_k}(q,1)+1).$$ -Now, noting that $\partial/\partial \tau$ here is -$-\partial/\partial t$ of Proposition~, we apply -Proposition~ to see that +$$ +l^{1/2}_{x_k}(q',\tau)\le +l_{x_k}^{1/2}(q,1)+\frac{\sqrt{3}}{2\sqrt{\tau}}r_0(\tau), +$$ + + so that + +$$ +l_{x_k}^{1/2}(q',\tau)\le \left(\tau^{-1/4} ++\frac{\sqrt{3}}{2}\tau^{1/4}\right)\sqrt{l_{x_k}(q,1)+1}, +$$ + and + +$$ +|\mathit{Ric}(q',\tau)\le R(q',\tau)\le +\frac{3}{\tau}\left(\tau^{-1/4}+\frac{\sqrt{3}}{2}\tau^{1/4}\right)^2(l_{x_k}(q,1)+1). +$$ + +We set + +$$ +K=\frac{3}{\tau}\left(\tau^{-1/4}+\frac{\sqrt{3}}{2}\tau^{1/4}\right)^2(l_{x_k}(q,1)+1). $$ +Now, noting that $\partial/\partial \tau$ here is +$-\partial/\partial t$ of \entryref{5d895f9c3918}, we apply +\entryref{5d895f9c3918} to see that + $$ \begin{aligned} \left|\frac{\partial}{\partial @@ -921,15 +1116,11 @@ $$ \end{aligned} $$ -$$ - where $C_1,C_2,C_3$ are constants depending only on the dimension $n$. -Now plugging Equation~() and~() and the above -inequality into Equation~() we see that - -$$ +Now plugging Equation (9.11) and (9.12) and the above +inequality into Equation (9.10) we see that $$ \begin{aligned} @@ -940,28 +1131,45 @@ d_{g(-1)}(p,q) & \le & \end{aligned} $$ +This implies that + +$$ +d_{g(-1)}(p,q)\le +C\left(\sqrt{l_{x_k}(q,1)+1}+\sqrt{l_{x_k}(p,1)+1}\right), $$ -This implies that -$$d_{g(-1)}(p,q)\le -C\left(\sqrt{l_{x_k}(q,1)+1}+\sqrt{l_{x_k}(p,1)+1}\right),$$ for + for some constant depending only on the dimension. Thus, since we are assuming that $l_{x_k}(p,1)\ge l_{x_k}(q,1)$ we have -$$d^2_{g(-1)}(p,q)\le + +$$ +d^2_{g(-1)}(p,q)\le C^2\left(3(l_{x_k}(p,1)+1)+(l_{x_k}(q,1)+1)\right)\le -4C^2(l_{x_k}(p,1)+1+l_{x_k}(q,1)),$$ for some constant $C<\infty$ +4C^2(l_{x_k}(p,1)+1+l_{x_k}(q,1)), +$$ + + for some constant $C<\infty$ depending only on the dimension. The result now follows immediately. -**Corollary.** For any $q'\in M$ and any $0<\tau_0\le\tau'$ we have -$$l_{x_k}(q',\tau') \ge -\frac{n}{2(\tau')^2}-\frac{\tau'}{2}-1+c_1\frac{d_{g^2_k(-\tau_0)}(q_k,q')}{\tau'},$$ -where $c_1$ is the constant from Lemma~. + +**Corollary.** For any $q'\in M$ and any $0<\tau_0\le\tau'$ we have -**Proof.** By Claim~ -$$l_{x_k}(q_k,\tau')\le \frac{n}{2(\tau')^2}+\frac{n\tau'}{2}.$$ -Now applying Lemma~ we see that for any $0<\tau'$ and any $q'\in -M_k$ we have +$$ +l_{x_k}(q',\tau') \ge -\frac{n}{2(\tau')^2}-\frac{\tau'}{2}-1+c_1\frac{d_{g^2_k(-\tau_0)}(q_k,q')}{\tau'}, $$ +where $c_1$ is the constant from \entryref{a6ee3b2ab615}. + + +**Proof.** By \entryref{88571a22a2bc} + +$$ +l_{x_k}(q_k,\tau')\le \frac{n}{2(\tau')^2}+\frac{n\tau'}{2}. +$$ + +Now applying \entryref{a6ee3b2ab615} we see that for any $0<\tau'$ and any $q'\in +M_k$ we have + $$ \begin{aligned} l_{x_k}(q',\tau') & \ge & @@ -972,13 +1180,12 @@ l_{x_k}(q',\tau') & \ge & \end{aligned} $$ -$$ In the last inequality, we use the fact that the Ricci curvature is positive so that the metric is decreasing under the Ricci flow. Since the time slices of all the flows in question have non-negative -curvature, by Theorem~ the volume of the ball of +curvature, by \entryref{1a1b2c4fdc9b} the volume of the ball of radius $s$ is at most $\omega s^{n}$ where $\omega$ is the volume of the ball of radius one in $\Ar^n$. Since the $l_k$ converge uniformly to $l_\infty$ on compact sets and since the metrics $h_k$ @@ -986,26 +1193,38 @@ converge uniformly in the $C^\infty$-topology on compact sets to $g_\infty$, it follows that for any $\epsilon>0$, for any $0<\tau_0\le \tau'<\infty$ there is a radius $r$ such that for every $k$ and any $\tau\in [\tau_0,\tau']$ the integral -$$\int_{M_\infty\setminus B_{h_k(-\tau_0)}(q_k,r)}e^{-l_k(q,\tau)}dq<\epsilon.$$ + +$$ +\int_{M_\infty\setminus B_{h_k(-\tau_0)}(q_k,r)}e^{-l_k(q,\tau)}dq<\epsilon. +$$ + It follows by Lebesgue dominated convergence that -$$\int_{M_\infty\setminus + +$$ +\int_{M_\infty\setminus B_{g_\infty(-\tau_0)}(q_\infty,r)}e^{-l_\infty(q,\tau)}dq\le -\epsilon.$$ +\epsilon. +$$ -**Claim.** - Fix a compact interval $[-\tau,-\tau_0]\subset + +**Claim.** Fix a compact interval $[-\tau,-\tau_0]\subset (-\infty,0)$. Let $f$ be a locally Lipschitz function that is defined on $M_\infty\times [-\tau,-\tau_0]$ and such that there is a -constant $C$ with the property that $f(q,\tau')$ by $C$ times $max(l_\infty(q,\tau'),1)$ . Then the distribution ${\mathcal +constant $C$ with the property that $f(q,\tau')$ by $C$ times $\mathit{max}(l_\infty(q,\tau'),1)$ . Then the distribution ${\mathcal D}_1=fe^{-l_\infty}$ is absolutely convergent in the following sense. For any bounded smooth function $\varphi$ defined on all of $M_\infty\times [-\tau,-\tau_0]$ and any sequence of compactly supported, non-negative smooth functions $\psi_k$, bounded above by $1$ everywhere that are eventually $1$ on every compact subset, the following limit exists and is finite: -$$lim_{k\rightarrow\infty}{\mathcal D}_1(\varphi\psi_k).$$ + +$$ +\mathit{lim}_{k\rightarrow\infty}{\mathcal D}_1(\varphi\psi_k). +$$ + Furthermore, the limit is independent of the choice of the $\psi_k$ with the given properties. + **Proof.** It follows from the above discussion that there are constants $c>0$ and a ball $B\subset M_\infty$ centered at $q_\infty$ such that on @@ -1018,45 +1237,62 @@ every $\tau'$, the flow is distance decreasing, and there is a fixed polynomial upper bound to the growth rate of volume at infinity. This leads to the claimed convergence property. -**Corollary.** -The distributions $|\nabla l_\infty|^2e^{-l_\infty}$, + +**Corollary.** The distributions $|\nabla l_\infty|^2e^{-l_\infty}$, $Re^{-l_\infty}$, $|(\partial l_\infty/\partial \tau)|e^{-l_\infty}$ are absolutely convergent in the sense of the above claim. + **Proof.** By -Corollary~, each of the Lipschitz functions $|\nabla +\entryref{8b8e3a0f3e36}, each of the Lipschitz functions $|\nabla l_\infty|^2$, $|\partial l_\infty/\partial \tau|$ and $R$ - is at most a constant multiple of $l_\infty$. + is at most a constant multiple of $l_\infty$. Hence, the corollary follows from the previous claim. There is a slightly weaker statement that is true for $\triangle e^{-l_\infty}$. -**Claim.** -Suppose that $\varphi$ and $\psi_k$ are as in Claim~, + +**Claim.** Suppose that $\varphi$ and $\psi_k$ are as in \entryref{1766f37a13cf}, but in addition $\varphi$ and all the $\psi_k$ are uniformly Lipschitz. Then -$$lim_{k\rightarrow\infty}\int_{M_\infty}\varphi\psi_k\triangle -e^{-l_\infty}dvol_{g_\infty}$$ converges absolutely. + +$$ +\mathit{lim}_{k\rightarrow\infty}\int_{M_\infty}\varphi\psi_k\triangle +e^{-l_\infty}d\mathit{vol}_{g_\infty} +$$ + + converges absolutely. + **Proof.** This time the value of the distribution on a compactly supported -smooth function $\rho$ is given by the integral of $$-\langle +smooth function $\rho$ is given by the integral of + +$$ +-\langle \nabla\rho,\nabla e^{-l_\infty}\rangle=\langle \nabla \rho,\nabla -l_\infty\rangle e^{-l_\infty}.$$ Since $|\nabla l_\infty|$ is less +l_\infty\rangle e^{-l_\infty}. +$$ + + Since $|\nabla l_\infty|$ is less than or equal to the maximum of $1$ and $|\nabla l_\infty|^2$, it follows immediately, that if $|\nabla \rho|$ is bounded, then the integral is absolutely convergent. From this the claim follows easily. -**Corollary.** -Fix $0<\tau_0<\tau_1<\infty$. Let $f$ be a non-negative, smooth + +**Corollary.** Fix $0<\tau_0<\tau_1<\infty$. Let $f$ be a non-negative, smooth bounded function on $M_\infty\times [\tau_0,\tau_1]$ with (spatial) gradient of bounded norm. Then -$$\int_{\tau_0}^{\tau_1}\int_{M_\infty\times\{-\tau\}}\left(\frac{\partial l_\infty}{\partial \tau}+|\nabla + +$$ +\int_{\tau_0}^{\tau_1}\int_{M_\infty\times\{-\tau\}}\left(\frac{\partial l_\infty}{\partial \tau}+|\nabla l_\infty|^2-R+\frac{n}{2\tau}-\triangle -l_\infty\right)f\tau^{-n/2}e^{-l_\infty}dvol_{g_\infty}d\tau\ge 0.$$ +l_\infty\right)f\tau^{-n/2}e^{-l_\infty}d\mathit{vol}_{g_\infty}d\tau\ge 0. +$$ + -**Proof.** For the interval $[\tau_0,\tau']$ we construct a sequence of +**Proof.** For the interval $[\tau_0,\tau']$ we construct a sequence of uniformly Lipschitz functions $\psi_k$ on $M_\infty\times [\tau_0,\tau']$ that are non-negative, bounded above by one and eventually one on every compact set. Let $\rho(x)$ be a smooth bump @@ -1075,23 +1311,25 @@ Take a family $\psi_k$ of uniformly Lipschitz functions, each bounded between $0$ and $1$ and eventually one of every compact subset of $M_\infty\times [\tau_0,\tau_1]$. Then the family $f\psi_k$ is a uniformly Lipschitz family of compactly supported -functions. Hence, we can apply Claims~ -and~ to establish that the integral in question is the +functions. Hence, we can apply Claims \entryref{1766f37a13cf} +and \entryref{5b4c94102f41} to establish that the integral in question is the limit of an absolutely convergence sequence. By -Corollary~ each term in the sequence is non-positive. +\entryref{a8e10db32f90} each term in the sequence is non-positive. -### Completion of the proof of -Proposition~\protect{} +### Completion of the proof of Proposition \entryref{2e9e6c89f40c} Lebesgue dominated convergence implies that the following limit exists -$$lim_{k\rightarrow\infty}\widetilde V_k(\tau)\equiv\widetilde V_\infty(\tau)=\int_{M_\infty\times\{-\tau\}}\tau^{-n/2}e^{- -l_\infty(q,\tau)}dvol_{g_\infty(\tau)}.$$ By Corollary~, the function $\tau\rightarrow -\widetilde V_\infty(\tau)$ is constant. On the other hand, note that -for any $0< \tau_0<\tau_1<\infty$, we have +$$ +\mathit{lim}_{k\rightarrow\infty}\widetilde V_k(\tau)\equiv\widetilde V_\infty(\tau)=\int_{M_\infty\times\{-\tau\}}\tau^{-n/2}e^{- +l_\infty(q,\tau)}d\mathit{vol}_{g_\infty(\tau)}. $$ + By \entryref{e08895f2b0d8}, the function $\tau\rightarrow +\widetilde V_\infty(\tau)$ is constant. On the other hand, note that +for any $0< \tau_0<\tau_1<\infty$, we have + $$ \begin{aligned} \widetilde V_\infty(\tau_1)-\widetilde V_\infty(\tau_0) & = & @@ -1099,83 +1337,95 @@ $$ V_\infty}{d\tau}d\tau \\ & = & \int_{\tau_0}^{\tau_1}\int_{M_\infty}\left(\frac{\partial l_\infty}{\partial \tau}-R+\frac{n}{2\tau}\right) -\left(\tau^{-n/2}e^{-l_\infty(q,\tau)}dvol_{g_\infty(\tau)}\right). +\left(\tau^{-n/2}e^{-l_\infty(q,\tau)}d\mathit{vol}_{g_\infty(\tau)}\right). \end{aligned} $$ -$$ According to -Corollary~ this is an absolutely convergent integral, -and so this integral is zero. +\entryref{0e9703c15036} this is an absolutely convergent integral, +and so this integral is zero. -**Claim.** $$ + +**Claim.** $$ \begin{aligned} \int_{\tau_0}^{\tau_1}\int_{M_\infty\times\{-\tau\}}\triangle -e^{-l_\infty}dvol_{g_\infty}d\tau & = +e^{-l_\infty}d\mathit{vol}_{g_\infty}d\tau & = &\int_{\tau_0}^{\tau_1}\int_{M_\infty\times\{-\tau\}}\left(|\nabla -l_\infty|^2-\triangle l_\infty\right)e^{-l_\infty}dvol_{g_\infty}d\tau \\ +l_\infty|^2-\triangle l_\infty\right)e^{-l_\infty}d\mathit{vol}_{g_\infty}d\tau \\ & = & 0. \end{aligned} $$ - -$$ + **Proof.** Since we are integrating against the constant function $1$, this result is clear, given the convergence result, -Corollary~, necessary to show that this integral is +\entryref{0e9703c15036}, necessary to show that this integral is well defined. Adding these two results together gives us the following $$ - \int_{\tau_0}^{\tau_1}\int_{M_\infty\times\{-\tau\}}\left(\frac{\partial l_\infty}{\partial \tau}+|\nabla l_\infty|^2 -R+\frac{n}{2\tau}-\triangle -l_\infty\right)\tau^{-n/2}e^{-l_\infty}dvol_{g_\infty}=0. +l_\infty\right)\tau^{-n/2}e^{-l_\infty}d\mathit{vol}_{g_\infty}=0. $$ Now let $\varphi$ be any compactly supported, non-negative smooth function. By scaling by a positive constant, we can assume that $\varphi\le 1$ everywhere. Let $\widetilde{\mathcal D}$ denote the distribution given by -$$\widetilde{\mathcal D}(\varphi)=\int_{\tau_0}^{\tau_1}\int_{M_\infty\times\{-\tau\}} + +$$ +\widetilde{\mathcal D}(\varphi)=\int_{\tau_0}^{\tau_1}\int_{M_\infty\times\{-\tau\}} \varphi\left(\frac{\partial l_\infty}{\partial \tau}+|\nabla l_\infty|^2 -R+\frac{n}{2\tau}-\triangle -l_\infty\right)\tau^{-n/2}e^{-l_\infty}dvol_{g_\infty}.$$ Then +l_\infty\right)\tau^{-n/2}e^{-l_\infty}d\mathit{vol}_{g_\infty}. +$$ + + Then we have seen that $\widetilde{\mathcal D}$ extends to a functional on bounded smooth functions of bounded gradient. Furthermore, -according to Equation~(), we have $\widetilde {\mathcal +according to Equation (9.13), we have $\widetilde {\mathcal D}(1)=0$. Thus, -$$0={\mathcal D}(1)={\mathcal D}(\varphi)+{\mathcal D}(1-\varphi).$$ + +$$ +0={\mathcal D}(1)={\mathcal D}(\varphi)+{\mathcal D}(1-\varphi). +$$ + Since both $\varphi$ and $1-\varphi$ are non-negative, it follows from -Corollary~, that ${\mathcal D}(\varphi)$ and ${\mathcal +\entryref{42d9f91e59f3}, that ${\mathcal D}(\varphi)$ and ${\mathcal D}(1-\varphi)$ are each $\ge 0$. Since their sum is zero, it must be the case that each is individually zero. -This proves that the Inequality~() is actually an +This proves that the Inequality (9.9) is actually an equality in the weak sense, i.e., an equality of distributions on $M_\infty\times [\tau_0,\tau')$. Taking limits we see: $$ - \widetilde{\mathcal D}=\left(\frac{\partial l_\infty}{\partial \tau}+|\nabla l_\infty|^2-R+\frac{n}{2\tau}-\triangle l_\infty\right)\tau^{-n/2}e^{-l_\infty}=0, $$ + in the weak sense on all of $M\times(-\infty,0)$. Of course, this implies that -$$\frac{\partial l_\infty}{\partial \tau}+|\nabla -l_\infty|^2-R+\frac{n}{2\tau}-\triangle l_\infty=0$$ in the weak + +$$ +\frac{\partial l_\infty}{\partial \tau}+|\nabla +l_\infty|^2-R+\frac{n}{2\tau}-\triangle l_\infty=0 +$$ + + in the weak sense. It now follows by parabolic regularity that $l_\infty$ is a smooth function on $M_\infty\times(-\infty,0)$ and that -Equation~() holds in the usual sense. +Equation (9.14) holds in the usual sense. -Now from the last two equations in Corollary~ and the +Now from the last two equations in \entryref{967c0f8bb774} and the convergence of the $l_{x_k}$ to $l_\infty$, we conclude that the following equation also holds: @@ -1184,108 +1434,181 @@ $$ l_\infty|^2+R+\frac{l_\infty-n}{\tau}=0. $$ -This completes the proof of Proposition~. +This completes the proof of \entryref{2e9e6c89f40c}. ### The gradient shrinking soliton equation -Now we return to the proof of Theorem~. We have shown that the -limiting Ricci flow referred to in that result exists, and we have +Now we return to the proof of \entryref{671402eb454b}. We have shown that the +limiting Ricci flow referred to in that result exists, and we have established that the limit $l_\infty$ of the length functions $l_{x_k}$ is a smooth function and satisfies the differential equalities given in -Proposition~. We shall use these to establish the gradient +\entryref{2e9e6c89f40c}. We shall use these to establish the gradient shrinking soliton equation, -Equation~(), for the limit for $f=l_\infty$. +Equation (9.2), for the limit for $f=l_\infty$. + + +**Proposition.** The following equation holds on $M_\infty\times (-\infty,0)$: -**Proposition.** -The following equation holds on $M_\infty\times (-\infty,0)$: -$$ Ric_{g_{\infty}(t)} + Hess^{g_\infty(t)}(l_\infty(\cdot,\tau)) --\frac{1}{2\tau}g_{\infty}(t) = 0,$$ where $\tau=-t$, +$$ +\mathit{Ric}_{g_{\infty}(t)} + \mathit{Hess}^{g_\infty(t)}(l_\infty(\cdot,\tau)) +-\frac{1}{2\tau}g_{\infty}(t) = 0, +$$ + + where $\tau=-t$, + **Proof.** This result will follow immediately from: -**Lemma.** -Let $(M,g(t)),\ 0\le t\le T$, be an $n$-dimensional Ricci flow, and + +**Lemma.** Let $(M,g(t)),\ 0\le t\le T$, be an $n$-dimensional Ricci flow, and let $f\colon M\times [0,T]\to \Ar$ be a smooth function. As usual set $\tau=T-t$. Then the function -$$u=(4\pi\tau)^{-\frac{n}{2}}e^{-f}$$ satisfies the conjugate heat + +$$ +u=(4\pi\tau)^{-\frac{n}{2}}e^{-f} +$$ + + satisfies the conjugate heat equation -$$-\frac{\partial u}{\partial t}-\triangle u+Ru=0,$$ if and only if we have -$$\frac{\partial f}{\partial t}+\triangle f-|\nabla -f|^2+R-\frac{n}{2\tau}=0.$$ Assuming that $u$ satisfies the + +$$ +-\frac{\partial u}{\partial t}-\triangle u+Ru=0, +$$ + + if and only if we have + +$$ +\frac{\partial f}{\partial t}+\triangle f-|\nabla +f|^2+R-\frac{n}{2\tau}=0. +$$ + + Assuming that $u$ satisfies the conjugate heat equation, then setting -$$v=\left[\tau\left(2\triangle f-|\nabla -f|^2+R\right)+f-n\right]u,$$ we have -$$-\frac{\partial v}{\partial t}-\triangle v+Rv=-2\tau -\bigl|Ric_g+Hess^g(f)-\frac{1}{2\tau}g\bigr|^2u.$$ + +$$ +v=\left[\tau\left(2\triangle f-|\nabla +f|^2+R\right)+f-n\right]u, +$$ + + we have + +$$ +-\frac{\partial v}{\partial t}-\triangle v+Rv=-2\tau +\bigl|\mathit{Ric}_g+\mathit{Hess}^g(f)-\frac{1}{2\tau}g\bigr|^2u. +$$ + Let us assume the lemma for a moment and use it to complete the proof of the proposition. We apply the lemma to the limiting Ricci flow $(M_\infty,g_\infty(t))$ with the function $f=l_\infty$. According -to Proposition~ and the first statement in -Lemma~, the function $u$ satisfies the conjugate heat +to \entryref{2e9e6c89f40c} and the first statement in +\entryref{ba1adbac0a75}, the function $u$ satisfies the conjugate heat equation. Thus, according to the second statement in -Lemma~, setting -$$v=\left[\tau\left(2\triangle f-|\nabla -f|^2+R\right)+f-n\right]u,$$ we have -$$\frac{\partial v}{\partial\tau}-\triangle v+Rv=-2\tau -\bigl|Ric_g+Hess(f)-\frac{1}{2\tau}g\bigr|^2u.$$ On the -other hand, the second equality in Proposition~ shows +\entryref{ba1adbac0a75}, setting + +$$ +v=\left[\tau\left(2\triangle f-|\nabla +f|^2+R\right)+f-n\right]u, +$$ + + we have + +$$ +\frac{\partial v}{\partial\tau}-\triangle v+Rv=-2\tau +\bigl|\mathit{Ric}_g+\mathit{Hess}(f)-\frac{1}{2\tau}g\bigr|^2u. +$$ + + On the +other hand, the second equality in \entryref{2e9e6c89f40c} shows that $v=0$. Since $u$ is nowhere zero, this implies that -$$Ric_{g_\infty}+Hess^{g_\infty}(f)-\frac{1}{2\tau}g_\infty=0.$$ + +$$ +\mathit{Ric}_{g_\infty}+\mathit{Hess}^{g_\infty}(f)-\frac{1}{2\tau}g_\infty=0. +$$ + This completes the proof of the proposition assuming the lemma. Now we turn to the proof of the lemma. -**Proof.** {\em (of Lemma~)} -Direct computation shows that $$-\frac{\partial u}{\partial +**Proof.** *(of \entryref{ba1adbac0a75})* +Direct computation shows that + +$$ +-\frac{\partial u}{\partial t}-\triangle u+Ru= \left(-\frac{n}{2\tau}+\frac{\partial f}{\partial -t}+\triangle f-|\nabla f|^2+R\right)u.$$ From this, the first +t}+\triangle f-|\nabla f|^2+R\right)u. +$$ + + From this, the first statement of the lemma is clear. Let -$$H=\bigl[\tau(2\triangle f-|\nabla f|^2+R)+f-n\bigr]$$ so that $v=Hu$. + +$$ +H=\bigl[\tau(2\triangle f-|\nabla f|^2+R)+f-n\bigr] +$$ + + so that $v=Hu$. Then, of course, -$$\frac{\partial v}{\partial t}=\frac{\partial H}{\partial t}u+H\frac{\partial u}{\partial t}\ \ \ and$$ -$$\triangle v=\triangle H\cdot u+2\langle\nabla H,\nabla u\rangle+H\triangle u.$$ + +$$ +\frac{\partial v}{\partial t}=\frac{\partial H}{\partial t}u+H\frac{\partial u}{\partial t}\ \ \ \mathit{and} +$$ + +$$ +\triangle v=\triangle H\cdot u+2\langle\nabla H,\nabla u\rangle+H\triangle u. +$$ + Since $u$ satisfies the conjugate heat equation, we have -$$-\frac{\partial v}{\partial t}-\triangle v+Rv=\left(-\frac{\partial H}{\partial t}-\triangle -H\right)u-2\langle\nabla H,\nabla u\rangle.$$ Differentiating the -definition of $H$ yields + $$ +-\frac{\partial v}{\partial t}-\triangle v+Rv=\left(-\frac{\partial H}{\partial t}-\triangle +H\right)u-2\langle\nabla H,\nabla u\rangle. +$$ + + Differentiating the +definition of $H$ yields +$$ \frac{\partial H}{\partial t} = -(2\triangle f-|\nabla f|^2+R)+\frac{\partial f}{\partial t}+ \tau \left(2\frac{\partial }{\partial t}\triangle f -\frac{\partial }{\partial t}(|\nabla f|^2)+\frac{\partial R}{\partial t}\right) $$ + **Claim.** -$$\frac{\partial }{\partial t}\triangle -f=\triangle(\frac{\partial f}{\partial t})+2\langle Ric,Hess(f)\rangle.$$ -\begin{proof} -We work in local coordinates. We have $$\triangle -f=g^{ij}\nabla_i\nabla_j -f=g^{ij}(\partial_i\partial_jf-\Gamma_{ij}^k\partial_kf),$$ so that -from the Ricci flow equation we have +$$ +\frac{\partial }{\partial t}\triangle +f=\triangle(\frac{\partial f}{\partial t})+2\langle \mathit{Ric},\mathit{Hess}(f)\rangle. +$$ + + +**Proof.** We work in local coordinates. We have $$ +\triangle +f=g^{ij}\nabla_i\nabla_j +f=g^{ij}(\partial_i\partial_jf-\Gamma_{ij}^k\partial_kf), +$$ + + so that +from the Ricci flow equation we have $$ \begin{aligned} -\frac{\partial }{\partial t}\triangle f & = & 2Ric^{ij}Hess(f)_{ij}+g^{ij}\frac{\partial}{\partial t}\left(Hess(f)_{ij}\right) +\frac{\partial }{\partial t}\triangle f & = & 2\mathit{Ric}^{ij}\mathit{Hess}(f)_{ij}+g^{ij}\frac{\partial}{\partial t}\left(\mathit{Hess}(f)_{ij}\right) \\ -& = & 2Ric^{ij}Hess(f)_{ij}+g^{ij}Hess\left({\frac{\partial f}{\partial +& = & 2\mathit{Ric}^{ij}\mathit{Hess}(f)_{ij}+g^{ij}\mathit{Hess}\left({\frac{\partial f}{\partial t}}\right)_{ij}-g^{ij}\frac{\partial \Gamma_{ij}^k}{\partial t}\partial_kf. \end{aligned} $$ -$$ - -Since the first term is $2\langle Ric,Hess(f)\rangle$ +Since the first term is $2\langle \mathit{Ric},\mathit{Hess}(f)\rangle$ and the second is $\triangle(\frac{\partial f}{\partial t})$, to complete the proof of the claim, we must show that the last term of this equation vanishes. In order to simplify the computations, we @@ -1295,34 +1618,39 @@ definition of the Christoffel symbols in terms of the metric, and the fact that $g_{ij}$ is the identity matrix at the given point and time and that its covariant derivatives in all spatial directions vanish at this point and time, we get -$$g^{ij}\frac{\partial \Gamma_{ij}^k}{\partial t}=g^{kl}g^{ij}\left(-(\nabla_j Ric)_{li}-(\nabla_i - Ric)_{lj}+(\nabla_lRic)_{ij}\right).$$ - This expression vanishes by the second Bianchi identity (Claim~). + +$$ +g^{ij}\frac{\partial \Gamma_{ij}^k}{\partial t}=g^{kl}g^{ij}\left(-(\nabla_j \mathit{Ric})_{li}-(\nabla_i + \mathit{Ric})_{lj}+(\nabla_l\mathit{Ric})_{ij}\right). +$$ + + This expression vanishes by the second Bianchi identity (\entryref{c4848a1b6f80}). This completes the proof of the claim. We also have -$$\frac{\partial }{\partial t}(|\nabla f|^2)=2Ric(\nabla f,\nabla f)+2\langle -\nabla\frac{\partial f}{\partial t},\nabla f\rangle.$$ (Here $\nabla + +$$ +\frac{\partial }{\partial t}(|\nabla f|^2)=2\mathit{Ric}(\nabla f,\nabla f)+2\langle +\nabla\frac{\partial f}{\partial t},\nabla f\rangle. +$$ + + (Here $\nabla f$ is a one-form, which explains the positive sign in the Ricci term.) -Plugging this and Claim~ into Equation~() +Plugging this and \entryref{8283cc66c217} into Equation (9.16) yields -$$ - $$ \begin{aligned} \frac{\partial H}{\partial t} & = & -2\triangle f+|\nabla f|^2-R+\frac{\partial f}{\partial t} \\ -& & + \tau \left(4\langle Ric,Hess(f)\rangle +2\triangle -\frac{\partial f}{\partial t} -2Ric(\nabla f,\nabla f) +& & + \tau \left(4\langle \mathit{Ric},\mathit{Hess}(f)\rangle +2\triangle +\frac{\partial f}{\partial t} -2\mathit{Ric}(\nabla f,\nabla f) -2\langle \nabla\frac{\partial f}{\partial t},\nabla f\rangle+\frac{\partial R}{\partial t}\right). \end{aligned} $$ -$$ - Also, $$ @@ -1340,41 +1668,35 @@ f|^2-R+\frac{n}{2\tau}. $$ Putting all this together and -using the Equation~() for $\partial R/\partial t$ yields - -$$ +using the Equation (3.5) for $\partial R/\partial t$ yields $$ \begin{aligned} \frac{\partial H}{\partial t}+\triangle H & = & -\triangle f+|\nabla -f|^2+\frac{\partial f}{\partial t}-R \\ & & +\tau\bigl(4\langle Ric,Hess(f)\rangle+2\triangle \frac{\partial f}{\partial t} -+2\triangle^2f-2Ric(\nabla f,\nabla f)\bigr. \\ +f|^2+\frac{\partial f}{\partial t}-R \\ & & +\tau\bigl(4\langle \mathit{Ric},\mathit{Hess}(f)\rangle+2\triangle \frac{\partial f}{\partial t} ++2\triangle^2f-2\mathit{Ric}(\nabla f,\nabla f)\bigr. \\ & & \bigl.-\triangle(|\nabla f|^2) -2\langle \nabla\frac{\partial f}{\partial t},\nabla f\rangle+2\triangle -R +2|Ric|^2\bigr) \\ +R +2|\mathit{Ric}|^2\bigr) \\ & = & -\triangle f+|\nabla f|^2+\frac{\partial f}{\partial t}-R -+\tau\bigl(4\langle Ric,Hess(f)\rangle+2\triangle(|\nabla f|^2-R)\bigr. ++\tau\bigl(4\langle \mathit{Ric},\mathit{Hess}(f)\rangle+2\triangle(|\nabla f|^2-R)\bigr. \\ -& & \bigl.-2Ric(\nabla f,\nabla f)-\triangle(|\nabla +& & \bigl.-2\mathit{Ric}(\nabla f,\nabla f)-\triangle(|\nabla f|^2)-2\langle\nabla\frac{\partial f}{\partial t},\nabla -f\rangle+2\triangle R+2|Ric|^2\bigr) +f\rangle+2\triangle R+2|\mathit{Ric}|^2\bigr) \\ & = & -\triangle f+|\nabla f|^2+\frac{\partial f}{\partial -t}-R+\tau\Bigl[4\langle Ric,Hess(f)\rangle+\triangle(|\nabla f|^2)\Bigr. +t}-R+\tau\Bigl[4\langle \mathit{Ric},\mathit{Hess}(f)\rangle+\triangle(|\nabla f|^2)\Bigr. \\ -& & -2Ric(\nabla f,\nabla f) +2\langle\nabla(\triangle +& & -2\mathit{Ric}(\nabla f,\nabla f) +2\langle\nabla(\triangle f),\nabla f\rangle -2\langle \nabla(|\nabla f|^2),\nabla f\rangle \\ -& & \Bigl. +2\langle\nabla R,\nabla f\rangle+2|Ric|^2\Bigr]. +& & \Bigl. +2\langle\nabla R,\nabla f\rangle+2|\mathit{Ric}|^2\Bigr]. \end{aligned} $$ -$$ - Similarly, we have -$$ - $$ \begin{aligned} \frac{2\langle\nabla u,\nabla H\rangle}{u} & = & -2\langle\nabla f,\nabla H\rangle \\ @@ -1386,117 +1708,120 @@ f,\nabla R\rangle\right). \end{aligned} $$ -$$ - Thus, -$$ - $$ \begin{aligned} \frac{\partial H}{\partial t} +\triangle H+\frac{2\langle\nabla u,\nabla H\rangle}{u} & = & -\triangle f-|\nabla f|^2+\frac{\partial -f}{\partial t}-R + \tau\Bigl[4\langle Ric,Hess(f)\rangle +f}{\partial t}-R + \tau\Bigl[4\langle \mathit{Ric},\mathit{Hess}(f)\rangle \Bigr. \\ -& & +\triangle(|\nabla f|^2)-2Ric(\nabla f,\nabla f)+2|Ric|^2 +& & +\triangle(|\nabla f|^2)-2\mathit{Ric}(\nabla f,\nabla f)+2|\mathit{Ric}|^2 \\ & & \Bigl.-2\langle\nabla f,\nabla(\triangle f)\rangle\Bigr]. \end{aligned} $$ -$$ - + **Claim.** The following equality holds: -$$ - $$ \begin{aligned} \triangle(|\nabla f|^2) & = & -2\langle\nabla(\triangle f),\nabla f\rangle+2Ric(\nabla -f,\nabla f)+2|Hess(f)|^2, +2\langle\nabla(\triangle f),\nabla f\rangle+2\mathit{Ric}(\nabla +f,\nabla f)+2|\mathit{Hess}(f)|^2, \end{aligned} $$ + +**Proof.** We have + +$$ +\triangle(|\nabla f|^2)=\triangle \langle \nabla f,\nabla f\rangle =\triangle \langle df,df\rangle=2\langle \triangle +df,df\rangle+2\langle \nabla df,\nabla df\rangle. $$ -**Proof.** We have -$$\triangle(|\nabla f|^2)=\triangle \langle \nabla f,\nabla f\rangle =\triangle \langle df,df\rangle=2\langle \triangle -df,df\rangle+2\langle \nabla df,\nabla df\rangle.$$ The last term is -$|Hess(f)|^2$. According to Lemma~ we have -$\triangle df=d(\triangle f)+Ric(\nabla f,\cdot)$. Plugging + The last term is +$|\mathit{Hess}(f)|^2$. According to \entryref{8d03f3ca0aea} we have +$\triangle df=d(\triangle f)+\mathit{Ric}(\nabla f,\cdot)$. Plugging this in gives -$$\triangle(|\nabla f|^2)=2\langle d(\triangle f),df\rangle+2\langle -Ric(\nabla f,\cdot),df\rangle+2|Hess(f)|^2,$$ which is + +$$ +\triangle(|\nabla f|^2)=2\langle d(\triangle f),df\rangle+2\langle +\mathit{Ric}(\nabla f,\cdot),df\rangle+2|\mathit{Hess}(f)|^2, +$$ + + which is clearly another way of writing the claimed result. Using this we can simplify the above to -$$ - $$ \begin{aligned} \frac{\partial H}{\partial t} +\triangle H+\frac{2\langle\nabla u,\nabla H\rangle}{u} & = &-\triangle f-|\nabla f|^2+\frac{\partial f}{\partial t}-R \\ -& & +\tau\left(4\langle Ric,Hess(f)\rangle +2|Hess(f)|^2+2|Ric|^2\right). +& & +\tau\left(4\langle \mathit{Ric},\mathit{Hess}(f)\rangle +2|\mathit{Hess}(f)|^2+2|\mathit{Ric}|^2\right). \end{aligned} $$ -$$ - -Now using Equation~() we have - -$$ +Now using Equation (9.17) we have $$ \begin{aligned} \frac{\partial H}{\partial t} +\triangle H+\frac{2\langle\nabla u,\nabla H\rangle}{u} & = & -2\triangle f-2R+\frac{n}{2\tau} \\ -& & +\tau\left(4\langle Ric,Hess(f)\rangle+2|Ric|^2+2|Hess(f)|^2\right) \\ -& = & 2\tau\bigl(2\langle Ric,Hess(f)\rangle+|Ric|^2+|Hess(f)|^2\bigr. \\ +& & +\tau\left(4\langle \mathit{Ric},\mathit{Hess}(f)\rangle+2|\mathit{Ric}|^2+2|\mathit{Hess}(f)|^2\right) \\ +& = & 2\tau\bigl(2\langle \mathit{Ric},\mathit{Hess}(f)\rangle+|\mathit{Ric}|^2+|\mathit{Hess}(f)|^2\bigr. \\ & & -\frac{\triangle f}{\tau}-\frac{R}{\tau}+\frac{n}{4\tau^2}\bigr) \\ -& = & 2\tau\bigl|Ric+Hess(f)-\frac{1}{2\tau}g_\infty\bigr|^2 +& = & 2\tau\bigl|\mathit{Ric}+\mathit{Hess}(f)-\frac{1}{2\tau}g_\infty\bigr|^2 \end{aligned} $$ +Since + +$$ +-\frac{\partial v}{\partial t}-\triangle v+Rv=-u\left(\frac{\partial H}{\partial t} + +\triangle H+\frac{2\langle\nabla u,\nabla H\rangle}{u}\right), $$ -Since -$$-\frac{\partial v}{\partial t}-\triangle v+Rv=-u\left(\frac{\partial H}{\partial t} - +\triangle H+\frac{2\langle\nabla u,\nabla H\rangle}{u}\right),$$ this proves the lemma. -\end{proof} At this point, setting $f=l_\infty$, we have established all the -results claimed in Theorem~ except for the fact that +results claimed in \entryref{671402eb454b} except for the fact that the limit is not flat. This we establish in the next chapter. -### Completion of the proof of -Theorem~\protect{} +### Completion of the proof of Theorem \entryref{671402eb454b} To complete the -proof of Theorem~ it remains to show that for no $t\in +proof of \entryref{671402eb454b} it remains to show that for no $t\in (-\infty,0)$ is $(M_\infty,g_\infty(t))$ flat. + **Claim.** If, for some $t\in (-\infty,0)$, the Riemannian manifold $(M_\infty,g_\infty(t))$ is flat, then there is an isometry from -$\Ar^n$ to $(M_\infty,g_\infty(t))$ and the pullback under this +$\Ar^n$ to $(M_\infty,g_\infty(t))$ and the pullback under this isometry of the function $l_\infty(x,\tau)$ is the function $|x|^2/4\tau+\langle x,a\rangle+b\cdot \tau$ for some $a\in \Ar^n$ and $b\in \Ar$. + **Proof.** We know that $f=l_\infty(\cdot,\tau)$ solves the equation given in -Lemma~ and hence by the above argument, $f$ also satisfies -the equation given in Proposition~. If the limit is +\entryref{ba1adbac0a75} and hence by the above argument, $f$ also satisfies +the equation given in \entryref{07232fc42f3a}. If the limit is flat, then the equation becomes -$$Hess(f)=\frac{1}{2\tau}g.$$ The universal covering of $(M_\infty,g_\infty(t))$ + +$$ +\mathit{Hess}(f)=\frac{1}{2\tau}g. +$$ + + The universal covering of $(M_\infty,g_\infty(t))$ is isometric to $\Ar^n$. Choose an identification with $\Ar^n$, and lift $f$ to the universal cover. Call the result $\tilde f$. Then $\tilde f$ satisfies -$Hess(\tilde f)=\frac{1}{2\tau}\tilde g$, where $\tilde g$ is the usual +$\mathit{Hess}(\tilde f)=\frac{1}{2\tau}\tilde g$, where $\tilde g$ is the usual Euclidean metric on $\Ar^n$. This means that $\tilde f-|x|^2/4\tau$ is an affine linear function. Clearly, then $\tilde f$ is not invariant under any free action of a non-trivial group, so that the universal covering in question @@ -1504,14 +1829,14 @@ is the trivial cover. This completes the proof of the claim. If $(M_\infty,g_\infty(t))$ is flat for some $t\in (-\infty,0)$, then by the above $(M_\infty,g_\infty(t))$ is isometric to $\Ar^n$. According to -Theorem~ this implies that $\widetilde -V_\infty(\tau)=(4\pi)^{n/2}$. This contradicts Corollary~, and +\entryref{613fb3994956} this implies that $\widetilde +V_\infty(\tau)=(4\pi)^{n/2}$. This contradicts \entryref{e08895f2b0d8}, and the contradiction establishes that $(M_\infty,g_\infty(t))$ is not flat for any -$t<0$. Together with Proposition~, this completes the proof of -Theorem~, namely of the fact that $(M_\infty,g_\infty(t)),\ +$t<0$. Together with \entryref{07232fc42f3a}, this completes the proof of +\entryref{671402eb454b}, namely of the fact that $(M_\infty,g_\infty(t)),\ -\infty0$ be + +**Lemma.** Let $(M,g)$ be a Riemannian manifold and let $p\in M$ and $r>0$ be given. Suppose that $B(p,2r)$ has compact closure in $M$ and suppose that $f\colon B(p,2r)\times (-2r,0]\to \Ar$ is a continuous, bounded function with $f(p,0)>0$. Then there is a point $(q,t)\in B(p,2r)\times (-2r,0]$ with the following properties: -\begin{enumerate} -\item[(1)] $f(q,t)\ge f(p,0)$. -\item[(2)] Setting $\alpha=f(p,0)/f(q,t)$ we have $d(p,q)\le 2r(1-\alpha)$ + +- **(1)** $f(q,t)\ge f(p,0)$. +- **(2)** Setting $\alpha=f(p,0)/f(q,t)$ we have $d(p,q)\le 2r(1-\alpha)$ and $t\ge -2r(1-\alpha)$. -\item[(3)] $f(q',t')<2f(q,t)$ for all $(q',t')\in B(q,\alpha r)\times (t-\alpha r,t]$. -\end{enumerate} +- **(3)** $f(q',t')<2f(q,t)$ for all $(q',t')\in B(q,\alpha r)\times (t-\alpha r,t]$. + **Proof.** Consider sequences of points $x_0=(p,0),x_1=(p_1,t_1),\ldots,x_j=(p_j,t_j)$ in $B(p,2r)\times (-2r,0]$ with the following properties: -\begin{enumerate} -\item[(1)] $f(x_i)\ge 2f(x_{i-1})$; -\item[(2)] Setting $r_i=rf(x_0)/f(x_{i-1})$, then $r_i\le 2^{i-1}r$, and - we have that $$x_i\in -B(p_{i-1},r_i)\times (t_{i-1}-r_i,t_{i-1}].$$ -\end{enumerate} - -Of course, there is exactly one such sequence with $j=0$: it has + +- **(1)** $f(x_i)\ge 2f(x_{i-1})$; +- **(2)** Setting $r_i=rf(x_0)/f(x_{i-1})$, then $r_i\le 2^{i-1}r$, and + we have that + +$$ +x_i\in +B(p_{i-1},r_i)\times (t_{i-1}-r_i,t_{i-1}]. +$$ + +Of course, there is exactly one such sequence with $j=0$: it has $x_0=(p,0)$. Suppose we have such a sequence defined for some $j\ge 0$. If follows immediately from the properties of the sequence that $f(p_j,t_j)\ge 2^jf(p,0)$, that -$$t_j\ge -r(1+2^{-1}+\cdots+2^{1-j}),$$ + +$$ +t_j\ge -r(1+2^{-1}+\cdots+2^{1-j}), +$$ + and that $r_{j+1}\le 2^{-j}r$. It also follows immediately from the triangle inequality that $d(p,p_j)\le r(1+2^{-1}+\cdots +2^{1-j})$. This means that - $$B(p_j,r_{j+1})\times(t_j-r_{j+1},t_j]\subset B(p,2r)\times (-2r,0].$$ - Either the point + +$$ +B(p_j,r_{j+1})\times(t_j-r_{j+1},t_j]\subset B(p,2r)\times (-2r,0]. +$$ + + Either the point $x_j$ satisfies the conclusion of the lemma, or we can find $x_{j+1}\in B(p_j,r_{j+1})\times (t_j-r_{j+1},t_j]$ with $f(x_{j+1})\ge 2f(x_j)$. In the latter case we extend our sequence @@ -1569,16 +1905,16 @@ $f$ is bounded on $B(p,2r)\times (-2r,0]$. One special case worth stating separately is when $f$ is independent of $t$. -**Corollary.** -Let $(M,g)$ be a Riemannian manifold and let $p\in M$ and $r>0$ be given. + +**Corollary.** Let $(M,g)$ be a Riemannian manifold and let $p\in M$ and $r>0$ be given. Suppose that $B(p,2r)$ has compact closure in $M$ and suppose that $f\colon B(p,2r)\to \Ar$ is a continuous, bounded function with $f(p)>0$. Then there is a point $q\in B(x,2r)$ with the following properties: -\begin{enumerate} -\item[(1)] $f(q)\ge f(p)$. -\item[(2)] Setting $\alpha=f(p)/f(q)$ we have $d(p,q)\le 2r(1-\alpha)$ + +- **(1)** $f(q)\ge f(p)$. +- **(2)** Setting $\alpha=f(p)/f(q)$ we have $d(p,q)\le 2r(1-\alpha)$ and $f(q')<2f(q)$ for all $q'\in B(q,\alpha r)$. -\end{enumerate} + **Proof.** Apply the previous lemma to $\widehat f\colon B(p,2r)\times (-2r,0]\to \Ar$ defined by $\widehat f(p,t)=f(p)$. @@ -1586,48 +1922,59 @@ and $f(q')<2f(q)$ for all $q'\in B(q,\alpha r)$. ### Splitting results Here we prove a splitting result for ancient solutions -of non-negative curvature. They are both based on Theorem~. +of non-negative curvature. They are both based on \entryref{e9a9172d16e2}. -**Proposition.** -Suppose that $(M,g(t)),\ -\infty +**Proposition.** Suppose that $(M,g(t)),\ -\infty -**Corollary.** -There is no two-dimensional flow satisfying the hypotheses of -Proposition~. + +**Corollary.** There is no two-dimensional flow satisfying the hypotheses of +\entryref{703d5ca9da05}. + -**Proof.** (of Proposition~) +**Proof.** (of \entryref{703d5ca9da05}) Take a sequence $p_i\in M$ such that -$$d_{g(-1)}^2(p,p_i)R(p_i,-1)\rightarrow\infty$$ as $i\rightarrow \infty$. - We set $d_i=d_{g(-1)}(p,p_i)$ and we set + +$$ +d_{g(-1)}^2(p,p_i)R(p_i,-1)\rightarrow\infty +$$ + + as $i\rightarrow \infty$. + We set $d_i=d_{g(-1)}(p,p_i)$ and we set $B_i=B(p_i,-1,d_i/2)$, and we let $f\colon B_i\to \Ar$ be the square root of the scalar curvature. Since $(M,g(-1))$ is complete, $B_i$ has compact closure in $M$, and consequently $f$ is a bounded -continuous function on $B_i$. Applying Corollary~ to +continuous function on $B_i$. Applying \entryref{b71a179769b1} to $(B_i,g(-1))$ and $f$, we conclude that there is a point $q_i\in B_i$ with the following properties: -\begin{enumerate} -\item[(1)] $R(q_i,-1)\ge R(p_i,-1)$ -\item[(2)] $B'_i=B(q_i,-1,(d_iR(p_i,-1)^{1/2})/(4R(q_i,t_i)^{1/2})\subset B(p_i,-1,d_i/2)$. -\item[(3)] $R(q',-1)\le 4R(q_i,-1)$ for + +- **(1)** $R(q_i,-1)\ge R(p_i,-1)$ +- **(2)** $B'_i=B(q_i,-1,(d_iR(p_i,-1)^{1/2})/(4R(q_i,t_i)^{1/2})\subset B(p_i,-1,d_i/2)$. +- **(3)** $R(q',-1)\le 4R(q_i,-1)$ for all $(q',-1)\in B'_i$. -\end{enumerate} + Since $d_{g(-1)}(p,q_i)\ge d_i/2$, it is also the case that $d^2_{g(-1)}(p,q_i)R(q_i,-1)$ tends to infinity as $i$ tends to infinity. Because of our assumption on the time derivative of $R$, it follows that @@ -1641,7 +1988,7 @@ $(M_i,g_i(t),x_i),\ -\infty **Definition.** Let $(M,g)$ be an $n$-dimensional complete Riemannian manifold and fix $\kappa>0$. We say that $(M,g)$ is *$\kappa$-non-collapsed* -if for every $p\in M$ and any $r>0$, if $|Rm_g|\le r^{-2}$ on -$B(p,r)$ then $Vol\, B(p,r)\ge \kappa r^n$. +if for every $p\in M$ and any $r>0$, if $|\mathit{Rm}_g|\le r^{-2}$ on +$B(p,r)$ then $\mathit{Vol} B(p,r)\ge \kappa r^n$. + Here is the theorem that we shall prove: -**Theorem.** -Let $(M,g)$ be a complete, non-flat Riemannian manifold of bounded + +**Theorem.** Let $(M,g)$ be a complete, non-flat Riemannian manifold of bounded non-negative curvature of dimension $2$ or $3$. Suppose that the Riemannian manifold $(M,g)$ is $\kappa$-non-collapsed. Lastly, suppose that there is a $C^2$-function $f\colon M \to \Ar$ such that -$$Ric_{g}+Hess^{g}(f)=\frac{1}{2}g.$$ + +$$ +\mathit{Ric}_{g}+\mathit{Hess}^{g}(f)=\frac{1}{2}g. +$$ + Then there is a Ricci flow $(M,G(t)),\ -\infty -Now let us begin the proof of Theorem~ +Now let us begin the proof of \entryref{2d68880771ff} ### Integrating $\nabla f$ Since the curvature of $(M,g)$ is bounded, it follows immediately from the -gradient shrinking soliton equation that $Hess^g(f)$ is bounded. Fix a +gradient shrinking soliton equation that $\mathit{Hess}^g(f)$ is bounded. Fix a point $p\in M$. For any $q\in M$ let $\gamma(s)$ be a minimal geodesic from $p$ to $q$ parameterized at unit length. Since -$$\frac{d}{ds}\left(|\nabla f(\gamma(s))|\right)^2=2\langle Hess(f)(\gamma'(s),\nabla f(\gamma(s))\rangle,$$ it follows that -$$\frac{d}{ds}\left(|\nabla f(\gamma(s))|\right)\le C,$$ -where $C$ is an upper bound for $|Hess(f)|$. + +$$ +\frac{d}{ds}\left(|\nabla f(\gamma(s))|\right)^2=2\langle \mathit{Hess}(f)(\gamma'(s),\nabla f(\gamma(s))\rangle, +$$ + + it follows that + +$$ +\frac{d}{ds}\left(|\nabla f(\gamma(s))|\right)\le C, +$$ + +where $C$ is an upper bound for $|\mathit{Hess}(f)|$. By integrating, it -follows that -$$|\nabla f(q)|\le Cd_g(p,q)+|\nabla f(p)|.$$ +follows that + +$$ +|\nabla f(q)|\le Cd_g(p,q)+|\nabla f(p)|. +$$ + This means that any flow line $\lambda(t)$ for $\nabla f$ satisfies -$$\frac{d}{dt}d_g(p,\lambda(t))\le Cd_g(p,\lambda (t))+|\nabla f(p)|,$$ + +$$ +\frac{d}{dt}d_g(p,\lambda(t))\le Cd_g(p,\lambda (t))+|\nabla f(p)|, +$$ + and hence these flow lines do not escape to infinity in finite time. It follows that there is a flow $\Phi_t\colon M\to M$ defined for -all time with $\Phi_0=Id$ and $\partial\Phi_t/\partial +all time with $\Phi_0=\mathit{Id}$ and $\partial\Phi_t/\partial t=\nabla f$. We consider the one-parameter family of diffeomorphisms -$\Phi_{-log(|t|)}\colon M\to M$ and define +$\Phi_{-\mathit{log}(|t|)}\colon M\to M$ and define $$ - G(t)=|t|\Phi_{-log(|t|)}^*g.\ \ +G(t)=|t|\Phi_{-\mathit{log}(|t|)}^*g.\ \ \ -\infty **Claim.** Suppose that $(M,g)$ and $f\colon M\to \Ar$ satisfies the hypotheses of -Theorem~ and that $M$ is compact and of positive curvature. Then -the Ricci flow $(M,G(t))$ with $G(-1)=g$ given in Equation~() is a +\entryref{2d68880771ff} and that $M$ is compact and of positive curvature. Then +the Ricci flow $(M,G(t))$ with $G(-1)=g$ given in Equation (9.19) is a shrinking family of compact round manifolds. + **Proof.** The -manifold $(M,G(t))$ given in Equation~() is equivalent up to +manifold $(M,G(t))$ given in Equation (9.19) is equivalent up to diffeomorphism and scaling by $|t|$ to $(M,g)$. If the dimension of $M$ is three, then according to Hamilton's pinching toward positive curvature result -(Theorem~), the Ricci flow becomes singular in finite time and +(\entryref{86e5a5766bda}), the Ricci flow becomes singular in finite time and as it becomes singular the metric approaches constant curvature in the sense that the ratio of the largest sectional curvature to the smallest goes to one. But this ratio is invariant under scaling and diffeomorphism, so that it must be the case that for each $t$, all the sectional curvatures of the metric $G(t)$ are equal; i.e., for each $t$ the metric $G(t)$ is round. If the dimension of $M$ is two, then the results go back to Hamilton in -. According to Proposition 5.21 on p.. 118 of -, $M$ is a shrinking family of constant positive curvature +[Hamiltonsurface]. According to Proposition 5.21 on p.. 118 of +[ChowKnopf], $M$ is a shrinking family of constant positive curvature surfaces, which must be either $S^2$ or $\Ar P^2$. This completes the analysis in the compact case. @@ -1805,11 +2184,12 @@ From this result, we can easily deduce a complete classification of $\kappa$-solutions with compact asymptotic gradient shrinking soliton. -**Corollary.** -Suppose that $(M,g(t))$ is a $\kappa$-solution of dimension $3$ with a compact + +**Corollary.** Suppose that $(M,g(t))$ is a $\kappa$-solution of dimension $3$ with a compact asymptotic gradient shrinking soliton. Then the Ricci flow $(M,g(t))$ is isomorphic to a time-shifted version of its asymptotic gradient shrinking soliton. + **Proof.** We suppose that the compact asymptotic gradient shrinking soliton is the limit of the $(M,g_{\tau_n}(t),(q_n,-1))$ for some sequence of @@ -1817,7 +2197,7 @@ $\tau_n\rightarrow\infty$. Since by the discussion in the compact case, this limit is of constant positive curvature. It follows that for all $n$ sufficiently large, $M$ is diffeomorphic to the limit manifold and the metric $g_{\tau_n}(-1)$ is close to a metric of -constant positive curvature. In particular, for all $n$ +constant positive curvature. In particular, for all $n$ sufficiently large, $(M,g_{\tau_n}(-1))$ is compact and of strictly positive curvature. Furthermore, as $n\rightarrow\infty$ $\tau_n\rightarrow\infty$ and Riemannian manifolds @@ -1827,7 +2207,7 @@ sectional curvature goes to one. Since this is a scale invariant ratio, the same is true for the sequence of Riemannian manifolds $(M,g(-\tau_n))$. In the case when the dimension of $M$ is three, by Hamilton's pinching toward round result or Ivey's theorem (see -Theorem~), this implies that the $(M,g(t))$ are all +\entryref{86e5a5766bda}), this implies that the $(M,g(t))$ are all exactly round. This proves that $(M,g(t))$ is a shrinking family of round metrics. @@ -1839,25 +2219,26 @@ is equivalent to a time-shifted version of its asymptotic soliton. ### Case 2: Non-strictly positively curved. -**Claim.** -Suppose that $(M,g)$ and $f\colon M\to \Ar$ are as in the statement -of Theorem~ and that $(M,g)$ does not have strictly + +**Claim.** Suppose that $(M,g)$ and $f\colon M\to \Ar$ are as in the statement +of \entryref{2d68880771ff} and that $(M,g)$ does not have strictly positive curvature. Then $n=3$ and the Ricci flow $(M,G(t))$ with -$G(-1)=g$ given in Equation~() has a one- or two-sheeted +$G(-1)=g$ given in Equation (9.19) has a one- or two-sheeted covering that is a product of a two-dimensional $\kappa$-non-collapsed Ricci flow of positive curvature and a constant flat copy of $\Ar$. The curvature is bounded on each time-slice. + **Proof.** According to Hamilton's strong maximum principle -(Corollary~), the Ricci flow $(M,G(t))$ has a one- or +(\entryref{d21a675e4ea8}), the Ricci flow $(M,G(t))$ has a one- or two-sheeted covering that splits as a product of an evolving family of manifolds of one dimension less of positive curvature and a -constant one-manifold. It follows immediately that $n=3$. +constant one-manifold. It follows immediately that $n=3$. Let $\tilde f$ be the lifting of $f$ to this one- or two-sheeted covering. Let $Y$ be a unit tangent vector in the direction of the -one-manifold. Then it follows from Equation~() that the +one-manifold. Then it follows from Equation (9.18) that the value of the Hessian of $\tilde f$ of $(Y,Y)$ is one. If the flat one-manifold factor is a circle then there can be no such function $\tilde f$. Hence, it follows that the one- or two-sheeted covering @@ -1868,41 +2249,69 @@ $(M,G(t))$ is $\kappa$-non-collapsed and each time-slice has positive bounded curvature. These statements are also true for the flow of surfaces. -### Case 3: $M$ is non-compact and -strictly positively curved +### Case 3: $M$ is non-compact and strictly positively curved Here the main result is that this case does not occur. -**Proposition.** -There is no two- or three-dimensional Ricci flow satisfying the hypotheses of -Theorem~ with $(M,g)$ non-compact and of positive curvature. + +**Proposition.** There is no two- or three-dimensional Ricci flow satisfying the hypotheses of +\entryref{2d68880771ff} with $(M,g)$ non-compact and of positive curvature. + -We suppose that we have $(M,g)$ as in Theorem~ with +We suppose that we have $(M,g)$ as in \entryref{2d68880771ff} with $(M,g)$ being non-compact and of positive curvature. Let $n$ be the dimension of $M$, so that $n$ is either $2$ or $3$. Taking the trace of the gradient shrinking soliton equation yields -$$R+\triangle f-\frac{n}{2}=0,$$ -and consequently that $$ dR+d(\triangle f)=0.$$ Using -Lemma~ we rewrite this equation as +$$ +R+\triangle f-\frac{n}{2}=0, +$$ + +and consequently that + +$$ +dR+d(\triangle f)=0. $$ -dR+\triangle(df)-Ric(\nabla f,\cdot)=0. + Using +\entryref{8d03f3ca0aea} we rewrite this equation as + +$$ +dR+\triangle(df)-\mathit{Ric}(\nabla f,\cdot)=0. $$ On the other hand, taking the divergence of the gradient shrinking soliton equation and using the fact that $\nabla^* g=0$ gives -$$\nabla^*Ric+\nabla^*Hess(f)=0.$$ -Of course, $$\nabla^*Hess(f)=\nabla^*(\nabla\nabla -f)=(\nabla^*\nabla)\nabla f=\triangle (df),$$ so that -$$\triangle(df)=-\nabla^*Ric.$$ Plugging this into Equation~ + +$$ +\nabla^*\mathit{Ric}+\nabla^*\mathit{Hess}(f)=0. +$$ + +Of course, + +$$ +\nabla^*\mathit{Hess}(f)=\nabla^*(\nabla\nabla +f)=(\nabla^*\nabla)\nabla f=\triangle (df), +$$ + + so that + +$$ +\triangle(df)=-\nabla^*\mathit{Ric}. +$$ + + Plugging this into Equation 9.20 gives -$$dR-\nabla^*Ric-Ric(\nabla f,\cdot)=0.$$ -Now invoking Lemma~ we have + +$$ +dR-\nabla^*\mathit{Ric}-\mathit{Ric}(\nabla f,\cdot)=0. $$ -dR=2Ric(\nabla f,\cdot). +Now invoking \entryref{9c83686198e8} we have + +$$ +dR=2\mathit{Ric}(\nabla f,\cdot). $$ Fix a point $p\in M$. Let @@ -1910,9 +2319,14 @@ $\gamma(s);\ 0\le s\le \bar s$, be a shortest geodesic (with respect to the metric $g$), parameterized at unit speed, emanating from $p$, and set $X(s)=\gamma'(s)$. + **Claim.** There is a constant $C$ independent of the choice of $\gamma$ and of $\bar s$ such that -$$\int_0^{\bar s}Ric(X,X)ds\le C.$$ + +$$ +\int_0^{\bar s}\mathit{Ric}(X,X)ds\le C. +$$ + **Proof.** Since the curvature is bounded, clearly it suffices to assume that $\bar s>>1$. Since $\gamma$ is length-minimizing and parameterized @@ -1922,126 +2336,175 @@ among all paths with the same end points. Thus, letting $\gamma_u(s)=\gamma(s,u)$ be a one-parameter family of variations (fixed at the endpoints) with $\gamma_0=\gamma$ and with $d\gamma/du|_{u=0}=Y$, we see -$$0\le \delta_{Y}^2E(\gamma_u)=\int_0^{\bar -s}|\nabla_XY|^2+\langle {\mathcal R}(Y,X)Y,X\rangle ds.$$ We -conclude that $$ +0\le \delta_{Y}^2E(\gamma_u)=\int_0^{\bar +s}|\nabla_XY|^2+\langle {\mathcal R}(Y,X)Y,X\rangle ds. +$$ + + We +conclude that +$$ \int_0^{\bar s}\langle -{\mathcal R}(Y,X)Y,X\rangle ds\le \int_0^{\bar s}|\nabla_XY|^2ds. $$ - Fix an orthonormal basis $\{E_i\}_{i=1}^n$ at $p$ with $E_n=X$, and let + + Fix an orthonormal basis $\{E_i\}_{i=1}^n$ at $p$ with $E_n=X$, and let $\widetilde E_i$ denote the parallel translation of $E_i$ along $\gamma$. (Of course, $\widetilde E_n=X$.) Then, for $i\le n-1$, we define -$$Y_i=\left\{\begin{array}{ll} s\widetilde E_i & \mbox{if $0\le s\le -1$} \\ \widetilde E_i & \mbox{if $1\le s\le \bar s-1$} \\ -(\bar s-s)E_i & \mbox{if $\bar s-1\le s\le \bar s$.} \end{array} -\right. $$ Adding up Equation~() for each $i$ gives -$$-\sum_{i=1}^{n-1}\int_0^{\bar s}\langle{\mathcal R}(Y_i,X)Y_i,X\rangle ds\le -\sum_{i=1}^{n-1}\int_0^{\bar s}|\nabla_XY_i|^2ds.$$ Of course, since the + +$$ +Y_i=\left\{\begin{array}{ll} s\widetilde E_i & \text{if }0\le s\le +1 \\ \widetilde E_i & \text{if }1\le s\le \bar s-1 \\ +(\bar s-s)E_i & \text{if }\bar s-1\le s\le \bar s\text{.} \end{array} +\right. +$$ + + Adding up Equation (9.22) for each $i$ gives + +$$ +-\sum_{i=1}^{n-1}\int_0^{\bar s}\langle{\mathcal R}(Y_i,X)Y_i,X\rangle ds\le +\sum_{i=1}^{n-1}\int_0^{\bar s}|\nabla_XY_i|^2ds. +$$ + + Of course, since the $\widetilde E_i$ are parallel along $\gamma$, we have -$$|\nabla_XY_i|^2=\left\{\begin{array}{ll} 1 & \mbox{if $0\le s\le 1$} -\\ 0 & \mbox{if $1\le s\le \bar s-1$} \\ 1 & \mbox{if $\bar s-1\le s\le -\bar s$}\end{array}\right. ,$$ so that -$$\sum_{i=1}^{n-1}\int_0^{\bar s}|\nabla_XY_i|^2=2(n-1).$$ + +$$ +|\nabla_XY_i|^2=\left\{\begin{array}{ll} 1 & \text{if }0\le s\le 1 +\\ 0 & \text{if }1\le s\le \bar s-1 \\ 1 & \text{if }\bar s-1\le s\le +\bar s\end{array}\right. , +$$ + + so that + +$$ +\sum_{i=1}^{n-1}\int_0^{\bar s}|\nabla_XY_i|^2=2(n-1). +$$ + On the other hand, -$$-\sum_{i=1}^{n-1}\langle{\mathcal R}(Y_i,X)(Y_i),X\rangle =\left\{ -\begin{array}{ll} s^2Ric(X,X) & \mbox{if $0\le s\le 1$} \\ -Ric(X,X) & \mbox{if $1\le s\le \bar s-1$} \\ (\bar s-s)^2Ric(X,X) & \mbox{if $\bar s-1\le s\le \bar s$.} \end{array} \right. -$$ Since the curvature is bounded and $|X|=1$, we see that -$\int_0^{\bar s}(1-s^2)Ric(X,X)ds+\int_{\bar s-1}^{\bar -s}(\bar s-s)^2Ric(X,X)$ is bounded independent of $\gamma$ and + +$$ +-\sum_{i=1}^{n-1}\langle{\mathcal R}(Y_i,X)(Y_i),X\rangle =\left\{ +\begin{array}{ll} s^2\mathit{Ric}(X,X) & \text{if }0\le s\le 1 \\ +\mathit{Ric}(X,X) & \text{if }1\le s\le \bar s-1 \\ (\bar s-s)^2\mathit{Ric}(X,X) & \text{if }\bar s-1\le s\le \bar s\text{.} \end{array} \right. +$$ + + Since the curvature is bounded and $|X|=1$, we see that +$\int_0^{\bar s}(1-s^2)\mathit{Ric}(X,X)ds+\int_{\bar s-1}^{\bar +s}(\bar s-s)^2\mathit{Ric}(X,X)$ is bounded independent of $\gamma$ and of $\bar s$. This concludes the proof of the claim. -**Claim.** $|Ric(X,\cdot)|^2\le R \cdot Ric(X,X)$. + +**Claim.** $|\mathit{Ric}(X,\cdot)|^2\le R \cdot \mathit{Ric}(X,X)$. + **Proof.** This is obvious if $n=2$, so we may as well assume that $n=3$. We -diagonalize $Ric$ in an orthonormal basis $\{e_i\}$. Let +diagonalize $\mathit{Ric}$ in an orthonormal basis $\{e_i\}$. Let $\lambda_i\ge 0$ be the eigenvalues. Write $X=X^ie_i$ with $\sum_i(X^i)^2=1$. Then -$$Ric(X,\cdot)=X^i\lambda_i(e_i)^*,$$ so that -$|Ric(X,\cdot)|^2=\sum_i(X^i)^2\lambda_i^2$. Of course, since -the $\lambda_i\ge 0$, this gives -$$R\cdot Ric(X,X)=(\sum_i\lambda_i)\sum_i\lambda_i(X^i)^2\ge -\sum_i\lambda_i^2(X^i)^2,$$ establishing the claim. -Now we compute, using Cauchy-Schwarz, +$$ +\mathit{Ric}(X,\cdot)=X^i\lambda_i(e_i)^*, +$$ + + so that +$|\mathit{Ric}(X,\cdot)|^2=\sum_i(X^i)^2\lambda_i^2$. Of course, since +the $\lambda_i\ge 0$, this gives $$ +R\cdot \mathit{Ric}(X,X)=(\sum_i\lambda_i)\sum_i\lambda_i(X^i)^2\ge +\sum_i\lambda_i^2(X^i)^2, +$$ + + establishing the claim. + +Now we compute, using Cauchy-Schwarz, $$ \begin{aligned} -\left(\int_0^{\bar s}|Ric(X,\widetilde E_i)|ds\right)^2 & \le -&\bar s\int_0^{\bar s}|Ric(X,\widetilde E_i)|^2ds\le \bar -s\int_0^{\bar s}|Ric(x,\cdot)|^2ds\\ & \le & \bar -s\int_0^{\bar s}R\cdot Ric(X,X)ds. +\left(\int_0^{\bar s}|\mathit{Ric}(X,\widetilde E_i)|ds\right)^2 & \le +&\bar s\int_0^{\bar s}|\mathit{Ric}(X,\widetilde E_i)|^2ds\le \bar +s\int_0^{\bar s}|\mathit{Ric}(x,\cdot)|^2ds\\ & \le & \bar +s\int_0^{\bar s}R\cdot \mathit{Ric}(X,X)ds. \end{aligned} $$ -$$ Since $R$ is bounded, it follows from the first claim that there is a constant $C'$ independent of $\gamma$ and $\bar s$ with $$ - -\int_0^{\bar s}|Ric(X,\widetilde E_i)|ds\le C'\sqrt{\bar +\int_0^{\bar s}|\mathit{Ric}(X,\widetilde E_i)|ds\le C'\sqrt{\bar s}. $$ + Since $\gamma$ is a geodesic in the metric $g$, we have $\nabla_XX=0$. Hence, -$$\frac{d^2f(\gamma(s))}{ds^2}=X(X(f))=Hess(f)(X,X).$$ -Applying the gradient shrinking soliton equation to the pair + +$$ +\frac{d^2f(\gamma(s))}{ds^2}=X(X(f))=\mathit{Hess}(f)(X,X). +$$ + +Applying the gradient shrinking soliton equation to the pair $(X,X)$ gives -$$\frac{d^2f(\gamma(s))}{ds^2}=\frac{1}{2}-Ric_{g}(X,X).$$ + +$$ +\frac{d^2f(\gamma(s))}{ds^2}=\frac{1}{2}-\mathit{Ric}_{g}(X,X). +$$ + Integrating we see -$$\frac{df(\gamma(s))}{ds}|_{s=\bar -s}=\frac{df(\gamma(s))}{ds}|_{s=0}+\frac{\bar s}{2}-\int_0^{\bar -s}Ric(X,X)ds.$$ It follows that +$$ +\frac{df(\gamma(s))}{ds}|_{s=\bar +s}=\frac{df(\gamma(s))}{ds}|_{s=0}+\frac{\bar s}{2}-\int_0^{\bar +s}\mathit{Ric}(X,X)ds. $$ + It follows that + +$$ X(f)(\gamma(\bar s))\ge \frac{\bar s}{2}-C'', $$ + for some constant $C''$ depending only on $(M,g)$ and $f$. Similarly, applying the gradient shrinking soliton equation to the pair -$(X,\widetilde E_i)$, using Equation~() and the fact that +$(X,\widetilde E_i)$, using Equation (9.23) and the fact that $\nabla_X\widetilde E_i=0$ gives $$ - |\widetilde E_i(f)(\gamma(\bar s))|\le C''(\sqrt{\bar s}+1). $$ These two inequalities imply that for $\bar s$ sufficiently large, -$f$ has no critical points and that $\nabla f$ makes a small angle +$f$ has no critical points and that $\nabla f$ makes a small angle with the gradient of the distance function from $p$, and $|\nabla f|$ goes to infinity as the distance from $p$ increases. In particular, $f$ is a proper function going off to $+\infty$ as we approach infinity in $M$. -Now apply Equation~() to see that $R$ is increasing along the gradient +Now apply Equation (9.21) to see that $R$ is increasing along the gradient curves of $f$. Hence, there is a sequence $p_k$ tending to infinity in $M$ with -$lim_kR(p_k)=limsup_{q\in M} R_g(q)>0$. +$\mathit{lim}_kR(p_k)=\mathit{limsup}_{q\in M} R_g(q)>0$. The Ricci flow $(M,G(t)),\ -\infty +**Corollary.** There is no non-compact, two-dimensional Riemannian manifold $(M,g)$ +satisfying the hypotheses of \entryref{2d68880771ff}. For any non-compact three-manifold $(M,g)$ of positive curvature satisfying -the hypotheses of Theorem~, there is a sequence of -points $q_i\in M$ tending to infinity such that $lim_{i\rightarrow \infty}R_g(q_i)=sup_{p\in M}$ such that the +the hypotheses of \entryref{2d68880771ff}, there is a sequence of +points $q_i\in M$ tending to infinity such that $\mathit{lim}_{i\rightarrow \infty}R_g(q_i)=\mathit{sup}_{p\in M}$ such that the based Ricci flows $(M,G(t),(q_i,-1))$ converge to a Ricci flow $(M_\infty,G_\infty(t),(q_\infty,-1))$ defined for $-\infty -**Proof.** In Claim~ we saw that every two-dimensional $(M,g)$ satisfying -the hypotheses of Theorem~ has strictly positive curvature. The +**Proof.** In \entryref{afc86a351f0c} we saw that every two-dimensional $(M,g)$ satisfying +the hypotheses of \entryref{2d68880771ff} has strictly positive curvature. The argument that we just completed shows that there is no non-compact two-dimensional example of strictly positive curvature. -The final statement is exactly what we just established. +The final statement is exactly what we just established. -**Corollary.** -\begin{enumerate} -\item[(1)] Let $(M,g(t))$ be a two-dimensional Ricci flow satisfying all -the hypotheses of Proposition~ except possible the + +**Corollary.** - **(1)** Let $(M,g(t))$ be a two-dimensional Ricci flow satisfying all +the hypotheses of \entryref{da767c71996b} except possible the non-compactness hypothesis. Then $M$ is compact and for any $a>0$ the restriction of the flow to any interval of the form $(-\infty,-a]$ followed by a shift of time by $+a$ is a $\kappa$-solution. -\item[(2)] Any asymptotic gradient +- **(2)** Any asymptotic gradient shrinking soliton for a two-dimensional $\kappa$-solution is a shrinking family of round surfaces. -\item[(3)] Let $(M,g(t)), -\infty **Proof.** Let $(M,g(t))$ be a two-dimensional Ricci flow satisfying all the hypotheses of -Proposition~ except possibly non-compactness. It then follows -from Corollary~ that $M$ is compact. This proves the first item. +\entryref{703d5ca9da05} except possibly non-compactness. It then follows +from \entryref{18ee66757f32} that $M$ is compact. This proves the first item. Now suppose that $(M,g(t))$ is an asymptotic soliton for a $\kappa$-solution of dimension two. If $(M,g(-1))$ does not have bounded curvature, then there is a sequence $p_i\rightarrow \infty$ -so that $lim_{i\rightarrow \infty}R(p_i,-1)=\infty$. By this +so that $\mathit{lim}_{i\rightarrow \infty}R(p_i,-1)=\infty$. By this and - Theorem~ the Ricci flow $(M,g(t))$ satisfies all the hypotheses - of Proposition ~. But this contradicts Corollary~. + \entryref{671402eb454b} the Ricci flow $(M,g(t))$ satisfies all the hypotheses + of Proposition \entryref{703d5ca9da05}. But this contradicts \entryref{18ee66757f32}. We conclude that $(M,g(-1))$ has bounded curvature. According to - Corollary~ this means that + \entryref{fd11f1f8def3} this means that $(M,g(t))$ is compact. -Results going back to Hamilton in imply that +Results going back to Hamilton in [Hamiltonsurface] imply that this compact asymptotic shrinking soliton is a shrinking family of compact, round surfaces. For example, this result is contained in -Proposition 5.21 on p. 118 of . - This proves the second item. +Proposition 5.21 on p. 118 of [ChowKnopf]. + This proves the second item. Now suppose that $(M,g(t))$ is a two-dimensional $\kappa$-solution. By the second item any asymptotic gradient shrinking soliton for @@ -2116,9 +2579,9 @@ Proposition 5.21 on p. 118 of . It follows that $M$ is compact. We know that as $t$ goes to $-\infty$ the Riemannian surfaces $(M,g(t))$ are converging to compact, round surfaces. Extend the flow forward from $0$ to a maximal time $\Omega<\infty$. By Theorem -5.64 on p. 149 of the surfaces $(M,g(t))$ are also becoming +5.64 on p. 149 of [ChowKnopf] the surfaces $(M,g(t))$ are also becoming round as $t$ approaches $\Omega$ from below. Also, according to Proposition -5.39 on p. 134 of the entropy of the flow is weakly monotone +5.39 on p. 134 of [ChowKnopf] the entropy of the flow is weakly monotone decreasing and is strictly decreasing unless the flow is a gradient shrinking soliton. But we have seen that the limits at both $-\infty$ and $\Omega$ are round manifolds, and hence of the same entropy. It follows that the @@ -2126,52 +2589,62 @@ $\kappa$-solution is a shrinking family of compact, round surfaces. Now that we have shown that every two-dimensional $\kappa$-solution is a shrinking family of round surfaces, we can complete the proof -of Proposition~. Let $(M,g)$ be a non-compact +of \entryref{03957890f08f}. Let $(M,g)$ be a non-compact manifold of positive curvature satisfying the hypotheses of -Theorem~. According to Corollary~ the +\entryref{2d68880771ff}. According to \entryref{da767c71996b} the limiting Ricci flow $(M_\infty,G_\infty(t))$ referred to in -Corollary~ is the product of a line and a +\entryref{fd11f1f8def3} is the product of a line and a shrinking family of round surfaces. Since $(M,g)$ is non-compact and has positive curvature, it is diffeomorphic to $\Ar ^3$ and hence does not contain an embedded copy of a projective plane. It follows -that the round surfaces are in fact round two-spheres. Thus, -$(M_\infty,G_\infty(t)),\ -\infty **Claim.** The scalar curvature of $(S^2,h(-1))$ is equal to $1$. + **Proof.** Since the shrinking family of round two-spheres $(S^2,h(t))$ exists for all $-\infty0.$$ -(On a manifold with non-negative curvature $Ric(Y,Y)\le R/2$ for any unit + +$$ +\mathit{Hess}(f)(Y,Y)=\frac{1}{2}-\mathit{Ric}(Y,Y)\ge \frac{1}{2}-\frac{R}{2}>0. +$$ + +(On a manifold with non-negative curvature $\mathit{Ric}(Y,Y)\le R/2$ for any unit tangent vector $Y$.) This means that for $u$ sufficiently large the level surfaces of $N_u=f^{-1}(u)$ are convex and hence have increasing area as $u$ increases. -According to Equations~() and~() the angle between $\nabla +According to Equations (9.24) and (9.25) the angle between $\nabla f$ and the gradient of the distance function from $p$ goes to zero as we go to -infinity. According to Theorem~ the gradient of the distance +infinity. According to \entryref{e9a9172d16e2} the gradient of the distance function from $p$ converges to the unit vector field in the $\Ar$-direction of the product structure. It follows that the unit vector in the $\nabla f$-direction converges to the unit vector in the $\Ar$-direction. Hence, as $u$ tends to $\infty$ the level surfaces $f^{-1}(u)$ converge in the $C^1$-sense to $\Sigma\times \{0\}$. Thus, the areas of these level surfaces converge to the area of $(\Sigma,h(-1))$ which is $8\pi$ since the scalar curvature of this -limiting surface is $limsup_{p\in M} R(p,-1)= 1$. It follows that the +limiting surface is $\mathit{limsup}_{p\in M} R(p,-1)= 1$. It follows that the area of $f^{-1}(u)$ is less than $8\pi$ for all $u$ sufficiently large. Now let us estimate the intrinsic curvature of $N=N_u=f^{-1}(u)$. @@ -2180,87 +2653,107 @@ $N$, whereas $K_M$ is the sectional curvature of $M$. We also denote by $R_N$ the scalar curvature of the induced metric on $N$. Fix an orthonormal basis $\{e_1,e_2,e_3\}$ at a point of $N$, where $e_3=\nabla f/|\nabla f|$. Then by the Gauss-Codazzi formula we have -$$R_N=2K_N(e_1,e_2)=2(K_M(e_1,e_2)+det\, S)$$ + +$$ +R_N=2K_N(e_1,e_2)=2(K_M(e_1,e_2)+\mathit{det} S) +$$ + where $S$ is the shape operator -$$S=\frac{Hess(f|TN)}{|\nabla f|}.$$ -Clearly, we have $R-2Ric(e_3,e_3)=2K_M(e_1,e_2)$, so that -$$R_N=R-2Ric(e_3,e_3)+2det\, S.$$ -We can assume that the basis is chosen so that $Ric|_{TN}$ is + +$$ +S=\frac{\mathit{Hess}(f|TN)}{|\nabla f|}. +$$ + +Clearly, we have $R-2\mathit{Ric}(e_3,e_3)=2K_M(e_1,e_2)$, so that + +$$ +R_N=R-2\mathit{Ric}(e_3,e_3)+2\mathit{det} S. +$$ + +We can assume that the basis is chosen so that $\mathit{Ric}|_{TN}$ is diagonal; i.e., in the given basis we have -$$Ric=\begin{pmatrix} r_1 & 0 & c_1 \\ 0 & r_2 & c_2 \\ c_1 & c_2 & -r_3\end{pmatrix}.$$ From the gradient shrinking soliton equation we -have $Hess(f)=(1/2)g-Ric$ so that + +$$ +\mathit{Ric}=\begin{pmatrix} r_1 & 0 & c_1 \\ 0 & r_2 & c_2 \\ c_1 & c_2 & +r_3\end{pmatrix}. $$ + From the gradient shrinking soliton equation we +have $\mathit{Hess}(f)=(1/2)g-\mathit{Ric}$ so that + $$ \begin{aligned} -det(Hess(f|TN)) & = & \left(\frac{1}{2}-r_1\right)\left(\frac{1}{2}- r_2\right) \\ +\mathit{det}(\mathit{Hess}(f|TN)) & = & \left(\frac{1}{2}-r_1\right)\left(\frac{1}{2}- r_2\right) \\ & \le & \frac{1}{4}(1-r_1-r_2)^2 \\ -& = & \frac{1}{4} (1-R+Ric(e_3,e_3))^2. +& = & \frac{1}{4} (1-R+\mathit{Ric}(e_3,e_3))^2. \end{aligned} $$ -$$ - Thus, it follows that + $$ - R_N\le R-2Ric(e_3,e_3)+\frac{(1-R+Ric(e_3,e_3))^2}{2|\nabla f|^2}. +R_N\le R-2\mathit{Ric}(e_3,e_3)+\frac{(1-R+\mathit{Ric}(e_3,e_3))^2}{2|\nabla f|^2}. $$ - It follows from Equation~() that $|\nabla + It follows from Equation (9.24) that $|\nabla f(x)|\rightarrow\infty$ as $x$ goes to infinity in $M$. Thus, since the curvature of $(M,g(-1))$ is bounded, provided that $u$ is -sufficiently large, we have $1-R+Ric(e_3,e_3)<2|\nabla f|^2$. +sufficiently large, we have $1-R+\mathit{Ric}(e_3,e_3)<2|\nabla f|^2$. Since the left-hand side of this inequality is positive (since $R< 1$), it follows that $$ -(1-R+Ric(e_3,e_3))^2 < 2(1-R+Ric(e_3,e_3))|\nabla +(1-R+\mathit{Ric}(e_3,e_3))^2 < 2(1-R+\mathit{Ric}(e_3,e_3))|\nabla f|^2. $$ -Plugging this into Equation~() gives that -$$R_N<1-Ric(e_3,e_3)\le 1,$$ -assuming that $u$ is sufficiently large. +Plugging this into Equation (9.26) gives that + +$$ +R_N<1-\mathit{Ric}(e_3,e_3)\le 1, +$$ + +assuming that $u$ is sufficiently large. This contradicts the Gauss-Bonnet theorem for the surface $N$: Its -area is less than $8\pi$, and the scalar curvature of the induced +area is less than $8\pi$, and the scalar curvature of the induced metric is less than $1$, meaning that its Gaussian curvature is less than $1/2$; yet $N$ is diffeomorphic to a $2$-sphere. This completes -the proof of Proposition~, that is to say this shows +the proof of \entryref{03957890f08f}, that is to say this shows that there are no non-compact positive curved examples satisfying -the hypotheses of Theorem~. +the hypotheses of \entryref{2d68880771ff}. ### Case of non-positive curvature revisited -We return now to the second case of Theorem~. We +We return now to the second case of \entryref{2d68880771ff}. We extend $(M,g)$ to a Ricci flow $(M,G(t))$ defined for $-\infty +**Corollary.** Let $(M,g(t))$ be a three-dimensional Ricci flow satisfying the hypotheses of +\entryref{703d5ca9da05}. Then the limit constructed in that proposition splits as a product of a shrinking family of compact round surfaces with a line. In particular, for any non-compact gradient shrinking soliton of a three-dimensional $\kappa$-solution the limit constructed in -Proposition~ is the product of a shrinking family of round +\entryref{703d5ca9da05} is the product of a shrinking family of round surfaces and the real line. + **Proof.** Let $(M,g(t))$ be a three-dimensional Ricci flow satisfying the -hypotheses of Proposition~ and let $(N^2,h(t))\times +hypotheses of \entryref{703d5ca9da05} and let $(N^2,h(t))\times (\Ar,ds^2)$ be the limit constructed in that proposition. Since this limit is $\kappa$-non-collapsed, $(N,h(t))$ is $\kappa'$-non-collapsed for some $\kappa'>0$ depending only on @@ -2268,77 +2761,80 @@ $\kappa$. Since the limit is not flat and has non-negative curvature, the same is true for $(N,h(t))$. Since $\partial R/\partial t\ge 0$ for the limit, the same is true for $(N,h(t))$. That is to say $(N,h(t))$ satisfies all the hypotheses of -Proposition~ except possibly non-compactness. It now -follows from Corollary~ that $(N,h(t))$ is a shrinking +\entryref{703d5ca9da05} except possibly non-compactness. It now +follows from \entryref{da767c71996b} that $(N,h(t))$ is a shrinking family of compact, round surfaces. -**Corollary.** -Let $(M,g(t)),\ -\infty +**Corollary.** Let $(M,g(t)),\ -\infty0$ the flow $(M,g(t)),\ -\infty **Proof.** If an asymptotic gradient shrinking soliton $(M,g(t))$ of a three-dimensional $\kappa$-solution does not have strictly positive curvature, then according to -Corollary~, $(M,g(t))$ has a covering that splits as a product +\entryref{d21a675e4ea8}, $(M,g(t))$ has a covering that splits as a product of a a two-dimensional Ricci flow and a line. The two-dimensional Ricci flow -satisfies all the hypotheses of Proposition~ except possibly -compactness, and hence by Corollary~ it is a shrinking family of +satisfies all the hypotheses of \entryref{703d5ca9da05} except possibly +compactness, and hence by \entryref{da767c71996b} it is a shrinking family of round surfaces. In this case, it is clear that each time-slice of $(M,g(t))$ has bounded curvature. Now we consider the remaining case when $(M,g(t))$ has strictly positive -curvature. Assume that $(M,g(t))$ has unbounded curvature. Then there is a +curvature. Assume that $(M,g(t))$ has unbounded curvature. Then there is a sequence of points $p_i$ tending to infinity in $M$ such that $R(p_i,t)$ tends -to infinity. By Corollary~ we can replace the points $p_i$ by +to infinity. By \entryref{f8ff249efb81} we can replace the points $p_i$ by points $q_i$ with $Q_i=R(q_i,t)\ge R(p_i,t)$ so that the based Riemannian manifolds $(M,Q_ig(t),q_i)$ converge to a product of a round surface $(N,h(t))$ with $\Ar$. The surface $N$ is either diffeomorphic to $S^2$ or $\Ar P^2$. -Since $(M,g(t))$ has positive curvature, by Theorem~, it is +Since $(M,g(t))$ has positive curvature, by \entryref{2b5f18292e62}, it is diffeomorphic to $\Ar ^3$, and hence it contains no embedded $\Ar P^2$. It follows that $(N,h(t))$ is a round two-sphere. -Fix $\epsilon>0$ sufficiently small as in Proposition~. Then the +Fix $\epsilon>0$ sufficiently small as in \entryref{cb9ca6d6db9f}. Then the limiting statement means that, for every $i$ sufficiently large, there is an $\epsilon$-neck in $(M,g(t))$ centered at $q_i$ with scale $Q_i^{-1/2}$. This -contradicts Proposition~, establishing that for each $t<0$ the +contradicts \entryref{cb9ca6d6db9f}, establishing that for each $t<0$ the curvature of $(M,g(t))$ is bounded. + **Corollary.** Let $(M,g(t)),\ -\infty **Proof.** Let $(M_\infty,g_\infty(t)),\ -\infty0$ depending only on $\kappa$. On the other hand, according -to Theorem~ there is a function $f(\cdot, -1)$ from +to \entryref{671402eb454b} there is a function $f(\cdot, -1)$ from $M_\infty$ to $\Ar$ satisfying the gradient shrinking soliton equation at the time-slice $-1$. Thus, -Theorem~ applies to $(M_\infty,g_\infty(-1))$ to +\entryref{2d68880771ff} applies to $(M_\infty,g_\infty(-1))$ to produce a Ricci flow $G(t),\ -\infty **Definition.** Let $(M,g)$ be a complete, connected, non-compact Riemannian manifold of non-negative curvature. Fix a point $p\in M$. We define the *asymptotic scalar curvature* -$${\mathcal R}(M,g)=limsup_{x\rightarrow\infty}R(x)d^2(x,p).$$ Clearly, this limit is + +$$ +{\mathcal R}(M,g)=\mathit{limsup}_{x\rightarrow\infty}R(x)d^2(x,p). +$$ + + Clearly, this limit is independent of $p$. + -**Proposition.** -Suppose that $(M,g(t)),\ -\infty +**Proposition.** Suppose that $(M,g(t)),\ -\infty -**Proof.** By Corollary~ the only two-dimensional $\kappa$-solutions are +**Proof.** By \entryref{da767c71996b} the only two-dimensional $\kappa$-solutions are compact, so that the result is vacuously true in this case. Suppose that $(M,g(t))$ is three-dimensional If $(M,g(t))$ does not have strictly positive curvature, then, since it is not flat, by -Corollary~ it must be three-dimensional and it has a +\entryref{d21a675e4ea8} it must be three-dimensional and it has a finite-sheeted covering space that splits as a product $(Q,h(0))\times (\Ar,ds^2)$ with $(Q,h(0))$ being a surface of strictly positive curvature and $T$ being a flat one-manifold. Clearly, in this case the asymptotic curvature @@ -2378,12 +2882,23 @@ ${\mathcal R}(M,g(t))$ has a finite, nonzero value. Fix a point $p\in M$. Take a sequence of points $x_n$ tending to infinity and set $\lambda_n=d^2_0(x_n,p)$ and $Q_n=R(x_n,t)$. We choose this sequence such that -$$lim_{n\rightarrow\infty}Q_n\lambda_n={\mathcal R}(M,g(t)).$$ + +$$ +\mathit{lim}_{n\rightarrow\infty}Q_n\lambda_n={\mathcal R}(M,g(t)). +$$ + We consider the sequence of Ricci flows $(M,h_n(t),(x_n,0))$, where -$$h_n(t)=Q_ng(Q_n^{-1}t).$$ - Fix $00$, independent of $n$, such that for all $n$ sufficiently large, every ball of radius $\delta$ centered at any point of $B_{h_n}(x_n,0,\alpha/2)$ has volume at least $\kappa \delta^3$, Now -applying Theorem~ we see that a subsequence +applying \entryref{74c7682a68ed} we see that a subsequence converges geometrically to a limit which will automatically be a metric ball $B_{g_\infty}(x_\infty,0,\alpha/2)$. In fact, by -Hamilton's result (Proposition~) there is a +Hamilton's result (\entryref{4673eb9f4424}) there is a limiting flow on $B_{g_\infty}(x_\infty,0,\alpha/4)\times (-\infty,0]$. Notice that the limiting flow is not flat since $R(x_\infty,0)=1$. -On the other hand, according to Lemma~ the +On the other hand, according to \entryref{88a75d6dd09e} the Gromov-Hausdorff limit of a subsequence $(M,\lambda_n^{-1}g_n(0),x_n)$ is the Tits cone, i.e., the cone over $S_\infty(M,p)$. Since $Q_n={\mathcal R}(M,g(t))\lambda_n^{-1}$, the rescalings -$(M,Q_ng_n(0),x_n)$ also converge to a cone, say $(C,h,y_\infty)$, +$(M,Q_ng_n(0),x_n)$ also converge to a cone, say $(C,h,y_\infty)$, which is in fact simply a rescaling of the Tits cone by a factor -${\mathcal R}(M,g(t))$. Pass to a subsequence so that both the +${\mathcal R}(M,g(t))$. Pass to a subsequence so that both the geometric limit on the ball of radius $\alpha/2$ and the Gromov-Hausdorff limit exist. Then the geometric limit $B_{g_\infty}(x_\infty,0,\alpha/2)$ is isometric to an open ball in the cone. Since we have a limiting Ricci flow -$$(B_{g_\infty}(x_\infty,0,\alpha/2),g_\infty(t)),\ -\infty **Claim.** $S_\infty(M,p)$ is isometric to a round $2$-sphere. + -\begin{proof} -Since $M$ is orientable the complement of the cone point in the Tits +**Proof.** Since $M$ is orientable the complement of the cone point in the Tits cone is an orientable manifold and hence $S_\infty(M,p)$ is an orientable surface. Since we have already established that it has a metric of constant positive curvature, it must be diffeomorphic to @@ -2466,13 +2990,22 @@ isometric to a round sphere.) It follows that the Tits cone is a smooth flat manifold even at the origin, and hence is isometric to Euclidean $3$-space. This means that in the limit, for any $r>0$ the volume of the ball of radius $r$ centered at the cone point - is exactly $\omega_3 r^3$, where + is exactly $\omega_3 r^3$, where $\omega_3$ is the volume of the unit ball in $\Ar^3$. Consequently, -$$lim_{n\rightarrow\infty}Vol\left(B_{g}(p,0,\sqrt{\lambda_n}r)\setminus -B_g(p,0,1)\right)\rightarrow \omega_3\lambda_n^{3/2}r^3.$$ By -Theorem~ and the fact that the Ricci curvature is + +$$ +\mathit{lim}_{n\rightarrow\infty}\mathit{Vol}\left(B_{g}(p,0,\sqrt{\lambda_n}r)\setminus +B_g(p,0,1)\right)\rightarrow \omega_3\lambda_n^{3/2}r^3. +$$ + + By +\entryref{1a1b2c4fdc9b} and the fact that the Ricci curvature is non-negative, this implies that -$$Vol\, B_g(p,0,R)=\omega_3R^3$$ + +$$ +\mathit{Vol} B_g(p,0,R)=\omega_3R^3 +$$ + for all $R<\infty$. Since the Ricci curvature is non-negative, this means that $(M,g(t))$ is Ricci-flat, and hence flat. But this contradicts the fact that $(M,g(t))$ is a $\kappa$-solution and @@ -2480,7 +3013,6 @@ hence is not flat. Having ruled out the other two cases, we are left with only one possibility: ${\mathcal R}(M,g(t))=\infty$. -\end{proof} ## Universal $\kappa$ @@ -2489,14 +3021,15 @@ shrinking soliton is that there is a universal $\kappa$ for all $3$-dimensional $\kappa$-solutions, except those of constant positive curvature. -**Proposition.** -There is a $\kappa_0>0$ such that any non-round $3$-dimensional + +**Proposition.** There is a $\kappa_0>0$ such that any non-round $3$-dimensional $\kappa$-solution is a $\kappa_0$-solution. + **Proof.** Let $(M,g(t))$ be a non-round $3$-dimensional $\kappa$-solution. By -Corollary~ since $(M,g(t))$ is not a family of +\entryref{01028c93347d} since $(M,g(t))$ is not a family of round manifolds, the asymptotic soliton for the $\kappa$-solution -cannot be compact. Thus, according to Corollary~ there +cannot be compact. Thus, according to \entryref{2d68880771ff} there are only two possibilities for the asymptotic soliton $(M_\infty,g_\infty(t))$ -- either $(M_\infty,g_\infty(t))$ is the product of a round $2$-sphere of Gaussian curvature $1/2|t|$ with a @@ -2513,24 +3046,28 @@ is a gradient shrinking soliton $(M_\infty,g_\infty(t))$ and a ball $B$ of radius $1$ in $(M_\infty,g_\infty(-1))$ centered at a point $q_\infty\in M_\infty$ and a sequence of embeddings $\psi_k\colon B\to M$ such that $\psi_k(q_\infty)=q_k$ and such that the map -$$B\times [-2,-1]\to M\times [-2\bar\tau_k,-\bar\tau_k]$$ + +$$ +B\times [-2,-1]\to M\times [-2\bar\tau_k,-\bar\tau_k] +$$ + given by $(b,t)\mapsto (\psi_k(b),\bar\tau_kt)$ has the property that the pullback of $\bar\tau^{-1}_kg(\bar\tau_k t)$ converges smoothly and uniformly as $k\rightarrow\infty$ to the restriction of $g_\infty(t),\ -2\le t\le -1$, to $B$. Let $(M_k,g_k(t))$ be this rescaling of the the $\kappa$-solution by $\tau_k$. Then the -embeddings $\psi_k\times id\colon B\times (-2,-1]\to +embeddings $\psi_k\times \mathit{id}\colon B\times (-2,-1]\to (M_k\times [-2,-1]$ converge as $k\rightarrow \infty$ to a one-parameter family of isometries. That is to say, the image $\psi_k(B\times [-2,-1])\subset M_k\times [-2,-1]$ is an almost isometric embedding. Since the reduced length function from $x$ to $(\psi_k(a),-1)$ is at most $3/2$ (from the invariance of reduced -length under rescalings, see Corollary~), it follows easily +length under rescalings, see \entryref{541f71387f84}), it follows easily that the reduced length function on $\psi_k(B\times \{-2\})$ is bounded independent of $k$. Similarly, the volume of $\psi_k(B\times\{-2\})$ is bounded independent of $k$. This means the reduced volume of $\psi_k(B\times\{-2\})$ in $(M_k,g_k(t))$ is -bounded independent of $k$. Now by Theorem~ this implies +bounded independent of $k$. Now by \entryref{3defa9b2209c} this implies that $(M_k,g_k(t))$ is $\kappa_0$-non-collapsed at $(p,0)$ on scales $\le \sqrt{2}$ for some $\kappa_0$ depending only on the geometry of the three possibilities for $(M_\infty,g_\infty(t)),\ -2\le t\le @@ -2551,32 +3088,38 @@ that we can apply the previous results. ## Asymptotic volume Let $(M,g(t))$ be an $n$-dimensional $\kappa$-solution. For any -$t\le 0$ and any point $p\in M$ we consider $(Vol\,B_{g(t)}(p,r))/r^n$. According to the Bishop-Gromov Theorem -(Theorem~), this is a non-increasing function of +$t\le 0$ and any point $p\in M$ we consider $(\mathit{Vol} B_{g(t)}(p,r))/r^n$. According to the Bishop-Gromov Theorem +(\entryref{1a1b2c4fdc9b}), this is a non-increasing function of $r$. We define the *asymptotic volume* ${\mathcal V}_\infty(M,g(t))$, or ${\mathcal V}_\infty(t)$ if the flow is clear from the context, to be the limit as $r\rightarrow\infty$ of this function. Clearly, this limit is independent of $p\in M$. -**Theorem.** -For\footnote{This theorem and all the other results of this section are valid -in all dimensions. Our proofs use Theorem~ and -Proposition~ which are also valid in all dimensions but which we + +**Theorem.** For (This theorem and all the other results of this section are valid +in all dimensions. Our proofs use \entryref{5d9f1b75170d} and +\entryref{703d5ca9da05} which are also valid in all dimensions but which we proved only in dimensions $2$ and $3$. Thus, while we state the results of this section for all dimensions, strictly speaking we give proofs only for -dimensions $2$ and $3$. These are the only cases we need in what follows.} any +dimensions $2$ and $3$. These are the only cases we need in what follows.) any $\kappa>0$ and any $\kappa$-solution $(M,g(t))$ the asymptotic volume ${\mathcal V}_\infty(M,g(t))$ is identically zero. + **Proof.** The proof is by induction on the dimension $n$ of the solution. For $n=2$ by -Corollary~ there are only compact $\kappa$-solutions, which clearly +\entryref{da767c71996b} there are only compact $\kappa$-solutions, which clearly have zero asymptotic volume. Suppose that we have established the result for $n-1\ge 2$ and let us prove it for $n$. -According to Proposition~ there is a sequence of points $p_n\in +According to \entryref{703d5ca9da05} there is a sequence of points $p_n\in M$ tending to infinity such that setting $Q_n=R(p_n,0)$ the sequence of Ricci flows -$$(M,Q_ng(Q_n^{-1}t),(q_n,0))$$ converges geometrically to a limit + +$$ +(M,Q_ng(Q_n^{-1}t),(q_n,0)) +$$ + + converges geometrically to a limit $(M_\infty,g_\infty(t),(q_\infty,0))$, and this limit splits off a line: $(M_\infty,g_\infty(t))= (N,h(t))\times \Ar$. Since the ball of radius $R$ about a point $(x,t)\in N\times \Ar$ is contained in @@ -2591,7 +3134,8 @@ $(M,g(t))$. One important consequence of the asymptotic volume result is a volume comparison result. -**Proposition.** Fix the dimension $n$. + +**Proposition.** Fix the dimension $n$. For every $\nu>0$ there is $A<\infty$ such that the following holds. Suppose that $(M_k,g_k(t)),\ -t_k\le t\le 0$, is a sequence of (not necessarily complete) $n$-dimensional Ricci flows of non-negative @@ -2600,110 +3144,147 @@ and radii $r_k>0$ with the property that for each $k$ the ball $B(p_k,0,r_k)$ has compact closure in $M_k$. Let $Q_k=R(p_k,0)$ and suppose that $R(q,t)\le 4Q_k$ for all $q\in B(p_k,0,r_k)$ and for all $t\in [-t_k,0]$, and suppose that $t_kQ_k\rightarrow\infty$ and -$r_k^2Q_k\rightarrow\infty$ as $k\rightarrow\infty$. Then $Vol\,B(p_k,0,A/\sqrt{Q_k})< \nu(A/\sqrt{Q_k})^n$ for all $k$ +$r_k^2Q_k\rightarrow\infty$ as $k\rightarrow\infty$. Then $\mathit{Vol} B(p_k,0,A/\sqrt{Q_k})< \nu(A/\sqrt{Q_k})^n$ for all $k$ sufficiently large. + **Proof.** Suppose that the result fails for some $\nu>0$. Then there is a sequence $(M_k,g_k(t)),\ -t_k\le t\le 0$, of $n$-dimensional Ricci flows, points $p_k\in M_k$, and radii $r_k$ as in the statement of the lemma such that for every $A<\infty$ there is an arbitrarily -large $k$ with $Vol\,B(p_k,0,A/\sqrt{Q_k})\ge +large $k$ with $\mathit{Vol} B(p_k,0,A/\sqrt{Q_k})\ge \nu(A/\sqrt{Q_k})^n$. Pass to a subsequence so that for each $A<\infty$ we have -$$Vol\,B(p_k,0,A/\sqrt{Q_k})\ge\nu(A/\sqrt{Q_k})^n$$ for all $k$ + +$$ +\mathit{Vol} B(p_k,0,A/\sqrt{Q_k})\ge\nu(A/\sqrt{Q_k})^n +$$ + + for all $k$ sufficiently large. Consider now the flows $h_k(t)=Q_kg_k(Q_k^{-1}t)$, defined for $-Q_kt_k\le t\le 0$. Then for every $A<\infty$ for all $k$ sufficiently large we have $R_{h_k}(q,t)\le 4$ for all $q\in B_{h_k}(p_k,0,A)$ and all $t\in (-t_kQ_k,0]$. Also, for every $A<\infty$ for all $k$ sufficiently -large we have $Vol\,B(p_k,0,A)\ge\nu A^n$. According to -Theorem~ we can then pass to a subsequence that has a +large we have $\mathit{Vol} B(p_k,0,A)\ge\nu A^n$. According to +\entryref{a0212d0eefcf} we can then pass to a subsequence that has a geometric limit which is an ancient flow of complete Riemannian manifolds. Clearly, the time-slices of the limit have non-negative curvature operator, and the scalar curvature is bounded (by $4$) and is equal to $1$ at the base point of the limit. Also, the asymptotic volume ${\mathcal V}(0)\ge \nu$. + **Claim.** Suppose that $(M,g(t))$ is an ancient Ricci flow such that for each $t\le 0$ the Riemannian manifold $(M,g(t))$ is complete and has -bounded, non-negative curvature operator. Let ${\mathcal V}(t)$ be +bounded, non-negative curvature operator. Let ${\mathcal V}(t)$ be the asymptotic volume of the manifold $(M,g(t))$. -\begin{enumerate} -\item[(1)] The asymptotic volume ${\mathcal V}(t)$ is a non-increasing function -of $t$. \item[(2)] If ${\mathcal V}(t)=V>0$ then every metric ball + +- **(1)** The asymptotic volume ${\mathcal V}(t)$ is a non-increasing function +of $t$. +- **(2)** If ${\mathcal V}(t)=V>0$ then every metric ball $B(x,t,r)$ has volume at least $Vr^n$. -\end{enumerate} + -\begin{proof} -We begin with the proof of the first item. Fix $a0$ we have -$$B(p_0,b,r)\subset B(p_0,a,r+A(b-a)).$$ -On the other hand, since $dVol/dt=-RdVol$, it follows +$$ +B(p_0,b,r)\subset B(p_0,a,r+A(b-a)). +$$ + +On the other hand, since $d\mathit{Vol}/dt=-Rd\mathit{Vol}$, it follows that in the case of non-negative curvature that the volume of any open set is non-increasing in time. Consequently, -$$Vol_{g(b)}B(p_0,b,r)\le Vol_{g(a)}B(p_0,a,r+A(b-a)),$$ and hence -$$\frac{Vol_{g(b)}B(p_0,b,r)}{r^n}\le\frac{Vol_{g(a)}B(p_0,a,r+A(b-a))}{(r+A(b-a))^n}\frac{(r+A(b-a))^n}{r^n}.$$ + +$$ +\mathit{Vol}_{g(b)}B(p_0,b,r)\le \mathit{Vol}_{g(a)}B(p_0,a,r+A(b-a)), +$$ + + and hence + +$$ +\frac{\mathit{Vol}_{g(b)}B(p_0,b,r)}{r^n}\le\frac{\mathit{Vol}_{g(a)}B(p_0,a,r+A(b-a))}{(r+A(b-a))^n}\frac{(r+A(b-a))^n}{r^n}. +$$ + Taking the limit as $r\rightarrow \infty$ gives -$${\mathcal V}(b)\le {\mathcal V}(a).$$ + +$$ +{\mathcal V}(b)\le {\mathcal V}(a). +$$ The second item of the claim is immediate from the Bishop-Gromov -inequality (Theorem~). +inequality (\entryref{1a1b2c4fdc9b}). Now we return to the proof of the proposition. Under the assumption that there is a counterexample to the proposition for some $\nu>0$, we have constructed a limit that is an ancient Ricci flow with bounded, non-negative curvature with ${\mathcal V}(0)\ge \nu$. Since ${\mathcal V}(0)\ge \nu$, it follows from the claim that ${\mathcal -V}(t)\ge \nu$ for all $t\le 0$ and hence, also by the claim, we see +V}(t)\ge \nu$ for all $t\le 0$ and hence, also by the claim, we see that $(M,g(t))$ is $\nu$-non-collapsed for all $t$. This completes the proof that the limit is a $\nu$-solution. This contradicts -Theorem~ applied with $\kappa=\nu$, and proves the +\entryref{7468282bd017} applied with $\kappa=\nu$, and proves the proposition. -\end{proof} This proposition has two useful corollaries about balls in $\kappa$-solutions with volumes bounded away from zero. The first says that the normalized curvature is bounded on such balls. -**Corollary.** -For any $\nu>0$ there is a $C=C(\nu)<\infty$ depending only on the + +**Corollary.** For any $\nu>0$ there is a $C=C(\nu)<\infty$ depending only on the dimension $n$ such that the following holds. Suppose that $(M,g(t)),\ -\infty0$ are such that $Vol\,B(p,0,r)\ge \nu r^n$. Then $r^2R(q,0)\le C$ for all $q\in +curvature operator. Suppose $p\in M$, and $r>0$ are such that $\mathit{Vol} B(p,0,r)\ge \nu r^n$. Then $r^2R(q,0)\le C$ for all $q\in B(p,0,r)$. + **Proof.** Suppose that the result fails for some $\nu>0$. Then there is a sequence $(M_k,g_k(t))$ of $n$-dimensional Ricci flows, complete, -with bounded non-negative curvature operator and points $p_k\in +with bounded non-negative curvature operator and points $p_k\in M_k$, constants $r_k>0$, and points $q_k\in B(p_k,0,r_k)$ such that: -\begin{enumerate} -\item[(1)] $Vol\,B(p_k,0,r_k)\ge \nu r_k^n$, and -\item[(2)] setting $Q_k=R(q_k,0)$ we have $r_k^2Q_k\rightarrow\infty$ as $k\rightarrow\infty$. -\end{enumerate} -Using Lemma~ we can find points $q_k'\in B(p_k,0,2r_k)$ + +- **(1)** $\mathit{Vol} B(p_k,0,r_k)\ge \nu r_k^n$, and +- **(2)** setting $Q_k=R(q_k,0)$ we have $r_k^2Q_k\rightarrow\infty$ as $k\rightarrow\infty$. + +Using \entryref{351728d82639} we can find points $q_k'\in B(p_k,0,2r_k)$ and constants $s_k\le r_k$, such that setting $Q_k'=R(q_k',0)$ we -have $Q_k's_k^2=Q_kr_k^2$ and $R(q,0)<4Q_k'$ for all $q\in -B(q_k',0,s_k)$. Of course, $Q'_ks_k^2\rightarrow\infty$ as +have $Q_k's_k^2=Q_kr_k^2$ and $R(q,0)<4Q_k'$ for all $q\in +B(q_k',0,s_k)$. Of course, $Q'_ks_k^2\rightarrow\infty$ as $k\rightarrow\infty$. Since $d_0(p_k,q'_k)<2r_k$, we have $B(p_k,0,r_k)\subset B(q_k',0,3r_k)$ so that -$$Vol\,B(q_k',0,3r_k)\ge Vol\,B(p_k,0,r_k)\ge \nu -r_k^n=(\nu/3^n)(3r_k)^n.$$ Since the sectional curvatures of + +$$ +\mathit{Vol} B(q_k',0,3r_k)\ge \mathit{Vol} B(p_k,0,r_k)\ge \nu +r_k^n=(\nu/3^n)(3r_k)^n. +$$ + + Since the sectional curvatures of $(M,g_k(0))$ are non-negative, it follows from the Bishop-Gromov -inequality (Theorem~) that $Vol\,B(q_k',0,s)\ge (\nu/3^n)s^n$ for any $s\le s_k$. +inequality (\entryref{1a1b2c4fdc9b}) that $\mathit{Vol} B(q_k',0,s)\ge (\nu/3^n)s^n$ for any $s\le s_k$. -Of course, by Corollary~, we have $R(q,t)<4Q_k'$ for +Of course, by \entryref{36dd2f6fa9c2}, we have $R(q,t)<4Q_k'$ for all $t\le 0$ and all $q\in B(q'_k,0,s_k)$. Now consider the sequence -of based, rescaled flows $$(M_k,Q'_kg(Q_k'^{-1}t),(q_k',0)).$$ In +of based, rescaled flows + +$$ +(M_k,Q'_kg(Q_k'^{-1}t),(q_k',0)). +$$ + + In these manifolds all balls centered at $(q_k',0)$ of radii at most $\sqrt{Q_k}s_k$ are $(\nu/3^n)$ non-collapsed. Also, the curvatures of these manifolds @@ -2714,127 +3295,162 @@ geometric limit. as $k\rightarrow\infty$ the asymptotic volume of this limit is at least $\nu/3^n$. But this geometric limit is a $\nu/3^n$-non-collapsed ancient solution with non-negative curvature -operator with scalar curvature bounded by $4$. This contradicts -Theorem~. +operator with scalar curvature bounded by $4$. This contradicts +\entryref{7468282bd017}. The second corollary gives curvature bounds at all points in terms of the distance to the center of the ball. -**Corollary.** -Fix the dimension $n$. Given $\nu>0$, there is a function + +**Corollary.** Fix the dimension $n$. Given $\nu>0$, there is a function $K(A)<\infty$, defined for $A\in (0,\infty)$, such that if $(M,g(t)),\ -\infty **Proof.** Fix $q\in M$ and let $d=d_0(p,q)$. We have -$$Vol\,B(q,0,r+d)\ge Vol\,B(p,0,r)\ge \nu r^n=\frac{\nu}{(1+(d/r))^n}(r+d)^n.$$ Let + +$$ +\mathit{Vol} B(q,0,r+d)\ge \mathit{Vol} B(p,0,r)\ge \nu r^n=\frac{\nu}{(1+(d/r))^n}(r+d)^n. +$$ + + Let $K(A)=C(\nu/^n)$, where $C$ is the constant provided by the previous corollary. The result is immediate from the previous corollary. -## Compactness of the space of $3$-dimensional -$\kappa$-solutions +## Compactness of the space of $3$-dimensional $\kappa$-solutions This section is devoted to proving the following result. -**Theorem.** -Let $(M_k,g_k(t),(p_k,0))$ be a sequence of based $3$-dimensional + +**Theorem.** Let $(M_k,g_k(t),(p_k,0))$ be a sequence of based $3$-dimensional $\kappa$-solutions satisfying $R(p_k,0)=1$. Then there is a subsequence converging smoothly to a based $\kappa$-solution. + The main point in proving this theorem is to establish the uniform curvature bounds given in the next lemma. -**Lemma.** -For each $r<\infty$ there is a constant $C(r)<\infty$, such that the -following holds. Let $(M,g(t),(p,0))$ be a based $3$-dimensional + +**Lemma.** For each $r<\infty$ there is a constant $C(r)<\infty$, such that the +following holds. Let $(M,g(t),(p,0))$ be a based $3$-dimensional $\kappa$-solution satisfying $R(p,0)=1$. Then $R(q,0)\le C(r)$ for all $q\in B(p,0,r)$. + **Proof.** Fix a based $3$-dimensional $\kappa$-solution $(M,g(t),(p,0))$. By -Theorem~ we have -$$sup_{q\in M}d_{0}(p,q)^2R(q,0)=\infty.$$ +\entryref{5d9f1b75170d} we have + +$$ +\mathit{sup}_{q\in M}d_{0}(p,q)^2R(q,0)=\infty. +$$ + Let $q$ be a closest point to $p$ satisfying -$$d_0(p,q)^2R(q,0)=1.$$ + +$$ +d_0(p,q)^2R(q,0)=1. +$$ We set $d=d_{0}(p,q)$, and we set $Q=R(q,0)$. Of course, $d^2Q=1$. We carry this notation and these assumptions through the next five claims. The goal of these claims is to show that $R(q',0)$ is uniformly bounded for $q'$ near $(p,0)$ so that in fact the distance $d$ from the point $q$ to $p$ is uniformly bounded from below by a -positive constant (see Claim~ for a more precise +positive constant (see \entryref{7c6182738b78} for a more precise statement). Once we have this the lemma will follow easily. To establish this uniform bound requires a sequence of claims. -**Claim.** -There is a universal (i.e., independent of the $3$-dimensional + +**Claim.** There is a universal (i.e., independent of the $3$-dimensional $\kappa$-solution) upper bound $C$ for $R(q',0)/R(q,0)$ for all $q'\in B(q,0,2d)$. + -\begin{proof} -Suppose not. Then there is a sequence $(M_k,g_k(t),(p_k,0))$ of +**Proof.** Suppose not. Then there is a sequence $(M_k,g_k(t),(p_k,0))$ of $3$-dimensional $\kappa$-solutions with $R(p_k,0)=1$, points $q_k$ in $(M_k,g_k(0))$ closest to $p_k$ satisfying $d_k^2R(q_k,0)=1$, where $d_k=d_0(p_k,q_k)$, and points $q_k'\in B(q_k,0,2d_k)$ with -$$lim_{k\rightarrow\infty}(2d_k)^2R(q_k',0)=\infty.$$ Then according -to Corollary~ for every $\nu>0$ for all $k$ + +$$ +\mathit{lim}_{k\rightarrow\infty}(2d_k)^2R(q_k',0)=\infty. +$$ + + Then according +to \entryref{3227c39e1892} for every $\nu>0$ for all $k$ sufficiently large, we have + $$ -Vol\,B(q_k,0,2d_k)<\nu (2d_k)^3. +\mathit{Vol} B(q_k,0,2d_k)<\nu (2d_k)^3. $$ + Therefore, by passing to a subsequence, we can assume that for each $\nu>0$ $$ - -Vol\,B(q_k,0,2d_k)< \nu (2d_k)^3 +\mathit{Vol} B(q_k,0,2d_k)< \nu (2d_k)^3 $$ + for all $k$ sufficiently large. Let $\omega_3$ be the volume of the unit ball in -$\Ar^3$. Then for all $k$ sufficiently large, $Vol\,B(q_k,0,2d_k)<[\omega_3/2](2d_k)^3$. Since the sectional +$\Ar^3$. Then for all $k$ sufficiently large, $\mathit{Vol} B(q_k,0,2d_k)<[\omega_3/2](2d_k)^3$. Since the sectional curvatures of $(M_k,g_k(0))$ are non-negative, by the Bishop-Gromov -inequality (Theorem~), it follows that for every +inequality (\entryref{1a1b2c4fdc9b}), it follows that for every $k$ sufficiently large there is $r_k<2d_k$ such that $$ - -Vol\,B(q_k,0,r_k)=[\omega_3/2]r_k^3. +\mathit{Vol} B(q_k,0,r_k)=[\omega_3/2]r_k^3. $$ + Of course, -because of Equation~() we see that $lim_{k\rightarrow\infty}r_k/d_k=0$. Then, according to -Corollary~, we have for all $q\in M_k$ +because of Equation (9.28) we see that $\mathit{lim}_{k\rightarrow\infty}r_k/d_k=0$. Then, according to +\entryref{329d0175aed8}, we have for all $q\in M_k$ + +$$ +(r_k+d_{g_k(0)}(q_k,q))^2R(q,0)\le K(d_{g_k(0)}(q_k,q)/r_k), +$$ -$$(r_k+d_{g_k(0)}(q_k,q))^2R(q,0)\le K(d_{g_k(0)}(q_k,q)/r_k),$$ -where $K$ is as given in Corollary~. Form the sequence +where $K$ is as given in \entryref{329d0175aed8}. Form the sequence $(M_k,g'_k(t),(q_k,0))$, where $g'_k(t)=r_k^{-2}g_k(r_k^2t)$. This -is a sequence of based Ricci flows. For each $A<\infty$ we have -$$(1+A)^2R_{g'_k}(q,0)\le K(A)$$ for all +is a sequence of based Ricci flows. For each $A<\infty$ we have + +$$ +(1+A)^2R_{g'_k}(q,0)\le K(A) +$$ + + for all $q\in B_{g_k'(0)}(q_k,0,A)$. Hence, by the consequence of Hamilton's -Harnack inequality (Corollary~) -$$R_{g'_k}(q,t)\le K(A),$$ -for all $(q,t)\in B_{g_k'(0)}(q_k,0,A)\times(-\infty,0]$. Using +Harnack inequality (\entryref{36dd2f6fa9c2}) + +$$ +R_{g'_k}(q,t)\le K(A), +$$ + +for all $(q,t)\in B_{g_k'(0)}(q_k,0,A)\times(-\infty,0]$. Using this and the fact that all the flows are $\kappa$-non-collapsed, -Theorem~ implies that, after passing to a +\entryref{a0212d0eefcf} implies that, after passing to a subsequence, the sequence $(M_k,g_k'(t),(q_k,0))$ converges geometrically to a limiting Ricci flow $(M_\infty,g_\infty(t),(q_\infty,0))$ consisting of non-negatively curved, complete manifolds $\kappa$-non-collapsed on all scales (though possibly with unbounded curvature). -Furthermore, Equation~() passes to the limit to give +Furthermore, Equation (9.29) passes to the limit to give $$ - -Vol\,B_{g_\infty}(q_\infty,0,1)=\omega_3/2. +\mathit{Vol} B_{g_\infty}(q_\infty,0,1)=\omega_3/2. $$ Since $r_k/d_k\rightarrow 0$ as $k\rightarrow\infty$ and since $R_{g_k}(q_k,0)=d_k^{-2}$, we see that $R_{g_\infty}(q_\infty,0)=0$. By the strong maximum principle for -scalar curvature (Theorem~), this implies that +scalar curvature (\entryref{016cf36686b2}), this implies that the limit $(M_\infty,g_\infty(0))$ is flat. But -Equation~() tells us that this limit is not $\Ar^3$. Since +Equation (9.30) tells us that this limit is not $\Ar^3$. Since it is complete and flat, it must be a quotient of $\Ar^3$ by an action of a non-trivial group of isometries acting freely and properly discontinuously. But the quotient of $\Ar^3$ by any @@ -2853,27 +3469,30 @@ volume of the quotient is zero.] We have now shown that $(M_\infty,g_\infty(0))$ has zero curvature and zero asymptotic volume. But this implies that it is not $\kappa$-non-collapsed on all scales, which is a contradiction. This contradiction completes the proof of -Claim~. +\entryref{ea9b285e40cb}. This claim establishes the existence of a universal constant $C<\infty$ (universal in the sense that it is independent of the $3$-dimensional $\kappa$-solution) such that $R(q',0)\le CQ$ for all $q'\in B(q,0,2d)$. Since the curvature of $(M,g(t))$ is non-negative and bounded, we know from the -Harnack inequality (Corollary~) that $R(q',t)\le CQ$ for all -$q'\in B(q,0,2d)$ and all $t\le 0$. Hence, the Ricci curvature $Ric(q',t)\le CQ$ for all $q'\in B(q,0,2d)$ and all $t\le 0$. +Harnack inequality (\entryref{36dd2f6fa9c2}) that $R(q',t)\le CQ$ for all +$q'\in B(q,0,2d)$ and all $t\le 0$. Hence, the Ricci curvature $\mathit{Ric}(q',t)\le CQ$ for all $q'\in B(q,0,2d)$ and all $t\le 0$. + **Claim.** Given any constant $c>0$ there is a constant $\widetilde C=\widetilde C(c)$, depending only on $c$ and not on the $3$-dimensional $\kappa$-solution, so that -$$d_{g(-cQ^{-1})}(p,q)\le \tilde CQ^{-1/2}.$$ + +$$ +d_{g(-cQ^{-1})}(p,q)\le \tilde CQ^{-1/2}. +$$ + **Proof.** Let $\gamma\colon[0,d]\to M$ be a $g(0)$-geodesic from $p$ to $q$, parameterized at unit speed. Denote by $\ell_t(\gamma)$ the length of $\gamma$ under the metric $g(t)$. We have $d_t(p,q)\le \ell_t(\gamma)$. We estimate $\ell_t(\gamma)$ using the fact that -$|Ric|\le CQ$ on the image of $\gamma$ at all times. - -$$ +$|\mathit{Ric}|\le CQ$ on the image of $\gamma$ at all times. $$ \begin{aligned} @@ -2881,103 +3500,133 @@ $$ \frac{d}{dt}\left(\int_0^d\sqrt{\langle\gamma'(s),\gamma'(s)\rangle_{g(t)}}ds\right) \Bigl|_{t=t_0}\Bigr. \\ -& = & \int_0^{d}\frac{-Ric_{g(t_0)}(\gamma'(s),\gamma'(s))} +& = & \int_0^{d}\frac{-\mathit{Ric}_{g(t_0)}(\gamma'(s),\gamma'(s))} {\sqrt{\langle\gamma'(s),\gamma'(s)\rangle_{g(t_0)}}}ds \\ & \ge & -CQ\int_0^d|\gamma'(s)|_{g(t_0)}ds=-CQ\ell_{t_0}(\gamma). \end{aligned} $$ +Integrating yields + +$$ +\ell_{-t}(\gamma)\le e^{CQt}\ell_0(\gamma)=e^{CQt}Q^{-1/2}. $$ -Integrating yields -$$\ell_{-t}(\gamma)\le e^{CQt}\ell_0(\gamma)=e^{CQt}Q^{-1/2}.$$ (Recall $d^2Q=1$.) Plugging in $t=cQ^{-1}$ gives us -$$d_{-cQ^{-1}}(p,q)\le \ell_{-cQ^{-1}}(\gamma)\le e^{cC}Q^{-1/2}.$$ + +$$ +d_{-cQ^{-1}}(p,q)\le \ell_{-cQ^{-1}}(\gamma)\le e^{cC}Q^{-1/2}. +$$ + Setting $\widetilde C=e^{cC}$ completes the proof of the claim. -The integrated form of Hamilton's Harnack inequality, -Theorem~, tells us that -$$log\left(\frac{R(p,0)}{R(q,-cQ^{-1})}\right)\ge --\frac{d_{-cQ^{-1}}^2(p,q)}{2cQ^{-1}}.$$ According to the above +The integrated form of Hamilton's Harnack inequality, +\entryref{6b21448ff214}, tells us that + +$$ +\mathit{log}\left(\frac{R(p,0)}{R(q,-cQ^{-1})}\right)\ge +-\frac{d_{-cQ^{-1}}^2(p,q)}{2cQ^{-1}}. +$$ + + According to the above claim, this in turn tells us -$$log\left(\frac{R(p,0)}{R(q,-cQ^{-1})}\right)\ge -\widetilde -C^2/2c.$$ Since $R(p,0)=1$, it immediately follows that -$R(q,-cQ^{-1})\le exp(\widetilde C^2/(2c))$. -**Claim.** -There is a universal (i.e., independent of the $3$-dimensional -$\kappa$-solution) upper bound for $Q=R(q,0)$. +$$ +\mathit{log}\left(\frac{R(p,0)}{R(q,-cQ^{-1})}\right)\ge -\widetilde +C^2/2c. +$$ + + Since $R(p,0)=1$, it immediately follows that +$R(q,-cQ^{-1})\le \mathit{exp}(\widetilde C^2/(2c))$. + + +**Claim.** There is a universal (i.e., independent of the $3$-dimensional +$\kappa$-solution) upper bound for $Q=R(q,0)$. + **Proof.** Let $G'=QG$ and $t'=Qt$. Then $R_{G'}(q',0)\le C$ for all $q'\in B_{G'}(q,0,2)$. Consequently, $R_{G'}(q',t')\le C$ for all $q'\in -B_{G'}(q,0,2)$ and all $t'\le 0$. Thus, by Shi's derivative -estimates (Theorem~)applied with $T=2$ and $r=2$, there is +B_{G'}(q,0,2)$ and all $t'\le 0$. Thus, by Shi's derivative +estimates (\entryref{ed4b72caf9f9})applied with $T=2$ and $r=2$, there is a universal constant $C_1$ such that for all $-1\le t'\le 0$ -$$|\triangle Rm_{G'}(q,t')|_{G'}\le C_1,$$ + +$$ +|\triangle \mathit{Rm}_{G'}(q,t')|_{G'}\le C_1, +$$ + (where the Laplacian is taken with respect to the metric $G'$). Rescaling by $Q^{-1}$ we see that for all $-Q^{-1}\le t\le 0$ we have -$$|\triangle Rm_{G}(q,t)|\le C_1Q^2,$$ + +$$ +|\triangle \mathit{Rm}_{G}(q,t)|\le C_1Q^2, +$$ + where the Laplacian is taken with respect to the metric $G$. Since -the metric is non-negatively curved, by Corollary~ we -have $2|Ric(q,t)|^2\le 2Q^2$ for all $t\le 0$. +the metric is non-negatively curved, by \entryref{36dd2f6fa9c2} we +have $2|\mathit{Ric}(q,t)|^2\le 2Q^2$ for all $t\le 0$. From these two facts we conclude from -the flow equation~() that there is a constant $10$ and $C_1<\infty$ + +**Claim.** There are universal constants $\delta>0$ and $C_1<\infty$ (independent of the based $3$-dimensional $\kappa$-solution $(M,g(t),(p,0))$ with $R(p,0)=1$) such that $d(p,q)\ge \delta$. In -addition, $R(q',t)\le C_1$ for all $q'\in B(p,0,d)$ and all $t\le +addition, $R(q',t)\le C_1$ for all $q'\in B(p,0,d)$ and all $t\le 0$. + **Proof.** Since, according to the previous claim, $Q$ is universally bounded above and $d^2Q=1$, the existence of $\delta>0$ as required is clear. Since $B(p,0,d)\subset B(q,0,2d)$, since $R(q',0)/R(q,0)$ is -universally bounded on $B(q,0,2d)$ by Claim~, and since -$R(q,0)$ is universally bounded by Lemma~, the second +universally bounded on $B(q,0,2d)$ by \entryref{ea9b285e40cb}, and since +$R(q,0)$ is universally bounded by \entryref{f61f72020102}, the second statement is clear for all $(q',0)\in B(p,0,d)\subset B(q,0,2d)$. Given this, the fact that the second statement holds for all $(q',t)\in B(p,0,d)\times(-\infty,0]$ then follows immediately from the derivative inequality for $\partial R(q,t)/\partial t$, -Corollary~. +\entryref{36dd2f6fa9c2}. This, in turn, leads immediately to: -**Corollary.** Fix $\delta>0$ the universal constant of the last + +**Corollary.** Fix $\delta>0$ the universal constant of the last claim. Then $R(q',t)\le \delta^{-2}$ for all $q'\in B(p,0,\delta)$ and all $t\le 0$. + -Now we return to the proof of Lemma~. Since +Now we return to the proof of \entryref{ee314aacd207}. Since $(M,g(t))$ is $\kappa$-non-collapsed, it follows from the previous -corollary that $Vol\,B(p,0,\delta)\ge \kappa \delta^3$. Hence, -according to Corollary~ for each $A<\infty$ there is a +corollary that $\mathit{Vol} B(p,0,\delta)\ge \kappa \delta^3$. Hence, +according to \entryref{329d0175aed8} for each $A<\infty$ there is a constant $K(A)$ such that $R(q',0)\le K(A/\delta)/(\delta+A)^2$ for all $q'\in B(p,0,A)$. Since $\delta$ is a universal positive -constant, this completes the proof of Lemma~. -\end{proof} +constant, this completes the proof of \entryref{ee314aacd207}. -Now let us turn to the proof of Theorem~, the +Now let us turn to the proof of \entryref{49e865228b19}, the compactness result for $\kappa$-solutions. **Proof.** Let $(M_k,g_k(t),(p_k,0))$ be a sequence of based $3$-dimensional $\kappa$-solutions with $R(p_k,0)=1$ for all $k$. The immediate -consequence of Lemma~ and Corollary~ +consequence of \entryref{ee314aacd207} and \entryref{36dd2f6fa9c2} is the following. For every $r<\infty$ there is a constant $C(r)<\infty$ such that $R(q,t)\le C(r)$ for all $q\in B(p_k,0,r)$ and for all $t\le 0$. Of course, since, in addition, the elements in -the sequence are $\kappa$-non-collapsed, by Theorem~ +the sequence are $\kappa$-non-collapsed, by \entryref{a0212d0eefcf} this implies that there is a subsequence of the $(M_k,g_k(t),(p_k,0))$ that converges geometrically to an ancient flow $(M_\infty,g_\infty(t),(p_\infty,0))$. Being a geometric limit @@ -2987,53 +3636,45 @@ curvature. Also, it is not flat since, by construction, $R(p_\infty,0)=1$. Of course, it also follows from the limiting procedure that $\partial R(q,t)/\partial t\ge 0$ for every $(q,t)\in M_\infty\times(-\infty,0]$. Thus, according to -Corollary~ the limit +\entryref{351e4ac14e4e} the limit $(M_\infty,g_\infty(t))$ has bounded curvature for each $t\le 0$. Hence, the limit is a $\kappa$-solution. This completes the proof of -Theorem~. +\entryref{49e865228b19}. -**Corollary.** -Given $\kappa>0$, there is $C<\infty$ such that for any + +**Corollary.** Given $\kappa>0$, there is $C<\infty$ such that for any $3$-dimensional $\kappa$-solution $(M,g(t)),\ -\infty **Proof.** Notice that the two inequalities are scale invariant. Thus, this result is immediate from the compactness theorem, -Theorem~. +\entryref{49e865228b19}. -Because of Proposition~, and the fact that the previous +Because of \entryref{9c54e0d0c7d8}, and the fact that the previous corollary obviously holds for any shrinking family of round metrics, we can take the constant $C$ in the above corollary to be independent of $\kappa>0$. -Notice that, using Equation~(), we can rewrite the second +Notice that, using Equation (3.5), we can rewrite the second inequality in the above corollary as -$$ - $$ \begin{aligned} -sup_{(x,t)}\frac{|\triangle R+2|Ric|^2|}{R(x,t)^2} +**Definition.** Fix constants $0<\epsilon<1/2$ and $C<\infty$. Let $(M,g)$ be a Riemannian $3$-manifold. A $(C,\epsilon)$-cap in $(M,g)$ is an open submanifold $({\mathcal C},g|_{\mathcal C})$ together with an open submanifold $N\subset {\mathcal C}$ with the following properties: -\begin{enumerate} -\item[(1)] ${\mathcal C}$ is diffeomorphic either to an open $3$-ball or to a + +- **(1)** ${\mathcal C}$ is diffeomorphic either to an open $3$-ball or to a punctured $\Ar P^3$. -\item[(2)] $N$ is an $\epsilon$-neck with compact complement in +- **(2)** $N$ is an $\epsilon$-neck with compact complement in ${\mathcal C}$. -\item[(3)] $\overline Y={\mathcal C}\setminus N$ is a compact submanifold with boundary. -Its interior, $Y$, is called +- **(3)** $\overline Y={\mathcal C}\setminus N$ is a compact submanifold with boundary. +Its interior, $Y$, is called the *core* of ${\mathcal C}$. The frontier of $Y$, which is $\partial \overline Y$, - is a central + is a central $2$-sphere of an $\epsilon$-neck contained in ${\mathcal C}$. -\item[(4)] The scalar curvature $R(y)>0$ for every $y\in {\mathcal C}$ and -$$ diam({\mathcal C},g|_{\mathcal C})< C \left(sup_{y\in -{\mathcal C}}R(y)\right)^{-1/2}.$$ -\item[(5)] $sup_{x,y\in {\mathcal C}}\left[ R(y)/R(x)\right]< C$. -\item[(6)] $Vol\,{\mathcal C} < C(sup_{y\in {\mathcal C}}R(y))^{-3/2}$. -\item[(7)] For any $y\in Y$ let $r_y$ be defined so -that $sup_{y'\in B(y,r_y)}R(y')=r_y^{-2}$. Then for each $y\in +- **(4)** The scalar curvature $R(y)>0$ for every $y\in {\mathcal C}$ and + +$$ +\mathit{diam}({\mathcal C},g|_{\mathcal C})< C \left(\mathit{sup}_{y\in +{\mathcal C}}R(y)\right)^{-1/2}. +$$ + +- **(5)** $\mathit{sup}_{x,y\in {\mathcal C}}\left[ R(y)/R(x)\right]< C$. +- **(6)** $\mathit{Vol} {\mathcal C} < C(\mathit{sup}_{y\in {\mathcal C}}R(y))^{-3/2}$. +- **(7)** For any $y\in Y$ let $r_y$ be defined so +that $\mathit{sup}_{y'\in B(y,r_y)}R(y')=r_y^{-2}$. Then for each $y\in Y$, the ball $B(y,r_y)$ lies in ${\mathcal C}$ and indeed has compact closure in ${\mathcal C}$. Furthermore, -$$C^{-1} + + **Remark.** If the ball $B(y,r_y)$ meets the complement of the core of ${\mathcal C}$ then it contains a point whose scalar curvature is close to $R(x)$, and hence $r_y$ is bounded above by, say @@ -3089,54 +3746,70 @@ C}$. Implicitly, we always orient the $\epsilon$-neck structure on $N$ so that the closure of its negative end meets the core of ${\mathcal -C}$. See Fig.~ in the Introduction. +C}$. See Fig. 0.1 in the Introduction. + Condition (8) in the above definition may seem unnatural, but here is the reason for it. -**Claim.** -Suppose that $(M,g(t))$ is a Ricci flow and that $\left({\mathcal + +**Claim.** Suppose that $(M,g(t))$ is a Ricci flow and that $\left({\mathcal C},g(t)|_{\mathcal C}\right)$ is a subset of a $t$ time-slice. Then Condition (8) above is equivalent to -$$sup_{(x,t)\in {\mathcal C}}\frac{\left|\frac{\partial R(x,t)}{\partial -t}\right|}{R^2(x,t)} + +**Proof.** This is immediate from Equation (3.5). Notice that the definition of a $(C,\epsilon)$-cap is a scale invariant notion. -**Definition.** -Fix a positive constant $C$. A compact connected Riemannian manifold + +**Definition.** Fix a positive constant $C$. A compact connected Riemannian manifold $(M,g)$ is called *a $C$-component* if -\begin{enumerate} -\item[(1)] $M$ is diffeomorphic to either $S^3$ or $\Ar P^3$. -\item[(2)] $(M,g)$ has positive sectional curvature. -\item[(3)] $$C^{-1}<\frac{inf_{P}K(P)}{sup_{y\in M}R(y)}$$ + +- **(1)** $M$ is diffeomorphic to either $S^3$ or $\Ar P^3$. +- **(2)** $(M,g)$ has positive sectional curvature. +- **(3)** + +$$ +C^{-1}<\frac{\mathit{inf}_{P}K(P)}{\mathit{sup}_{y\in M}R(y)} +$$ + where $P$ varies over all $2$-planes in $TX$ (and $K(P)$ denotes the sectional curvature in the $P$-direction). -\item[(4)] $$C^{-1}sup_{y\in M}\left(R(y)^{-1/2}\right) -**Definition.** -Fix $\epsilon>0$. Let $(M,g)$ be a compact, connected $3$-manifold. -Then $(M,g)$ is {\em within $\epsilon$ of round in the -$C^{[1/\epsilon]}$-topology} if there exist a constant $R>0$, a + +**Definition.** Fix $\epsilon>0$. Let $(M,g)$ be a compact, connected $3$-manifold. +Then $(M,g)$ is *within $\epsilon$ of round in the +$C^{[1/\epsilon]}$-topology* if there exist a constant $R>0$, a compact manifold $(Z,g_0)$ of constant curvature $+1$, and a diffeomorphism $\varphi\colon Z\to M$ with the property that the pull back under $\varphi$ of $Rg$ is within $\epsilon$ in the $C^{[1/\epsilon]}$-topology of $g_0$. + Notice that both of these notions are scale invariant notions. + **Definition.** Fix $C<\infty$ and $\epsilon>0$. For any Riemannian manifold -$(M,g)$, an open neighborhood $U$ of a point $x\in M$ is a *$(C,\epsilon)$-canonical neighborhood* if one of the following holds: -\begin{enumerate} -\item[(1)] $U$ is an $\epsilon$-neck in $(M,g)$ centered at $x$. -\item[(2)] $U$ a $(C,\epsilon)$-cap in $(M,g)$ whose core contains $x$. -\item[(3)] $U$ is a $C$-component of $(M,g)$. -\item[(4)] $U$ is an $\epsilon$-round component of $(M,g)$. -\end{enumerate} +$(M,g)$, an open neighborhood $U$ of a point $x\in M$ is a *$(C,\epsilon)$-canonical neighborhood* if one of the following holds: + +- **(1)** $U$ is an $\epsilon$-neck in $(M,g)$ centered at $x$. +- **(2)** $U$ a $(C,\epsilon)$-cap in $(M,g)$ whose core contains $x$. +- **(3)** $U$ is a $C$-component of $(M,g)$. +- **(4)** $U$ is an $\epsilon$-round component of $(M,g)$. + Whether or not a point $x\in M$ has a $(C,\epsilon)$-canonical neighborhood in $M$ is a scale invariant notion. @@ -3144,61 +3817,61 @@ neighborhood in $M$ is a scale invariant notion. The notion of $(C,\epsilon)$-canonical neighborhoods is sufficient for some purposes, but often we need a stronger notion. -**Definition.** -Fix constants $C<\infty$ and $\epsilon>0$. Let $({\mathcal M},G)$ be -a generalized Ricci flow. An *evolving $\epsilon$-neck defined + +**Definition.** Fix constants $C<\infty$ and $\epsilon>0$. Let $({\mathcal M},G)$ be +a generalized Ricci flow. An *evolving $\epsilon$-neck defined for an interval of normalized time of length $t'>0$* centered at a point $x\in -{\mathcal M}$ with $**t**(x)=t$ is an embedding $\psi\colon +{\mathcal M}$ with $\mathbf{t}(x)=t$ is an embedding $\psi\colon S^2\times (-\epsilon^{-1},\epsilon^{-1})\stackrel{\cong}{\to} N\subset M_t$ with $x\in \psi(S^2\times\{0\})$ satisfying the following properties: -\begin{enumerate} -\item[(1)] There is an embedding $N\times (t-R(x)^{-1}t',t]\to {\mathcal + +- **(1)** There is an embedding $N\times (t-R(x)^{-1}t',t]\to {\mathcal M}$ compatible with time and the vector field. -\item[(2)] The pullback under $\psi$ of the one-parameter family of +- **(2)** The pullback under $\psi$ of the one-parameter family of metrics on $N$ determined by restricting $R(x)G$ to the image of this embedding is within $\epsilon$ in the $C^{[1/\epsilon]}$-topology of the standard family $(h(t),ds^2), -t' Whether or not a point $x$ in a generalized Ricci flow has a strong $(C,\epsilon)$-canonical neighborhood is a scale invariant notion. -**Proposition.** -The following holds for any $\epsilon<1/4$ and any $C<\infty$. Let + +**Proposition.** The following holds for any $\epsilon<1/4$ and any $C<\infty$. Let $({\mathcal M},G)$ be a generalized Ricci flow and let $x\in -{\mathcal M}$ be a point with $**t**(x)=t$. +{\mathcal M}$ be a point with $\mathbf{t}(x)=t$. -(1) Suppose that $U\subset M_t$ is a $(C,\epsilon)$-canonical +(1) Suppose that $U\subset M_t$ is a $(C,\epsilon)$-canonical neighborhood for $x$. Then for any horizontal metric $G'$ sufficiently close to $G|_U$ in the $C^{[1/\epsilon]}$-topology, -$(U,G'|_U)$ is a $(C,\epsilon)$ neighborhood for any $x'\in U$ +$(U,G'|_U)$ is a $(C,\epsilon)$ neighborhood for any $x'\in U$ sufficiently close to $x$. (2) Suppose that in $({\mathcal M},G)$ there is an evolving $\epsilon$-neck $U$ centered at $(x,t)$ defined for an interval of -normalized time of length $a>1$. Then any Ricci flow on $U\times +normalized time of length $a>1$. Then any Ricci flow on $U\times (t-aR(x,t)^{-1},t]$ sufficiently close in the $C^{[1/\epsilon]}$-topology to the pullback of $G$ contains a strong $\epsilon$-neck centered at $(x,t)$. @@ -3215,8 +3888,9 @@ metrics on $(U,g)$ that is a strong $\epsilon$-neck centered at $(x,0)$ and $R_g(x,0)=1$. Then any one-parameter family $g'(t)$ within $\delta$ in the $C^{[1/\epsilon]}$-topology of $g$ with $R_{g'}(x,0)=1$ is a strong $\epsilon'$-neck. + -**Proof.** Since $\epsilon<1/4$, the diameter of $(U,g)$, the volume of +**Proof.** Since $\epsilon<1/4$, the diameter of $(U,g)$, the volume of $(U,g)$, the supremum over $x\in U$ of $R(x)$, the supremum over $x$ and $y$ in $U$ of $R(y)/R(x)$, and the infimum over all $2$-planes $P$ in ${\mathcal H}TU$ of $K(P)$ are all continuous functions of @@ -3276,7 +3950,7 @@ $\epsilon$-necks. Also, since $R(x)\le 2$ the result is clear for $\epsilon$-round components and $C$-components as well. Lastly, we consider a $(C,\epsilon)$-cap $U$ whose core $Y$ contains $x$. Clearly, -since $R(x)$ is bounded above by $2$, for $\delta>0$ sufficiently +since $R(x)$ is bounded above by $2$, for $\delta>0$ sufficiently small, any metric $g'$ within $\delta$ of $g$ will satisfy the diameter, volume and curvature and the derivative of the curvature inequalities with $C'$ replacing $C$. Let $N$ be the $\epsilon$-neck @@ -3294,9 +3968,10 @@ $(C',\epsilon')$-cap with the same core. The fourth statement is immediate. -**Corollary.** -In an ancient solution $(M,g(t))$ the set of points that are centers + +**Corollary.** In an ancient solution $(M,g(t))$ the set of points that are centers of strong $\epsilon$-necks is an open subset + **Proof.** Let $T$ be the final time of the flow. Suppose that $(x,t)$ is the center of a strong $\epsilon$-neck $U\times (t-R(x,t)^{-1},t]\subset @@ -3305,26 +3980,25 @@ forwards until the final time $T$ giving an embedding $U\times (-\infty,T]\to M\times (-\infty,T]$. There is $a>1$ such that for all $t'$ sufficiently close to $t$ the restriction of this embedding determines an evolving $\epsilon$-neck centered at $(x,t')$ defined -for an interval of normalized time of length $a$. Composing this +for an interval of normalized time of length $a$. Composing this neck structure with a self-diffeomorphism of $U$ moving $x'$ to $x$, as described above, shows that all $(x',t')$ sufficiently close to $(x,t)$ are centers of strong $\epsilon$-necks. -**Definition.** -{\em An $\epsilon$-tube ${\mathcal T}$} in + +**Definition.** *An $\epsilon$-tube ${\mathcal T}$* in a Riemannian $3$-manifold $M$ is a submanifold diffeomorphic to the product of $S^2$ with a non-degenerate interval with the following properties: -\begin{enumerate} -\item[(1)] Each boundary component $S$ of ${\mathcal T}$ is the central + +- **(1)** Each boundary component $S$ of ${\mathcal T}$ is the central $2$-sphere of an $\epsilon$-neck $N(S)$ in $M$. -\item[(2)] ${\mathcal T}$ +- **(2)** ${\mathcal T}$ is a union of $\epsilon$-necks and the closed half $\epsilon$-necks whose boundary sphere is a component of $\partial{\mathcal T}$. Furthermore, the central $2$-sphere of each of the $\epsilon$-necks is isotopic in ${\mathcal T}$ to the $S^2$-factors of the product structure. -\end{enumerate} An *open $\epsilon$-tube* is one without boundary. It is a union of $\epsilon$-necks with the central spheres that are isotopic to @@ -3334,16 +4008,16 @@ A *$C$-capped $\epsilon$-tube* in $M$ is a connected submanifold that is the union of a $(C,\epsilon)$-cap ${\mathcal C}$ and an open $\epsilon$-tube where the intersection of ${\mathcal C}$ with the -$\epsilon$-tube is diffeomorphic to $S^2\times (0,1)$ and contains +$\epsilon$-tube is diffeomorphic to $S^2\times (0,1)$ and contains an end of the $\epsilon$-tube and an end of the cap. A *doubly $C$-capped $\epsilon$-tube* in $M$ is a closed, connected submanifold of $M$ that is the union of two -$(C,\epsilon)$-caps ${\mathcal C}_1$ and ${\mathcal C}_2$ and an +$(C,\epsilon)$-caps ${\mathcal C}_1$ and ${\mathcal C}_2$ and an open $\epsilon$-tube. Furthermore, we require (i) that the cores $Y_1$ and $Y_2$ of ${\mathcal C}_1$ and ${\mathcal C}_2$ have disjoint closures, (ii) that the union of either ${\mathcal C}_i$ -with the $\epsilon$-tube is a capped $\epsilon$-tube and ${\mathcal +with the $\epsilon$-tube is a capped $\epsilon$-tube and ${\mathcal C}_1$ and ${\mathcal C}_2$ contain the opposite ends of the $\epsilon$-tube. There is one further closely related notion, that of an *$\epsilon$-fibration*. By @@ -3354,37 +4028,40 @@ each neck is isotopic to a fiber of the fibration structure. We shall not see this notion again until the appendix, but because it is clearly closely related to the notion of an $\epsilon$-tube, we introduce it here. + -See Fig.~. +See Fig. 9.2. + **Definition.** A *strong $\epsilon$-tube* in a generalized Ricci flow is an $\epsilon$-tube with the property that each point of the tube is the center of a strong $\epsilon$-neck in the generalized flow. + ### Canonical neighborhoods for $\kappa$-solutions -**Proposition.** -Let $(M,g(t))$ be a $3$-dimensional $\kappa$-solution. Then one of + +**Proposition.** Let $(M,g(t))$ be a $3$-dimensional $\kappa$-solution. Then one of the following hold: -\begin{enumerate} -\item[(1)] For every $t\le 0$ the manifold $(M,g(t))$ has positive + +- **(1)** For every $t\le 0$ the manifold $(M,g(t))$ has positive curvature. -\item[(2)] $(M,g(t))$ is the product of an evolving family of +- **(2)** $(M,g(t))$ is the product of an evolving family of round $S^2$'s with a line. -\item[(3)] $M$ is diffeomorphic to a line bundle over $\Ar P^2$, and -there is a finite covering of $(M,g(t))$ that is a flow as in (2). -\end{enumerate} +- **(3)** $M$ is diffeomorphic to a line bundle over $\Ar P^2$, and +there is a finite covering of $(M,g(t))$ that is a flow as in (2). + **Proof.** Suppose that $(M,g(t))$ does not have positive curvature for some $t$. Then, by the application of the strong maximum principle given -in Corollary~, there is a covering $\widetilde M$ of +in \entryref{d21a675e4ea8}, there is a covering $\widetilde M$ of $M$, with either one or $2$-sheets, such that $(\widetilde M,\tilde g(t))$ is the product of an evolving family of round surfaces with a -flat one-manifold (either a circle or the real line). Of course, +flat one-manifold (either a circle or the real line). Of course, the covering must be $\kappa$-solution. In the case in which $(\widetilde M,\tilde g(t))$ is isometric to the product of an -evolving family of round surfaces and a circle, that circle has a +evolving family of round surfaces and a circle, that circle has a fixed length, say $L<\infty$. Since the curvature of the surface in the $t$ time-slice goes to zero as $t\rightarrow -\infty$, we see that the flow is not $\kappa$-non-collapsed on all scales for any @@ -3395,16 +4072,18 @@ established the result. If the round surface is $\Ar P^2$ a further double covering is a product of round two-spheres with $\Ar$. This proves the proposition. + **Lemma.** Let $(M,g(t))$ be a non-compact $3$-dimensional $\kappa$-solution of positive curvature and let $p\in M$. Then there is $D'<\infty$, possibly depending on $(M,g(0))$ and $p$, such that every point of $M\times\{0\}\setminus B(p,0,D'R(p,0)^{-1/2})$ is the center of an evolving $\epsilon$-neck in $(M,g(t))$ defined for an interval of -normalized time of length $2$. Furthermore, there is $D'_1<\infty$ +normalized time of length $2$. Furthermore, there is $D'_1<\infty$ such that for any point $x\in B(p,0,D'R(p,0)^{-1/2})$ and any $2$-plane $P_x$ in $T_xM$ we have $(D'_1)^{-1}< K(P_x)/R(p,0)< D'_1$ where $K(P_x)$ denotes the sectional curvature in the direction of the $2$-plane $P_x$. + **Proof.** Given $(M,g(t))$ and $p$, suppose that no such $D'<\infty$ exists. Because the statement is scale invariant, we can arrange that @@ -3413,24 +4092,24 @@ $R(p,0)=1$. can find a sequence of points $p_k\in M$ with $d_0(p,p_k)\rightarrow \infty$ as $k\rightarrow \infty$ such that no $p_k$ is the center of an evolving $\epsilon$-neck in $(M,g(0))$ defined for an interval of -normalized time of length $2$. By passing to a subsequence we can +normalized time of length $2$. By passing to a subsequence we can assume that one of two possibilities holds: either $d_0^2(p,p_k)R(p_k,0)\rightarrow\infty$ as $k\rightarrow\infty$ or -$lim_{k\rightarrow\infty}d_0^2(p,p_k)R(p_k,0)=\ell<\infty$. In -the first case, set $\lambda_k=R(p_k,0)$ and consider the based +$\mathit{lim}_{k\rightarrow\infty}d_0^2(p,p_k)R(p_k,0)=\ell<\infty$. In +the first case, set $\lambda_k=R(p_k,0)$ and consider the based flows $(M,\lambda_kg(\lambda_k^{-1}t),(p_k,0))$. According to -Theorem~, after passing to a subsequence there is -a geometric limit. Thus, by Theorem~ and -Corollary~ the limit splits as a product of a +\entryref{49e865228b19}, after passing to a subsequence there is +a geometric limit. Thus, by \entryref{e9a9172d16e2} and +\entryref{8ec855a2aab5} the limit splits as a product of a $2$-dimensional $\kappa$-solution and $\Ar$. By -Corollary~ it follows that the limit is the standard +\entryref{da767c71996b} it follows that the limit is the standard round evolving cylinder. This implies that for all $k$ sufficiently large $(p_k,0)$ is the center of an evolving $\epsilon$-neck in $(M,g(t))$ defined for an interval of normalized time of length $2$. This contradiction establishes the existence of $D'$ as required in this case. -Now suppose that $lim_{k\rightarrow\infty}d_0^2(p,p_k)R(p_k,0)=\ell<\infty$. Of +Now suppose that $\mathit{lim}_{k\rightarrow\infty}d_0^2(p,p_k)R(p_k,0)=\ell<\infty$. Of course, since $d_0(p,p_k)\rightarrow\infty$, it must be the case that $R(p_k,0)\rightarrow 0$ as $k\rightarrow \infty$. Set $Q_k=R(p_k,0)$. By passing to a subsequence we can arrange that @@ -3438,7 +4117,7 @@ $d_0^2(p,p_k)Q_k< \ell+1$ for all $k$. Consider the $\kappa$-solutions $(M_k,g_k(t))=(M,Q_kg(Q_k^{-1}t))$. For each $k$ we have $p\in B_{g_k}(p_k,0,\ell+1)$, and $R_{g_k}(p,0)=Q_k^{-1}\rightarrow\infty$ as $k\rightarrow\infty$. -This contradicts Lemma~, and completes the proof of +This contradicts \entryref{ee314aacd207}, and completes the proof of the existence of $D$ as required in this case as well. The existence of $D'_1$ is immediate since the @@ -3449,20 +4128,29 @@ In fact a much stronger result is true. The constants $D'$ and $D_1'$ in the above lemma can be chosen independent of the non-compact $\kappa$-solutions. -**Proposition.** For any $0<\epsilon$ sufficiently small there are constants + +**Proposition.** For any $0<\epsilon$ sufficiently small there are constants $D=D(\epsilon)<\infty$ and $D_1=D_1(\epsilon)<\infty$ - such that the following holds for any non-compact $3$-dimensional + such that the following holds for any non-compact $3$-dimensional $\kappa$-solution $(M,g(t))$ of positive curvature. Let $p\in M$ be a soul of $(M,g(0))$. Then: -\begin{enumerate} -\item[(1)] Every point in $M\setminus B(p,0,DR(p,0)^{-1/2})$ is the center + +- **(1)** Every point in $M\setminus B(p,0,DR(p,0)^{-1/2})$ is the center of a strong $\epsilon$-neck in $(M,g(t))$. Furthermore, for any $x\in B(p,0,DR(p,0)^{-1/2})$ and any $2$-plane $P_x$ in $T_xM$ we have -$$D_1^{-1} -**Remark.** In Part 1 of this theorem one can replace $p$ by any point $p'\in + +**Remark.** In Part 1 of this theorem one can replace $p$ by any point $p'\in M$ that is not the center of a strong $\epsilon$-neck. + **Proof.** First suppose that no $D$ exists so that the first statement holds. Then there is a sequence of such solutions $(M_k,g_k(t))$, with @@ -3484,14 +4174,14 @@ $k\rightarrow\infty$ such that $q_k$ is not the center of a strong $\epsilon$-neck in $(M_k,g_k(0))$. By rescaling we can assume that $R(p_k,0)=1$ for all $k$, and hence that $d_0(p_k,q_k)\rightarrow\infty$. Then, according to -Theorem~, by passing to a subsequence we can +\entryref{49e865228b19}, by passing to a subsequence we can assume that there is a geometric limit $(M_\infty,g_\infty(t),(p_\infty,0))$ with $R(p_\infty,0)=1$. By -Lemma~, provided that $\epsilon$ is sufficiently small +\entryref{04dd3c3ea1c8}, provided that $\epsilon$ is sufficiently small for all $k$ the soul $(p_k,0)$ is not the center of a strong -$2\epsilon$-neck in $(M_k,g_k(t))$. Hence, invoking Part 4 of -Proposition~ and using the fact that $R(p_k,0)=1$ for +$2\epsilon$-neck in $(M_k,g_k(t))$. Hence, invoking Part 4 of +\entryref{7c2758f0e2b3} and using the fact that $R(p_k,0)=1$ for all $k$ and hence $R(p_\infty,0)=1$, we see that $(p_\infty,0)$ is not the center of a strong $\epsilon$-neck in $(M_\infty,g_\infty(t))$. Since the manifolds $M_k$ are non-compact @@ -3499,7 +4189,7 @@ and have metrics of positive curvature they are diffeomorphic to $\Ar^3$ and in particular, do not contain embedded copies of $\Ar P^2$. Thus, the limit $(M_\infty,g_\infty(t))$ is a non-compact $\kappa$-solution containing no embedded copy of $\Ar P^2$. Thus, by -Proposition~ either it is positively curved or it is a +\entryref{25e0946e7222} either it is positively curved or it is a Riemannian product $S^2$ times $\Ar$. In the second case every point is the center of a strong $\epsilon$-neck. Since we have seen that the point $(p_\infty,0)$ is not the center of a strong @@ -3514,16 +4204,16 @@ length $2$. Now since $(M_k,g_k(t),(p_k,0))$ converge geometrically to $(M_\infty,g_\infty(t),(p_\infty,0))$, by Part 2 of -Proposition~ for any $L<\infty$, for all $k$ +\entryref{7c2758f0e2b3} for any $L<\infty$, for all $k$ sufficiently large, all points of $B(p_k,0,L)\setminus B(p_k,0,2D')$ are centers of strong $\epsilon$-necks in $(M_k,g_k(t))$. In particular, for all $k$ sufficiently large, $d_0(p_k,q_k)>L$. Let $L_k$ be a sequence tending to infinity as $k\rightarrow \infty$. Passing to a subsequence, we can suppose that every point of $\left(B(p_k,0,L_k)\setminus B(p_k,0,2D')\right)\subset M_k$ is the -center of a strong $\epsilon$-neck in $(M_k,g_k(0))$. Of course, +center of a strong $\epsilon$-neck in $(M_k,g_k(0))$. Of course, for all $k$ sufficiently large, $q_k\in M_k\setminus B(p_k,0,2D'))$. -By Corollary~ the subset of points in $M_k\times +By \entryref{75b4c67c9141} the subset of points in $M_k\times \{0\}$ that are centers of strong $\epsilon$-necks is an open set. Thus, replacing $q_k$ with another point if necessary we can suppose that it $q_k$ is a closest point to $p_k$ contained in $M_k\setminus @@ -3531,28 +4221,33 @@ B(p_k,0,2D')$ with the property $q_k$ is not the center of a strong $\epsilon$-neck. Then $q_k\in M_k\setminus B(p_k,0,L_k)$ and $(q_k,0)$ is in the closure of the set of points in $M_k$ that are centers of strong $\epsilon$-necks in $(M_k,g_k(t))$, and hence by -Part 3 of Proposition~ each $(q_k,0)$ is the center +Part 3 of \entryref{7c2758f0e2b3} each $(q_k,0)$ is the center of a $2\epsilon$-neck in $(M_k,g_k(t))$. Let $\gamma_k$ be a minimizing geodesic connecting $(p_k,0)$ to $(q_k,0)$, and let $\mu_k$ be a minimizing geodesic ray from $(q_k,0)$ to infinity. Set $Q_k=R(q_k,0)$. Since $(q_k,0)$ is the -center of a $2\epsilon$-neck, from Lemma~ we see -that, provided that $\epsilon$ is sufficiently small, the +center of a $2\epsilon$-neck, from \entryref{818772530e5a} we see +that, provided that $\epsilon$ is sufficiently small, the $2\epsilon$-neck centered at $q_k$ separates $p$ from $\infty$, so that $\gamma_k$ and $\mu_k$ exit this $2\epsilon$-neck at opposite -ends. According to Theorem~, after passing to a +ends. According to \entryref{49e865228b19}, after passing to a subsequence, the based, rescaled flows -$$(M_k,Q_kg(Q_k^{-1}t),(q_k,0))$$ + +$$ +(M_k,Q_kg(Q_k^{-1}t),(q_k,0)) +$$ + converge geometrically to a limit. Let $(q_\infty,0)$ be the base point of the resulting limit. By Part 3 of -Proposition~, it is the center of a $4\epsilon$-neck +\entryref{7c2758f0e2b3}, it is the center of a $4\epsilon$-neck in the limit. + **Claim.** $d_0^2(p_k,q_k)Q_k\rightarrow \infty$ as $k\rightarrow \infty$. + -\begin{proof} -Suppose not. Then by passing to a subsequence we can suppose that +**Proof.** Suppose not. Then by passing to a subsequence we can suppose that these products are bounded independent of $k$. Then since $d_0(p_k,q_k)\rightarrow\infty$, we see that $Q_k\rightarrow 0$. Thus, in the rescaled flows $(M_k,Q_kg_k(Q_k^{-1}t))$ the curvature @@ -3561,25 +4256,25 @@ $Q_kg_k$-distance from $(p_k,0)$ to $(q_k,0)$ is $\sqrt{Q_k}d_0(p_k,q_k)$ which is bounded independent of $k$ and the scalar curvature of $(p,0)$ in the metric $Q_kg_k(0)$ is $R(p_k,0)Q_k^{-1}=Q_k^{-1}$ tends to $\infty$. Unbounded curvature -at bounded distance contradicts Lemma~, and this +at bounded distance contradicts \entryref{ee314aacd207}, and this establishes the claim. A subsequence of the based flows $(M_k,Q_kg_k(Q_k^{-1}t),(q_k,0))$ converge geometrically to a $\kappa$-solution. According -Theorem~ and Corollary~, this limiting +\entryref{e9a9172d16e2} and \entryref{8ec855a2aab5}, this limiting flow is the product of a $2$-dimensional $\kappa$-solution with a line. Since $M$ is orientable, this $2$-dimensional $\kappa$-solution is an evolving family of round $2$-spheres. This implies that for all $k$ sufficiently large, $(q_k,0)$ is the center of a strong $\epsilon$-neck in $(M_k,g_k(t))$. This is a -contradiction and proves the existence of $D<\infty$ as stated in +contradiction and proves the existence of $D<\infty$ as stated in the proposition. -Let $(M_k,g_k(t),(p_k,0))$ be a sequence of non-compact Ricci flows -based at a soul $p_k$ of $(M_k,g_k(0))$. We rescale so that -$R(p_k,0)=1$. By Lemma~, if $\epsilon$ is sufficiently +Let $(M_k,g_k(t),(p_k,0))$ be a sequence of non-compact Ricci flows +based at a soul $p_k$ of $(M_k,g_k(0))$. We rescale so that +$R(p_k,0)=1$. By \entryref{04dd3c3ea1c8}, if $\epsilon$ is sufficiently small, then $p_k$ cannot be the center of an $\epsilon$-neck. It -follows from Proposition~ that for any limit of a +follows from \entryref{7c2758f0e2b3} that for any limit of a subsequence the point $p_\infty$, which is the limit of the $p_k$, is not the center of an $2\epsilon$-neck in the limit. Since the limit manifold is orientable, it is either contractible with @@ -3587,31 +4282,35 @@ strictly positive curvature or is a metric product of a round $2$-sphere and the line. It follows that the limit manifold has strictly positive curvature at $(p_\infty,0)$, and hence positive curvature everywhere. The existence of $D_1<\infty$ as required is -now immediate from Theorem~. +now immediate from \entryref{49e865228b19}. The fact that any soul is disjoint from the middle two-thirds of any $\epsilon$-neck and the fact that the central $2$-spheres of all $\epsilon$-necks are isotopic in $M\setminus\{p\}$ are contained in -Lemma~ and Corollary~. -\end{proof} +\entryref{04dd3c3ea1c8} and \entryref{818772530e5a}. -**Corollary.** -There is $\bar\epsilon_2>0$ such that for any $0<\epsilon\le \bar + +**Corollary.** There is $\bar\epsilon_2>0$ such that for any $0<\epsilon\le \bar \epsilon_2$ the following holds. There is $C_0=C_0(\epsilon)$ such that for any $\kappa>0$ and any non-compact $3$-dimensional $\kappa$-solution not containing an embedded $\Ar P^2$ with trivial normal bundle, the zero time-slice is either a strong $\epsilon$-tube or a $C_0$-capped strong $\epsilon$-tube. + **Proof.** For $\epsilon>0$ sufficiently small let $D(\epsilon)$ and $D_1(\epsilon)$ be as in the previous corollary. At the expenses of increasing these, we can assume that they are at least the constant -$C$ in Corollary~. We set -$$C_0(\epsilon)=max(D(\epsilon),D_1(\epsilon)).$$ +$C$ in \entryref{8c090fbd21a0}. We set + +$$ +C_0(\epsilon)=\mathit{max}(D(\epsilon),D_1(\epsilon)). +$$ + If the non-compact $\kappa$-solution has positive curvature, then the corollary follows immediately from -Proposition~ and Corollary~. If the +\entryref{c2c339703eb1} and \entryref{8c090fbd21a0}. If the $\kappa$-solution is the product of an evolving round $S^2$ with the line, then every point of the zero time-slice is the center of a strong $\epsilon$-neck for every $\epsilon>0$ so that the zero @@ -3631,19 +4330,19 @@ proves the corollary in this case and hence completes the proof. Now let us consider compact $\kappa$-solutions. -**Theorem.** -There is $\bar \epsilon_3>0$ such that for every $0<\epsilon\le \bar -\epsilon_3$ there is $C_1=C_1(\epsilon)<\infty$ such that one of + +**Theorem.** There is $\bar \epsilon_3>0$ such that for every $0<\epsilon\le \bar +\epsilon_3$ there is $C_1=C_1(\epsilon)<\infty$ such that one of the following holds for any $\kappa>0$ and any compact $3$-dimensional $\kappa$-solution $(M,g(t))$. -\begin{enumerate} -\item[(1)] The manifold $M$ is compact and of constant positive sectional curvature. -\item[(2)] The diameter of $(M,g(0))$ is less than $C_1\cdot (max_{x\in + +- **(1)** The manifold $M$ is compact and of constant positive sectional curvature. +- **(2)** The diameter of $(M,g(0))$ is less than $C_1\cdot (\mathit{max}_{x\in M}R(x,0))^{-1/2}$, and $M$ is diffeomorphic to either $S^3$ or $\Ar P^3$. -\item[(3)] $(M,g(0))$ - is a double $C_1$-capped strong $\epsilon$-tube. -\end{enumerate} +- **(3)** $(M,g(0))$ + is a double $C_1$-capped strong $\epsilon$-tube. + **Proof.** First notice that if $(M,g(t))$ is not of strictly positive curvature, then the universal covering of $(M,g(0))$ is a Riemannian @@ -3661,133 +4360,141 @@ surface in $M$ induces a surjective homomorphism of $H_1(M)$ onto $\Zee$. We conclude from this that $M$ does not contain an embedded $\Ar P^2$ with trivial normal bundle. - We assume that $(M,g(0))$ is not round so that by Proposition~ + We assume that $(M,g(0))$ is not round so that by \entryref{9c54e0d0c7d8} there is a universal $\kappa_0>0$ such that $(M,g(0))$ is a $\kappa_0$-solution. Let $C_0(\epsilon)$ be the constant from - Corollary~. + \entryref{274cda9be677}. + **Claim.** Assuming that $(M,g(0))$ is compact but not of constant positive sectional curvature, for each $\epsilon>0$ there is $C_1$ such that -if the diameter of $(M,g(0))$ is greater than $C_1(max_{x\in +if the diameter of $(M,g(0))$ is greater than $C_1(\mathit{max}_{x\in M}R(x,0))^{-1/2}$ then every point of $(M,g(0))$ is either contained in the core of $(C_0(\epsilon),\epsilon)$-cap or is the center of a strong $\epsilon$-neck in $(M,g(t))$. + -\begin{proof} -Suppose that for some $\epsilon>0$ there is no such $C_1$. Then we +**Proof.** Suppose that for some $\epsilon>0$ there is no such $C_1$. Then we take a sequence of constants $C'_k$ that diverges to $+\infty$ as $k\rightarrow \infty$ and a sequence $(M_k,g_k(t), (p_k,0))$ of based $\kappa_0$-solutions such that the diameter of $(M_k,0)$ is greater than $C'_kR^{-1/2}(p_k,0)$ and yet $(p_k,0)$ is not contained in the core of a $(C_0(\epsilon),\epsilon)$-cap nor is the -center of a strong $\epsilon$-neck. We scale $(M_k,g_k(t))$ by +center of a strong $\epsilon$-neck. We scale $(M_k,g_k(t))$ by $R(p_k,0)$. This allows us to assume that $R(p_k,0)=1$ for all $k$. -According to Theorem~, after passing to a +According to \entryref{49e865228b19}, after passing to a subsequence we can assume these based $\kappa$-solutions converge to a based $\kappa$-solution $(M_\infty,g_\infty(t),(p_\infty,0))$. Since the diameters of the $(M_k,g_k(0))$ go to infinity, $M_\infty$ -is non-compact. According to Corollary~ the point +is non-compact. According to \entryref{274cda9be677} the point $p_\infty$ is either the center of a strong $\epsilon$-neck, or is -contained in the core of a $(C_0(\epsilon),\epsilon)$-cap. Since +contained in the core of a $(C_0(\epsilon),\epsilon)$-cap. Since $R(p_k,0)=1$ for all $k$, it follows from Parts 1 and 4 of -Proposition~ that for all $k$ sufficiently large, +\entryref{7c2758f0e2b3} that for all $k$ sufficiently large, $(p_k,0)$ is either the center of a strong $\epsilon$-neck in $(M_k,g_k(t))$ or is contained in the core of a $(C_0(\epsilon),\epsilon)$-cap. This is a contradiction, proving the claim. -Now it follows from Proposition~ that if the -diameter of $(M,g(0))$ is greater than $C_1(max_{x\in +Now it follows from \entryref{fa8d9a43f34a} that if the +diameter of $(M,g(0))$ is greater than $C_1(\mathit{max}_{x\in M}R(x,0))^{-1/2}$ and if it is not of constant positive curvature, -then $M$ is diffeomorphic to either $S^3$, $\Ar P^3$, $\Ar P^3\#\Ar +then $M$ is diffeomorphic to either $S^3$, $\Ar P^3$, $\Ar P^3\#\Ar P^3$ or is a $S^2$-fibration over $S^1$. On the other hand, since $M$ is compact of positive curvature its fundamental group is -finite, see Theorem 4.1 on p. 154 of . This rules out +finite, see Theorem 4.1 on p. 154 of [Petersen]. This rules out the last two cases. This implies that when $(M,g(0))$ has diameter -greater than $C_1(max_{x\in M}R(x,0))^{-1/2}$ and is not of +greater than $C_1(\mathit{max}_{x\in M}R(x,0))^{-1/2}$ and is not of constant positive curvature, it is a double $C_0$-capped $\epsilon$-tube. We must consider the case when $(M,g(0))$ is not of constant positive curvature and its diameter is less than or equal to -$C_1(max_{x\in M}R(x,0))^{-1/2}$. Since $(M,g(0))$ is not -round, by Corollary~ its asymptotic soliton is not -compact. Thus, by Theorem~ its asymptotic soliton is +$C_1(\mathit{max}_{x\in M}R(x,0))^{-1/2}$. Since $(M,g(0))$ is not +round, by \entryref{01028c93347d} its asymptotic soliton is not +compact. Thus, by \entryref{2d68880771ff} its asymptotic soliton is either $S^2\times \Ar$ or is double covered by this product. This means that for $t$ sufficiently negative the diameter of $(M,g(t))$ -is greater than $C_1(max_{x\in M}R(x,0))^{-1/2}$. Invoking the +is greater than $C_1(\mathit{max}_{x\in M}R(x,0))^{-1/2}$. Invoking the previous result for this negative time tells us that $M$ is diffeomorphic to $S^3$ or $\Ar P^3$. -\end{proof} + **Proposition.** Let $\bar\epsilon_2$ and $\bar \epsilon_3$ be as in -Corollary~ and Theorem~, respectively. -For each $0<\epsilon\le min(\bar \epsilon_2,\bar\epsilon_3)$ -let $C_1=C_1(\epsilon)$ be as in Theorem~. There is +\entryref{274cda9be677} and \entryref{a4ea411d0765}, respectively. +For each $0<\epsilon\le \mathit{min}(\bar \epsilon_2,\bar\epsilon_3)$ +let $C_1=C_1(\epsilon)$ be as in \entryref{a4ea411d0765}. There is $C_2=C_2(\epsilon)<\infty$ such that for any $\kappa>0$ and any compact $\kappa$-solution $(M,g(t))$ the following holds. If $(M,g(0))$ is not of constant positive curvature and if $(M,g(0))$ -is of diameter less than $C_1(max_{x\in M}R(x,0))^{-1/2}$ then +is of diameter less than $C_1(\mathit{max}_{x\in M}R(x,0))^{-1/2}$ then for any $x\in M$ we have -$$C_2^{-1}R(x,0)^{-3/2}< Vol(M,g(0))< C_2R(x,0)^{-3/2}.$$ -In addition, for any $y\in M$ and any $2$-plane $P_y$ in $T_yM$ we + +$$ +C_2^{-1}R(x,0)^{-3/2}< \mathit{Vol}(M,g(0))< C_2R(x,0)^{-3/2}. +$$ + +In addition, for any $y\in M$ and any $2$-plane $P_y$ in $T_yM$ we have -$$C_2^{-1}< \frac{K(P_y)}{R(x,0)}< C_2,$$ + +$$ +C_2^{-1}< \frac{K(P_y)}{R(x,0)}< C_2, +$$ + where $K(P_y)$ is the sectional curvature in the $P_y$-direction. + -**Proof.** The result is immediate from Corollary~ and -Theorem~. +**Proof.** The result is immediate from \entryref{9c54e0d0c7d8} and +\entryref{49e865228b19}. + **Remark.** For a round $\kappa$-solution $(M,g(t))$ we have $R(x,0)=R(y,0)$ for all $x,y\in M$, and the volume of $(M,g(0))$ is bounded above by a constant times $R(x,0)^{-3/2}$. There is no universal lower bound to the volume in terms of the curvature. The lower bound takes the form $C_2|\pi_1(M)|^{-1}R(x,0)^{-3/2}$, where $|\pi_1(M)|$ is the order of the fundamental group $\pi_1(M)$. + Let us summarize our results. -**Theorem.** -There is $\bar \epsilon>0$ such that the following is true for any -$0<\epsilon<\bar \epsilon$. There is $C=C(\epsilon)$ such that for + +**Theorem.** There is $\bar \epsilon>0$ such that the following is true for any +$0<\epsilon<\bar \epsilon$. There is $C=C(\epsilon)$ such that for any $\kappa>0$ and any $\kappa$-solution $(M,g(t))$ one of the following holds. -\begin{enumerate} -\item[(1)] $(M,g(t))$ is round for all $t\le 0$. In this case $M$ is + +- **(1)** $(M,g(t))$ is round for all $t\le 0$. In this case $M$ is diffeomorphic to the quotient of $S^3$ by a finite subgroup of $SO(4)$ acting freely. -\item[(2)] $(M,g(0))$ is compact and of positive curvature. For any $x,y\in M$ +- **(2)** $(M,g(0))$ is compact and of positive curvature. For any $x,y\in M$ and any $2$-plane $P_y$ in $T_yM$ we have -$$ $$ \begin{aligned} -C^{-1/2}R(x,0)^{-1} & An immediate consequence of this result is: -**Corollary.** -For every $0<\epsilon\le\bar\epsilon'$ there is + +**Corollary.** For every $0<\epsilon\le\bar\epsilon'$ there is $C=C(\epsilon)<\infty$ such that every point in a $\kappa$-solution has a strong $(C,\epsilon)$-canonical neighborhood unless the $\kappa$-solution is a product $\Ar P^2\times \Ar$. + -**Corollary.** -Fix $0<\epsilon\le \bar\epsilon'$, and let $C(\epsilon)$ be as in + +**Corollary.** Fix $0<\epsilon\le \bar\epsilon'$, and let $C(\epsilon)$ be as in the last corollary. Suppose that $({\mathcal M}_n,G_n,x_n)$ is a -sequence of based, generalized Ricci flows with $**t**(x_n)=0$ for +sequence of based, generalized Ricci flows with $\mathbf{t}(x_n)=0$ for all $n$. Suppose that none of the time-slices of the ${\mathcal M}_n$ contain embedded $\Ar P^2$'s with trivial normal bundle. Suppose also that there is a smooth limiting flow @@ -3828,6 +4534,7 @@ $(M_\infty,g_\infty(t),(x_\infty,0))$ defined for $-\infty **Proof.** The limiting manifold $M_\infty$ cannot contain an embedded $\Ar P^2$ with trivial normal bundle. Hence, by the previous corollary, @@ -3841,15 +4548,15 @@ sufficiently large $x_n$ is contained in a $C$-component of the zero time-slice of ${\mathcal M}_n$. Suppose that $(x_\infty,0)$ is the center of a strong $\epsilon$-neck in the limiting flow. This neck extends backwards in the limiting solution some amount past an -interval of normalized time of length $1$, where by continuity it +interval of normalized time of length $1$, where by continuity it is an evolving $\epsilon$-neck defined backwards for an interval of -normalized time of length greater than $1$. Then by Part 2 of -Proposition~, any family of metrics on this neck +normalized time of length greater than $1$. Then by Part 2 of +\entryref{7c2758f0e2b3}, any family of metrics on this neck sufficiently close to the limiting metric will determine an strong $\epsilon$-neck. This implies that for all $n$ sufficiently large $x_n$ is the center of a strong $\epsilon$-neck in $({\mathcal M}_n,G_n)$. Lastly, if $(x_\infty,0)$ is contained in the core of a $(C,\epsilon)$-cap in the limiting flow, then by Part 1 of -Proposition~ for all $n$ sufficiently large $x_n$ is +\entryref{7c2758f0e2b3} for all $n$ sufficiently large $x_n$ is contained in the core of a $(C,\epsilon)$-cap in $({\mathcal M}_n,G_n)$. diff --git a/projects/poincare-conjecture/.astrolabe/docs/10-bddcurvbdddist.mdx b/projects/poincare-conjecture/.astrolabe/docs-src/11-bounded-curvature-at-bounded-distance.mdx similarity index 75% rename from projects/poincare-conjecture/.astrolabe/docs/10-bddcurvbdddist.mdx rename to projects/poincare-conjecture/.astrolabe/docs-src/11-bounded-curvature-at-bounded-distance.mdx index 1b2d15ca..463e6bbe 100644 --- a/projects/poincare-conjecture/.astrolabe/docs/10-bddcurvbdddist.mdx +++ b/projects/poincare-conjecture/.astrolabe/docs-src/11-bounded-curvature-at-bounded-distance.mdx @@ -1,4 +1,6 @@ -# Bounded curvature at bounded distance + + +# Chapter 10 -- Bounded curvature at bounded distance This chapter is devoted to Perelman's result about bounded curvature at bounded distance for blow-up limits. Crucial to the argument is that each @@ -12,32 +14,42 @@ generalized Ricci flow to have curvature pinched toward positive. This is the obvious generalization of the corresponding notion for Ricci flows. -**Definition.** -Let $({\mathcal M},G)$ be a generalized three-dimensional Ricci flow + +**Definition.** Let $({\mathcal M},G)$ be a generalized three-dimensional Ricci flow whose domain of definition is contained in $[0,\infty)$. For each $x\in {\mathcal M}$, let $\nu(x)$ be the smallest eigenvalue of -$Rm(x)$ on $\wedge^2T_xM_{**t**(x)}$, as measured with +$\mathit{Rm}(x)$ on $\wedge^2T_xM_{\mathbf{t}(x)}$, as measured with respect to a $G(x)$-orthonormal basis for the horizontal space at -$x$, and set $X(x)=max(0,-\nu(x))$. We say that $({\mathcal +$x$, and set $X(x)=\mathit{max}(0,-\nu(x))$. We say that $({\mathcal M},G)$ has curvature *pinched toward positive* if, for all $x\in {\mathcal M}$, if the following two inequalities hold: -\begin{enumerate} -\item[(1)] $$R(x)\ge \frac{-6}{1+4**t**(x)},$$ -\item[(2)] $$R(x)\geq 2X(x)\left(logX(x)+log(1+**t**(x))-3\right), -$$ whenever + +- **(1)** + +$$ +R(x)\ge \frac{-6}{1+4\mathbf{t}(x)}, +$$ + +- **(2)** + +$$ +R(x)\geq 2X(x)\left(\mathit{log}X(x)+\mathit{log}(1+\mathbf{t}(x))-3\right), +$$ + + whenever $0 -According to Theorem~ if $(M,g(t)),\ 0\le a\le t0$ sufficiently small such that for any $0<\epsilon\le \epsilon_0$ all the results of the Appendix hold for -$2\epsilon$ and $\alpha=10^{-2}$, and Proposition~ +$2\epsilon$ and $\alpha=10^{-2}$, and \entryref{cb9ca6d6db9f} holds for $2\epsilon$. ## The statement of the theorem @@ -46,38 +58,43 @@ Here is the statement of the main theorem of this chapter, the theorem that establishes bounded curvature at bounded distance for blow-up limits. -**Theorem.** -Fix $0<\epsilon\le \epsilon_0$ and $C<\infty$. Then for each + +**Theorem.** Fix $0<\epsilon\le \epsilon_0$ and $C<\infty$. Then for each $A<\infty$ there are $D_0<\infty$ and $D<\infty$ depending on $A$, $\epsilon$ and $C$ such that the following holds. Suppose that $({\mathcal M},G)$ is a generalized three-dimensional Ricci flow whose interval of definition is contained in $[0,\infty)$, and -suppose that $x\in {\mathcal M}$. Set $t=**t**(x)$. We suppose +suppose that $x\in {\mathcal M}$. Set $t=\mathbf{t}(x)$. We suppose that these data satisfy the following: -\begin{enumerate} -\item[(1)] $({\mathcal M},G)$ has curvature pinched toward positive. -\item[(2)] Every point $y\in{\mathcal M}$ with -$R(y)\ge 4R(x)$ and $**t**(y)\le t$ has a strong + +- **(1)** $({\mathcal M},G)$ has curvature pinched toward positive. +- **(2)** Every point $y\in{\mathcal M}$ with +$R(y)\ge 4R(x)$ and $\mathbf{t}(y)\le t$ has a strong $(C,\epsilon)$-canonical neighborhood. -\end{enumerate} + If $R(x)\ge D_0$, then $R(y)\le DR(x)$ for all $y\in B(x,t,AR(x)^{-1/2})$. + This chapter is devoted to the proof of this theorem. The proof is by contradiction. Suppose that there is some $A_0<\infty$ for which the result fails. Then there are a sequence of generalized -three-dimensional Ricci flows $({\mathcal M}_n,G_n)$ whose +three-dimensional Ricci flows $({\mathcal M}_n,G_n)$ whose intervals of definition are contained in $[0,\infty)$ and whose curvatures are pinched toward positive. Also, there are points $x_n\in {\mathcal M}_n$ satisfying the second condition given in the theorem and points $y_n\in {\mathcal M}_n$ such that for all $n$ we have: -\begin{enumerate} -\item[(1)] $lim_{n\rightarrow \infty}R(x_n)=\infty$. -\item[(2)] $**t**(y_n)=**t**(x_n)$, -\item[(3)] $d(x_n,y_n)< A_0R(x_n)^{-1/2}$, -\item[(4)] $$lim_{n\rightarrow\infty}\frac{R(y_n)}{R(x_n)}=\infty.$$ -\end{enumerate} + +- **(1)** $\mathit{lim}_{n\rightarrow \infty}R(x_n)=\infty$. +- **(2)** $\mathbf{t}(y_n)=\mathbf{t}(x_n)$, +- **(3)** $d(x_n,y_n)< A_0R(x_n)^{-1/2}$, +- **(4)** + +$$ +\mathit{lim}_{n\rightarrow\infty}\frac{R(y_n)}{R(x_n)}=\infty. +$$ + For the rest of this chapter we assume that such a sequence of generalized Ricci flows exists. We shall eventually derive a contradiction. @@ -110,7 +127,7 @@ of the rescaling factors used to construct them converges to a finite, non-zero limit. This means that the two limits differ only by an overall constant factor. That is to say the geometric blow-up limit is isometric to an open subset of a non-flat cone. This -contradicts Hamilton's result (Theorem~) which says +contradicts Hamilton's result (\entryref{92b5c664399e}) which says that it is not possible to flow under the Ricci flow to an open subset of a non-flat cone. Now we carry out all the steps in this argument. @@ -118,26 +135,21 @@ argument. ## The incomplete geometric limit We fix a sequence $({\mathcal M}_n,G_n,x_n)$ of generalized Ricci -flows as above. The first step is to shift and rescale this +flows as above. The first step is to shift and rescale this sequence of generalized Ricci flows so that we can form an (incomplete) geometric limit which will be a tube of finite length with scalar curvature going to infinity at one end. -We shift the time parameter of $({\mathcal M}_n,G_n)$ by $-**t**(x_n)$. We change notation and denote these shifted flows by -$({\mathcal M}_n,G_n)$. This allows us to arrange that $**t**(x_n)=0$ for all $n$. Since shifting leaves the curvature -unchanged, the shifted flows satisfy a weaker version of curvature +We shift the time parameter of $({\mathcal M}_n,G_n)$ by $-\mathbf{t}(x_n)$. We change notation and denote these shifted flows by +$({\mathcal M}_n,G_n)$. This allows us to arrange that $\mathbf{t}(x_n)=0$ for all $n$. Since shifting leaves the curvature +unchanged, the shifted flows satisfy a weaker version of curvature pinched toward positive. Namely, for the shifted flows we have -$$ - $$ \begin{aligned} - R(x) & \ge & -6 \nonumber \\ -R(x) & \ge & 2X(x)\left(log(X(x))-3\right). +R(x) & \ge & 2X(x)\left(\mathit{log}(X(x))-3\right). \end{aligned} -$$ - $$ We set $Q_n=R(x_n)$, and we denote by $M_n$ the $0$-time-slice of @@ -159,7 +171,7 @@ to the second condition in the statement of the theorem, every point of $\gamma_n$ has a strong $(C,\epsilon)$ canonical neighborhood. As $n$ tends to infinity the ratio of $R(y_n)/R(z_n)$ tends to infinity. This means that for -all $n$ sufficiently large, no point of $\gamma_n$ can be contained +all $n$ sufficiently large, no point of $\gamma_n$ can be contained in an $\epsilon$-round component or a $C$-component, because if it were then all of $\gamma_n$ would be contained in that component, contradicting @@ -168,20 +180,19 @@ $n$. Hence, for $n$ sufficiently large, every point of $\gamma_n$ is either contained in the core of a $(C,\epsilon)$-cap or is the center of a strong $\epsilon$-neck. According to -Proposition~, for all $n$ sufficiently large +\entryref{0e9a1a167caa}, for all $n$ sufficiently large $\gamma_n$ is contained an open submanifold $X_n$ of the zero time-slice of ${\mathcal M}'_n$ that is one of the following: -\begin{enumerate} -\item[(1)] an $\epsilon$-tube and both endpoints of $\gamma_n$ are centers + +- **(1)** an $\epsilon$-tube and both endpoints of $\gamma_n$ are centers of $\epsilon$-necks contained in $X_n$, -\item[(2)] a $C$-capped $\epsilon$-tube with cap ${\mathcal C}$, +- **(2)** a $C$-capped $\epsilon$-tube with cap ${\mathcal C}$, and each endpoint of $\gamma_n$ either is contained in the core $Y$ of ${\mathcal C}$ or is the center of an $\epsilon$-neck contained in $X_n$, -\item[(3)] a double $C$-capped $\epsilon$-tube, or finally -\item[(4)] the union of two $(C,\epsilon)$-caps. -\end{enumerate} +- **(3)** a double $C$-capped $\epsilon$-tube, or finally +- **(4)** the union of two $(C,\epsilon)$-caps. The fourth possibility is incompatible with the fact that the ratio of the curvatures at the endpoints of $\gamma_n$ grows arbitrarily @@ -190,9 +201,11 @@ cannot occur for $n$ sufficiently large. Thus, for all $n$ sufficiently large $X_n$ is one of the first three types listed above. + **Claim.** There is a geodesic $\hat\gamma_n$ in $X_n$ with endpoints $z_n$ and $y_n$. This geodesic is minimizing among all paths in $X_n$ from $z_n$ to $y_n$. + **Proof.** This is clear in the third case since $X_n$ is a closed manifold. @@ -201,14 +214,14 @@ and $N(y_n)$ centered at $z_n$ and $y_n$ and contained in $X_n$. Suppose first that the central $2$-spheres $S(z_n)$ and $S(y_n)$ of these necks are disjoint. Then they are the boundary of a compact sumanifold $X_n'$ of $X_n$. It follows easily from -Lemma~ that any sequence of minimizing paths from -$z_n$ to $y_n$ is contained in the union of $X_n'$ with the middle +\entryref{fc6c6ae605d2} that any sequence of minimizing paths from +$z_n$ to $y_n$ is contained in the union of $X_n'$ with the middle halves of $N(z_n)$ and $N(y_n)$. Since this manifold has compact closure in $X_n$, the usual arguments show that one can extract a limit of a subsequence which is a minimizing geodesic in $X_n$ from $z_n$ to $y_n$. If $S(z_n)\cap S(y_n)\not=\emptyset$, then $y_n$ is contained in the middle half of $N(z_n)$, and again it follows -immediately from Lemma~ that there is a minimizing +immediately from \entryref{fc6c6ae605d2} that there is a minimizing geodesic in $N(z_n)$ between these points. Now let us consider the second case. If each of $z_n$ and $y_n$ is @@ -219,14 +232,14 @@ the result is again immediate. Lastly, suppose that one of the points, we can assume by the symmetry of the roles of the points that it is $z_n$, is the center of an $\epsilon$-neck $N(z_n)$ in $X_n$ and the other is contained in the core of ${\mathcal C}$. -Suppose that the central $2$-sphere $S(z_n)$ of $N(z_n)$ meets the -core $Y$ of ${\mathcal C}$. Then $z_n$ lies in the half of the neck +Suppose that the central $2$-sphere $S(z_n)$ of $N(z_n)$ meets the +core $Y$ of ${\mathcal C}$. Then $z_n$ lies in the half of the neck $N={\mathcal C}\setminus \overline Y$ whose closure contains the frontier of $Y$. Orient $s_N$ so that this half is the positive -half. Thus, by Lemma~ any minimizing sequence of +half. Thus, by \entryref{fc6c6ae605d2} any minimizing sequence of paths from $z_n$ to $y_n$ is eventually contained in the union of the core of ${\mathcal C}$ and the the positive three-quarters of -this neck. Hence, as before we can pass to a limit and construct a +this neck. Hence, as before we can pass to a limit and construct a minimizing geodesic in $X_n$ connecting $z_n$ to $y_n$. On the other hand, if $S(z_n)$ is disjoint from $Y$, then $S(z_n)$ separates $X_n$ into a compact complementary component and a non-compact @@ -242,23 +255,24 @@ geodesic in $X_n$. This claim allows us to assume (as we now shall) that $\gamma_n$ is a minimizing geodesic in $X_n$ from $z_n$ to $y_n$. + **Claim.** For every $n$ sufficiently large, there is a sub-geodesic $\gamma'_n$ of $\gamma_n$ with end points $z_n'$ and $y_n'$ such that the following hold: -\begin{enumerate} -\item[(1)] The length of $\gamma'_n$ is bounded independent of $n$. -\item[(2)] $R(z_n')$ is bounded independent of $n$. -\item[(3)] $R(y_n')$ tends to infinity as $n$ tends to infinity. -\item[(4)] $\gamma_n'$ is contained in a strong $\epsilon$-tube + +- **(1)** The length of $\gamma'_n$ is bounded independent of $n$. +- **(2)** $R(z_n')$ is bounded independent of $n$. +- **(3)** $R(y_n')$ tends to infinity as $n$ tends to infinity. +- **(4)** $\gamma_n'$ is contained in a strong $\epsilon$-tube $T_n$ that is the union of a balanced chain of strong $\epsilon$-necks centered at points of $\gamma'_n$. The first element in this chain is a strong $\epsilon$-neck $N(z'_n)$ centered at $z'_n$. The last element is a strong $\epsilon$-neck containing $y'_n$. -\item[(5)] For every $x\in T_n$, we have $R(x)>3$ and $x$ is the center of a strong +- **(5)** For every $x\in T_n$, we have $R(x)>3$ and $x$ is the center of a strong $\epsilon$-neck in the flow $({\mathcal M}'_n,G'_n)$. -\end{enumerate} + **Proof.** The first item is clear since, for all $n$, the geodesic $\gamma_n$ has $G_n$-length at most $A_0Q_n^{-1/2}$ and hence $G'_n$-length at @@ -267,47 +281,54 @@ whose core $Y$ contains a point of $\gamma_n$. Let $N$ be the $\epsilon$-neck that is the complement of the closure of $Y$ in ${\mathcal C}$, and let $\widehat Y$ be the union of $Y$ and the closed negative half of $N$. We claim that $\widehat Y$ contains -either $z_n$ or $y_n$. By Corollary~, since $Y$ +either $z_n$ or $y_n$. By \entryref{5ef5906bb370}, since $Y$ contains a point of $\gamma_n$, the intersection of $\widehat Y$ with $\gamma_n$ is a subinterval containing one of the end points of $\gamma_n$, i.e., either $z_n$ or $y_n$. This means that any point -$w$ which is contained in a $(C,\epsilon)$-cap whose core contains +$w$ which is contained in a $(C,\epsilon)$-cap whose core contains a point of $\gamma_n$ must satisfy one of the following: -$$R(w)< CR(z'_n)\ \ \ or\ \ \ R(w)> C^{-1}R(y'_n).$$ + +$$ +R(w)< CR(z'_n)\ \ \ \mathit{or}\ \ \ R(w)> C^{-1}R(y'_n). +$$ We pass to a subsequence so that $R(y_n)/R(z_n)>4C^2$ for all $n$, and we pass to a subinterval $\gamma'_n$ of $\gamma_n$ with endpoints $z_n'$ and $y_n'$ such that: -\begin{enumerate} -\item[(1)] $R(z'_n)=2CR(z_n)$ -\item[(2)] $R(y'_n)=(2C)^{-1}R(y_n)$ -\item[(3)] $R(z'_n)\le R(w)\le R(y'_n)$ for all $w\in \gamma'_n$. -\end{enumerate} + +- **(1)** $R(z'_n)=2CR(z_n)$ +- **(2)** $R(y'_n)=(2C)^{-1}R(y_n)$ +- **(3)** $R(z'_n)\le R(w)\le R(y'_n)$ for all $w\in \gamma'_n$. + Clearly, with these choices $R(z'_n)$ is bounded independent of $n$ and $R(y'_n)$ tends to infinity as $n$ tends to infinity. Also, no point of $\gamma'_n$ is contained in the core of a $(C,\epsilon)$-cap. Since every point of $\gamma'_n$ has a strong $(C,\epsilon)$-canonical neighborhood, it follows that every point of $\gamma'_n$ is the center of a strong $\epsilon$-neck. It now -follows from Proposition~ that there is a balanced +follows from \entryref{ba681e15e1a6} that there is a balanced $\epsilon$-chain consisting of strong $\epsilon$-necks centered at points of $\gamma'_n$ whose union contains $\gamma'_n$. (Even if the $2$-spheres of these necks do not separate the zero time-slice of ${\mathcal M}'_n$, as we build the balanced $\epsilon$-chain as -described in Proposition~ the new necks we add can +described in \entryref{ba681e15e1a6} the new necks we add can not meet the negative end of $N(z'_n)$ since the geodesic $\gamma'_n$ is minimal.) We can take the first element in the balanced chain to be a strong $\epsilon$-neck $N(z'_n)$ centered at $z'_n$, and the last element to be a strong $\epsilon$-neck $N^+_n$ -containing $y'_n$. The union of this chain is $T_n$. (See Fig.~.) +containing $y'_n$. The union of this chain is $T_n$. (See Fig. 10.1.) Next, we show that every point of $T_n$ is the center of a strong $\epsilon$-neck in $({\mathcal M}_n,G_n)$. We must rule out the possibility that there is a point of $T_n$ that is contained in the core of a $(C,\epsilon)$-cap. Since $T_n$ is a union of $\epsilon$-necks centered at points of -$\gamma_n'$ we see that every point $w\in T_n$ has -$$(3C/2)R(z_n)3$. + This implies that $R(x)>3$. This completes the proof of the claim. ### Extracting a limit of a subsequence of the tubes @@ -376,65 +401,91 @@ non-empty since $R(z'_n)$ is bounded independent of $n$ and for every $n$ there is a strong $\epsilon$-neck $N(z'_n)$ centered at $z'_n$ contained in $T_n$. On the other hand, since $d_{G_n'}(z_n',y_n')$ is uniformly bounded and $R(y_n')\rightarrow -\infty$, there is a finite upper bound for ${\mathcal A}$. Let +\infty$, there is a finite upper bound for ${\mathcal A}$. Let $A_1$ be the least upper bound of ${\mathcal A}$. We set $U_n=T_n\cap B(z'_n,A_1)$. This is an open subset of $T_n$ containing $z'_n$. We let $g_n'=G_n'|U_n$. -**Claim.** For all $n$ sufficiently large, $3R(z'_n)^{-1/2}\epsilon^{-1}/2$ is + +**Claim.** For all $n$ sufficiently large, $3R(z'_n)^{-1/2}\epsilon^{-1}/2$ is less than $A_1$, and hence $U_n$ contains the strong $\epsilon$-neck $N(z_n')$ centered at $z'_n$. + -**Proof.** The curvature on $N(z'_n)$ is bounded independent of $n$. Consider +**Proof.** The curvature on $N(z'_n)$ is bounded independent of $n$. Consider a point $w$ near the end of $N(z'_n)$ that separates $y_n'$ from $z_n'$. It is also the center of a strong $\epsilon$-neck $N(w)$. By -Proposition~ and our assumption that +\entryref{a46d72b87e07} and our assumption that $\epsilon\le \bar\epsilon(10^{-2})$, the scalar curvature on $N(z'_n)\cup N(w)$ is between $(0.9)R(z'_n)$ and $(1.1)R(z'_n)$. Since, by construction, the negative end of $N(z'_n)$ contains an end of $T_n$, this implies that -$$N(z'_n)\cup N(w)\supset B(z'_n,7R(z'_n)^{-1/2}\epsilon^{-1}/4)\cap T_n,$$ + +$$ +N(z'_n)\cup N(w)\supset B(z'_n,7R(z'_n)^{-1/2}\epsilon^{-1}/4)\cap T_n, +$$ + so that we see that -$A_1\ge 7\epsilon^{-1}lim_{n\rightarrow\infty} +$A_1\ge 7\epsilon^{-1}\mathit{lim}_{n\rightarrow\infty} R(z'_n)^{-1/2}/4$. Thus, $A_1>3R(z'_n)^{-1}\epsilon^{-1}/2$ for all $n$ sufficiently large. Obviously then $U_n$ contains $N(z'_n)$. -The next claim uses terminology from Definition~. +The next claim uses terminology from \entryref{1588b6b37e8c}. + **Claim.** For any $\delta>0$ there is a uniform bound, independent of $n$, for -the curvature on $Reg_\delta(U_n,g'_n)$. +the curvature on $\mathit{Reg}_\delta(U_n,g'_n)$. + **Proof.** To prove this it suffices to show that given $\delta>0$ there is -$A0$ sufficiently small, then it follows for all -$\delta>0$. First of all, by Corollary~ and -Lemma~, the fact that $\epsilon\le +$\delta>0$. First of all, by \entryref{b74cd360f72f} and +\entryref{dc5709757703}, the fact that $\epsilon\le \bar\epsilon(10^{-2})$ implies that any point $w$ with the property that the strong $\epsilon$-neighborhood centered at $w$ contains $z_n'$ is contained in the ball of radius $(1.1)R(z'_n)^{-1/2}\epsilon^{-1}\delta/4$. Since $w_n'\in U_n$ and consequently that $d(z_n',w_n')0$ and any $\ell\ge -0$ there is a uniform bound for $|\nabla^\ell Rm|$ on $Reg_\delta(U_n,g'_n)$. Clearly, since the base point $z_n'$ has -bounded curvature it lies in $Reg_\delta(U_n,g'_n)$ for +0$ there is a uniform bound for $|\nabla^\ell \mathit{Rm}|$ on $\mathit{Reg}_\delta(U_n,g'_n)$. Clearly, since the base point $z_n'$ has +bounded curvature it lies in $\mathit{Reg}_\delta(U_n,g'_n)$ for sufficiently small $\delta$ (how small being independent of $n$). Lastly, the fact that every point in $U_n$ is the center of an $\epsilon$-neighborhood implies that $(U_n,g'_n)$ is $\kappa$ @@ -462,7 +513,7 @@ universal. Since the $\gamma'_n$ have uniformly bounded lengths, the $\epsilon$-tubes $T_n'$ have uniformly bounded diameter. Also, we have seen that their have curvatures are bounded from below by $3$. It follows that their volumes are uniformly bounded. Now invoking -Theorem~ we see that after passing to a subsequence +\entryref{74c7682a68ed} we see that after passing to a subsequence we have a geometric limit $(U_\infty,g_\infty,z_\infty)$ of a subsequence of $(U_n,g'_n,z'_n)$. @@ -471,15 +522,15 @@ subsequence of $(U_n,g'_n,z'_n)$. Now we come to a result establishing all the properties we need for the limiting manifold. -**Proposition.** -The geometric limit $(U_\infty,g_\infty,z_\infty)$ is an incomplete + +**Proposition.** The geometric limit $(U_\infty,g_\infty,z_\infty)$ is an incomplete Riemannian $3$-manifold of finite diameter. There is a diffeomorphism $\psi\colon U_\infty\to S^2\times (0,1)$. There is a $2\epsilon$-neck centered at $z_\infty$ whose central $2$-sphere $S^2(z_\infty)$ maps under $\psi$ to a $2$-sphere isotopic to a -$2$-sphere factor in the product decomposition. The scalar +$2$-sphere factor in the product decomposition. The scalar curvature is bounded at one end of $U_\infty$ but tends to infinity -at the other end, the latter end which is denoted ${\mathcal E}$. +at the other end, the latter end which is denoted ${\mathcal E}$. Let ${\mathcal U}_\infty\subset U_\infty\times(-\infty,0]$ be the open subset consisting of all $(x,t)$ for which $-R(x)^{-1} The proof of this proposition occupies the rest of -Chapter~. +Chapter 10.3. **Proof.** Let $V_1\subset V_2 \subset \cdots \subset U_\infty$ be the open subsets and $\varphi_n\colon V_n\to U_n$ be the maps having all the -properties stated in Definition~ so as to exhibit -$(U_\infty,g_\infty,z_\infty)$ as the geometric limit of the +properties stated in \entryref{467f4b920031} so as to exhibit +$(U_\infty,g_\infty,z_\infty)$ as the geometric limit of the $(U_n,g_n',z_n')$. Since the $U_n$ are all contained in $B(z_n',A_1)$, it follows that any point of $U_\infty$ is within $A_1$ of the limiting base point $z_\infty$. This proves that the diameter of $U_\infty$ is bounded. -For each $n$ there is the $\epsilon$-neck $N(z_n')$ centered at +For each $n$ there is the $\epsilon$-neck $N(z_n')$ centered at $z_n'$ contained in $U_n$. The middle two-thirds, $N_n'$, of this -neck has closure contained in $Reg_\delta(U_n,g_n)$ for some +neck has closure contained in $\mathit{Reg}_\delta(U_n,g_n)$ for some $\delta>0$ independent of $n$ (in fact, restricting to $n$ sufficiently large, $\delta$ can be taken to be approximately equal to $R(z_\infty)^{-1/2}\epsilon^{-1}/3$). This means that for some $n$ sufficiently large and for all $m\ge n$ the image $\varphi_m(V_n)\subset U_m$ contains $N'_m$. For any fixed $n$ as -$m$ tends to infinity the metrics $\varphi_m^*g_m|_{V_n}$ converge +$m$ tends to infinity the metrics $\varphi_m^*g_m|_{V_n}$ converge uniformly in the $C^\infty$-topology to $g_\infty|_{V_n}$. Thus, it -follows from Proposition~ that for all $m$ +follows from \entryref{7c2758f0e2b3} that for all $m$ sufficiently large, $\varphi^{-1}_m(N'_m)$ is a $3\epsilon/2$-neck centered at $z_\infty$. We fix such a neck $N'(z_\infty)\subset U_\infty$. Let $S(z_\infty)$ be the central $2$-sphere of @@ -530,34 +582,35 @@ where the curvature is bounded (and where, in fact, the curvature is close to $R(z_\infty)$) and the other, denoted $W^+_\infty$, where it is unbounded. + **Claim.** Any point $q\in W^+_\infty$ is the center of a $2\epsilon$-neck in $U_\infty$. + -\begin{proof} -Fix a point $q\in W^+_\infty$. +**Proof.** Fix a point $q\in W^+_\infty$. For all $n$ sufficiently large denote by $q_n=\varphi_n(q)$. Then for all $n$ sufficiently large, $q_n\in W^+_n$ and -$lim_{n\rightarrow\infty}R(q_n)=R(q)$. This means that for all +$\mathit{lim}_{n\rightarrow\infty}R(q_n)=R(q)$. This means that for all $n$ sufficiently large $R(y_n')>>R(q_n))$, and hence the $3\epsilon/2$-neck centered at $q_n\in U_n$ is disjoint from $N(y'_n)$. Thus, by the rescaled version of - Corollary~, we see that the distance from + \entryref{b74cd360f72f}, we see that the distance from the $3\epsilon/2$-neck centered at $q_n$ to $N(y'_n)$ is bounded below by $(0.99)\epsilon^{-1}R(q_n)^{-1/2}/4\ge \epsilon^{-1}R(q_\infty)^{-1/2}/12$. Also, since $q_n\in W_n$, this $3\epsilon/2$-neck $N'(q_n)$ centered at $q_n$ does not extend past the $2$-sphere at $s^{-1}(-3\epsilon^{-1}/4)$ in the $ \epsilon$-neck $N(z'_n)$. It follows that for all $n$ sufficiently large that this $3\epsilon/2$-neck has - compact closure contained in $Reg_\delta(U_n,g_n)$ for some $\delta$ + compact closure contained in $\mathit{Reg}_\delta(U_n,g_n)$ for some $\delta$ independent of $n$, and hence there is $m$ such that for all $n$ sufficiently large $N'(q_n)$ is contained in the image $\varphi_n(V_m)$. Again using the fact that $\varphi_n^*(g_n|_{V_m})$ converges in the $C^\infty$-topology to - $g_\infty|_{V_m}$ as $n$ tends to infinity, we see, by Proposition~ + $g_\infty|_{V_m}$ as $n$ tends to infinity, we see, by \entryref{7c2758f0e2b3} that for all $n$ sufficiently large $\varphi_n^{-1}(N_m)$ contains a $2\epsilon$-neck in $U_\infty$ centered at $q$. -It now follows from Proposition~ that $W^+_\infty$ +It now follows from \entryref{0e9a1a167caa} that $W^+_\infty$ is contained in an $2\epsilon$-tube $T_\infty$ that is contained in $U_\infty$. Furthermore, the frontier of $W^+_\infty$ in $T_\infty$ is the $2$-sphere $S(z_\infty)$ which is isotopic to the central @@ -566,17 +619,19 @@ the closure $\overline W^+_\infty$ of $W^+_\infty$ is a $2\epsilon$-tube with boundary $S(z_\infty)$. In particular, $\overline W^+_\infty$ is diffeomorphic to $S^2\times [0,1)$. -Now we consider the closure $\overline W^-_\infty$ of $W^-_\infty$. +Now we consider the closure $\overline W^-_\infty$ of $W^-_\infty$. Since the closure of each $W^-_n$ is the closed negative half of the $\epsilon$-neck $N(z_n')$ and the curvatures of the $z_n'$ have a finite, positive limit, the limit $\overline W^-_\infty$ is diffeomorphic to a product $S^2\times (-1,0]$. Hence, $U_\infty$ is -the union of $\overline W^+_\infty$ and $\overline W^-_\infty$ +the union of $\overline W^+_\infty$ and $\overline W^-_\infty$ along their common boundary. It follows immediately that $U_\infty$ is diffeomorphic to $S^2\times (0,1)$. + **Claim.** The curvature is bounded in a neighborhood of one end of $U_\infty$ and goes to infinity at the other end. + **Proof.** A neighborhood of one end of $U_\infty$, the end $\overline W^-_\infty$, is the limit of the @@ -592,16 +647,18 @@ the scales of these $\epsilon$-necks are converging to zero as $k$ goes to infinity. This is equivalent to the statement that $R(x_k)$ tends to $\infty$ as $k$ goes to infinity. -The next step in the proof of Proposition~ is to extend +The next step in the proof of \entryref{e2e64cdb0b75} is to extend the flow backwards a certain amount. As stated in the proposition, the amount of backward time that we can extend the flow is not uniform over all of $U_\infty$, but rather depends on the curvature of the point at time zero. + **Claim.** For each $x\in U_n\subset M_n$ there is a flowline $\{x\}\times (-R(x)^{-1},0]$ in ${\mathcal M}_n$. Furthermore, the scalar curvature at any point of this flow line is less than or equal to the scalar curvature at $x$. + **Proof.** Since $x\in U_n\subset T_n$, there is a strong $\epsilon$-neck in ${\mathcal M}_n$ centered at $x$. Both statements follow immediately @@ -609,11 +666,16 @@ from that. Let $X\subset U_\infty$ be an open submanifold with compact closure and set -$$t_0(X)= sup_{x\in X}(-R_{g_\infty}(x)^{-1}).$$ Then for + +$$ +t_0(X)= \mathit{sup}_{x\in X}(-R_{g_\infty}(x)^{-1}). +$$ + + Then for all $n$ sufficiently large $\varphi_n$ is defined on $X$ and the scalar curvature of the flow $g_n(t)$ on $\varphi_n(X)\times (t_0,0]$ is uniformly bounded independent of $n$. Thus, according to -Proposition~ by passing to a subsequence we +\entryref{4673eb9f4424} by passing to a subsequence we can arrange that there is a limiting flow defined on $X\times (t_0,0]$. Let ${\mathcal U}_\infty\subset U_\infty\times (-\infty,0]$ consist of all pairs $(x,t)$ with the property that @@ -624,61 +686,62 @@ diagonal subsequence. This allows us to pass to a subsequence so that the limiting flow exists (as a generalized Ricci flow) on ${\mathcal U}_\infty$. + **Claim.** The curvature of the generalized Ricci flow on ${\mathcal U}_\infty$ is non-negative. + **Proof.** This claim follows from the fact that the original sequence $({\mathcal M}_n,G_n)$ consists of generalized flows whose curvatures are pinched toward positive in the weak sense -given in Equation~ and the fact that $Q_n\rightarrow -\infty$ as $n\rightarrow \infty$. (See Theorem~.) +given in Equation 10.1 and the fact that $Q_n\rightarrow +\infty$ as $n\rightarrow \infty$. (See \entryref{2ddfbc74b081}.) This completes the proof that all the properties claimed in -Proposition~ hold for the geometric limit -$(U_\infty,g_\infty,z_\infty)$. This completes the proof of that -proposition. \end{proof} +\entryref{e2e64cdb0b75} hold for the geometric limit +$(U_\infty,g_\infty,z_\infty)$. This completes the proof of that +proposition. -## Cone limits near the end ${\mathcal E -$ for rescalings of $U_\infty$} +## Cone limits near the end ${\mathcal E}$ for rescalings of $U_\infty$ -The next step is to study the nature of the limit $U_\infty$ given -in Proposition~. We shall show that an appropriate +The next step is to study the nature of the limit $U_\infty$ given +in \entryref{e2e64cdb0b75}. We shall show that an appropriate blow-up limit (limit in the Gromov-Hausdorff sense) around the end is a cone. -Let $(X,d_X)$ be a metric space. Recall that the cone on $X$, +Let $(X,d_X)$ be a metric space. Recall that the cone on $X$, denoted $C(X)$, is the quotient space $X\times [0,\infty)$ under the identification $(x,0)\cong (y,0)$ for all $x,y\in X$. The image of $X\times\{0\}$ is the *cone point* of the cone. The metric on $C(X)$ is given by $$ - -d((x,s_1),(y,s_2))=s_1^2+s_2^2-2s_1s_2cos(min(d_X(x,y),\pi)). +d((x,s_1),(y,s_2))=s_1^2+s_2^2-2s_1s_2\mathit{cos}(\mathit{min}(d_X(x,y),\pi)). $$ + The open cone $C'(X)$ is the complement of the cone point in $C(X)$ with the induced metric. The purpose of this section is to prove the following result. -**Proposition.** -Let $(U_\infty,g_\infty,z_\infty)$ be as in the conclusion of -Proposition~. Let $Q_\infty=R_{g_\infty}(z_\infty)$ and + +**Proposition.** Let $(U_\infty,g_\infty,z_\infty)$ be as in the conclusion of +\entryref{e2e64cdb0b75}. Let $Q_\infty=R_{g_\infty}(z_\infty)$ and let ${\mathcal E}$ be the end of $U_\infty$ where the scalar curvature is unbounded. Let $\lambda_n$ be any sequence of positive -numbers with $lim_{n\rightarrow\infty}\lambda_n=+\infty$. Then +numbers with $\mathit{lim}_{n\rightarrow\infty}\lambda_n=+\infty$. Then there is a sequence $x_n$ in $U_\infty$ such that for each $n$ the distance from $x_n$ to ${\mathcal E}$ is $\lambda_n^{-1/2}$, and such that the pointed Riemannian manifolds $(U_\infty,\lambda_ng_\infty,x_n)$ converge in the Gromov-Hausdorff sense to an open cone, an open cone not homeomorphic to an open ray (i.e., not homeomorphic to the open cone on a point). - (see Fig.~). + (see Fig. 10.2). + The rest of this section is devoted to the proof of this result. -### Directions at ${\mathcal E -$} +### Directions at ${\mathcal E}$ We orient the direction down the tube $U_\infty$ so that ${\mathcal E}$ is at the positive end. This gives an $s_N$-direction @@ -690,8 +753,10 @@ every $y\in \gamma$ the segment on $\gamma$ from $x$ to $y$ is a minimizing geodesic segment; i.e., the length of this geodesic segment is equal to $d(x,y)$. + **Claim.** There is a minimizing geodesic ray to ${\mathcal E}$ from each $x\in U_\infty$ with $R(x)\ge 2Q_\infty$. + **Proof.** Fix $x$ with $R(x)\ge 2Q_\infty$ and fix a $2\epsilon$-neck $N_x$ centered at $x$. Let $S_x^2$ be the central $2$-sphere of this neck. @@ -699,10 +764,10 @@ Take a sequence of points $q_n$ tending to the end ${\mathcal E}$, each being closer to the end than $x$ in the sense that $S^2_x$ does not separate any $q_n$ from the end ${\mathcal E}$. We claim that there is a minimizing geodesic from $x$ to each $q_n$. The reason is -that by Lemma~ any minimizing sequence of arcs from $x$ +that by \entryref{bb69d9bc83dc} any minimizing sequence of arcs from $x$ to $q_n$ cannot exit from the minus end of $N_x$ nor the plus end of a $2\epsilon$-neck centered at $q_n$. Consider a sequence of paths -from $x$ to $q_n$ minimizing the distance. Hence these paths all +from $x$ to $q_n$ minimizing the distance. Hence these paths all lie in a fixed compact subset of $U_n$. After replacing the sequence by a subsequence, we can pass to a limit, which is clearly a minimizing geodesic from $x$ to $q_n$. Consider minimizing geodesics @@ -710,19 +775,21 @@ $\mu_n$ from $x$ to $q_n$. The same argument shows that, after passing to a subsequence, the $\mu_n$ converge to a minimizing geodesic ray from $x$ to ${\mathcal E}$. + **Claim.** (1) Any minimizing geodesic ray from $x$ to the end ${\mathcal E}$ is a shortest ray from $x$ to the end ${\mathcal E}$, and conversely any shortest ray from $x$ to the end ${\mathcal E}$ is a minimizing geodesic ray. (2) The length of a shortest ray from $x$ to ${\mathcal E}$ is the -distance (see Section~) from $x$ to ${\mathcal E}$. +distance (see Section 2.4) from $x$ to ${\mathcal E}$. + **Proof.** The implication in (1) in one direction is clear: If $\gamma$ is a ray from $x$ to the end ${\mathcal E}$, and for some $y\in \gamma$ the segment on $\gamma$ from $x$ to $y$ is not minimizing, then there is a shorter geodesic segment $\mu$ from $x$ to $y$. The union -of this together with the ray on $\gamma$ from $y$ to the end is a +of this together with the ray on $\gamma$ from $y$ to the end is a shorter ray from $x$ to the end. Let us establish the opposite implication. Suppose that $\gamma$ is @@ -735,7 +802,7 @@ $S^2_i$ be the central $2$-sphere in the $2\epsilon$-neck centered at $q_i$. Of course, for all $i$ sufficiently large, both $\gamma'$ and $\gamma$ must cross $S^2_i$. Since the scalar curvature tends to infinity at the end ${\mathcal E}$, it follows from -Lemma~ for all $i$ sufficiently large, the extrinsic +\entryref{88a5a2233418} for all $i$ sufficiently large, the extrinsic diameter of $S^2_i$ is less than $\delta/3$. Let $p_i$ be a point of intersection of $\gamma$ with $S^2_i$. For all $i$ sufficiently large the length $d_i$ of the sub-ray in $\gamma$ from $p_i$ to the @@ -747,27 +814,39 @@ $\lambda'$ the sub-geodesic of $\gamma'$ from $x$ to $p'_i$. Let $\beta$ be a minimizing geodesic from $p'_i$ to $p_i$. Of course, $|\beta|<\delta/3$ so that by the minimality of $\lambda$ and $\lambda'$ we have -$$-\delta/3<|\lambda|-|\lambda'|<\delta/3.$$ Since + +$$ +-\delta/3<|\lambda|-|\lambda'|<\delta/3. +$$ + + Since $|\lambda'|+d'_i=|\lambda|+d_i-\delta$, we have -$$2\delta/3\le d_i-d'_i.$$ + +$$ +2\delta/3\le d_i-d'_i. +$$ + This is absurd since $d'_i>0$ and $d_i<\delta/3$. (2) follows immediately from (1) and the definition. Given this result, the usual arguments show: -**Corollary.** -If $\gamma$ is a minimizing geodesic ray from $x$ to the end + +**Corollary.** If $\gamma$ is a minimizing geodesic ray from $x$ to the end ${\mathcal E}$, then for any $y\in \gamma\setminus\{x\}$ the sub-ray of $\gamma$ from $y$ to the end, is the unique shortest geodesic from $y$ to the end. + Also, we have a version of the triangle inequality for distances to ${\mathcal E}$. + **Lemma.** Let $x$ and $y$ be points of $M$. Then the three distances $d(x,y)$, $d(x,{\mathcal E})$ and $d(y,{\mathcal E})$ satisfy the triangle inequality. + **Proof.** From the definitions it is clear that $d(x,y)+d(y,{\mathcal E})\ge d(x,{\mathcal E})$, and symmetrically, reversing the roles of $x$ @@ -775,19 +854,21 @@ and $y$. The remaining inequality that we must establish is the following: $d(x,{\mathcal E})+d(y,{\mathcal E})\ge d(x,y)$. Let $q_n$ be any sequence of points converging to ${\mathcal E}$. Since the end is at finite distance, it is clear that $d(x,{\mathcal E})= -lim_{n\rightarrow\infty}d(x,q_n)$. The remaining inequality -follows from this and the usual triangle inequality applied to +\mathit{lim}_{n\rightarrow\infty}d(x,q_n)$. The remaining inequality +follows from this and the usual triangle inequality applied to $d(x,q_n)$, $d(y,q_n)$ and $d(x,y)$. + **Definition.** We say that two minimizing geodesic rays limiting to ${\mathcal E}$ are *equivalent* if one is contained in the other. From the unique continuation of geodesics it is easy to see that this -generates an equivalence relation. An equivalence class is a {\sl -direction at ${\mathcal E}$}, and the set of equivalence classes is -the {\sl set of directions at ${\mathcal E}$}. +generates an equivalence relation. An equivalence class is a *direction at ${\mathcal E}$*, and the set of equivalence classes is +the *set of directions at ${\mathcal E}$*. + -**Lemma.** -There is more than one direction at ${\mathcal E}$. + +**Lemma.** There is more than one direction at ${\mathcal E}$. + **Proof.** Take a minimal geodesic ray $\gamma$ from a point $x$ limiting to the end and let $y$ be a point closer to ${\mathcal E}$ than $x$ and @@ -795,16 +876,16 @@ not lying on $\gamma$. Then a minimal geodesic ray from $y$ to ${\mathcal E}$ gives a direction at ${\mathcal E}$ distinct from the direction determined by $\gamma$. + **Remark.** In fact, the general theory of positively curved spaces implies that the space of directions is homeomorphic to $S^2$. Since we do not need this stronger result we do not prove it. + -### The Metric on the space of directions at ${\mathcal -E -$} +### The Metric on the space of directions at ${\mathcal E}$ -**Definition.** -Let $\gamma$ and $\mu$ be minimizing geodesic rays limiting to + +**Definition.** Let $\gamma$ and $\mu$ be minimizing geodesic rays limiting to ${\mathcal E}$, of lengths $a$ and $b$, parameterized by the distance from the end. For $0 + + +**Lemma.** For all $\gamma,s,\mu,s'$ as in the previous definition we have -**Lemma.** -For all $\gamma,s,\mu,s'$ as in the previous definition we have -$$0\le \theta(\gamma,s,\mu,s')\le \pi.$$ Furthermore, $\theta(\gamma,s,\mu,s')$ -is a non-increasing function of $s$ when $\gamma,\mu,s'$ are held +$$ +0\le \theta(\gamma,s,\mu,s')\le \pi. +$$ + + Furthermore, $\theta(\gamma,s,\mu,s')$ +is a non-increasing function of $s$ when $\gamma,\mu,s'$ are held fixed, and symmetrically it is a non-increasing function of $s'$ when $\gamma,s,\mu$ are held fixed. In particular, fixing $\gamma$ and $\mu$, the function $\theta(\gamma,s,\mu,s')$ is non-decreasing @@ -828,6 +915,7 @@ $0\le\theta(\gamma,\mu)\le \pi$. The angle $\theta(\gamma,\mu)=0$ if and only if $\gamma$ and $\mu$ are equivalent. Furthermore, if $\gamma$ is equivalent to $\gamma'$ and $\mu$ is equivalent to $\mu'$, then $\theta(\gamma,\mu)=\theta(\gamma',\mu')$. + **Proof.** By restricting $\gamma$ and $\mu$ to slightly smaller rays, we can assume that each is the unique shortest ray from its endpoint to the @@ -837,7 +925,7 @@ limiting to the end ${\mathcal E}$, and consider minimizing geodesic rays $\gamma_n$ from $q_n$ to $x$ and $\mu_n$ from $q_n$ to $y$, each parameterized by the distance from $q_n$. By passing to a subsequence we can assume that each of the sequences $\{\gamma_n\}$ -and $\{\mu_n\}$ converge to a minimizing geodesic ray, which by +and $\{\mu_n\}$ converge to a minimizing geodesic ray, which by uniqueness, implies that the first sequence limits to $\gamma$ and the second to $\mu$. For $s,s'$ sufficiently small, let $\theta_n(s,s')$ be the angle at $\tilde q_n$ of the Euclidean @@ -846,7 +934,7 @@ q_n|=d(\gamma_n(s),q_n)$, $|\beta_n\tilde q_n|=d(\mu_n(s'),q_n)$ and $|\alpha_n\beta_n|=d(\gamma_n(s),\mu_n(s'))$. Clearly, for fixed $s$ and $s'$ sufficiently small, $\theta_n(s,s')$ converges as $n\rightarrow\infty$ to $\theta(\gamma,s,\mu,s')$. By the Toponogov -property (Theorem~) for manifolds with +property (\entryref{04518fa1ac2f}) for manifolds with non-negative curvature, for each $n$ the function $\theta_n(s,s')$ is a non-increasing function of each variable, when the other is held fixed. This property then passes to the limit, giving the first @@ -864,80 +952,110 @@ symmetric, non-negative function which is positive off of the diagonal. The next lemma shows that it is a metric by establishing the triangle inequality for $\theta$. + **Lemma.** If $\gamma,\mu,\nu$ are minimizing geodesic rays limiting to ${\mathcal E}$, then -$$\theta(\gamma,\mu)+\theta(\mu,\nu)\ge \theta(\gamma,\nu).$$ -**Proof.** By Corollary~, after replacing $\gamma, \mu,\nu$ by +$$ +\theta(\gamma,\mu)+\theta(\mu,\nu)\ge \theta(\gamma,\nu). +$$ + + +**Proof.** By \entryref{7ef7dfbbe151}, after replacing $\gamma, \mu,\nu$ by equivalent, shorter geodesic arcs, we can assume that they are the unique minimizing geodesics from their end points, say $x,y,z$ respectively, to ${\mathcal E}$. Let $q_n$ be a sequence of points limiting to ${\mathcal E}$, and let $\gamma_n,\mu_n,\nu_n$ be minimizing geodesics from $x,y,z$ to $q_n$. Denote by $\theta_n(x,y), \theta_n(y,z)$, and $\theta_n(x,z)$, respectively, -the angles at $\tilde q_n$ of the Euclidean triangles with the +the angles at $\tilde q_n$ of the Euclidean triangles with the following edge lengths: $\{d(x,y), d(x,q_n), d(y,q_n)\}$, $\{d(y,z),d(y,q_n),d(z,q_n)\}$, and $\{d(z,x),d(z,q_n),d(x,q_n)\}$. -According to Corollary~ we have +According to \entryref{dc263832416c} we have $\theta_n(x,y)+\theta_n(y,z)\ge \theta_n(x,z)$. Passing to the limit as $n$ goes to $\infty$ and then the limit as $x$, $y$ and $z$ tend to ${\mathcal E}$, gives the result. + **Definition.** Let $X({\mathcal E})$ denote the set of directions at ${\mathcal E}$. We define the metric on $X({\mathcal E})$ by setting $d([\gamma],[\mu])=\theta(\gamma,\mu)$. We call this the *(metric) space of realized directions at* ${\mathcal E}$. The *metric space of directions at* ${\mathcal E}$ is the completion $\bar X({\mathcal E})$ of $X({\mathcal E})$ with respect to the given metric. We denote by $(C_{\mathcal E},g_{\mathcal E})$ the cone on $\bar X({\mathcal E})$ with the cone metric as given in -Equation~(). (See Fig.~.) +Equation (10.2). (See Fig. 10.2.) + -**Proposition.** -$(C_{\mathcal E},g_{\mathcal E})$ is a metric cone that is not + +**Proposition.** $(C_{\mathcal E},g_{\mathcal E})$ is a metric cone that is not homeomorphic to a ray. + **Proof.** By construction $(C_{\mathcal E},g_{\mathcal E})$ is a metric cone. That it is not homeomorphic to a ray follows immediately from -Lemma~. +\entryref{dda8a7c7acb6}. ### Comparison results for distances -**Lemma.** -Suppose that $\gamma$ and $\mu$ are unique shortest geodesic rays + +**Lemma.** Suppose that $\gamma$ and $\mu$ are unique shortest geodesic rays from points $x$ and $y$ to the end ${\mathcal E}$. Let $[\gamma]$ and $[\mu]$ be the points of $X({\mathcal E})$ represented by these two geodesics rays. Let $a$, resp. $b$, be the distance from $x$, -resp. $y$, to ${\mathcal E}$. Denote by $x'$, resp. $y'$, the image +resp. $y$, to ${\mathcal E}$. Denote by $x'$, resp. $y'$, the image in $C_{\mathcal E}$ of the point $([\gamma],a)$, resp. $([\mu],b)$, of $X({\mathcal E})\times [0,\infty)$. Then -$$d_{g_\infty}(x,y)\le d_{g_{\mathcal E}}(x',y').$$ + +$$ +d_{g_\infty}(x,y)\le d_{g_{\mathcal E}}(x',y'). +$$ + **Proof.** By the definition of the cone metric we have -$$d_{g_{\mathcal E}}(x',y')=a^2+b^2-2ab\,cos(\theta(\gamma,\mu)).$$ -On the other hand by Definition~ and the law of + +$$ +d_{g_{\mathcal E}}(x',y')=a^2+b^2-2ab \mathit{cos}(\theta(\gamma,\mu)). +$$ + +On the other hand by \entryref{f4ee075394ef} and the law of cosines for Euclidean triangles, we have -$$d_{g_\infty}(x,y)=a^2+b^2-2ab\,cos(\theta(\gamma,a,\mu,b)).$$ + +$$ +d_{g_\infty}(x,y)=a^2+b^2-2ab \mathit{cos}(\theta(\gamma,a,\mu,b)). +$$ + The result is now immediate from the fact, proved in -Lemma~ that -$$0\le \theta(\gamma,a,\mu,b)\le \theta(\gamma,\mu)\le \pi,$$ +\entryref{f4ee075394ef} that + +$$ +0\le \theta(\gamma,a,\mu,b)\le \theta(\gamma,\mu)\le \pi, +$$ + and the fact that the cosine is a monotone decreasing function on the interval $[0,\pi]$. -**Corollary.** -Let $\gamma,\mu,x,y$ be as in the previous lemma. Fix $\lambda>0$. + +**Corollary.** Let $\gamma,\mu,x,y$ be as in the previous lemma. Fix $\lambda>0$. Let $a=d_{\lambda g_\infty}(x,{\mathcal E})$ and $b=d_{\lambda g_\infty}(y,{\mathcal E})$. Set $x'_\lambda$ and $y'_\lambda$ equal to the points in the cone $([\gamma],a)$ and $([\mu],b)$. Then we have -$$d_{\lambda g_\infty}(x,y)\le d_{g_{\mathcal E}}(x'_\lambda,y'_\lambda).$$ + +$$ +d_{\lambda g_\infty}(x,y)\le d_{g_{\mathcal E}}(x'_\lambda,y'_\lambda). +$$ + **Proof.** This is immediate by applying the previous lemma to the rescaled manifold $(U_\infty,\lambda g_\infty)$, and noticing that rescaling does not affect the cone $C_{\mathcal E}$ nor its metric. + **Lemma.** For any $\delta>0$ there is $K=K(\delta)<\infty$ so that for any set of realized directions at ${\mathcal E}$ of cardinality $K$, $\ell_1,\ldots,\ell_{K}$, it must be the case that there are $j$ and $j'$ with $j\not=j'$ such that $\theta(\ell_j,\ell_{j'})<\delta$. + **Proof.** Let $K$ be such that, given $K$ points in the central $2$-sphere of any $2\epsilon$-tube of scale $1$, at least two are within distance @@ -965,27 +1083,27 @@ points such that $X$ is contained in the union of the $\delta$-neighborhoods of these points. The above lemma immediately yields: -**Corollary.** -The metric completion $\bar X({\mathcal E})$ of the space of + +**Corollary.** The metric completion $\bar X({\mathcal E})$ of the space of directions at ${\mathcal E}$ is a compact space. For every -$\delta>0$ this space has a $\delta$-net consisting of realized +$\delta>0$ this space has a $\delta$-net consisting of realized directions. For every $0 -### Completion of the proof of a cone limit at ${\mathcal E -$} +### Completion of the proof of a cone limit at ${\mathcal E}$ -Now we are ready to prove Proposition~. In fact, we +Now we are ready to prove \entryref{9cf6ad839dbb}. In fact, we prove a version of the proposition that identifies the sequence of points $x_n$ and also identifies the cone to which the rescaled manifolds converge. -**Proposition.** -Let $(U_\infty,g_\infty)$ be an incomplete Riemannian $3$-manifold + +**Proposition.** Let $(U_\infty,g_\infty)$ be an incomplete Riemannian $3$-manifold of non-negative curvature with an end ${\mathcal E}$ as in the -hypothesis of Proposition~. Fix a minimizing geodesic +hypothesis of \entryref{9cf6ad839dbb}. Fix a minimizing geodesic ray $\gamma$ limiting to ${\mathcal E}$. Let $\lambda_n$ be any sequence of positive numbers tending to infinity. For each $n$ sufficiently large let $x_n\in \gamma$ be the point at distance @@ -996,6 +1114,7 @@ $\left( C'_{\mathcal E},g_{\mathcal E},([\gamma],1)\right)$. Under this convergence the distance function from the end ${\mathcal E}$ in $(U_\infty,\lambda_ng_\infty)$ converges to the distance function from the cone point in the open cone. + **Proof.** It suffices to prove that given any subsequence of the original sequence, the result holds for a further subsequence. So let us @@ -1009,9 +1128,9 @@ point, for all $n$ sufficiently large, there are compact subspaces $K_n\subset (U_\infty,\lambda_ng_\infty)$ containing $x_n$ with the property that the $(K_n,x_n)$ converge in the Gromov-Hausdorff topology to $(K,x)$ (see Section D of Chapter 3, p. 39, of -). +[Gromov]). -Because of this, it suffices to fix $0 **Remark.** Notice that since the manifolds $(U_\infty,\lambda_ng_\infty,x_n)$ are not complete, there can be more than one Gromov-Hausdorff limit. For example we could take the full cone as a limit. Indeed, the cone is the only Gromov-Hausdorff limit that is complete as a metric space. + -## Comparison of the Gromov-Hausdorff limit and the smooth -limit +## Comparison of the Gromov-Hausdorff limit and the smooth limit Let us recap the progress to date. We constructed an incomplete geometric blow-up limit $({\mathcal U}_\infty,G_\infty,z_\infty)$ @@ -1110,7 +1239,7 @@ showed that the zero time-slice $U_\infty$ of the limit is diffeomorphic to a tube $S^2\times (0,1)$ and that at one end of the tube the scalar curvature goes to infinity. Also, any point sufficiently near this end is the center of an evolving -$2\epsilon$-neck defined for an interval of normalized time of +$2\epsilon$-neck defined for an interval of normalized time of length $1/2$ in the limiting flow. Then we took a further blow-up limit. We chose a sequence of points $x_n\in U_\infty$ tending to the end ${\mathcal E}$ where the scalar curvature goes to infinity. @@ -1130,11 +1259,11 @@ metric ball of radius $\epsilon^{-1}/3$ centered at $x_n$ in $(U_\infty,\lambda'_ng_\infty(0))$. Since this ball is contained in a $2\epsilon$-neck centered at $x_n$, the curvature on this ball is bounded, and this ball has compact closure in $U_\infty$. Also, for -each $y\in B_n$, there is a rescaled flow $\lambda'g(t)$ defined on -$\{y\}\times (-1/2,0]$ whose curvature on $B_n\times (-1/2,0]$ is -bounded. Hence, by Theorem~ we can pass to a +each $y\in B_n$, there is a rescaled flow $\lambda'g(t)$ defined on +$\{y\}\times (-1/2,0]$ whose curvature on $B_n\times (-1/2,0]$ is +bounded. Hence, by \entryref{d9f3522dec12} we can pass to a subsequence and extract a geometric limit. In fact, by -Proposition~ there is even a geometric +\entryref{4673eb9f4424} there is even a geometric limiting flow defined on the time interval $(-1/2,0]$. We must compare the zero time-slice of this geometric limiting flow @@ -1153,22 +1282,27 @@ the sequence $(U_\infty,\lambda_ng_\infty,x_n)$ and (ii) the geometric limit of the sequence $(U_\infty,\lambda_n'g_\infty,x_n)$ constructed above. + **Claim.** The ratio $\rho_n=\lambda_n'/\lambda_n$ is bounded above and below by positive constants. + **Proof.** First of all, since there is a $2\epsilon$-neck centered at $x_n$, -by Proposition~ we see that the distance +by \entryref{a46d72b87e07} we see that the distance $\lambda_n^{-1/2}$ from $x_n$ to ${\mathcal E}$ is at least $R(x_n)^{-1/2}\epsilon^{-1}/2=(\lambda'_n)^{-1/2}\epsilon^{-1}/2$. Thus, -$$\rho_n^{-1}=\lambda_n/\lambda'_n\le 4\epsilon^2.$$ + +$$ +\rho_n^{-1}=\lambda_n/\lambda'_n\le 4\epsilon^2. +$$ On the other hand, suppose that $\rho_n=\lambda_n'/\lambda_n\rightarrow \infty$ as $n\rightarrow \infty$. Rescale by $\lambda'_n$ so that $R(x_n)=1$. The distance from $x_n$ to ${\mathcal E}$ is $\sqrt{\rho_n}$. Then by -Lemma~ with respect to this metric there is a sphere -of diameter at most $2\pi$ through $x_n$ that separates all points +\entryref{88a5a2233418} with respect to this metric there is a sphere +of diameter at most $2\pi$ through $x_n$ that separates all points at distance at most $\sqrt{\rho_n}-\epsilon^{-1}$ from ${\mathcal E}$ from all points at distance at least $\sqrt{\rho_n}+\epsilon^{-1}$ from ${\mathcal E}$. Now rescale the @@ -1189,16 +1323,16 @@ Once we know that the $\lambda_n/\lambda_n'$ are bounded above and below by positive constants, we can pass to a subsequence so that these ratios converge to a finite positive limit. This means that the Gromov-Hausdorff limit of the sequence of based metric spaces -$(U_\infty,\lambda_n'g_\infty,x_n)$ is a cone, namely the -Gromov-Hausdorff limiting cone constructed is Section~ -rescaled by $lim_{n\rightarrow\infty}\rho_n$. In particular, +$(U_\infty,\lambda_n'g_\infty,x_n)$ is a cone, namely the +Gromov-Hausdorff limiting cone constructed is Section 10.4 +rescaled by $\mathit{lim}_{n\rightarrow\infty}\rho_n$. In particular, the balls of radius $\epsilon^{-1}/2$ around the base points in this sequence converge in the Gromov-Hausdorff sense to the ball of radius $\epsilon^{-1}/2$ about the base point of a cone. But we have already seen that the balls of radius $\epsilon^{-1}/2$ centered at the base points converge geometrically to a limiting -manifold. That is to say, on every ball of radius less than +manifold. That is to say, on every ball of radius less than $\epsilon^{-1}/2$ centered at the base point the metrics converge uniformly in the $C^\infty$-topology to a limiting smooth metric. Thus, on every ball of radius less than $\epsilon^{-1}/2$ centered @@ -1216,9 +1350,9 @@ $\epsilon^{-1}/2$ centered at the base points of $(U_\infty,\lambda_n'g_\infty,x_n)$ is isometric to an open subset of a non-flat cone, and is also the zero time-slice of a Ricci flow defined for the time interval $(-1/2,0]$. This -contradicts Proposition~, one of the consequences of +contradicts \entryref{92b5c664399e}, one of the consequences of the maximum principle established by Hamilton. The contradiction shows that the limit $(U_\infty,g_\infty,x_\infty)$ cannot exist. The only assumption that we made in order to construct this limit -was that Theorem~ did not hold for some $A_0<\infty$. -Thus, we have established Theorem~ by contradiction. +was that \entryref{ecd62fc2ea2b} did not hold for some $A_0<\infty$. +Thus, we have established \entryref{ecd62fc2ea2b} by contradiction. diff --git a/projects/poincare-conjecture/.astrolabe/docs/11-singlimit2.mdx b/projects/poincare-conjecture/.astrolabe/docs-src/12-geometric-limits-of-generalized-ricci-flows.mdx similarity index 73% rename from projects/poincare-conjecture/.astrolabe/docs/11-singlimit2.mdx rename to projects/poincare-conjecture/.astrolabe/docs-src/12-geometric-limits-of-generalized-ricci-flows.mdx index da474f1c..424c1a76 100644 --- a/projects/poincare-conjecture/.astrolabe/docs/11-singlimit2.mdx +++ b/projects/poincare-conjecture/.astrolabe/docs-src/12-geometric-limits-of-generalized-ricci-flows.mdx @@ -1,4 +1,6 @@ -# Geometric limits of generalized Ricci flows + + +# Chapter 11 -- Geometric limits of generalized Ricci flows In this chapter we apply the main result of the last section, bounded curvature at bounded distance, to blow-up limits in order to @@ -13,40 +15,39 @@ of a generalized Ricci flow satisfying appropriate conditions. We characterize the ends of the components of these limits. We show that they are $\epsilon$-horns -- the ends are diffeomorphic to $S^2\times [0,1)$ and the scalar curvature goes to infinity at the -end. In the fourth section we prove for any $\delta>0$ that there +end. In the fourth section we prove for any $\delta>0$ that there are $\delta$-necks sufficiently deep in any $\epsilon$-horn, provided that the curvature at the other end of the horn is not too large. Throughout this chapter we fix $\epsilon>0$ sufficiently small such that all the results of the Appendix hold for $2\epsilon$ -and $\alpha=10^{-2}$, and Proposition~ holds for +and $\alpha=10^{-2}$, and \entryref{cb9ca6d6db9f} holds for $2\epsilon$. ## A smooth blow-up limit defined for a small time We begin with a theorem that produces a blow-up limit flow defined on some small time interval. -**Theorem.** -Fix canonical neighborhood constants $(C,\epsilon)$, and + +**Theorem.** Fix canonical neighborhood constants $(C,\epsilon)$, and non-collapsing constants $r>0,\kappa>0$. Let $({\mathcal M}_n,G_n,x_n)$ be a sequence of based generalized $3$-dimensional -Ricci flows. We set $t_n=**t**(x_n)$ and $Q_n=R(x_n)$. We denote +Ricci flows. We set $t_n=\mathbf{t}(x_n)$ and $Q_n=R(x_n)$. We denote by $M_n$ the $t_n$ time-slice of ${\mathcal M}_n$. We suppose that: -\begin{enumerate} -\item[(1)] Each $({\mathcal M}_n,G_n)$ either has a time interval of definition contained in $[0,\infty)$ -and has curvature pinched toward positive, or has non-negative curvature. -\item[(2)] Every point $y_n\in ({\mathcal M}_n,G_n)$ with $**t**(y_n)\le + +- **(1)** Each $({\mathcal M}_n,G_n)$ either has a time interval of definition contained in $[0,\infty)$ +and has curvature pinched toward positive, or has non-negative curvature. +- **(2)** Every point $y_n\in ({\mathcal M}_n,G_n)$ with $\mathbf{t}(y_n)\le t_n$ and with $R(y_n)\ge 4R(x_n)$ has a strong $(C,\epsilon)$-canonical neighborhood. -\item[(3)] $lim_{n\rightarrow\infty}Q_n=\infty$. -\item[(4)] For each $A<\infty$ the following holds for all $n$ sufficiently large. The ball +- **(3)** $\mathit{lim}_{n\rightarrow\infty}Q_n=\infty$. +- **(4)** For each $A<\infty$ the following holds for all $n$ sufficiently large. The ball $B(x_n,t_n,AQ_n^{-1/2})$ has compact closure in $M_n$ and the flow is $\kappa$-non-collapsed on scales $\le r$ at each point of $B(x_n,t_n,AQ_n^{-1/2})$. -\item[(5)] There is $\mu>0$ such that for every $A<\infty$ the following holds +- **(5)** There is $\mu>0$ such that for every $A<\infty$ the following holds for all $n$ sufficiently large. For every $y_n\in B(x_n,t_n,AQ_n^{-1/2})$ the maximal flow line -through $y_n$ extends backwards for a time at least $\mu\left(max(Q_n,R(y_n))\right)^{-1}$. -\end{enumerate} +through $y_n$ extends backwards for a time at least $\mu\left(\mathit{max}(Q_n,R(y_n))\right)^{-1}$. Then, after passing to a subsequence and shifting the times of each of the generalized flows so that $t_n=0$ for every $n$, there is a @@ -57,30 +58,41 @@ non-negative curvature. Furthermore, for some $t_0>0$ which depends on the curvature bound for $(M_\infty,g_\infty)$ and on $\mu$, there is a geometric limit Ricci flow defined on $(M_\infty,g_\infty(t)), -t_0\le t\le 0$, with $g_\infty(0)=g_\infty$. + Before beginning the proof of this theorem we establish a lemma that we shall need both in its proof and also for later applications. -**Lemma.** -Let $({\mathcal M},G)$ be a generalized $3$-dimensional Ricci flow. + +**Lemma.** Let $({\mathcal M},G)$ be a generalized $3$-dimensional Ricci flow. Suppose that $r_0>0$ and that any $z\in {\mathcal M}$ with $R(z)\ge r_0^{-2}$ has a strong $(C,\epsilon)$-canonical neighborhood. Suppose $z\in -{\mathcal M}$ and $**t**(z)=t_0$. Set -$$r=\frac{1}{2C\sqrt{max(R(z),r_0^{-2})}}$$ and -$$\Delta t=\frac{1}{16C\left(R(z)+r_0^{-2}\right)}.$$ +{\mathcal M}$ and $\mathbf{t}(z)=t_0$. Set + +$$ +r=\frac{1}{2C\sqrt{\mathit{max}(R(z),r_0^{-2})}} +$$ + + and + +$$ +\Delta t=\frac{1}{16C\left(R(z)+r_0^{-2}\right)}. +$$ + Suppose that $r'\le r$ and that $|t'-t_0|\le \Delta t$ and let $I$ be the interval with endpoints $t_0$ and $t'$. Suppose that there is an embedding of $j\colon B(z,t_0,r')\times I$ into ${\mathcal M}$ compatible with time and with the vector field. Then $R(y)\le 2\left(R(z)+r_0^{-2}\right)$ for all $y$ in the image of $j$. + -**Proof.** We first prove that for any $y\in B(z,t_0,r)$ we have +**Proof.** We first prove that for any $y\in B(z,t_0,r)$ we have $$ - R(y)\le \frac{16}{9}(R(z)+r_0^{-2}). $$ + Let $\gamma\colon [0,s_0]\to B(z,t_0,r)$ be a path of length $s_00$ and, for each $n$ sufficiently large, an embedding of $B_{Q_nG_n}(x_n,0,A)\times [-t_0(A),0]$ into ${\mathcal M}_n$ compatible with time and with the vector field. In fact, we can choose $t_0(A)$ so that more is true. -**Corollary.** -For each $A<\infty$, let $Q(A)$ be a bound on the scalar curvature -of the restriction of $Q_nG_n$ to $B_{Q_nG_n}(x_n,0,A)$ for all $n$ -sufficiently large. Then there exist a constant $t'_0(A)>0$ + +**Corollary.** For each $A<\infty$, let $Q(A)$ be a bound on the scalar curvature +of the restriction of $Q_nG_n$ to $B_{Q_nG_n}(x_n,0,A)$ for all $n$ +sufficiently large. Then there exist a constant $t'_0(A)>0$ depending on $t_0(A)$ and $Q(A)$, and a constant $Q'(A)<\infty$ depending only on $Q(A)$, and, for all $n$ sufficiently large, an embedding -$$B_{Q_nG_n}(x_n,0,A)\times (-t'_0(A),0]\to{\mathcal -M}_n$$ compatible with time and with the vector field with the -property that the scalar curvature of the restriction of $Q_nG_n$ + +$$ +B_{Q_nG_n}(x_n,0,A)\times (-t'_0(A),0]\to{\mathcal +M}_n +$$ + + compatible with time and with the vector field with the +property that the scalar curvature of the restriction of $Q_nG_n$ to the image of this subset is bounded by $Q'(A)$. + -\begin{proof} -This is immediate from Lemma~ and Assumption (5) +**Proof.** This is immediate from \entryref{0ddfe578961e} and Assumption (5) in the hypothesis of the theorem. Now since the curvatures of the $Q_nG_n$ are pinched toward positive or are non-negative, bounding the scalar curvature above gives a -bound on $|Rm_{Q_nG_n}|$ on the product +bound on $|\mathit{Rm}_{Q_nG_n}|$ on the product $B_{Q_nG_n}(x_n,0,A)\times (-t'_0(A),0]$. Now we invoke Shi's -theorem (Theorem~): +theorem (\entryref{ed4b72caf9f9}): + **Corollary.** For each $A<\infty$ and for each integer $\ell\ge 0$, there is a constant $C_2$ such that for all $n$ sufficiently large we have -$$|\nabla^\ellRm_{Q_nG_n}(x)|\le C_2$$ for all $x\in + +$$ +|\nabla^\ell{\mathit{Rm}}_{Q_nG_n}(x)|\le C_2 +$$ + + for all $x\in B_{Q_nG_n}(x_n,0,A) $. + -Also, by the curvature bound and the $\kappa$-non-collapsed +Also, by the curvature bound and the $\kappa$-non-collapsed hypothesis we have the following: + **Claim.** There is $\eta>0$ such that for all $n$ sufficiently large -$$Vol(B_{Q_nG_n}(x_n,0,\eta))\ge \kappa\eta^3.$$ -Now we are in a position to apply Corollary~. This +$$ +\mathit{Vol}(B_{Q_nG_n}(x_n,0,\eta))\ge \kappa\eta^3. +$$ + + +Now we are in a position to apply \entryref{b45765a3070e}. This implies that, after passing to a subsequence, there is a geometric limit $(M_\infty,g_\infty,x_\infty)$ of the sequence of based Riemannian manifolds $(M_n,Q_nG_n(0),x_n)$. The geometric limit is a complete Riemannian manifold. If the $({\mathcal M}_n,G_n)$ satisfy the curvature pinched toward positive hypothesis, by -Theorem~, the limit Riemannian manifold +\entryref{2ddfbc74b081}, the limit Riemannian manifold $(M_\infty,g_\infty)$ has non-negative curvature. If the $({\mathcal M}_n,G_n)$ have non-negative curvature, then it is obvious that the limit has non-negative curvature. By construction $R(x_\infty)=1$. -In fact, by Proposition~ for each $A<\infty$, +In fact, by \entryref{4673eb9f4424} for each $A<\infty$, there is $t(A)>0$ and, after passing to a subsequence, geometrically limit flow defined on $B(x_\infty,0,A)\times (-t(A),0]$. + **Claim.** Any point in $(M_\infty,g_\infty)$ of curvature greater than $4$ has a $(2C,2\epsilon)$-canonical neighborhood. + **Proof.** The fact that $(M_\infty,g_\infty,x_\infty)$ is the geometric limit of the $(M_n,Q_nG_n(0),x_n)$ means that we have the following. There -is an exhausting sequence $V_1\subset V_2\subset\cdots \subset +is an exhausting sequence $V_1\subset V_2\subset\cdots \subset M_\infty$ of open subsets of $M_\infty$, with compact closure, each -containing $x_\infty$, and for each $n$ an embedding $\varphi_n$ of +containing $x_\infty$, and for each $n$ an embedding $\varphi_n$ of $V_n$ into the zero time-slice of ${\mathcal M}_n$ such that $\varphi_n(x_\infty)=x_n$ and such that the Riemannian metrics $\varphi_n^*G_n$ converge uniformly on compact sets to $g_\infty$. @@ -198,26 +229,28 @@ Let $q\in M_\infty$ be a point with $R_{g_\infty}(q)>4$. Then for all $n$ sufficiently large, $q\in V_n$, so that $q_n=\varphi_n(q)$ is defined, and $R_{Q_nG_n}(q_n)>4$. Thus, $q_n$ has an $(C,\epsilon)$-canonical neighborhood, $U_n$, in ${\mathcal M}_n$; -and, since $R(q_n)>4$ for all $n$, there is a uniform bound to the +and, since $R(q_n)>4$ for all $n$, there is a uniform bound to the distance from any point of $U_n$ to $q_n$. Thus, there exists $m$ such that for all $n$ sufficiently large $\varphi_n(V_m)$ contains $U_n$. Clearly as $n$ goes to infinity the Riemannian metrics $\varphi_n^*(G_n)|_{\varphi_n^{-1}(U_m)}$ converge smoothly to $g_\infty|_{\varphi_n^{-1}(U_n)}$. Thus, by -Proposition~ for all $n$ sufficiently large the +\entryref{7c2758f0e2b3} for all $n$ sufficiently large the restriction of $g_\infty$ to $\varphi_n^{-1}(U_n)$ contains a $(2C,2\epsilon)$-canonical neighborhood of $q$. + **Claim.** The limit Riemannian manifold $(M_\infty,g_\infty)$ has bounded curvature. + **Proof.** First, suppose that $(M_\infty,g_\infty)$ does not have strictly positive curvature. Suppose that $y\in M_\infty$ has the property -that $Rm(y)$ has a zero eigenvalue. Fix $A<\infty$ greater +that $\mathit{Rm}(y)$ has a zero eigenvalue. Fix $A<\infty$ greater than $d_{g_\infty}(x_\infty,y)$. Then applying -Corollary~ to the limit flow on -$B(x_\infty,0,A)\times (-t(A),0]$, we see that the Riemannian -manifold $(B(x_\infty,0,A),g_\infty)$ is locally a Riemannian +\entryref{8ec855a2aab5} to the limit flow on +$B(x_\infty,0,A)\times (-t(A),0]$, we see that the Riemannian +manifold $(B(x_\infty,0,A),g_\infty)$ is locally a Riemannian product of a compact surface of positive curvature with a one-manifold. Since this is true for every $A<\infty$ sufficiently large, the same is true for $(M_\infty,g_\infty)$. Hence @@ -229,22 +262,22 @@ bounded. If $M_\infty$ is compact, then it is clear that the curvature is bounded. -It remains to consider the case where $(M_\infty,g_\infty)$ is +It remains to consider the case where $(M_\infty,g_\infty)$ is non-compact and of strictly positive curvature. Since any point of curvature greater than $4$ has a $(2C,2\epsilon)$-canonical neighborhood, and since $M_\infty$ is non-compact, it follows that -the only possible canonical neighborhoods for $x\in M_\infty$ are a -$2\epsilon$-neck centered at $x$ or $(2C,2\epsilon)$-cap whose core -contains $x$. Each of these canonical neighborhoods contains a +the only possible canonical neighborhoods for $x\in M_\infty$ are a +$2\epsilon$-neck centered at $x$ or $(2C,2\epsilon)$-cap whose core +contains $x$. Each of these canonical neighborhoods contains a $2\epsilon$-neck. Thus, if $(M_\infty,g_\infty)$ has unbounded and -positive Riemann curvature or equivalently, it has unbounded scalar +positive Riemann curvature or equivalently, it has unbounded scalar curvature, then it has $(2C,2\epsilon)$-canonical neighborhoods of arbitrarily small scale, and hence $2\epsilon$-necks of arbitrarily -small scale. But this contradicts Proposition~. It +small scale. But this contradicts \entryref{cb9ca6d6db9f}. It follows from this contradiction that the curvature of $(M_\infty,g_\infty)$ is bounded. -To complete the proof of Theorem~ it remains to +To complete the proof of \entryref{d51aab8a8065} it remains to extend the limit for the $0$ time-slices of the $({\mathcal M}_n,G_n)$ that we have just constructed to a limit flow defined for some positive amount of time backward. Since the curvature of @@ -253,23 +286,22 @@ bound, $Q$, such that for any $A<\infty$ the curvature of the restriction of $Q_nG_n$ to $B_{Q_nG_n}(x_n,0,A)$ is bounded by $Q$ for all $n$ sufficiently large. Thus, we can take the constant $Q(A)$ in -Corollary~ to be independent of $A$. According to that +\entryref{3989b6b34e08} to be independent of $A$. According to that corollary this implies that there is a $t'_0>0$ and $Q'<\infty$ such that for every $A$ there is an embedding $B_{Q_nG_n}(x_N,0,A)\times (-t'_0,0]\to {\mathcal M}_n$ compatible with time and with the vector field so that the scalar curvature of the restriction of $Q_nG_n$ to the image is bounded by $Q'$ for all $n$ sufficiently large. This uniform bound on the scalar curvature yields a uniform -bound, uniform in the sense of being independent of $n$, on $|Rm_{Q_nG_n}|$ on the image of the embedding +bound, uniform in the sense of being independent of $n$, on $|\mathit{Rm}_{Q_nG_n}|$ on the image of the embedding $B_{Q_nG_n}(x_N,0,A)\times (-t'_0,0]$. Then by Hamilton's result, -Proposition~, we see that, after passing to a +\entryref{4673eb9f4424}, we see that, after passing to a further subsequence, there is a limit flow defined on $(-t_0',0]$. Of course, the zero time-slice of this limit flow is the limit $(M_\infty,g_\infty)$. This completes the proof of -Theorem~. -\end{proof} +\entryref{d51aab8a8065}. ## Long-time blow-up limits @@ -278,11 +310,11 @@ passing to a further subsequence, establish the existence of a geometric limit flow defined on $-\infty +**Theorem.** Suppose that $\{({\mathcal M}_n,G_n,x_n)\}_{n=1}^\infty$ is a sequence of generalized $3$-dimensional Ricci flows satisfying all -the hypothesis of Theorem~. Suppose in addition that -there is $T_0$ with $0 + **Remark.** Let us point out the differences between this result and -Theorem~. The hypotheses of this theorem include all -the hypotheses of Theorem~. The main difference -between the conclusions is that in Theorem~ the +\entryref{d51aab8a8065}. The hypotheses of this theorem include all +the hypotheses of \entryref{d51aab8a8065}. The main difference +between the conclusions is that in \entryref{d51aab8a8065} the amount of backward time for which the limit flow is defined depends on the curvature bound for the final time-slice of the limit (as well as how far back the flows in the sequence are defined). This @@ -309,25 +347,30 @@ amount of backward time tends to zero as the curvature of the final time-slice limit tends to infinity. Here, the amount of backward time for which the limit flow is defined depends only on how far backwards the flows in the sequence are defined. + -**Proof.** In Theorem~ we proved that, after passing to a +**Proof.** In \entryref{d51aab8a8065} we proved that, after passing to a subsequence, there is a geometric limit Ricci flow, complete of bounded non-negative curvature, -$$(M_\infty,g_\infty(t),x_\infty),\ -t_0\le t\le 0,$$ + +$$ +(M_\infty,g_\infty(t),x_\infty),\ -t_0\le t\le 0, +$$ + defined for some $t_0>0$. Our next step is to extend the limit flow all the way back to time $-T_0$. -**Proposition.** -With the notation of, and under the hypotheses of -Theorem~, suppose that there is a geometric limit flow + +**Proposition.** With the notation of, and under the hypotheses of +\entryref{a9e023d082dc}, suppose that there is a geometric limit flow $(M_\infty,g_\infty(t))$ defined for $-T0$. + -\begin{proof} - The +**Proof.** The argument is by contradiction, so we suppose that there is a $T 4$ has a $(2C,2\epsilon)$-canonical -neighborhood. Hence, applying Lemma~ as in the +neighborhood. Hence, applying \entryref{0ddfe578961e} as in the argument in the proof of - Corollary~ shows that for all $n$ sufficiently large, + \entryref{3989b6b34e08} shows that for all $n$ sufficiently large, every point in $B_{Q_nG_n}(x_n,-T',A)$ has a uniform size parabolic neighborhood on which the Riemann curvature is uniformly bounded, where both the time interval in the parabolic neighborhood and the curvature bound on this neighborhood depend only on $C$ and the curvature bound on $Q$ for the limit flow. According to Hamilton's -result (Proposition~) this implies that, by +result (\entryref{4673eb9f4424}) this implies that, by passing to a further subsequence, we can extend the limit flow backward beyond $-T'$ a uniform amount of time, say $2\delta$. Taking $T'>T-\delta$ @@ -360,45 +403,75 @@ It remains to show that, provided that $T +**Lemma.** Suppose that there is a geometric limit flow defined on $(-T,0]$ for some $0 -\begin{proof} -Let us begin with: +**Proof.** Let us begin with: -**Claim.** - Let $Q$ be an upper bound on $R(x,0)$ for all $x\in + +**Claim.** Let $Q$ be an upper bound on $R(x,0)$ for all $x\in M_\infty$. Then for any points $x,y\in M_\infty$ and any $t\in (-T,0]$ we have -$$d_{t}(x,y)\le d_0(x,y)+16\sqrt{\frac{Q}{3}}T.$$ -\begin{proof} -Fix $-t_0\in (-T,0]$. Then for any $\epsilon>0$ sufficiently small, +$$ +d_{t}(x,y)\le d_0(x,y)+16\sqrt{\frac{Q}{3}}T. +$$ + + +**Proof.** Fix $-t_0\in (-T,0]$. Then for any $\epsilon>0$ sufficiently small, by the Harnack inequality -(the second result in Theorem~) +(the second result in \entryref{aa9d836895d8}) we have -$$\frac{\partial R}{\partial t}(x,t)\ge -\frac{R(x,t)}{t+T-\epsilon}.$$ + +$$ +\frac{\partial R}{\partial t}(x,t)\ge -\frac{R(x,t)}{t+T-\epsilon}. +$$ + Taking the limit as $\epsilon\rightarrow 0$ gives -$$\frac{\partial R}{\partial t}(x,t)\ge -\frac{R(x,t)}{t+T},$$ + +$$ +\frac{\partial R}{\partial t}(x,t)\ge -\frac{R(x,t)}{t+T}, +$$ + and hence, fixing $x$, -$$\frac{dR(x,t)}{R(x,t)}\ge \frac{-dt}{(t+T)}.$$ + +$$ +\frac{dR(x,t)}{R(x,t)}\ge \frac{-dt}{(t+T)}. +$$ + Integrating from $-t_0$ to $0$ shows that -$$log(R(x,0))-log(R(x,-t_0))\ge log(T-t_0)-log(T),$$ + +$$ +\mathit{log}(R(x,0))-\mathit{log}(R(x,-t_0))\ge \mathit{log}(T-t_0)-\mathit{log}(T), +$$ + and since $R(x,0)\le Q$, this implies -$$R(x,-t_0)\le Q\frac{T}{T-t_0}.$$ + +$$ +R(x,-t_0)\le Q\frac{T}{T-t_0}. +$$ + Recalling that $n=3$ and that the curvature is non-negative we see that -$$Ric(x,-t_0)\le (n-1)\frac{QT}{2}\frac{1}{T-t_0}.$$ -Hence by Corollary~, for all $-t_0\in (-T,0]$ we have + +$$ +\mathit{Ric}(x,-t_0)\le (n-1)\frac{QT}{2}\frac{1}{T-t_0}. +$$ + +Hence by \entryref{01a829e22cf6}, for all $-t_0\in (-T,0]$ we have that -$$dist_{-t_0}(x,y)\le dist_0(x,y)+8\int_{-t_0}^0\sqrt{\frac{QT}{3(T+t)}} -\le dist_0(x,y)+16\sqrt{\frac{Q}{3}}T.$$ + +$$ +\mathit{dist}_{-t_0}(x,y)\le \mathit{dist}_0(x,y)+8\int_{-t_0}^0\sqrt{\frac{QT}{3(T+t)}} +\le \mathit{dist}_0(x,y)+16\sqrt{\frac{Q}{3}}T. +$$ It follows immediately from this claim that any compact subset $X\subset M_\infty$ has uniformly bounded diameter under all the @@ -416,37 +489,38 @@ C'$. Since the sequence $({\mathcal M}_n,Q_nG_n,x_n)$ converges smoothly to $(M_\infty,g_\infty(t),x_\infty)$ for $t\in (-T,0]$, it follows that for each $m$ there are sequences $\{y_{m,n}\in {\mathcal M}_n\}_{n=1}^\infty$ and $\{z_{m,n}\in {\mathcal -M}_n\}_{n=1}^\infty$ with $**t**(y_{m,n})=**t**(z_{m,n})=t_m$ +M}_n\}_{n=1}^\infty$ with $\mathbf{t}(y_{m,n})=\mathbf{t}(z_{m,n})=t_m$ converging to $(y_m,t_m)$ and $(z_m,t_m)$ respectively. Thus, for all $m$ there is $n_0=n_0(m)$ such that for all $n\ge n_0$ we have: -\begin{enumerate} -\item[(1)] $1\le R_{Q_nG_n}(y_{m,n})\le 2C'$, -\item[(2)] $R_{Q_nG_n}(z_{m,n})\ge R_{g_\infty}(z_m,t_m)/2$, -\item[(3)] $d_{Q_nG_n}((y_{m,n}),(z_{m,n}))\le 2\,diam_{g_\infty(t_m)}(X).$ -\end{enumerate} + +- **(1)** $1\le R_{Q_nG_n}(y_{m,n})\le 2C'$, +- **(2)** $R_{Q_nG_n}(z_{m,n})\ge R_{g_\infty}(z_m,t_m)/2$, +- **(3)** $d_{Q_nG_n}((y_{m,n}),(z_{m,n}))\le 2 \mathit{diam}_{g_\infty(t_m)}(X).$ + Because of the third condition and the fact that $X$ has uniformly bounded diameter under all the metrics $g_\infty(t)$ for $t\in (-T,0]$, the distance $d_{Q_{m,n}G_{m,n}}(z_{m,n},y_{m,n})$ is bounded independent of $m$ and $n$ as long as $n\ge n_0$. Because of the fact that $R_{Q_nG_n}(y_{m,n})\ge 1=R(x_n)$, it follows that any -point $z\in {\mathcal M}_n$ with $**t**(z_m)\le t_{m}$ and with +point $z\in {\mathcal M}_n$ with $\mathbf{t}(z_m)\le t_{m}$ and with $R(z)\ge 4R(y_{m,n})$ has a strong $(C,\epsilon)$-canonical neighborhood. This then -contradicts Theorem~ and completes the proof of the lemma. -\end{proof} +contradicts \entryref{ecd62fc2ea2b} and completes the proof of the lemma. Clearly, this argument will be enough to handle the case when $M_\infty$ is compact. The case when $M_\infty$ is non-compact uses additional results. + **Lemma.** Let $(M,g)$ be a complete, connected, non-compact manifold of non-negative sectional curvature and let $x_0\in M$ be a point. Then there is $D>0$, such that for any $y\in M$ with $d(x_0,y)=d\ge D$, there is $x\in M$ with $d(y,x)=d$ and with $d(x_0,x)>3d/2$. + **Proof.** Suppose that the result is false for $(M,g)$ and $x\in M$. Then there is a sequence $y_n\in M$ such that setting $d_n=d(x,y_n)$ we -have $lim_{n\rightarrow\infty}d_n=\infty$ and yet +have $\mathit{lim}_{n\rightarrow\infty}d_n=\infty$ and yet $B(y_n,d_n)\subset B(x,3d_n/2)$ for every $n$. Let $\gamma_n$ be a minimal geodesic from $x$ to $y_n$. By passing to a subsequence we arrange that the $\gamma_n$ converge to a minimal geodesic ray @@ -454,24 +528,25 @@ $\gamma$ from $x$ to infinity in $M$. In particular, the angle at $x$ between $\gamma_n$ and $\gamma$ tends to zero as $n\rightarrow\infty$. Let $w_n$ be the point on $\gamma$ at distance $d_n$ from $x$, and let $\alpha_n=d(y_n,w_n)$. Because $(M,g)$ has -non-negative curvature, by Corollary~, $lim_{n\rightarrow\infty}\alpha_n/d_n=0$. In particular, for all $n$ +non-negative curvature, by \entryref{c81962d31706}, $\mathit{lim}_{n\rightarrow\infty}\alpha_n/d_n=0$. In particular, for all $n$ sufficiently large, $\alpha_n3d_n/2$. This contradiction proves the lemma. -**Claim.** -Fix $D<\infty$ greater than or equal to the constant given in the + +**Claim.** Fix $D<\infty$ greater than or equal to the constant given in the previous lemma for the Riemannian manifold $(M_\infty,g_\infty(0))$ and the point $x_\infty$. We also choose $D\ge 32\sqrt{\frac{Q}{3}}T$. Then for any $y\in M_\infty\setminus B(x_\infty,0,D)$ the scalar curvature $R_{g_\infty}(y,t)$ is uniformly bounded for all $t\in (-T,0]$. + **Proof.** Suppose this does not hold for some $y\in M_\infty\setminus B(x_\infty,0,D)$. Let $d=d_0(x_\infty,y)$. Of course, $d\ge D$. @@ -480,10 +555,10 @@ $d_0(x_\infty,z)>3d/2$. Since the scalar curvature $R(y,t)$ is not uniformly bounded for all $t\in (-T,0]$, there is $t$ for which $R(y,t)$ is arbitrarily large and hence $(y,t)$ has an $(2C,2\epsilon)$-canonical neighborhood of arbitrarily small scale. -By Claim~ we have $d_{t}(x_\infty,y)\le +By \entryref{002f3496c54c} we have $d_{t}(x_\infty,y)\le d+8\sqrt{\frac{Q}{3}}T$ and $d_{t}(y,z)\le d+8\sqrt{\frac{Q}{3}}T$. -Of course, since $Ric\ge 0$ the metric is non-increasing in -time and hence $d\le min(d_{t}(y,z),d_{t}(x_\infty,y))$ and +Of course, since $\mathit{Ric}\ge 0$ the metric is non-increasing in +time and hence $d\le \mathit{min}(d_{t}(y,z),d_{t}(x_\infty,y))$ and $3d/2\le d_{t}(x_0,z)$. Since $y$ has a $(2C,2\epsilon)$-canonical neighborhood in $(M_\infty,g_\infty(t))$, either $y$ is the center of an $2\epsilon$-neck in $(M_\infty,g_\infty(t))$ or $y$ is @@ -491,14 +566,15 @@ contained in the core of a $(2C,2\epsilon)$-cap in $(M_\infty,g_\infty(t))$. (The other two possibilities for canonical neighborhoods require that $M_\infty$ be compact.) + **Claim.** $y$ cannot lie in the core of a $(2C,2\epsilon)$-cap in $(M_\infty,g_\infty(t))$, and hence it is the center of a $2\epsilon$-neck $N$ in $(M_\infty,g_\infty(t))$. Furthermore, minimal $g(t)$-geodesics from -$y$ to $x_\infty$ and $z$ exit out of opposite ends of $N$ (see Fig.~). +$y$ to $x_\infty$ and $z$ exit out of opposite ends of $N$ (see Fig. 11.1). + -\begin{proof} -Let ${\mathcal C}$ be a $(2C,2\epsilon)$-canonical +**Proof.** Let ${\mathcal C}$ be a $(2C,2\epsilon)$-canonical neighborhood of $y$ in $(M_\infty,g_\infty(t))$. Since $R(y,t)$ can be arbitrarily large, we can assume that $d \gg 2CR(y)^{-1/2}$, which is a bound on the @@ -507,29 +583,40 @@ $\gamma_{x_\infty}$ and $\gamma_z$ connecting $y$ to $x_\infty$ and to $z$, respectively, must exit from ${\mathcal C}$. Let $a$ be a point on $\gamma_{x_\infty}\cap{\mathcal C}$ close to the complement of ${\mathcal C}$. Let $b$ be a point at the same $g(t)$-distance -from $y$ on $\gamma_z$. In the case that ${\mathcal C}$ is a cap +from $y$ on $\gamma_z$. In the case that ${\mathcal C}$ is a cap or that it is a $2\epsilon$-neck and $\gamma_{x_\infty}$ and $\gamma_z$ exit from the same end, then $d_t(b,y)/d_t(a,y)<4\pi\epsilon$. This means that the angle $\theta$ of the Euclidean triangle with these side lengths at the point corresponding to $y$ satisfies -$$cos(\theta)\ge 1-\frac{(4\pi\epsilon)^2}{2}.$$ -Recall that $Q$ is the maximum value of $R(x,0)$, and that by -Claim~ we have - $$d\le d_t(x_\infty,y)\le d+16\sqrt{\frac{Q}{3}}T,$$ +$$ +\mathit{cos}(\theta)\ge 1-\frac{(4\pi\epsilon)^2}{2}. +$$ + +Recall that $Q$ is the maximum value of $R(x,0)$, and that by +\entryref{002f3496c54c} we have + +$$ +d\le d_t(x_\infty,y)\le d+16\sqrt{\frac{Q}{3}}T, +$$ + with the same inequalities holding with $d_t(z,y)$ replacing $d_t(x_\infty,y)$. Also, by construction $d\ge 32\sqrt{\frac{Q}{3}}T$. We set $a_0=d_t(x_\infty,y)$ and $a_1=d_t(z,y)$. Then by the Toponogov property we have -$$d_t(x,z)^2\le a_0^2+a_1^2-2a_0a_1\left( -1-\frac{(4\pi\epsilon)^2}{2}\right)=(a_0-a_1)^2+(4\pi\epsilon)^2a_0a_1.$$ + +$$ +d_t(x,z)^2\le a_0^2+a_1^2-2a_0a_1\left( +1-\frac{(4\pi\epsilon)^2}{2}\right)=(a_0-a_1)^2+(4\pi\epsilon)^2a_0a_1. +$$ + Since $|a_0-a_1|\le d/2$ and $a_0,a_1\le 3d/2$ and $\epsilon<1/8\pi$, it follows that $d_t(x,z)4\pi R(y,t)^{-1/2}$, neither $z$ nor $x_\infty$ lies -in this ball.) Thus, if $R(y,t)$ is unbounded as $t\rightarrow -T$ +in this ball.) Thus, if $R(y,t)$ is unbounded as $t\rightarrow -T$ then arbitrarily small $g(0)$-balls centered at $y$ separate $z$ and $x_\infty$. Since $y$ is distinct from $x_\infty$ and $z$, this is clearly impossible. -\end{proof} Next we establish that the curvature near the base point $x_\infty$ is bounded for all $t\in (-T,0]$. -**Corollary.** -Suppose that there is a geometric limit flow + +**Corollary.** Suppose that there is a geometric limit flow $(M_\infty,g_\infty(t))$ of a subsequence defined on $(-T,0]$ for some $T<\infty$. Suppose that the limit flow is complete of non-negative curvature with the curvature locally bounded in time. Then for every $A<\infty$ the scalar curvature $R_{g_\infty}(y,t)$ is uniformly bounded for all $(y,t)\in B(x_\infty,0,A)\times (-T,0]$. + **Proof.** First we pass to a subsequence so that a geometric limit flow -$$(M_\infty,g_\infty(t),(x_\infty,0))$$ exists on $(-T,0]$. We let + +$$ +(M_\infty,g_\infty(t),(x_\infty,0)) +$$ + + exists on $(-T,0]$. We let $Q$ be the upper bound for $R(x,0)$ for all $x\in M_\infty$. We now divide the argument into two cases: (i) $M_\infty$ is compact, and (ii) $M_\infty$ is non-compact. -Suppose that $M_\infty$ is compact. By Proposition~ +Suppose that $M_\infty$ is compact. By \entryref{95cc838cfc6d} we know that -$$min_{x\in M_\infty}(R_{g_\infty}(x,t))$$ + +$$ +\mathit{min}_{x\in M_\infty}(R_{g_\infty}(x,t)) +$$ + is a non-decreasing function of $t$. Since $R_{g_\infty}(x_\infty,0)=1$, it follows that for each $t\in -(-T,0]$, we have $min_{x\in M_\infty}R(x,t)\le 1$, and hence +(-T,0]$, we have $\mathit{min}_{x\in M_\infty}R(x,t)\le 1$, and hence there is a point $x_{t}\in M_\infty$ with $R(x_{t},t)\le 1$. Now we -can apply Lemma~ to see that the scalar curvature of +can apply \entryref{cc3a26ca3809} to see that the scalar curvature of $g_\infty$ is bounded on all of $M_\infty\times (-T,0]$. -If $M_\infty$ is non-compact, choose $D$ as in Lemma~. +If $M_\infty$ is non-compact, choose $D$ as in \entryref{9a7e3fe2e5c4}. According to that lemma every point in the boundary of $B(x_\infty,0,D)$ has bounded curvature under $g_\infty(t)$ for all $t\in (-T,0]$. In particular, for each $t\in (-T,0]$ the minimum of $R(x,t)$ over $\overline B(x_\infty,0,D)$ is bounded independent of -$t$. Now apply Lemma~ to the closure of +$t$. Now apply \entryref{cc3a26ca3809} to the closure of $B(x_\infty,0,D)$. We conclude that the curvature of $B(x_\infty,0,D)$ is uniformly bounded for all $g_\infty(t)$ for all $t\in (-T,0]$. In particular, $R(x_\infty,t)$ is uniformly bounded for all $t\in (-T,0]$. -Now for any $A<\infty$ we apply Lemma~ to the compact +Now for any $A<\infty$ we apply \entryref{cc3a26ca3809} to the compact subset $\overline B(x_\infty,0,A)$ to conclude that the curvature is uniformly bounded on $B(x_\infty,0,A)\times (-T,0]$. This completes the proof of the corollary. -Now let us return to the proof of Proposition~. +Now let us return to the proof of \entryref{d6919616448d}. + **Claim.** For each $A<\infty$ and for all $n$ sufficiently large, there are $\delta>0$ with $\delta\le T_0-T$ and a bound, independent of $n$, on the scalar curvature of the restriction of $Q_nG_n$ to $B_{Q_nG_n}(x_n,0,A)\times [-(T+\delta),0]$. + **Proof.** Fix $A<\infty$ and let $K$ be the bound for the scalar curvature of $g_\infty$ on $B(x_\infty,0,2A)\times (-T,0]$ from -Corollary~. Lemma~ shows that +\entryref{a66fa0ea8720}. \entryref{0ddfe578961e} shows that there are $\delta>0$ and a bound in terms of $K$ and $C$ on the scalar curvature of the restriction of $Q_nG_n$ to $B_{Q_nG_n}(x_n,0,A)\times [-(T+\delta),0]$. @@ -611,7 +709,7 @@ subsequence we can arrange that the $-T$ time-slices of the $({\mathcal M}_n,G_n,x_n)$ converge to a limit $(M_\infty,g_\infty(-T))$. This limit manifold -satisfies the hypothesis of Proposition~ and hence, by +satisfies the hypothesis of \entryref{cb9ca6d6db9f} and hence, by that proposition, it has bounded sectional curvature. This means that there is a uniform $\delta>0$ such that for all $n$ sufficiently large and for any $A<\infty$ the scalar curvatures (and @@ -619,45 +717,43 @@ hence the Riemann curvatures) of the restriction of $Q_nG_n$ to $B_{Q_nG_n}(x_n,0,A)\times [-(T+\delta),0]$ are uniformly bounded. This allows us to pass to a further subsequence for which there is a geometric limit defined on $(-(T+\delta/2),-T]$. This geometric -limit is complete of bounded, non-negative curvature. Hence, we +limit is complete of bounded, non-negative curvature. Hence, we have now constructed a limit flow on $(-(T+\delta/2),0]$ with the property that for each $t\in (-(T+\delta/2),0]$ the Riemannian manifold $(M,g(t))$ is complete and of bounded non-negative curvature. (We still don't know whether the entire flow is of bounded curvature.) But now invoking Hamilton's Harnack inequality -(Theorem~), we see that the curvature is bounded on +(\entryref{aa9d836895d8}), we see that the curvature is bounded on $[-T,0]$. Since we already know it is bounded in $(-T+\delta/2,-T]$, this completes the proof of the proposition. -\end{proof} - It follows immediately from Proposition~ that there is a -geometric limit flow defined on $(-T_0,0]$. The geometric limit + It follows immediately from \entryref{d6919616448d} that there is a +geometric limit flow defined on $(-T_0,0]$. The geometric limit flow on $(-T_0,0]$ is complete of non-negative curvature, locally bounded in time. It remains to prove the last statement in the theorem. So let us suppose that $T_0=\infty$. We have just established the existence of -a geometric limit flow defined for $t\in (-\infty,0]$. Since the +a geometric limit flow defined for $t\in (-\infty,0]$. Since the $({\mathcal M}_n,G_n)$ either have curvature pinched toward positive or are of non-negative curvature, it follows from -Theorem~ that all time-slices of the limit flow +\entryref{2ddfbc74b081} that all time-slices of the limit flow are complete manifolds of non-negative curvature. Since points of scalar curvature greater than $4$ have $(2C,2\epsilon)$-canonical -neighborhoods, it follows from Proposition~ that the +neighborhoods, it follows from \entryref{cb9ca6d6db9f} that the curvature is bounded on each time-slice, and hence universally -bounded by the Harnack inequality (Theorem~). Since for +bounded by the Harnack inequality (\entryref{aa9d836895d8}). Since for any $A<\infty$ and every $T<\infty$ the parabolic neighborhoods $B_{Q_nG_n}(x_n,0,A)\times [-T,0]$ are $\kappa$-non-collapsed on -scales $Q_nr$ for every $n$ sufficiently large, the limit is -$\kappa$-non-collapsed on scales $\le lim_{n\rightarrow -\infty}Q_nr$. Since $r>0$ and $lim_{n\rightarrow +scales $Q_nr$ for every $n$ sufficiently large, the limit is +$\kappa$-non-collapsed on scales $\le \mathit{lim}_{n\rightarrow +\infty}Q_nr$. Since $r>0$ and $\mathit{lim}_{n\rightarrow \infty}Q_n=\infty$, it follows that the limit flow is $\kappa$-non-collapsed on all scales. Since $R_{Q_n G_n}(x_n)=1$, $R_{g_\infty}(x_\infty,0)=1$ and the limit flow is non-flat. This establishes all the properties need to show that the limit is a $\kappa$-solution. This completes the proof of -Theorem~. -\end{proof} +\entryref{a9e023d082dc}. ## Incomplete smooth limits at singular times @@ -669,74 +765,91 @@ incomplete, but we have strong control over their ends. We shall assume the following about the generalized Ricci flow $({\mathcal M},G)$: -\begin{assumption} -\begin{enumerate} -\item[(a)] The singular times form a discrete subset of $\Ar$, and + +**Assumption.** - **(a)** The singular times form a discrete subset of $\Ar$, and each time slice of the flow at a non-singular time is a compact -$3$-manifold. \item[(b)] The time interval of definition of the +$3$-manifold. +- **(b)** The time interval of definition of the generalized Ricci flow $({\mathcal M},G)$ is contained in $[0,\infty)$ and its curvature is pinched toward positive. -\item[(c)] There are $r_0>0$ and $C<\infty$, such that any point +- **(c)** There are $r_0>0$ and $C<\infty$, such that any point $x\in {\mathcal M}$ with $R(x)\ge r_0^{-2}$ has a strong $(C,\epsilon)$-canonical neighborhood. In particular, for every -$x\in {\mathcal M}$ with $R(x)\ge r_0^{-2}$ the following two +$x\in {\mathcal M}$ with $R(x)\ge r_0^{-2}$ the following two inequalities hold: -$$\left|\frac{\partial R(x)}{\partial t}\right|< CR^2(x),$$ -$$|\nabla R(x)|< CR^{3/2}(x).$$ -\end{enumerate} -\end{assumption} + +$$ +\left|\frac{\partial R(x)}{\partial t}\right|< CR^2(x), +$$ + +$$ +|\nabla R(x)|< CR^{3/2}(x). +$$ + + With these assumptions we can say quite a bit about the limit metric at time $T$. -**Theorem.** -Suppose that $({\mathcal M},G)$ is a generalized Ricci flow defined + +**Theorem.** Suppose that $({\mathcal M},G)$ is a generalized Ricci flow defined for $0\le tr_0^{-2}$ +- **(5)** Any point $x\in \Omega\times\{T\}$ with $R(x)>r_0^{-2}$ has a strong $(2C,2\epsilon)$-canonical neighborhood in $\widehat{\mathcal M}$. -\end{enumerate} + + **Remark.** Recall that by definition a function $f$ is proper if the pre-image under $f$ of every compact set is compact. + In order to prove this result we establish a sequence of lemmas. The first in the series establishes that $\Omega$ is an open subset and also establishes the first two of the above five conclusions. -**Lemma.** -Suppose that $({\mathcal M},G)$ is a generalized Ricci flow defined + +**Lemma.** Suppose that $({\mathcal M},G)$ is a generalized Ricci flow defined for $0\le t **Proof.** We pull back $G$ to $M\times [T^-,T)$ to define a Ricci flow $(M,g(t)),\ T^{-}\le t0$ such that $R(y,t)$ is +the \entryref{0ddfe578961e}, there is $r>0$ such that $R(y,t)$ is uniformly bounded for $y\in B(x,t_n,r)\times [t_n,T)$. This means that $B(x,t_n,r)\subset \Omega$, proving that $\Omega$ is open in $M$. Furthermore, since $R(y,t)$ is bounded on $B(x,t_n,r)\times [t_n,T)$, it follows from the curvature pinching toward positive -hypothesis that $|Rm(y,t)|$ is bounded on -$B(x,t_n,r)\times[t_n,T)$. Now applying Theorem~ we see -that in fact $Rm$ is bounded in the $C^\infty$-topology on +hypothesis that $|\mathit{Rm}(y,t)|$ is bounded on +$B(x,t_n,r)\times[t_n,T)$. Now applying \entryref{ed4b72caf9f9} we see +that in fact $\mathit{Rm}$ is bounded in the $C^\infty$-topology on $B(x,t_n,r)\times[(t_n+T)/2 ,T)$. The same is of course also true -for $Ric$ and hence for $\frac{\partial g}{\partial t}$ in the +for $\mathit{Ric}$ and hence for $\frac{\partial g}{\partial t}$ in the $C^\infty$-topology. It then follows that there is a continuous -extension of $g$ to $B(x,t_n,r)\times [t_n,T]$. Since this is true +extension of $g$ to $B(x,t_n,r)\times [t_n,T]$. Since this is true for every $x\in \Omega$ we see that $g(t)$ converges in the $C^\infty$-topology, uniformly on compact subsets of $\Omega$, to $g(T)$. @@ -769,21 +882,26 @@ hypothesis, this function is bounded below. We must show that it is proper. Since $M$ is compact, it suffices to show that if $x_n$ is a sequence in $\Omega\subset M$ converging to a point $x\in M\setminus \Omega$ then $R(x_n,T)$ is unbounded. Suppose that $R(x_n,T)$ is -bounded independent of $n$. It follows from Lemma~ +bounded independent of $n$. It follows from \entryref{0ddfe578961e} that there is a positive constant $\Delta t$ such that $R(x_n,t)$ is uniformly bounded for all $n$ and all $t\in [T-\Delta t,T)$, and hence, by the same result, there is $r>0$ such that $R(y_n, t)$ is -bounded for all $n$, all $y_n\in B(x_n,T-\Delta t,r)$, and all +bounded for all $n$, all $y_n\in B(x_n,T-\Delta t,r)$, and all $t\in [T-\Delta t,T)$. Since the $x_n\rightarrow x\in M$, it follows that for all $n$ sufficiently large that $x\in B(x_n,T-\Delta t,r)$, and hence $R(x,t)$ is uniformly bounded as $t\rightarrow T$. This contradicts the fact that $x\not\in \Omega$. -**Definition.** -Let -$$\widehat {\mathcal M}= {\mathcal + +**Definition.** Let + +$$ +\widehat {\mathcal M}= {\mathcal M}\cup_{\Omega\times[T^-,T)} \left(\Omega\times -[T^-,T]\right).$$$} Since both +[T^-,T]\right). +$$ + + Since both ${\mathcal M}$ and $\Omega\times [T^-,T]$ have the structure of space-times and the time functions and vector fields agree on the overlap, $\widehat {\mathcal M}$ inherits the structure of a @@ -798,46 +916,48 @@ extension of* $({\mathcal M},G)$ to time $T$. Notice that even though the time-slices $M_t$ of ${\mathcal M}$ are compact, it will not necessarily be the case that the time-slice $\Omega$ is complete. + At this point we have established the first three of the five -conclusions stated in Theorem~. Let us turn to the last +conclusions stated in \entryref{af00774dbaa9}. Let us turn to the last two. -### Canonical neighborhoods for $(\widehat{\mathcal M -,\widehat G)$} +### Canonical neighborhoods for $(\widehat{\mathcal M},\widehat G)$ We continue with the notation and assumptions of the previous subsection. Here we establish the fifth conclusion in -Theorem~, namely the existence of strong canonical +\entryref{af00774dbaa9}, namely the existence of strong canonical neighborhoods for $(\widehat{\mathcal M},\widehat G)$ -**Lemma.** -For any $x\in \Omega\times\{T\}$ with $R(x,T)>r_0^{-2}$ one of the + +**Lemma.** For any $x\in \Omega\times\{T\}$ with $R(x,T)>r_0^{-2}$ one of the following holds: -\begin{enumerate} -\item[(1)] $(x,T)$ is the center of a strong $2\epsilon$-neck in -$(\widehat{\mathcal M},\widehat G)$. \item[(2)] There is a + +- **(1)** $(x,T)$ is the center of a strong $2\epsilon$-neck in +$(\widehat{\mathcal M},\widehat G)$. +- **(2)** There is a $(2C,2\epsilon)$-cap in $(\Omega(T),\widehat G(T))$ whose core -contains $(x,T)$. \item[(3)] There is a $2C$-component of +contains $(x,T)$. +- **(3)** There is a $2C$-component of $\Omega(T)$ that contains $(x,T)$. -\item[(4)] There is a $2\epsilon$-round component of $\Omega(T)$ that +- **(4)** There is a $2\epsilon$-round component of $\Omega(T)$ that contains $(x,T)$. -\end{enumerate} + **Proof.** We fix $x\in \Omega(T)$ with $R(x,T)>r_0^{-2}$. First notice that for all $tr_0^{-2}$. Thus, for all such $t$ the point $(x,t)$ has a strong $(C,\epsilon)$-canonical neighborhood in $({\mathcal M},G)\subset (\widehat{\mathcal -M},\widehat G)$. Furthermore, since $lim_{t\rightarrow +M},\widehat G)$. Furthermore, since $\mathit{lim}_{t\rightarrow T}R(x,t)=R(x,T)<\infty$, for all $t +**Claim.** Fix a point $x\in\Omega$. Suppose that there is a sequence $t_n\rightarrow T$ such that for every $n$, the point $(x,t_n)$ is the center of a strong $\epsilon$-neck in $\widehat{\mathcal M}$. Then $(x,T)$ is the center of a strong $2\epsilon$-neck in $\widehat {\mathcal M}$. + -\begin{proof} -By an overall rescaling we can assume that $R(x,T)=1$. For each $n$ +**Proof.** By an overall rescaling we can assume that $R(x,T)=1$. For each $n$ let $N_n\subset \Omega$ and let $\psi_n\colon S^2\times (-\epsilon^{-1},\epsilon^{-1})\to N_n\times\{t\}$ be a strong $\epsilon$-neck centered at $(x,t_n)$. Let @@ -877,7 +997,11 @@ $R(x,t_n)\rightarrow R(x,T)=1$ as $n\rightarrow \infty$, it follows that there is a flow line through $y$ defined on $(T-1,T]$. Consider the maps -$$\alpha_n\colon B\times (-1,0]\to \widehat{\mathcal M}$$ + +$$ +\alpha_n\colon B\times (-1,0]\to \widehat{\mathcal M} +$$ + that send $(y,t)$ to the value at time $t_n-tR(x,t_n)^{-1}$ of the flow line through $y$. Pulling back the metric $R(x,t_n)\widehat G$ by $\alpha_n$ produces the restriction of a strong $\epsilon$-neck @@ -887,10 +1011,10 @@ $C^\infty$-topology to the map $\alpha\colon B\times (-1,0]\to through $(y,T)$ at the time $T-t$. Hence, the sequence of metrics $\alpha^*_n(R(x,t_n))\widehat G$ on $B\times (-1,0]$ converges uniformly on compact subsets of $B\times (-1,0]$ in the -$C^\infty$-topology to the family $\alpha^*(\widehat G)$. Then, for +$C^\infty$-topology to the family $\alpha^*(\widehat G)$. Then, for all $n$ sufficiently large, the image $\psi_n(S^2\times (-\epsilon^{-1}/2,\epsilon^{-1}/2))$ is contained in $B$ and has -compact closure in $B$. Since the family of metrics +compact closure in $B$. Since the family of metrics $\psi_n^*\widehat G$ on $B$ converge smoothly to $\psi^*\widehat G$, it follows that for every $n$ sufficiently @@ -902,28 +1026,29 @@ is a strong $2\epsilon$-neck at time $T$. This completes the proof of the lemma. -\end{proof} -The lemma tells us that every point $x\in\Omega\times\{T\}$ with +The lemma tells us that every point $x\in\Omega\times\{T\}$ with $R(x)>r_0^{-2}$ has a strong $(2C,2\epsilon)$-canonical neighborhood. Since, by assumption, points at time before $T$ with scalar curvature at least $r_0^{-2}$ have strong $(C,\epsilon)$-canonical neighborhoods, this completes the proof of -the fifth conclusion of Theorem~. It remains to establish +the fifth conclusion of \entryref{af00774dbaa9}. It remains to establish the fourth conclusion of that theorem. ### The ends of $(\Omega,g(T))$ + **Definition.** A *strong $2\epsilon$-horn* in $(\Omega,g(T))$ is a submanifold of $\Omega$ diffeomorphic to $S^2\times [0,1)$ with the following properties: -\begin{enumerate} -\item[(1)] The embedding $\psi$ of $S^2\times [0,1)$ into $\Omega$ is a proper map. -\item[(2)] Every point of the image of this map is the center of a strong + +- **(1)** The embedding $\psi$ of $S^2\times [0,1)$ into $\Omega$ is a proper map. +- **(2)** Every point of the image of this map is the center of a strong $2\epsilon$-neck in $(\widehat{\mathcal M},\widehat G)$. -\item[(3)] The image of the boundary $S^2\times \{0\}$ +- **(3)** The image of the boundary $S^2\times \{0\}$ is the central sphere of a strong $2\epsilon$-neck. -\end{enumerate} + + **Definition.** A *strong double $2\epsilon$-horn* in $(\Omega,g(T))$ is a component of $\Omega$ diffeomorphic to $S^2\times (0,1)$ with the property that every point of this @@ -938,38 +1063,43 @@ $2\epsilon$-horn* in $(\Omega,g(T))$ is a component of $\Omega$ that is a the union of a the core of a $(C',2\epsilon)$-cap and a strong $2\epsilon$-horn. Such a component is diffeomorphic to an open $3$-ball or to a punctured $\Ar P^3$. + -See Fig.~. +See Fig. 11.2. + **Definition.** Fix any $\rho,\ 0<\rho + **Lemma.** Suppose that $0<\rho **Proof.** Let $\Omega^0$ be a component of $\Omega$ containing no point of -$\Omega_\rho$. Then for every $x\in \Omega^0$, we have -$R(x,T)>r_0^{-2}$. Therefore, by Lemma~ $(x,T)$ +$\Omega_\rho$. Then for every $x\in \Omega^0$, we have +$R(x,T)>r_0^{-2}$. Therefore, by \entryref{0760630d3990} $(x,T)$ has a $(2C,2\epsilon)$-canonical neighborhood. Of course, this entire neighborhood is contained in $\widehat{\mathcal M}$ and hence is contained in $\Omega^0$ (or, more precisely, in the case of @@ -980,29 +1110,32 @@ component, then of course $\Omega^0$ is that $2C$-component or $2\epsilon$-round component. Otherwise, each point of $\Omega^0$ is either the center of a strong $2\epsilon$-neck or is contained in the core of a $(2C,2\epsilon)$-cap. We have chosen $2\epsilon$ -sufficiently small so that the result follows from -Proposition~. +sufficiently small so that the result follows from +\entryref{fa8d9a43f34a}. + **Remark.** We do not claim that there are only finitely many such components; in particular, as far as we know there may be infinitely double $2\epsilon$-horns. + It follows immediately from this lemma that if $X$ is a component of $\Omega$ not containing any point of $\Omega_\rho$, then every end of $X$ is contained -in a strong $2\epsilon$-tube. To complete the proof of Theorem~, it +in a strong $2\epsilon$-tube. To complete the proof of \entryref{af00774dbaa9}, it remains only to establish the same result for the components of $\Omega$ that meet $\Omega_\rho$. That is part of the content of the next lemma. -**Lemma.** -Let $({\mathcal M},G)$ be a generalized $3$-dimensional Ricci flow + +**Lemma.** Let $({\mathcal M},G)$ be a generalized $3$-dimensional Ricci flow defined for $0\le t **Proof.** Since $R\colon \Omega\times\{T\}\to \Ar$ is a proper function bounded below, $\Omega_\rho$ is compact. Hence, there are only @@ -1010,19 +1143,24 @@ finitely many components of $\Omega$ containing points of $\Omega_\rho$. Let $\Omega^0$ be a non-compact component of $\Omega$ containing a point of $\Omega_\rho$, and let ${\mathcal E}$ be an end of $\Omega^0$. Let -$$X=\{x\in \Omega^0\bigl|\bigr. -R(x)\ge 2C^2\rho^{-2}\}.$$ Then $X$ is a closed set and contains a + +$$ +X=\{x\in \Omega^0\bigl|\bigr. +R(x)\ge 2C^2\rho^{-2}\}. +$$ + + Then $X$ is a closed set and contains a neighborhood of the end ${\mathcal E}$. Since $\Omega^0$ contains a point of $\Omega_\rho$, $\Omega^0\setminus X$ is non-empty. Let $X_0$ be the connected component of $X$ that contains a neighborhood of ${\mathcal E}$. This is a closed, connected set every point of -which has a $(2C,2\epsilon)$-canonical neighborhood. Since $X_0$ +which has a $(2C,2\epsilon)$-canonical neighborhood. Since $X_0$ includes an end of $\Omega^0$, no point of $X_0$ can be contained in an $\epsilon$-round component nor in a $C$-component. Hence, every point of $X_0$ is either the center of a strong $2\epsilon$-neck or is contained in the core of a $(2C,2\epsilon)$-cap. Since $2\epsilon$ is sufficiently small to invoke -Proposition~, the latter implies that $X_0$ is +\entryref{0e9a1a167caa}, the latter implies that $X_0$ is contained either in a $2\epsilon$-tube which is a union of strong $2\epsilon$-necks centered at points of $X_0$ or $X_0$ is contained in a $2C$-capped $2\epsilon$-tube where the core of the cap contains @@ -1038,7 +1176,7 @@ centered at a point of $X_0$ has curvature at least $C^2\rho^{-2}$ and hence is also disjoint from $\Omega_\rho$. Hence, if $\Omega^0$ is a $2C$-capped $2\epsilon$-tube and there is a point of $X_0$ in the core of the cap, then this component is disjoint from -$\Omega_\rho$, which is a contradiction. Thus, $X_0$ is contained +$\Omega_\rho$, which is a contradiction. Thus, $X_0$ is contained in a $2\epsilon$-tube made up of strong $2\epsilon$-necks centered at points of $X_0$. @@ -1086,19 +1224,18 @@ boundaries in $\Omega^0\cap \Omega_{\rho/2C}$. This completes the proof of the fact that each component of $\Omega_{\rho/2C}$ has only finitely many ends. -This completes the proof of Theorem~. +This completes the proof of \entryref{af00774dbaa9}. -## Existence of strong $\delta$-necks sufficiently deep in -a $2\epsilon$-horn +## Existence of strong $\delta$-necks sufficiently deep in a $2\epsilon$-horn We keep the notation and assumptions of the previous section. -**Theorem.** -Fix $\rho>0$. Then for any $\delta>0$ there is an -$0 +**Theorem.** Fix $\rho>0$. Then for any $\delta>0$ there is an +$0 **Proof.** The proof of the first statement is by contradiction. Fix $\rho>0$ -and $\delta>0$ and suppose that there is no $00$ and suppose that there is no $0 +**Claim.** The sequence $(\widehat{\mathcal M}_n,\widehat G_n,x_n)$ satisfies +the five hypothesis of \entryref{d51aab8a8065}. + -\begin{proof} -By our assumptions, Hypotheses (1) and (3) of -Theorem~ hold for this sequence. +**Proof.** By our assumptions, Hypotheses (1) and (3) of +\entryref{d51aab8a8065} hold for this sequence. Also, we are assuming that any point $y\in {\mathcal M}_n$ with $R(y)\ge r_0^{-2}$ has a strong $(C,\epsilon)$-canonical neighborhood. Since $R(x_n)=Q_n\rightarrow \infty$ as $n\rightarrow\infty$ this means that for all $n$ sufficiently large, any point $y\in \widehat {\mathcal M}_n$ with $R(y)\ge R(x_n)$ has a strong $(C,\epsilon)$-canonical neighborhood. This establishes Hypothesis -(2) in the statement of Theorem~. +(2) in the statement of \entryref{d51aab8a8065}. Next, we have: + **Claim.** For any $A<\infty$ for all $n$ sufficiently large, $B(x_n,0,AQ_n^{-1/2})$ is contained in the $2\epsilon$-horn ${\mathcal H}_n$ and has compact closure in ${\mathcal M}_n$. + -\begin{proof} -Any point $z\in\partial {\mathcal H}_n$ has scalar curvature at +**Proof.** Any point $z\in\partial {\mathcal H}_n$ has scalar curvature at most $16C^2\rho^{-2}$ and there is a $2\epsilon$-neck centered at $z$. This means that for all $y$ with $d_{G_n}(z,y)<\epsilon^{-1}\rho/2C$ we have $R(y)\le @@ -1158,7 +1297,7 @@ That is to say, we must show that for every $A$ for all $n$ sufficiently large the distance from $x_n$ to the end of the horn ${\mathcal H}_n$ is greater than $AQ_n^{-1/2}$. If not, then since the curvature at the end of ${\mathcal H}_n$ goes to infinity for -each $n$, this sequence would violate Theorem~. +each $n$, this sequence would violate \entryref{ecd62fc2ea2b}. Because $B(x_n,0,AQ_n^{-1/2})$ is contained in a $2\epsilon$-horn, it is $\kappa$-non-collapsed on scales $\le r$ for a universal @@ -1168,27 +1307,28 @@ large and every $y\in B(x_n,0,AQ_n^{-1/2})$ the flow is defined on an interval through $y$ defined for backward time $R(y)^{-1}$. This completes the proof that all the hypotheses of -Theorem~ hold and establishes Claim~. -\end{proof} +\entryref{d51aab8a8065} hold and establishes \entryref{dff170186455}. We form a new sequence of generalized Ricci flows from the $({\mathcal M}_n, G_n)$ by translating by $-T_n$, so that the final -time-slice is at $**t**_n=0$, where $**t**_n$ is the time +time-slice is at $\mathbf{t}_n=0$, where $\mathbf{t}_n$ is the time function for ${\mathcal M}_n$. -Theorem~ implies that, after passing to a +\entryref{d51aab8a8065} implies that, after passing to a subsequence, there is a limit flow $(M_\infty,g_\infty(t),(x_\infty,0)),\ t\in[-t_0,0])$ defined for -some $t_0>0$ for the sequence $(Q_n\widehat{\mathcal +some $t_0>0$ for the sequence $(Q_n\widehat{\mathcal M}_n,Q_n\widehat G_n,x_n)$. Because of the curvature pinching toward -positive assumption, by Theorem~, the limit Ricci +positive assumption, by \entryref{2ddfbc74b081}, the limit Ricci flow has non-negative sectional curvature. Of course, $R(x_\infty)=1$ so that the limit $(M_\infty,g_\infty(0))$ is non-flat. + **Claim.** $(M_\infty,g_\infty(0))$ is isometric to the product $(S^2,h)\times (\Ar,ds^2)$, where $h$ is a metric of non-negative curvature on $S^2$ and $ds^2$ is the usual Euclidean metric on the real line. + **Proof.** Because of the fact that the $({\mathcal M}_n,G_n)$ have curvature pinched toward positive, and since $Q_n$ tend to @@ -1207,28 +1347,28 @@ smooth ray in ${\mathcal H_n}$ meeting the central $2$-sphere of a $2\epsilon$-neck centered at $x_n$ in a single point and at this point crossing from one side of this $2$-sphere to the other, the union of $\alpha$ and $\beta$ forms a proper, piecewise smooth map -of $\Ar$ to $M_\infty$ that meets the central $2$-sphere of a +of $\Ar$ to $M_\infty$ that meets the central $2$-sphere of a $2\epsilon$-neck centered at $x_\infty$ in a single point and crosses from one side to the other at the point. This means that $M_\infty$ has at least two ends. Since $(M_\infty,g_\infty)$ has -non-negative curvature, according to Theorem~, this +non-negative curvature, according to \entryref{e5b196773169}, this implies that $M_\infty$ is a product of a surface with $\Ar$. Since $M$ has non-negative curvature, the surface has non-negative curvature. Since $M$ has positive curvature at at least one point, the surface is diffeomorphic to the $2$-sphere. -According to Theorem~, after passing to a subsequence +According to \entryref{d51aab8a8065}, after passing to a subsequence there is a limit flow defined on some interval of the form $[-t_0,0]$ for $t_0>0$. Suppose that, after passing to a subsequence there is a limit flow defined on $[-T,0]$ for some $0 **Claim.** For any $z\in M_\infty$ for all $n$ sufficiently large, $z\in V_n$, so that $\varphi_n(z)$ is defined. Furthermore, for all $n$ sufficiently large, there is a backward flow line through @@ -1247,6 +1388,7 @@ $\varphi_n(z)$ in the generalized Ricci flow $(Q_n{\mathcal M}_n,Q_nG_n)$ defined on the interval $(-T-(R_{Q_nG_n}^{-1}(\varphi_n(z),0)/2),0]$. The scalar curvature is bounded above on this entire flow line by $R(\varphi_n(z),0)$. + **Proof.** Of course, for any compact subset $K\subset M_\infty$ and any $t' +**Corollary.** We can take the function $h(\rho,\delta)$ in the last lemma to be $\le \delta\rho$, to be a weakly monotone non-decreasing function of $\delta$ when $\rho$ is fixed, and to be a weakly monotone non-decreasing function of $\rho$ when $\delta$ is held fixed. + -**Proof.** If $h$ satisfies the conclusion of Theorem~ for $\rho$ +**Proof.** If $h$ satisfies the conclusion of \entryref{074e02b067b8} for $\rho$ and $\delta$ and if $\rho'\ge \rho$ and $\delta'\ge \delta$ then $h$ -also satisfies the conclusion of Theorem~ for $\rho'$ +also satisfies the conclusion of \entryref{074e02b067b8} for $\rho'$ and $\delta'$. Also, any $h'\le h$ also satisfies the conclusion of -Theorem~ for $\delta$ and $\rho$. Take a sequence +\entryref{074e02b067b8} for $\delta$ and $\rho$. Take a sequence $(\delta_n,\rho_n)$ where each of the sequences $\{\delta_n\}$ and $\{\rho_n\}$ is a monotone decreasing sequence with limit $0$. Then we choose $h_n=h(\rho_n,\delta_n)\le \rho_n\delta_n$ as in the -statement of Theorem~. We of course can assume that +statement of \entryref{074e02b067b8}. We of course can assume that $\{h_n\}_n$ is a non-increasing sequence of positive numbers with limit $0$. Then for any $(\rho,\delta)$ we take the largest $n$ such that $\rho\ge \rho_n$ and $\delta\ge \delta_n$, and we define diff --git a/projects/poincare-conjecture/.astrolabe/docs/12-stdsoln.mdx b/projects/poincare-conjecture/.astrolabe/docs-src/13-the-standard-solution.mdx similarity index 71% rename from projects/poincare-conjecture/.astrolabe/docs/12-stdsoln.mdx rename to projects/poincare-conjecture/.astrolabe/docs-src/13-the-standard-solution.mdx index a8431e05..cecf564f 100644 --- a/projects/poincare-conjecture/.astrolabe/docs/12-stdsoln.mdx +++ b/projects/poincare-conjecture/.astrolabe/docs-src/13-the-standard-solution.mdx @@ -1,64 +1,79 @@ -# The standard solution + + +# Chapter 12 -- The standard solution The process of surgery involves making a choice of the metric on a -three-ball to `glue in'. In order to match approximatively with the +three-ball to 'glue in'. In order to match approximatively with the metric coming from the flow, the metric we glue in must be asymptotic to the product of a round two-sphere and an interval near the boundary. There is no natural choice for this metric; yet it is crucial to the argument that we choose an initial metric so that the Ricci flow with these initial conditions has several properties. -Conditions on the initial metric that ensure the required +Conditions on the initial metric that ensure the required properties for the subsequence flow are contained in the following definition. -**Definition.** -A *standard initial metric* + +**Definition.** A *standard initial metric* is a metric $g_0$ on $\Ar^3$ with the following properties: -\begin{itemize} -\item $g_0$ is a complete metric. -\item $g_0$ has non-negative sectional curvature at every point. -\item $g_0$ is invariant under the usual $SO(3)$-action on $\Ar^3$. -\item there is a compact ball $B\subset \Ar^3$ so + +- $g_0$ is a complete metric. +- $g_0$ has non-negative sectional curvature at every point. +- $g_0$ is invariant under the usual $SO(3)$-action on $\Ar^3$. +- there is a compact ball $B\subset \Ar^3$ so that the restriction of the metric $g_0$ to the complement of this ball is isometric to the product $(S^2,h)\times (\Ar^+,ds^2)$ where $h$ is the round metric of scalar curvature $1$ on $S^2$. -\item $g_0$ has constant sectional curvature $1/4$ near the origin. (This point will be +- $g_0$ has constant sectional curvature $1/4$ near the origin. (This point will be denoted $p$ and is called the *tip* of the initial metric.) -\end{itemize} See Fig.~. + + See Fig. 12.1. + Actually, one can work with an alternative weaker version of the fourth condition, namely: (iv) $g_0$ is asymptotic at infinity in the $C^\infty$-topology to the product of the round metric $h_0$ on -$S^2$ of scalar curvature $1$ with the usual metric $ds^2$ on the -real line. By this we mean that if $x_n\in \Ar^3$ is any sequence +$S^2$ of scalar curvature $1$ with the usual metric $ds^2$ on the +real line. By this we mean that if $x_n\in \Ar^3$ is any sequence converging to infinity, then the based Riemannian manifolds $(\Ar^3,g_0,x_n)$ converge smoothly to $(S^2,h_0)\times (\Ar,ds^2)$. But we shall only use standard initial metrics as given in -Definition~. +\entryref{c5e007547ff3}. -**Lemma.** -There is a standard initial metric. + +**Lemma.** There is a standard initial metric. + **Proof.** We construct our Riemannian manifold as follows. Let $(x_0,x_1,x_2,x_3)$ be Euclidean coordinates on $\Ar^4$. Let $y=f(s)$ be a function defined for $s\ge 0$ and satisfying: -\begin{enumerate} -\item[(1)] $f$ is $C^\infty$ on $(0,\infty)$ -\item[(2)] $f(s)>0$ for all $s>0$. -\item[(3)] $f''(s)\le 0$ for all $s>0$. -\item[(4)] There is $s_1>0$ such that $f(s)=\sqrt{2}$ for all $s\ge + +- **(1)** $f$ is $C^\infty$ on $(0,\infty)$ +- **(2)** $f(s)>0$ for all $s>0$. +- **(3)** $f''(s)\le 0$ for all $s>0$. +- **(4)** There is $s_1>0$ such that $f(s)=\sqrt{2}$ for all $s\ge s_1$. -\item[(5)] There is $s_0>0$ such that +- **(5)** There is $s_0>0$ such that $f(s)=\sqrt{4s-s^2}$ for all $s\in [0,s_0]$. -\end{enumerate} + Given such a function $f$, consider the graph -$$\Gamma=\{(x_0,x_1)\bigl|\bigr. x_0\ge 0 \text{ and } -x_1=f(x_0)\}$$ in the $(x_0,x_1)$-plane. We define $\Sigma(f)$ by + +$$ +\Gamma=\{(x_0,x_1)\bigl|\bigr. x_0\ge 0 \text{ and } +x_1=f(x_0)\} +$$ + + in the $(x_0,x_1)$-plane. We define $\Sigma(f)$ by rotating $\Gamma$ about the $x_0$-axis in four-space: -$$\Sigma(f)=\{(x_0,x_1,x_2,x_3)\bigl|\bigr. x_0\ge 0 \text{ and } -x_1^2+x_2^2+x_3^2=f(x_0)^2\}.$$ Because of the last condition on + +$$ +\Sigma(f)=\{(x_0,x_1,x_2,x_3)\bigl|\bigr. x_0\ge 0 \text{ and } +x_1^2+x_2^2+x_3^2=f(x_0)^2\}. +$$ + + Because of the last condition on $f$, there is a neighborhood of $0\in \Sigma(f)$ that is isometric to a neighborhood of the north pole in the three-sphere of radius $2$. Because of this and the first item, we see that $\Sigma(f)$ is @@ -77,47 +92,81 @@ It remains to compute the sectional curvatures of $g_0$. Let $q\in $SO(3)$-action. Direct computation shows that the tangent plane to the two-dimensional $SO(3)$-orbit through $q$ is a principal direction for the curvature, and the sectional curvature on this - tangent two-plane is given by -$$\frac{1}{f(q)^2(1+f'(q)^2)}.$$ + tangent two-plane is given by + +$$ +\frac{1}{f(q)^2(1+f'(q)^2)}. +$$ + On the subspace in $\wedge^2T_q\Sigma(f)$ perpendicular to the line -given by this two-plane, the curvature is constant with eigenvalue -$$\frac{-f''(q)}{f(q)(1+f'(q)^2)^2}.$$ +given by this two-plane, the curvature is constant with eigenvalue + +$$ +\frac{-f''(q)}{f(q)(1+f'(q)^2)^2}. +$$ + Under our assumptions about $f$, it is clear that $\Sigma(f)$ has non-negative curvature and has constant sectional curvature $1/4$ near the origin. It remains to choose the function $f$ satisfying Items (1) -- (5) above. Consider the function $h(s)=(2-s)/\sqrt{4s-s^2}$. This function is -integrable from $0$ and the definite integral from zero to $s$ is +integrable from $0$ and the definite integral from zero to $s$ is equal to $\sqrt{4s-s^2}$. Let $\lambda(s)$ be a non-increasing $C^\infty$-function defined on $[0,1/2]$, with $\lambda$ identically one near $0$ and identically equal to $0$ near $1/2$. We extend $\lambda$ to be identically $1$ for $s<0$ and identically $0$ for $s>1/2$. Clearly, -$$\int_0^2h(s)\lambda(s-3/2)ds>\int_0^{3/2}h(s)ds>\sqrt{2}$$ -and $$\int_0^2h(s)\lambda(s)ds<\int_0^{1/2}h(s)<\sqrt{2}.$$ Hence, + +$$ +\int_0^2h(s)\lambda(s-3/2)ds>\int_0^{3/2}h(s)ds>\sqrt{2} +$$ + +and + +$$ +\int_0^2h(s)\lambda(s)ds<\int_0^{1/2}h(s)<\sqrt{2}. +$$ + + Hence, for some $s_0\in (1/2,3/2)$ we have -$$\int_0^2h(s)\lambda(s-s_0)ds=\sqrt{2}.$$ -We define $$f(s)=\int_0^sh(\sigma)\lambda(\sigma-s_0)d\sigma.$$ It + +$$ +\int_0^2h(s)\lambda(s-s_0)ds=\sqrt{2}. +$$ + +We define + +$$ +f(s)=\int_0^sh(\sigma)\lambda(\sigma-s_0)d\sigma. +$$ + + It is easy to see that $f$ satisfies all the above conditions. The following lemma is clear from the construction. -**Lemma.** -There is $A_0<\infty$ such that -$$(\Ar^3\setminus B(0,A_0),g(0))$$ + +**Lemma.** There is $A_0<\infty$ such that + +$$ +(\Ar^3\setminus B(0,A_0),g(0)) +$$ + is isometric to the product of a round metric on $S^2$ of scalar curvature $1$ with the Euclidean metric on $[0,\infty)$. There is a constant $K<\infty$ such that the volume of $B_{g(0)}(0,A_0)$ is at most $K$. Furthermore, there is a constant $D<\infty$ so that the scalar curvature of standard initial metric $(\Ar^3,g(0))$ is bounded above by $D$ and below by $D^{-1}$. + ### Uniqueness and properties: The statement **Fix once and for all a standard initial metric $g_0$ on $\Ar^3$.** + **Definition.** A *partial standard Ricci flow* is a Ricci flow $(\Ar^3,g(t)),\ 0\le tT$ with the property that the extension has curvature locally bounded in time. + Here is the main result of this chapter. -**Theorem.** There is a standard Ricci flow defined for some + +**Theorem.** There is a standard Ricci flow defined for some positive amount of time. Let $(\Ar^3,g(t)),\ 0\le t0$ there is a compact subset $X$ of $\Ar^3$ such that for any $x\in \Ar ^3\setminus X$ the restriction of the standard flow to an appropriate neighborhood of @@ -151,9 +202,9 @@ $C^{[1/\epsilon]}$-topology of the product Ricci flow $(S^2\times (-\epsilon^{-1},\epsilon^{-1})),h(t)\times ds^2,\ 0\le t\le t_0$, where $h(t)$ is the round metric with scalar curvature $1/(1-t)$ on $S^2$. -\item[(6)] **(Non-collapsing):** There are $r>0$ and $\kappa>0$ such that $(\Ar^3,g(t)),\ 0\le +- **(6) (Non-collapsing):** There are $r>0$ and $\kappa>0$ such that $(\Ar^3,g(t)),\ 0\le t<1$, is $\kappa$-non-collapsed on scales less than $r$. -\end{enumerate} + The proof of this result occupies the next few subsections. All the properties except the uniqueness are fairly straightforward to @@ -161,7 +212,7 @@ prove. We establish uniqueness by reducing the Ricci flow to the Ricci-DeTurck by establishing the existence of a solution to the harmonic map flow in this case. This technique can be made to work more generally in the case of complete manifolds of bounded -curvature, see , but we preferred to give the more +curvature, see [ChenZhu], but we preferred to give the more elementary argument that covers this case, where the symmetries allow us to reduce the existence problem for the harmonic map flow to a problem that is the essentially one-dimensional. Also, in the @@ -176,7 +227,7 @@ $\kappa$-solutions. ## Existence of a standard flow For any $R<\infty$, denote by $B_R\subset \Ar^3$, the ball of radius -$R$ about the origin in the metric $g_0$. For $R\ge A_0+1$, a +$R$ about the origin in the metric $g_0$. For $R\ge A_0+1$, a neighborhood of the boundary of this ball is isometric to $(S^2,h)\times ([0,1],ds^2)$. Thus, in this case, we can double the ball, gluing the boundary to itself by the identity, forming a @@ -189,24 +240,24 @@ $(\Ar^3,g_0,p)$. For each $n$, let $(S^3_{R_n},g_{R_n}(t)),\ 0\le tt_0$ for all $n$. Since the Riemann curvatures of the $(S^3_{R_n},g_{R_n}(t)),\ 0\le t +**Lemma.** For each $t_0\in [0,T)$ the Riemannian manifold $(\Ar^3,g(t_0))$ is complete and of positive curvature. + **Proof.** Fix $t_0\in [0,T)$. By hypothesis $(\Ar^3,g(t)),\ 0\le t\le t_0$ has bounded curvature. Hence, there is a constant $C<\infty$ such that $ g(0)\le Cg(t_0)$, @@ -237,30 +289,28 @@ so that for any points $x,y\in \Ar^3$, we have $d_0(x,y)\le also complete. Now let us show that $(M,g(t_0))$ has non-negative curvature. Here, -the argument is the analogue of the proof of Corollary~ +the argument is the analogue of the proof of \entryref{0c451aa3e5cc} with one additional step, the use of a function $\varphi$ to localize the argument. Suppose this is false, i.e., suppose that -there is $x\in M$ with $Rm(x,t_0)$ having an eigenvalue less +there is $x\in M$ with $\mathit{Rm}(x,t_0)$ having an eigenvalue less than zero. Since the restriction of the flow to $[0,t_0]$ is complete and of bounded curvature, according to - for any constants $C<\infty$ and $\eta>0$ and +[Hamiltonlimits] for any constants $C<\infty$ and $\eta>0$ and any compact subset $K\subset M\times [0,t_0]$ there is $\epsilon>0$ and a function $\varphi\colon M\times [0,t_0]\to \Ar$ with the following properties: -\begin{enumerate} -\item[(1)] $\varphi|_K\le \eta$. -\item[(2)] $\varphi\ge \epsilon$ everywhere. -\item[(3)] For each $t\in [0,t_0]$ the restriction of $\varphi$ to $M\times \{t\}$ -goes to infinity at infinity in the sense that for any $A<\infty$ +- **(1)** $\varphi|_K\le \eta$. +- **(2)** $\varphi\ge \epsilon$ everywhere. +- **(3)** For each $t\in [0,t_0]$ the restriction of $\varphi$ to $M\times \{t\}$ +goes to infinity at infinity in the sense that for any $A<\infty$ the pre-image $\varphi^{-1}([0,A]\cap (M\times\{t\})$ is compact. -\item[(4)] On all of $M\times [0,t_0]$ we have +- **(4)** On all of $M\times [0,t_0]$ we have $ \left(\frac{\partial}{\partial t}-\triangle\right)\varphi\ge C\varphi. $ -\end{enumerate} -Recall from Section~ that ${\mathcal T}$ is the curvature +Recall from Section 4.2.1 that ${\mathcal T}$ is the curvature tensor written with respect to an evolving orthonormal frame $\{F_\alpha\}$ for the tangent bundle. Consider the symmetric, horizontal two-tensor $\widehat{\mathcal T}={\mathcal T}+\varphi g$. @@ -271,8 +321,8 @@ follows from the third property of $\varphi$ that for each $t\in infinity at infinity in $M$. In particular, the subset of $(x,t)\in M\times [0,t_0]$ with the property that $\hat\mu(x,t)\le \hat\mu(y,t)$ for all $y\in M$ is a compact subset of $M\times -[0,t_0]$. It follows from Proposition~ that -$f(t)=min_{x\in M}\hat\mu(x,t)$ is a continuous function of +[0,t_0]$. It follows from \entryref{37d7d059a526} that +$f(t)=\mathit{min}_{x\in M}\hat\mu(x,t)$ is a continuous function of $t$. Choosing $\eta>0$ sufficiently small and $K$ to include $(x,t_0)$, then $\widehat{\mathcal T}$ will have a negative eigenvalue at $(x,t_0)$. Clearly, it has only positive eigenvalues @@ -284,48 +334,58 @@ Diagonalizing ${\mathcal T}$ at any point $(x,t)$ with $t\le t_1$, all its eigenvalues are at least $-\varphi(x,t_1)$. It follows immediately that on $M\times [0,t_1]$ the smallest eigenvalue of the symmetric form ${\mathcal T}^2+{\mathcal T}^\#$ is bounded below by -$2\varphi$. Thus, choosing $C\ge 4$ we see that for $t\le t_1$ +$2\varphi$. Thus, choosing $C\ge 4$ we see that for $t\le t_1$ every eigenvalue of ${\mathcal T}^2+{\mathcal T}^\#$ is at least $-C\varphi/2$. We compute the evolution equation using the formula in -Lemma~ for the evolution of ${\mathcal T}$ in an evolving +\entryref{0c451aa3e5cc} for the evolution of ${\mathcal T}$ in an evolving orthonormal frame: -$$ - $$ \begin{aligned} \frac{\partial \widehat{\mathcal T}}{\partial t} & = & \frac{\partial{\mathcal T}}{\partial t}+\frac{\partial -\varphi}{\partial t}g-2\varphi Ric(g) \\ -& = & \triangle {\mathcal T}+{\mathcal T}^2+{\mathcal T}^\#+\frac{\partial \varphi}{\partial t}g-2\varphi Ric(g) \\ +\varphi}{\partial t}g-2\varphi \mathit{Ric}(g) \\ +& = & \triangle {\mathcal T}+{\mathcal T}^2+{\mathcal T}^\#+\frac{\partial \varphi}{\partial t}g-2\varphi \mathit{Ric}(g) \\ & = & \triangle \widehat{\mathcal T}+{\mathcal T}^2+{\mathcal T}^\#+\left(\frac{\partial \varphi}{\partial t}-\triangle -\varphi\right)g-2\varphi Ric(g)\\ +\varphi\right)g-2\varphi \mathit{Ric}(g)\\ &\ge & \triangle \widehat{\mathcal T}+{\mathcal T}^2+{\mathcal -T}^\#+\left(Cg-2Ric(g)\right)\varphi. +T}^\#+\left(Cg-2\mathit{Ric}(g)\right)\varphi. \end{aligned} $$ -$$ - -Since every eigenvalue of ${\mathcal T}^2+{\mathcal T}^\#$ on +Since every eigenvalue of ${\mathcal T}^2+{\mathcal T}^\#$ on $M\times [0,t_1]$ is at least $-C\varphi/2$, it follows that on $M\times [0,t_1]$ -$$\frac{\partial \widehat{\mathcal T}}{\partial t}\ge - \triangle \widehat{\mathcal T}+(Cg/2-2Ric_g)\varphi.$$ Once again assuming that $C$ is + +$$ +\frac{\partial \widehat{\mathcal T}}{\partial t}\ge + \triangle \widehat{\mathcal T}+(Cg/2-2\mathit{Ric}_g)\varphi. +$$ + + Once again assuming that $C$ is sufficiently large, we see that for any $t\le t_1$ -$$\frac{\partial \widehat{\mathcal T}}{\partial t}\ge \triangle -\widehat {\mathcal T}.$$ Thus, at any local minimum $x\in M$ for + +$$ +\frac{\partial \widehat{\mathcal T}}{\partial t}\ge \triangle +\widehat {\mathcal T}. +$$ + + Thus, at any local minimum $x\in M$ for $\hat\mu(\cdot,t)$, we have -$$\frac{\partial \hat\mu}{\partial t}\ge 0.$$ -This immediately implies by Proposition~ that -$\psi(t)=min_{x\in M}\hat\mu(x,t)$ is a non-decreasing + +$$ +\frac{\partial \hat\mu}{\partial t}\ge 0. +$$ + +This immediately implies by \entryref{37d7d059a526} that +$\psi(t)=\mathit{min}_{x\in M}\hat\mu(x,t)$ is a non-decreasing function of $t$. Since its value at $t=0$ is at least $\epsilon>0$ and its value at $t_1$ is zero, this is a contradiction. This establishes that the solution has non-negative curvature everywhere. -Indeed, by Corollary~ it has strictly positive +Indeed, by \entryref{d21a675e4ea8} it has strictly positive curvature for every $t>0$. Now let us turn to the asymptotic behaviour of the flow. @@ -340,19 +400,19 @@ there is a uniform $C^\infty$ point-wise bound to the curvatures of $g_0$ restricted to the images of the $\psi_k$ for $k\ge k_0(R)$. Since the flow $g(t)$ has bounded curvature on $\Ar^3\times [0,T']$, it follows from -Theorem~ that there are uniform $C^\infty$ +\entryref{e0b6a1f646a7} that there are uniform $C^\infty$ point-wise bounds for the curvatures of $g(t)$ restricted to $\psi_k(S^2\times (-R,R))$. Thus, - by Theorem~, after passing to a subsequence, the flows -$\psi_k^*g(t)$ converge to a limiting flow on $S^2\times \Ar$. Of + by \entryref{4673eb9f4424}, after passing to a subsequence, the flows +$\psi_k^*g(t)$ converge to a limiting flow on $S^2\times \Ar$. Of course, since the curvature of $g(t)$ is everywhere $\ge 0$, the same is true of this limiting flow. Since the time-slices of this -flow have two ends, it follows from Theorem~ that +flow have two ends, it follows from \entryref{e5b196773169} that every manifold in the flow is a product of a compact surface with -$\Ar$. According to Corollary~ this implies that the +$\Ar$. According to \entryref{d21a675e4ea8} this implies that the flow is the product $(S^2,h(t))\times (\Ar,ds^2)$. This means that given $\epsilon>0$, for all $k$ sufficiently large, the restriction -of the flow to the cylinder of length $2R$ centered at $y_k$ is +of the flow to the cylinder of length $2R$ centered at $y_k$ is within $\epsilon$ in the $C^{[1/\epsilon]}$-topology of the shrinking cylindrical flow on time $[0,T']$. Given $\epsilon>0$ and $R<\infty$ this statement is true for all $y$ outside a compact ball @@ -360,15 +420,17 @@ $B$ centered at the origin. We have now established the following -**Proposition.** -Given $T'0$ there is a compact ball $B$ centered + +**Proposition.** Given $T'0$ there is a compact ball $B$ centered at the origin of $\Ar^3$ such that the restriction of the flow $(\Ar^3\setminus B,g(t)),\ 0\le t\le T'$, is within $\epsilon$ in the $C^{[1/\epsilon]}$-topology of the standard evolving cylinder $(S^2,h(t))\times (\Ar^+,ds^2)$. + -**Corollary.** -The maximal time $T$ is $\le 1$. + +**Corollary.** The maximal time $T$ is $\le 1$. + **Proof.** If $T>1$, then we can apply the above result to $T'$ with $1 +**Lemma.** With $X$ and its dual $X^*$ evolving by the above equations, set $V=\nabla X^*$, so that $V$ is a contravariant two-tensor. In local coordinates we have $V=V_{ij}dx^i\otimes dx^j$ with -$$V_{ij}=(\nabla_iX)_j=g_{jk}(\nabla_iX)^k.$$ + +$$ +V_{ij}=(\nabla_iX)_j=g_{jk}(\nabla_iX)^k. +$$ + This symmetric two-tensor satisfies $$ \frac{\partial}{\partial t}V=\triangle -V-\left(2{{R_k}^{rl}}_jV_{rl}+Ric_k^lV_{lj} +Ric_j^lV_{kl}\right)dx^k\otimes dx^j. +V-\left(2{{R_k}^{rl}}_jV_{rl}+\mathit{Ric}_k^lV_{lj} +\mathit{Ric}_j^lV_{kl}\right)dx^k\otimes dx^j. $$ + + **Remark.** The covariant derivative acts on one-forms $\omega$ in such a way that the following equation holds: -$$\langle\nabla(\omega),\xi\rangle=\langle -\omega,\nabla(\xi)\rangle$$ for every vector field $\xi$. This means + +$$ +\langle\nabla(\omega),\xi\rangle=\langle +\omega,\nabla(\xi)\rangle +$$ + + for every vector field $\xi$. This means that in local coordinates we have -$$\nabla_{\partial_r}(dx^k)=-\Gamma_{rl}^kdx^l.$$ Similarly, the Riemann + +$$ +\nabla_{\partial_r}(dx^k)=-\Gamma_{rl}^kdx^l. +$$ + + Similarly, the Riemann curvature acts on one-forms $\omega$ satisfying -$$Rm(\xi_1,\xi_2)(\omega)(\xi)=-\omega\left(Rm(\xi_1,\xi_2)(\xi)\right).$$ -Recall that in local coordinates $$R_{ijkl}=\langle Rm(\partial_i,\partial_j)(\partial_l),\partial_k\rangle.$$ Thus, we + +$$ +\mathit{Rm}(\xi_1,\xi_2)(\omega)(\xi)=-\omega\left(\mathit{Rm}(\xi_1,\xi_2)(\xi)\right). +$$ + +Recall that in local coordinates + +$$ +R_{ijkl}=\langle \mathit{Rm}(\partial_i,\partial_j)(\partial_l),\partial_k\rangle. +$$ + + Thus, we have -$$Rm(\partial_i,\partial_j)(dx^k)=-g^{ka}R_{ijal}dx^l=-{{R_{ij}}^k}_ldx^l,$$ + +$$ +\mathit{Rm}(\partial_i,\partial_j)(dx^k)=-g^{ka}R_{ijal}dx^l=-{{R_{ij}}^k}_ldx^l, +$$ + where as usual we use the inverse metric tensor to raise the index. -Also, notice that $\Delta X_{i}-Ric_{ik}X^{k}=-\Delta_{d}X_{i\text{ }}$, where by $\Delta_d$ we +Also, notice that $\Delta X_{i}-\mathit{Ric}_{ik}X^{k}=-\Delta_{d}X_{i\text{ }}$, where by $\Delta_d$ we mean the Laplacian associated to the operator $d$ from vector fields to one-forms with values in the vector field. Since -$$ - $$ \begin{aligned} -\left( d\delta+\delta d\right) X_{i} @@ -451,13 +548,12 @@ $$ =\nabla_{i}\nabla^{k}X_{k}+\nabla^{k}\nabla_{k}X_{i}-\nabla^{k}\nabla _{i}X_{k}\\ & ={{{R_{i}}^k}_k}^jX_{j}+\nabla^{k}\nabla_{k}X_{i}=\Delta -X_{i}-Ric_i^jX_{j}. +X_{i}-\mathit{Ric}_i^jX_{j}. \end{aligned} $$ + -$$ - -**Proof.** (of Lemma~) +**Proof.** (of \entryref{78cab9a6159b}) The computation is routine, if complicated. We make the computation at a point $(p,t)$ of space-time. We fix local $g(t)$-Gaussian coordinates $(x^1,\ldots,x^n)$ centered at $p$ for space, so that @@ -466,8 +562,6 @@ the Christoffel symbols vanish at $(p,t)$. We compute -$$ - $$ \begin{aligned} \nonumber @@ -476,17 +570,14 @@ t}\left( \nabla X^* \right) =-\left( \frac{\partial}{\partial t}\Gamma_{kj}^{l}\right) X^*_{l}dx^k\otimes dx^j +\nabla\left( \frac{\partial}{\partial t}X^*\right) \\ \nonumber &= & -\left(-\nabla^{l}Ric_{kj}+\nabla_{k}Ric_j^l+\nabla_{j}Ric_k^l\right)X^*_{l}dx^k\otimes +\left(-\nabla^{l}\mathit{Ric}_{kj}+\nabla_{k}\mathit{Ric}_j^l+\nabla_{j}\mathit{Ric}_k^l\right)X^*_{l}dx^k\otimes dx^j \\ -& & +\nabla\left( \Delta X^*-Ric(X,\cdot)\right) +& & +\nabla\left( \Delta X^*-\mathit{Ric}(X,\cdot)\right) . \end{aligned} $$ -$$ - We have -$$ $$ \begin{aligned} @@ -498,88 +589,73 @@ g^{rs}\left(\nabla\left(\nabla_r\nabla_s(X^*)-\Gamma_{rs}^l\nabla_lX^*\right)\ri \end{aligned} $$ -$$ - Let us recall the formula for commuting $\nabla$ and $\nabla_r$. The following is immediate from the definitions. + **Claim.** For any tensor $\phi$ we have -$$\nabla(\nabla_r\phi)=\nabla_r(\nabla\phi)+dx^k\otimes -Rm(\partial_k,\partial_r)(\phi)-\nabla_r(dx^l)\otimes -\nabla_l(\phi).$$ - -Applying this to our formula gives $$ +\nabla(\nabla_r\phi)=\nabla_r(\nabla\phi)+dx^k\otimes +\mathit{Rm}(\partial_k,\partial_r)(\phi)-\nabla_r(dx^l)\otimes +\nabla_l(\phi). +$$ + + +Applying this to our formula gives $$ \begin{aligned} \nabla(\triangle -X^*) & = & g^{rs}\left(\nabla_r\nabla\nabla_sX^*+dx^k\otimes Rm(\partial_k,\partial_r)(\nabla_sX^*)-\nabla_r(dx^l)\otimes +X^*) & = & g^{rs}\left(\nabla_r\nabla\nabla_sX^*+dx^k\otimes \mathit{Rm}(\partial_k,\partial_r)(\nabla_sX^*)-\nabla_r(dx^l)\otimes \nabla_l\nabla_sX^* \right. \\ & & \left. -\nabla(\Gamma_{rs}^l\nabla_lX^*)\right) . \end{aligned} -$$ - $$ Now we apply the same formula to commute $\nabla$ and $\nabla_s$. The result is -$$ - $$ \begin{aligned} \nabla(\triangle X^*) & = & g^{rs}\left(\nabla_r\nabla_s\nabla -X^*+\nabla_r\left(dx^k\otimes Rm(\partial_k,\partial_s)X^*-\nabla_sdx^l\otimes \nabla_lX^*\right)\right. \\ -& & \left. +dx^k\otimes Rm(\partial_k,\partial_r)(\nabla_sX^*)-\nabla_r(dx^l)\otimes +X^*+\nabla_r\left(dx^k\otimes \mathit{Rm}(\partial_k,\partial_s)X^*-\nabla_sdx^l\otimes \nabla_lX^*\right)\right. \\ +& & \left. +dx^k\otimes \mathit{Rm}(\partial_k,\partial_r)(\nabla_sX^*)-\nabla_r(dx^l)\otimes \nabla_l\nabla_sX^*-\nabla(\Gamma_{rs}^l\nabla_lX^*)\right). \end{aligned} $$ -$$ - Now we expand -$$ - $$ \begin{aligned} -\lefteqn{\nabla_r\left(dx^k\otimes Rm(\partial_k,\partial_s)X^*-\nabla_sdx^l\otimes -\nabla_lX^*\right)} +\nabla_r\left(dx^k\otimes \mathit{Rm}(\partial_k,\partial_s)X^*-\nabla_sdx^l\otimes +\nabla_lX^*\right) \\ & = & \nabla_r(dx^k)\otimes -Rm(\partial_k,\partial_s)X^*+dx^k\otimes\nabla_r(Rm(\partial_k,\partial_s))X^* +\mathit{Rm}(\partial_k,\partial_s)X^*+dx^k\otimes\nabla_r(\mathit{Rm}(\partial_k,\partial_s))X^* \\ -& & + dx^k\otimes Rm(\partial_k,\partial_s)\nabla_rX^*-\nabla_r\nabla_sdx^l\otimes\nabla_lX^*-\nabla_sdx^l\otimes +& & + dx^k\otimes \mathit{Rm}(\partial_k,\partial_s)\nabla_rX^*-\nabla_r\nabla_sdx^l\otimes\nabla_lX^*-\nabla_sdx^l\otimes \nabla_r\nabla_lX^* \end{aligned} $$ -$$ - Invoking the fact that the Christoffel symbols vanish at the point of space-time where we are making the computation, this above expression simplifies to -$$ - $$ \begin{aligned} -\lefteqn{\nabla_r\left(dx^k\otimes Rm(\partial_k,\partial_s)X^*-\nabla_sdx^l\otimes -\nabla_lX^*\right)} -\\ & = & dx^k\otimes\nabla_r(Rm(\partial_k,\partial_s))X^* + dx^k\otimes -Rm(\partial_k,\partial_s)\nabla_rX^*-\nabla_r\nabla_sdx^l\otimes\nabla_lX^*. +\nabla_r\left(dx^k\otimes \mathit{Rm}(\partial_k,\partial_s)X^*-\nabla_sdx^l\otimes +\nabla_lX^*\right) +\\ & = & dx^k\otimes\nabla_r(\mathit{Rm}(\partial_k,\partial_s))X^* + dx^k\otimes +\mathit{Rm}(\partial_k,\partial_s)\nabla_rX^*-\nabla_r\nabla_sdx^l\otimes\nabla_lX^*. \end{aligned} $$ -$$ - Also, expanding and using the vanishing of the Christoffel symbols we have -$$ - $$ \begin{aligned} -\nabla(\Gamma_{rs}^l\nabla_lX^*)& = & @@ -588,51 +664,43 @@ $$ \end{aligned} $$ -$$ - Plugging these computations into equation above and using once more the vanishing of the Christoffel symbols gives -$$ - $$ \begin{aligned} \nabla(\triangle X^*) & = & \triangle(\nabla X^*)+g^{rs}\left( -dx^k\otimes\nabla_r(Rm(\partial_k,\partial_s))X^* + +dx^k\otimes\nabla_r(\mathit{Rm}(\partial_k,\partial_s))X^* + dx^k\otimes -Rm(\partial_k,\partial_s)\nabla_rX^* \right.\\ +\mathit{Rm}(\partial_k,\partial_s)\nabla_rX^* \right.\\ & & \left. -\nabla_r\nabla_sdx^l\otimes\nabla_lX^* +dx^k\otimes -Rm(\partial_k,\partial_r)(\nabla_sX^*) +\mathit{Rm}(\partial_k,\partial_r)(\nabla_sX^*) -d\Gamma_{rs}^l\otimes\nabla_lX^*\right). \end{aligned} -$$ - $$ Now by the symmetry of $g^{rs}$ we can amalgamate the second and fourth terms on the right-hand side to give -$$ - $$ \begin{aligned} \nabla(\triangle X^*) & = & \triangle(\nabla X^*)+g^{rs}\left( -dx^k\otimes\nabla_r(Rm(\partial_k,\partial_s))X^* \right. \\ +dx^k\otimes\nabla_r(\mathit{Rm}(\partial_k,\partial_s))X^* \right. \\ & & \left. + 2dx^k\otimes -Rm(\partial_k,\partial_s)\nabla_rX^* +\mathit{Rm}(\partial_k,\partial_s)\nabla_rX^* -\nabla_r\nabla_sdx^l\otimes\nabla_lX^* -d\Gamma_{rs}^l\otimes\nabla_lX^*\right). \end{aligned} $$ -$$ - We expand -$$Rm(\partial_k,\partial_s)\nabla_rX^* =-{{R_{ks}}^{l}}_jV_{rl}dx^j.$$ -Also we have (again using the vanishing of the Christoffel symbols) $$ +\mathit{Rm}(\partial_k,\partial_s)\nabla_rX^* =-{{R_{ks}}^{l}}_jV_{rl}dx^j. +$$ + +Also we have (again using the vanishing of the Christoffel symbols) $$ \begin{aligned} @@ -642,24 +710,16 @@ $$ \end{aligned} $$ -$$ - Lastly, -$$ - $$ \begin{aligned} -\nabla_r(Rm(\partial_k,\partial_s))X^*=-{{(\nabla_rR)_{ks}}^l}_jX^*_ldx^j. +\nabla_r(\mathit{Rm}(\partial_k,\partial_s))X^*=-{{(\nabla_rR)_{ks}}^l}_jX^*_ldx^j. \end{aligned} $$ -$$ - Plugging all this in and raising indices yields -$$ - $$ \begin{aligned} \nabla(\triangle X^*) & = & \triangle(\nabla X^*)- @@ -669,67 +729,62 @@ dx^j-2{{R_k}^{rl}}_jV_{rl}dx^k\otimes dx^j \\ g^{rs}{{R_{rk}}^l}_sV_{lj}dx^k\otimes dx^j \\ & = & \triangle(\nabla X^*)- g^{rs}{{(\nabla_rR)_{ks}}^j}_lX^*_jdx^k\otimes dx^l-2{{R_k}^{rl}}_jV_{rl}dx^k\otimes dx^j \\ -& & -Ric_k^lV_{lj}dx^k\otimes dx^j. +& & -\mathit{Ric}_k^lV_{lj}dx^k\otimes dx^j. \end{aligned} $$ -$$ - Thus, we have -$$ - $$ \begin{aligned} -\lefteqn{\nabla(\triangle X^*) -\nabla(Ric(X,\cdot)^*)=} \\ & & +\nabla(\triangle X^*) -\nabla(\mathit{Ric}(X,\cdot)^*)= \\ & & \triangle(\nabla X^*)- g^{rs}{{(\nabla_rR)_{ks}}^l}_jX^*_ldx^k\otimes dx^j-2{{R_k}^{rl}}_jV_{rl}dx^k\otimes dx^j \\ -& & -\left(Ric_k^lV_{lj}+\nabla_k(Ric)_j^lX^*_l+Ric_j^lV_{kl}\right)dx^k\otimes dx^j, +& & -\left(\mathit{Ric}_k^lV_{lj}+\nabla_k(\mathit{Ric})_j^lX^*_l+\mathit{Ric}_j^lV_{kl}\right)dx^k\otimes dx^j, \end{aligned} $$ -$$ - -and consequently, plugging back into Equation~(), and +and consequently, plugging back into Equation (12.4), and canceling the two like terms appearing with opposite sign, we have -$$ - $$ \begin{aligned} -\frac{\partial}{\partial t}V & = & \left(-\nabla^{l}Ric_{kj}+\nabla_{j}Ric_k^l\right)X^*_{l}dx^k\otimes +\frac{\partial}{\partial t}V & = & \left(-\nabla^{l}\mathit{Ric}_{kj}+\nabla_{j}\mathit{Ric}_k^l\right)X^*_{l}dx^k\otimes dx^j +\triangle(\nabla X^*) \\ & & -g^{rs}{{(\nabla_rR)_{ks}}^l}_jX^*_ldx^k\otimes dx^j-2{{R_k}^{rl}}_jV_{rl}dx^k\otimes dx^j \\ -& & -\left(Ric_k^lV_{lj}+Ric_j^lV_{kl}\right)dx^k\otimes +& & -\left(\mathit{Ric}_k^lV_{lj}+\mathit{Ric}_j^lV_{kl}\right)dx^k\otimes dx^j. \end{aligned} $$ +The last thing we need to see in order to complete the proof is that + +$$ +-g^{rs}{{(\nabla_rR)_{ks}}^l}_j-\nabla^{l}\mathit{Ric}_{kj}+\nabla_{j}\mathit{Ric}_k^l=0. $$ -The last thing we need to see in order to complete the proof is that -$$-g^{rs}{{(\nabla_rR)_{ks}}^l}_j-\nabla^{l}Ric_{kj}+\nabla_{j}Ric_k^l=0.$$ -This is obtained by contracting $g^{rs}$ against the Bianchi +This is obtained by contracting $g^{rs}$ against the Bianchi identity -$$\nabla_r{{R_{ks}}^l}_j+\nabla^lR_{ksjr}+\nabla_j{R_{ksr}}^l=0.$$ + +$$ +\nabla_r{{R_{ks}}^l}_j+\nabla^lR_{ksjr}+\nabla_j{R_{ksr}}^l=0. +$$ Let $h_{ij}$ be defined by $h_{ij}= V_{ij}+V_{ji}$. It follows from -() that +(12.3) that $$ \frac{\partial}{\partial t}h_{ij}=\Delta_{L}h_{ij}, $$ where by definition $\Delta_{L}h_{ij}=\Delta -h_{ij}+2{{R_i}^{kl}j}h_{kl}-Ric_i^k h_{kj}-Ric_j^kh_{ki}$ is the **Lichnerowicz +h_{ij}+2{{R_i}^{kl}j}h_{kl}-\mathit{Ric}_i^k h_{kj}-\mathit{Ric}_j^kh_{ki}$ is the **Lichnerowicz Laplacian**. A simple calculation shows that there is a constant $C>0$ such that -$$ - $$ \begin{aligned} \left( \frac{\partial}{\partial t}-\Delta\right) \left\vert h_{ij} @@ -741,27 +796,25 @@ $$ \end{aligned} $$ -$$ - Note that $X(t)$ is a Killing vector field for $g(t)$ if and only if -$h_{ij}(t)=0$. Since Equation~() is linear and since the +$h_{ij}(t)=0$. Since Equation (12.1) is linear and since the curvature is bounded on each time-slice, for any given bounded Killing vector field $X(0)$ for metric $g(0)$, there is a bounded -solution $X^{i}\left( t\right) $ of Equation~() for -$t\in\lbrack0,T]$. Then $\left \vert h_{ij}\left( t\right) \right -\vert^2 $ is a bounded function satisfying () and $\left +solution $X^{i}\left( t\right) $ of Equation (12.1) for +$t\in[0,T]$. Then $\left \vert h_{ij}\left( t\right) \right +\vert^2 $ is a bounded function satisfying (12.6) and $\left \vert h_{ij} \right \vert^2 (0)=0$. One can apply the maximum -principle to () to conclude that $h_{ij}(t)=0$ for all +principle to (12.6) to conclude that $h_{ij}(t)=0$ for all $t\geq 0$. This is done as follows: Let $h(t)$ denote the maximum of $|h_{ij}(x,t)|^2$ on the $t$ time-slice. Note that, for any fixed $t$ the function $|h_{ij}(x,t)|^2$ approaches $0$ as $x$ tends to -infinity since the metric is asymptotic at infinity to the product +infinity since the metric is asymptotic at infinity to the product of a round metric on $S^2$ and the standard metric on the line. By -virtue of () and Proposition~, the +virtue of (12.6) and \entryref{37d7d059a526}, the function $h(t)$ satisfies $dh/dt\le Ch$ in the sense of forward difference quotients, so that $d(e^{-Ct}h)/dt\le 0$, also in the sense of forward difference quotients. Thus, by -Corollary~, since $h(0)=0$ and $h\ge 0$, it follows +\entryref{67c378f87457}, since $h(0)=0$ and $h\ge 0$, it follows that $e^{-Ct}h(t)=0$ for all $t\ge 0$, and consequently, $h(t)=0$ for all $t\ge 0$. @@ -770,7 +823,7 @@ for $g(t)$ for all $t\in [0,T)$. The following is a very nice observation of Bennett Chow; we thank him for allowing us to use it here. From $h_{ij}=0$ we have $\nabla_{j} X^i +\nabla_{i} X^{j}=0$. Taking the $\nabla_{j}$ derivative and summing over $j$ we get -$\Delta X^i +R^i_kX^k =0$ for all $t$. Hence () gives +$\Delta X^i +R^i_kX^k =0$ for all $t$. Hence (12.2) gives $\frac{\partial}{\partial t}X^{i}=0$ and $X(t)=X(0)$, i.e., the Killing vector fields are stationary and remain Killing vector fields for the entire flow $g(t)$. @@ -778,53 +831,76 @@ fields for the entire flow $g(t)$. consists of Killing vector fields, the same is true for all the metrics $g(t)$ in the standard solution. Thus, the -rotation group $SO(3)$ of $\mathbb{R}^3$ is contained in the +rotation group $SO(3)$ of $\mathbb{R}^3$ is contained in the isometry group of $g(t)$ for every $t\in [0,T)$. We have shown: + **Corollary.** The standard solution $g(t),\ t\in [0,T)$, consists of a family of metrics all of which are rotationally symmetric by the standard action of $SO(3)$ on $\mathbb{R}^3$. + ### Non-collapsing -**Proposition.** -For any $r>0$ sufficiently small, there is a $\kappa>0$ such that + +**Proposition.** For any $r>0$ sufficiently small, there is a $\kappa>0$ such that the standard flow is $\kappa$-non-collapsed on all scales $\le r$. + **Proof.** Since the curvature of the standard solution is non-negative, it -follows directly that $2|Ric|^2\le R^2$. By -Equation~() this gives -$$\frac{\partial R}{\partial t}=\triangle R+2|Ric|^2\le \triangle -R+R^2.$$ Let $C=max(2,max_{x\in \Ar^3}R(x,0))$. Suppose +follows directly that $2|\mathit{Ric}|^2\le R^2$. By +Equation (3.5) this gives + +$$ +\frac{\partial R}{\partial t}=\triangle R+2|\mathit{Ric}|^2\le \triangle +R+R^2. +$$ + + Let $C=\mathit{max}(2,\mathit{max}_{x\in \Ar^3}R(x,0))$. Suppose that $t_0< T$ and $t_0<1/C$. + **Claim.** For all $x\in \Ar^3$ and $t\in [0,t_0]$ we have -$$R(x,t)\le \frac{C}{1-Ct}.$$ -\begin{proof} -By the asymptotic condition, there is a compact subset $X\subset +$$ +R(x,t)\le \frac{C}{1-Ct}. +$$ + + +**Proof.** By the asymptotic condition, there is a compact subset $X\subset \Ar^3$ such that for any point $p\in \Ar^3\setminus X$ and for any $t\le t_0$ we have $R(p,t)<2/(1-t)$. Since $C\ge 2$, for all $t$ for -which $sup_{x\in \Ar^3}R(x,t)\le 2/(1-t)$, we also have -$$R(x,t)\le \frac{C}{1-Ct}.$$ +which $\mathit{sup}_{x\in \Ar^3}R(x,t)\le 2/(1-t)$, we also have + +$$ +R(x,t)\le \frac{C}{1-Ct}. +$$ Consider the complementary subset of $t$, that is to say the subset of $[0,t_0]$ for which there is $x\in \Ar ^3$ with $R(x,t)>C/(1-Ct)$. This is an open subset of $[0,t_0]$, and hence is a disjoint union of relatively open intervals. Let $\{t_10$ such that for any metric ball -$B(x,0,r)$ on which $|Rm|\le r^{-2}$ we have $Vol\,B(x,r)\ge V r^3$. Since there is a uniform bound on the +$B(x,0,r)$ on which $|\mathit{Rm}|\le r^{-2}$ we have $\mathit{Vol} B(x,r)\ge V r^3$. Since there is a uniform bound on the curvature on $[0,1/2C]$, it follows that there is $V'>0$ so that any -ball $B(q,t,r)$ with $t\le 1/2C$ on which $|Rm|\le r^{-2}$ -satisfies $Vol\,B(q,t,r)\ge V'r^3$. Set $t_0=1/4C$. For any +ball $B(q,t,r)$ with $t\le 1/2C$ on which $|\mathit{Rm}|\le r^{-2}$ +satisfies $\mathit{Vol} B(q,t,r)\ge V'r^3$. Set $t_0=1/4C$. For any point $x=(p,t)$ with $t\ge 1/2C$ there is a point $(q,t_0)$ such -that $l_x(q,t_0)\le 3/2$; this by Theorem~. Since -$B(q,0,1/\sqrt{R_max(0)})\subset \Ar^3$ has volume at least -$V/R_max(0)^{3/2}$, and clearly $l_x$ is bounded above on -$B(q,0,1/\sqrt{R_max(0)})$ by a uniform constant, we see that -the reduced volume of $B(q,0,1/\sqrt{R_max(0)})$ is uniformly -bounded from below. It now follows from Theorem~ that there -is $\kappa_0>0$ such that if $|Rm|$ is bounded by $r^{-2}$ on +that $l_x(q,t_0)\le 3/2$; this by \entryref{aebc3ff2bd3b}. Since +$B(q,0,1/\sqrt{R_\mathit{max}(0)})\subset \Ar^3$ has volume at least +$V/R_\mathit{max}(0)^{3/2}$, and clearly $l_x$ is bounded above on +$B(q,0,1/\sqrt{R_\mathit{max}(0)})$ by a uniform constant, we see that +the reduced volume of $B(q,0,1/\sqrt{R_\mathit{max}(0)})$ is uniformly +bounded from below. It now follows from \entryref{3defa9b2209c} that there +is $\kappa_0>0$ such that if $|\mathit{Rm}|$ is bounded by $r^{-2}$ on the parabolic neighborhood $P(p,t,r,-r^2)$ and $r\le \sqrt{1/4C}$, then the volume of this neighborhood is at least $\kappa_0r^3$. Putting all this together we see that there is a universal $\kappa>0$ such that the standard solution is $\kappa$-non-collapsed on all scales at most $\sqrt{1/4C}$. -\end{proof} ## Uniqueness @@ -860,9 +935,9 @@ Now we turn to the proof of uniqueness. The idea is to mimic the proof of uniqueness in the compact case, by replacing the Ricci flow by a strictly parabolic flow. The material we present here is closely related to and derived from the presentation given in -. The presentation here is the analogy in the context +[LuTian]. The presentation here is the analogy in the context of the standard solution of DeTurck's argument presented in -Section~. +Section 3.3. ### From Ricci flow to Ricci-DeTurck flow @@ -872,22 +947,18 @@ the Ricci flow and let $\psi_{t}\colon M \rightarrow M,\ t \in \left[ t_0,T_{1}\right]$ be a solution of the harmonic map flow -$$ - $$ \begin{aligned} \frac{\partial\psi_{t}}{\partial t} & =\Delta_{g\left( t\right) -,g(t_0) }\psi_{t}, \quad \psi_{t_0} =Id. +,g(t_0) }\psi_{t}, \quad \psi_{t_0} =\mathit{Id}. \end{aligned} $$ -$$ - Here, $\Delta_{g\left( t\right) ,g(t_0) }$ is the Laplacian for maps from the Riemannian manifold $(M,g(t))$ to the Riemannian manifold $(M,g(t_0))$. In local coordinates $(x^i)$ on the domain $M$ and $(y^\alpha)$ on the target $M$, the harmonic map flow -() can be written as +(12.7) can be written as $$ \left( \frac{\partial}{\partial t}-\Delta_{g\left( t\right) @@ -900,7 +971,7 @@ $$ where $\Gamma_{\beta\gamma}^{\alpha}$ are the Christoffel symbols of $g(t_0)$. Suppose $\psi\left(x, t\right)$ is a bounded smooth -solution of~ with $\psi_{t_0}=Id$. Then $\psi\left( +solution of 12.8 with $\psi_{t_0}=\mathit{Id}$. Then $\psi\left( t\right),\ t \in[t_0, T_1]$ are diffeomorphisms when $T_{1}>t_0$ is sufficiently close to $t_0$. For any such $T_1$ and for $t_0 \leq t \leq T_1$, define $\hat{g}\left( t\right) =\left( @@ -908,30 +979,27 @@ sufficiently close to $t_0$. For any such $T_1$ and for $t_0 \leq t satisfies the following equation: $$ - -\frac{\partial}{\partial t}\hat{g}_{ij} =-2\widehat{Ric}_{ij}+\hat{\nabla} _{i}W_{j}(t)+\hat{\nabla}_{j}W_{i}(t) \quad +\frac{\partial}{\partial t}\hat{g}_{ij} =-2\widehat{\mathit{Ric}}_{ij}+\hat{\nabla} _{i}W_{j}(t)+\hat{\nabla}_{j}W_{i}(t) \quad \quad \hat{h}\left( 0\right) =h(0), $$ -where $\widehat{Ric}_{ij}$ and $\hat{\nabla}_{i}$ are the +where $\widehat{\mathit{Ric}}_{ij}$ and $\hat{\nabla}_{i}$ are the Ricci curvature and Levi-Civita connection of $\hat{g}(t)$ respectively and $W(t)$ is the time-dependent $1$-form defined by $$ -W(t) _{j}=\hat{g}_{jk}(t)\hat{g}\,^{pq}(t)\left( +W(t) _{j}=\hat{g}_{jk}(t)\hat{g} ^{pq}(t)\left( \hat{\Gamma}_{pq}^{k}(t)-\Gamma_{pq}^{k}(t_0) \right). $$ Here, $\hat{\Gamma}_{pq}^{k}(t)$ denotes the Christoffel symbols of the metric $\hat g(t)$ and $\Gamma_{pq}^k(t_0)$ denotes the Christoffel symbols of the metric $g(t_0)$. (See, for example, -( Lemma 2.1).) +([Shi1] Lemma 2.1).) We call a solution to this flow equation a - **Ricci-DeTurck flow** (see , or + **Ricci-DeTurck flow** (see [DeTurck], or [ChowKnopf] Chapter 3 for details). In local coordinates we have -$$ - $$ \begin{aligned} & \frac{\partial\hat{g}_{ij}}{\partial t} =\hat{g}^{kl}\nabla @@ -951,50 +1019,75 @@ j}\hat{g}_{pk}\nabla_{l}\hat{g}_{iq}-2\nabla_{ i}\hat \end{aligned} $$ -$$ - where $\nabla$ is the Levi-Civita connection of $g(t_0)$. This is a strictly parabolic equation. -**Lemma.** -Suppose that $g(t)$ solves the Ricci flow equation and suppose that + +**Lemma.** Suppose that $g(t)$ solves the Ricci flow equation and suppose that $\psi_t$ solves the harmonic map flow equation, -Equation~(); then $\hat g(t)=(\psi_t^{-1})^*g(t)$ solves -the Ricci-DeTurck flow, Equation~() and $\psi_t$ +Equation (12.7); then $\hat g(t)=(\psi_t^{-1})^*g(t)$ solves +the Ricci-DeTurck flow, Equation (12.9) and $\psi_t$ satisfies the following ODE: -$$\frac{\partial \psi_t}{\partial t}=-\hat g^{ij}(t)W(t).$$ + +$$ +\frac{\partial \psi_t}{\partial t}=-\hat g^{ij}(t)W(t). +$$ + **Proof.** The first statement follows from the second statement and a standard Lie derivative computation. For the second statement, we need to show -$$\triangle_{g(t),g(0)} \psi^\alpha = -\hat -g^{pq}\left(\hat\Gamma_{pq}^\alpha(t)-\Gamma_{pq}^\alpha(t_0)\right).$$ + +$$ +\triangle_{g(t),g(0)} \psi^\alpha = -\hat +g^{pq}\left(\hat\Gamma_{pq}^\alpha(t)-\Gamma_{pq}^\alpha(t_0)\right). +$$ + Notice that this equation is a tensor equation, so that we can choose coordinates in the domain and range so that $\Gamma(t)$ vanishes at the point $p$ in question and $\Gamma(t_0)$ vanishes at -$\psi_t(p)$. With these assumptions we need to show -$$g^{pq}(t)\frac{\partial^2 \psi^\alpha}{\partial x^p\partial x^q}=-\hat g^{pq}(t)\hat\Gamma_{pq}^\alpha(t).$$ +$\psi_t(p)$. With these assumptions we need to show + +$$ +g^{pq}(t)\frac{\partial^2 \psi^\alpha}{\partial x^p\partial x^q}=-\hat g^{pq}(t)\hat\Gamma_{pq}^\alpha(t). +$$ + This is a direct computation using the change of variables formula relating $\hat \Gamma$ and $\Gamma$. -**Corollary.** -Suppose that $(M,g_1(t)),\ t_0\le t\le T$, and $(M,g_2(t)),\ t_0\le + +**Corollary.** Suppose that $(M,g_1(t)),\ t_0\le t\le T$, and $(M,g_2(t)),\ t_0\le t\le T$, are solutions to the Ricci flow equation for which there are solutions -$$\psi_{1,t}\colon (M,g_1(t))\to (M,g_1(0))$$ and -$$\psi_{2,t}\colon -(M,g_2(t))\to (M,g_2(0))$$ to the harmonic map equation with -$\psi_{1,t_0}=\psi_{2,t_0}=Id$. Let $\hat + +$$ +\psi_{1,t}\colon (M,g_1(t))\to (M,g_1(0)) +$$ + + and + +$$ +\psi_{2,t}\colon +(M,g_2(t))\to (M,g_2(0)) +$$ + + to the harmonic map equation with +$\psi_{1,t_0}=\psi_{2,t_0}=\mathit{Id}$. Let $\hat g_1(t)=(\psi_{1,t}^{-1})^*g_1(t)$ and $\hat g_2(t)=(\psi_{2,t}^{-1})^*g_2(t)$ be the corresponding solutions to the Ricci-DeTurck flow. Suppose that $\hat g_1(t)=\hat g_2(t)$ for all $t\in [t_0,T]$. Then $g_1(t)=g_2(t)$ for all $t\in [t_0,T]$. + **Proof.** Since $\psi_{a,t}$ satisfies the equation -$$\frac{\partial \psi_{a,t}}{\partial t}=-\hat g_a^{ij}W(t)_j$$ + +$$ +\frac{\partial \psi_{a,t}}{\partial t}=-\hat g_a^{ij}W(t)_j +$$ + where the time-dependent vector field $W(t)$ depends only on $\hat g_a$, we see that $\psi_{1,t}$ and $\psi_{2,t}$ both solve the same -time-dependent ODE and since $\psi_{1,t_0}=\psi_{2,t_0}=Id$, +time-dependent ODE and since $\psi_{1,t_0}=\psi_{2,t_0}=\mathit{Id}$, it follows that $\psi_{1,t}=\psi_{2,t}$ for all $t\in [t_0,T]$. On the other hand, $g_a(t)=\psi_{a,t}^*\hat g_a(t)$, so that it follows that $g_1(t)=g_2(t)$ for $t\in [t_0,T]$. @@ -1017,7 +1110,7 @@ with appropriate control at infinity and given initial condition $g_1(t_0)=g_2(t_0)$. This implies that the two Ricci-DeTurck flows we have constructed are in fact equal. Invoking the above corollary, we conclude that $g_1(t)$ and $g_2(t)$ agree on a longer interval -extending past $t_0$. From this it follows easily that $g_1(t)$ and +extending past $t_0$. From this it follows easily that $g_1(t)$ and $g_2(t)$ agree on their common domain of definition. Hence, if they are both maximal flows, they must be equal. @@ -1027,7 +1120,7 @@ In order to pass from a solution to the Ricci flow equation to a solution of the Ricci-DeTurck flow we must prove the existence of a solution of the harmonic map flow associated with the Ricci flow. In this section we study the -existence of the harmonic flow () and its asymptotic +existence of the harmonic flow (12.7) and its asymptotic behavior at the space infinity when $h(t)=g(t)$ is a standard solution. Here we use in an essential way the rotationally symmetric property and asymptotic property at infinity of $g(t)$. In this @@ -1041,26 +1134,23 @@ on the sphere. We denote by $\hat r$ the standard radial coordinate in $\Ar^n$. Since $g(t)$ is rotationally symmetric and $n \geq 3$, we can write -$$ - $$ \begin{aligned} g(t)=dr^2+f(r,t)^2 d \sigma \end{aligned} $$ -$$ - Here $r=r(\hat r,t)$ is the (time-dependent) radial coordinate on $\mathbb{R}^n$ for the metric $g(t)$. -**Claim.** -For any fixed $t$ the function $r\colon \Ar^n \to[0,\infty)$ is a + +**Claim.** For any fixed $t$ the function $r\colon \Ar^n \to[0,\infty)$ is a function only of $\hat r$. Considered as a function of two variables, $r(\hat r,t)$ is a smooth function defined for $\hat r\ge 0$. It is an odd function of $\hat r$. For fixed $t$ it is an increasing function of $\hat r$. + **Proof.** Write the metric $g(t)=g_{ij}dx^idx^j$ and let @@ -1096,47 +1186,60 @@ that $r(\hat r,t)$ is an odd function. It is also clear from this formula that $\partial r/\partial \hat r>0$. Since, for each $t_0$, the function $r(\hat -r,t_0)$ is an increasing function of $\hat r$, it can be inverted -to give a function $\hat r(r,t_0)$. In Equation~(), we +r,t_0)$ is an increasing function of $\hat r$, it can be inverted +to give a function $\hat r(r,t_0)$. In Equation (12.11), we have chosen to write $f$ as a function of $r$ and $t$, rather than a function of $\hat r$ and $t$. We look for rotationally symmetric -solutions to Equation~(), i.e., solutions of the form: - -$$ +solutions to Equation (12.7), i.e., solutions of the form: $$ \begin{aligned} - \psi(t)\colon \mathbb{R}^n \rightarrow \mathbb{R}^n \qquad -\psi(t)(r,\theta) & =(\rho(r,t),\theta) \ \ \ for\ \ t\ge t_0 \\ -\psi(r,t_0) & = Id \nonumber +\psi(t)(r,\theta) & =(\rho(r,t),\theta) \ \ \ \mathit{for\ \} t\ge t_0 \\ +\psi(r,t_0) & = \mathit{Id} \nonumber \end{aligned} $$ -$$ - We shall adopt the following conventions: we shall consider functions $f(w,t)$ defined in the closed half-plane $w\ge 0$. When we say that such a function is *smooth* we mean that for each $n,m\ge 0$ we have a continuous function $f_{nm}(w,t)$ defined for all $w\ge 0$ with the following properties: -\begin{enumerate} -\item[(1)] $$f_{00}=f$$ -\item[(2)] $$\frac{\partial f_{nm}}{\partial t}=f_{n(m+1)}$$ -\item[(3)] $$\frac{\partial f_{nm}}{\partial w}=f_{(n+1)m},$$ -\end{enumerate} + +- **(1)** + +$$ +f_{00}=f +$$ + +- **(2)** + +$$ +\frac{\partial f_{nm}}{\partial t}=f_{n(m+1)} +$$ + +- **(3)** + +$$ +\frac{\partial f_{nm}}{\partial w}=f_{(n+1)m}, +$$ + where in Item (3) the partial derivative along the boundary $w=0$ is a right-handed derivative only. We say such a function is *even* if $f_{(2k+1)m}(0,t)=0$ for all $k\ge 0$. We have the following elementary lemma: + **Lemma.** (a) Suppose that $f(w,t)$ is a smooth function defined for $w\ge 0$. Define $\phi(r,t)=f(r^2,t)$. Then $\phi(r,t)$ is a smooth function defined for all $r\in \Ar$. Now fix $k$ and let $\hat r\colon -\Ar^k\to [0,\infty)$ be the usual radial coordinate. Then we have a +\Ar^k\to [0,\infty)$ be the usual radial coordinate. Then we have a smooth family of smooth functions on $\Ar^k$ defined by -$$\hat\phi(x^1,\ldots,x^k,t)=\phi(\hat r(x^1,\ldots,x^k),t)=f(\sum_{i=1}^k(x^i)^2,t).$$ + +$$ +\hat\phi(x^1,\ldots,x^k,t)=\phi(\hat r(x^1,\ldots,x^k),t)=f(\sum_{i=1}^k(x^i)^2,t). +$$ (b) If $\psi(r,t)$ is a smooth function defined for $r\ge 0$ and if it is even in the sense that its Taylor expansion to all @@ -1145,14 +1248,23 @@ there is a smooth function $f(w,t)$ defined for $w\ge 0$ such that $\psi(r,t)=f(r^2,t)$. In particular, for any $k\ge 2$ the function $\hat\psi((x^1,\ldots,x^k),t)=\psi(r(x^1,\ldots,x^k),t)$ is a smooth family of smooth functions on $\Ar^k$. + **Proof.** Item (a) is obvious, and Item (b) is obvious away from $r=0$. We establish Item (b) along the line $r=0$. Consider the Taylor theorem with remainder to order $2N$ in the $r$-direction for $\psi(r,t)$ at a point $(0,t)$. By hypothesis it takes the form -$$\sum_{i=0}c_i(t)w^{2i}+w^{2N+1}R(w,t).$$ + +$$ +\sum_{i=0}c_i(t)w^{2i}+w^{2N+1}R(w,t). +$$ + Now we replace $w$ by $\sqrt{r}$ to obtain -$$f(r,t)=\sum_{i=0}c_ir^i+\sqrt{r}^{2N+1}R(\sqrt{r},t).$$ + +$$ +f(r,t)=\sum_{i=0}c_ir^i+\sqrt{r}^{2N+1}R(\sqrt{r},t). +$$ + Applying the usual chain rule and taking limits as $r\rightarrow 0^+$ we see that $f(r,t)$ is $N$ times differentiable along the line $r=0$. Since this is true for every $N<\infty$, the result follows. @@ -1170,7 +1282,7 @@ $r$-direction along the line $r=0$ involves only odd powers of $r$. These do not define smooth functions on $\Ar^n$. On the other hand, by the same argument as above with the Taylor expansion one sees that they can be written as $rg(r,t)$ where $g$ is even, and hence -define smoothly varying families of smooth functions on $\Ar^n$. +define smoothly varying families of smooth functions on $\Ar^n$. Notice also that the product of two odd functions $f_1(r,t)f_2(r,t)$ is an even function and hence this product defines a smoothly varying family of smooth function on $\Ar^n$. @@ -1187,47 +1299,59 @@ coordinate on $\Ar^n$ by $\hat r$ to distinguish it from the varying radial function $r=r(t)$ that measures the distance from the tip in the metric $g(t)$.) -As a corollary of Claim~ we have: +As a corollary of \entryref{19cc95ff1cb9} we have: -**Corollary.** -$r^2(\hat r,t)$ is a smoothly varying family of smooth functions on -$\Ar^n$. Also, $\hat r$ is a smooth function of $(r,t)$ defined for + +**Corollary.** $r^2(\hat r,t)$ is a smoothly varying family of smooth functions on +$\Ar^n$. Also, $\hat r$ is a smooth function of $(r,t)$ defined for $r\ge 0$ and odd in $r$. In particular, any smooth even function of $r$ is a smooth even function of $\hat r$ and thus defines a smooth function on $\Ar^n$. Moreover, there is a smooth function $\xi(w,t)$ -such that $d(log\, r)/dt=r^{-1}(dr/dt)=\xi(r^2,t)$. +such that $d(\mathit{log} r)/dt=r^{-1}(dr/dt)=\xi(r^2,t)$. + For future reference we define $$ - B(w,t)=\frac{1}{2}\int_0^w\xi(u,t)du. $$ + Then $B(r^2,t)$ is a smooth function even in $r$ and hence, as $t$ varies, defines a smoothly varying family of smooth functions on $\Ar^n$. Notice that -$$\frac{\partial B(r^2,t)}{\partial r}=2r\frac{\partial B}{\partial -w}(w,t)\left|_{w=r^2}\right.=2r\left(\frac{1}{2}\xi(r^2,t)\right)=\frac{dr}{dt}.$$ + +$$ +\frac{\partial B(r^2,t)}{\partial r}=2r\frac{\partial B}{\partial +w}(w,t)\left|_{w=r^2}\right.=2r\left(\frac{1}{2}\xi(r^2,t)\right)=\frac{dr}{dt}. +$$ Now let us consider $f(r,t)$. + **Claim.** $f(r,t)$ is a smooth function defined for $r\ge 0$. It is an odd function of $r$. + **Proof.** We have -$$f(r,t)=\hat r(r,t)\sqrt{g_{22}(\hat r(r,t),0,\ldots,0,t)}.$$ + +$$ +f(r,t)=\hat r(r,t)\sqrt{g_{22}(\hat r(r,t),0,\ldots,0,t)}. +$$ + Since $\sqrt{g_{22}(\hat r,0,\ldots,0,t)}$ is a smooth function of -$(\hat r,t)$ defined for $\hat r\ge 0$ and since it is an even +$(\hat r,t)$ defined for $\hat r\ge 0$ and since it is an even function of $\hat r$, it follows immediately from the fact that $\hat r$ is a smooth odd function of $r$, that $f(r,t)$ is a smooth odd function of $r$. + **Corollary.** There is a smooth function $h(w,t)$ defined for $w\ge 0$ so that $f(r,t)=rh(r^2,t)$. In particular, $h(r^2,t)$ defines a smooth function on all of $\Ar^n$. Clearly, $h(w,t)>0$ for all $w\ge 0$ and all $t$. + -We set $\tilde h(w,t)=log(h(w,t))$, so that $f(r,t)=re^{\tilde +We set $\tilde h(w,t)=\mathit{log}(h(w,t))$, so that $f(r,t)=re^{\tilde h(r^2,t)}$. Notice that $\tilde h(r^2,t)$ defines a smooth function of $\hat r^2$ and $t$ and hence is a smoothly varying family of smooth functions on $\Ar^n$. @@ -1236,13 +1360,11 @@ smooth functions on $\Ar^n$. . Let $\psi(t)\colon \Ar^n\to \Ar^n$ be a smoothly varying family of -smooth functions as given in Equation~(). Using -() and () it is easy to calculate the energy +smooth functions as given in Equation (12.13). Using +(12.11) and (12.13) it is easy to calculate the energy functional using spherical coordinates with $r$ as the radial coordinate. -$$ - $$ \begin{aligned} E(\psi(t)) & = \frac{1}{2} \int_{\mathbb{R}^n} \left \vert \nabla @@ -1254,14 +1376,10 @@ E(\psi(t)) & = \frac{1}{2} \int_{\mathbb{R}^n} \left \vert \nabla \end{aligned} $$ -$$ - If we have a compactly supported variation $\delta \rho =w$, then -letting $dvol_\sigma$ denote the standard volume element on +letting $d\mathit{vol}_\sigma$ denote the standard volume element on $S^{n-1}$, we have -$$ - $$ \begin{aligned} & \delta E(\psi(t))(w) = \frac{1}{2} \int_{\mathbb{R}^n} \left[2 @@ -1271,11 +1389,11 @@ $$ &= \int_0^{+\infty} \left[ f^{n-1}(r,t)\frac{\partial \rho}{\partial r} \frac{\partial w}{\partial r} +(n-1) f(\rho,t_0) \frac{\partial f(\rho,t_0)}{\partial -\rho}f^{n-3}(r,t)w \right] dr \cdot \int_{S^{n-1}} dvol_\sigma \\ +\rho}f^{n-3}(r,t)w \right] dr \cdot \int_{S^{n-1}} d\mathit{vol}_\sigma \\ & = \int_0^{+\infty} \left[-\frac{\partial}{\partial r}\left(f^{n-1}\frac{\partial\rho}{\partial r}\right)w+(n-1) f(\rho,t_0) \frac{\partial f(\rho,t_0)}{\partial \rho}f^{n-3}(r,t)w -\right] dr \cdot \int_{S^{n-1}} dvol_\sigma\\ &= +\right] dr \cdot \int_{S^{n-1}} d\mathit{vol}_\sigma\\ &= \int_{\mathbb{R}^n}\left[ - f^{1-n} \frac{\partial }{\partial r} \left( \frac{\partial \rho}{\partial r} f^{n-1}\right) +(n-1) f(\rho,t_0) \frac{\partial f(\rho,t_0)}{\partial \rho}f^{-2}(r,t) @@ -1283,26 +1401,32 @@ f(\rho,t_0) \frac{\partial f(\rho,t_0)}{\partial \rho}f^{-2}(r,t) \end{aligned} $$ -$$ - The usual argument shows that for a compactly supported variation $w$ we have -$$\delta_w\left(\frac{1}{2}\int_{\Ar^n}|\nabla_{g(t),g(t_0)}\psi|^2dvol=\int_{\Ar^n}\langle -w,-\triangle_{g(t),g(t_0)}\psi\rangle dvol\right).$$ Thus, -$$\triangle_{g(t),g(t_0)}\psi= \left[ + +$$ +\delta_w\left(\frac{1}{2}\int_{\Ar^n}|\nabla_{g(t),g(t_0)}\psi|^2d\mathit{vol}=\int_{\Ar^n}\langle +w,-\triangle_{g(t),g(t_0)}\psi\rangle d\mathit{vol}\right). +$$ + + Thus, + +$$ +\triangle_{g(t),g(t_0)}\psi= \left[ f^{1-n}\frac{\partial }{\partial r} \left( \frac{\partial \rho}{\partial r} f^{n-1}\right) -(n-1) f(\rho,t_0) \frac{\partial f(\rho,t_0)}{\partial \rho}f^{-2}(r,t) \right] -\frac{\partial}{\partial r}$$ where we have written this expression +\frac{\partial}{\partial r} +$$ + + where we have written this expression using the coordinates $(r,\theta)$ on the range $\Ar ^n$ (rather than the fixed coordinates $(\hat r,\theta)$). Now let us compute $\partial \psi/\partial t(\hat r,t)$ in these same coordinates. (We use $\hat r$ for the coordnates for $\psi$ in the domain to emphasize that this must be the time derivative at a -fixed point in the underlying space.) Of course, by the chain rule, - -$$ +fixed point in the underlying space.) Of course, by the chain rule, $$ \begin{aligned} @@ -1315,10 +1439,8 @@ t}. \end{aligned} $$ -$$ - -Consequently, for rotationally symmetric maps as in -Equation~() the harmonic map flow equation () +Consequently, for rotationally symmetric maps as in +Equation (12.13) the harmonic map flow equation (12.7) has the following form: $$ @@ -1332,7 +1454,6 @@ $$ or equivalently $$ - \frac{\partial \rho}{\partial t}=\frac{1}{f^{n-1}(r,t)} \frac{\partial }{\partial r} \left( f^{n-1}(r,t) \frac{\partial \rho}{\partial r} \right)-(n-1)f^{-2}(r,t) f(\rho,t_0) @@ -1340,35 +1461,31 @@ $$ \rho}{\partial r}\frac{\partial r}{\partial t} $$ -The point of rewriting the harmonic map equation in this way is to +The point of rewriting the harmonic map equation in this way is to find an equation for the functions $\rho(r,t), f(r,t)$ defined on $r\ge 0$. Even though the terms in this rewritten equation involve odd functions of $r$, as we shall see, solutions to these equations will be even in $r$ and hence will produce a smooth solution to the harmonic map flow equation on $\Ar ^n$. -### An equation equivalent to the harmonic map flow -equation +### An equation equivalent to the harmonic map flow equation -We will solve () for solutions of the form +We will solve (12.15) for solutions of the form $$ \rho(r,t)= re^{\tilde{\rho}(r,t)}, \ \ \ t\ge t_0;\ \ \ \tilde \rho(r,t_0)=0. $$ -For $\psi$ as in Equation~() to define a diffeomorphism, +For $\psi$ as in Equation (12.13) to define a diffeomorphism, it must be the case that $\rho(r,t)$ is a smooth function for $r\ge 0$ which is odd in $r$. It follows from the above expression that $\tilde\rho(r,t)$ is a smooth function of $r$ and $t$ defined for $r\ge 0$ and even in $r$, so that it defines a smoothly varying family of smooth functions on $\Ar^n$. Then some straightforward -calculation shows that () becomes - -$$ +calculation shows that (12.15) becomes $$ \begin{aligned} - \frac{\partial\tilde{\rho} }{\partial t} & = & \frac{\partial^{2}\tilde{\rho}}{\partial r^{2}}+\frac{n+1}{r}\frac {\partial\tilde{\rho}}{\partial r} @@ -1387,11 +1504,10 @@ t} \frac{\partial\tilde{\rho} }{\partial r} .\nonumber \end{aligned} $$ -$$ - Note that from the definition, $\tilde{h}\left( 0,t \right)=0$, we can write $\tilde{h}\left( w,t \right)=w \tilde{h}^*(w,t)$ where $\tilde{h}^*(w,t)$ is a smooth function of $w\ge 0$ and $t$. So + $$ \frac{n-1}{r^{2}}\left[ 1-e^{2\tilde{h}\left( \rho^{2},t_0\right) -2\tilde{h}\left( r^{2},t\right) }\right] =\frac{n-1}{r^{2}}\left[ @@ -1399,12 +1515,11 @@ $$ \rho^{2},t_0\right) -\tilde{h}^*\left( r^{2},t\right) \right] }\right] $$ + which is a smooth function of $\tilde{\rho},r^2,t$. Let -$$ - $$ \begin{aligned} G(\tilde{\rho},w,t) = & \frac{n-1}{w}\left[ 1-e^{2\tilde{h}\left( @@ -1417,18 +1532,14 @@ h}{\partial w}(\rho^2,t_0) -2\xi(w,t), \end{aligned} $$ -$$ - -where $\xi$ is the function from Corollary~. Then +where $\xi$ is the function from \entryref{e353993489e9}. Then $G(\tilde{\rho},w,t)$ is a smooth function defined for $w\ge 0$. Notice that when $r$ and $\tilde \rho$ are the functions associated with the varying family of metrics $g(t)$ and the solutions to the harmonic map flow, then $G(\tilde\rho,r^2,t)$ defines a smoothly varying family of smooth functions on $\Ar^n$. -We have the following form of equation (): - -$$ +We have the following form of equation (12.16): $$ \begin{aligned} @@ -1442,32 +1553,26 @@ r} +\left( \frac{\partial\tilde{\rho} }{\partial r}\right) ^{2} \end{aligned} $$ -$$ - Now we think of $\tilde{\rho}$ as a rotationally symmetric function defined on $\mathbb{R}^{n+2}$ and let $\widehat G(\tilde\rho,(x^1,\ldots,x^{n+2}),t)= G(\tilde\rho,\sum_{i=1}^{n+2}(x^i)^2,t)$ and then the above equation can be written as -$$ - $$ \begin{aligned} \frac{\partial\tilde{\rho} }{\partial t} = \Delta \tilde{\rho} + \nabla [ (n-1)\tilde{h} -B] \cdot \nabla\tilde{\rho} + \left | \nabla \tilde{\rho} \right |^2 +G(\tilde{\rho},x,t) - \end{aligned} $$ -$$ - where $\nabla$ and $\Delta$ are the Levi-Civita connection and -Laplacian defined by the Euclidean metric on $\mathbb{R}^{n+2}$ +Laplacian defined by the Euclidean metric on $\mathbb{R}^{n+2}$ respectively and where $B$ is the function defined in -Equation~(). +Equation (12.14). + **Remark.** The whole purpose of this rewriting of the PDE for $\tilde \rho$ is to present this equation in such a form that all its coefficients represent smooth functions of $\hat r$ and $t$ that are even in @@ -1475,30 +1580,30 @@ $\hat r$ and hence define smooth functions on Euclidean space of any dimension. We have chosen to work on $\Ar^{n+2}$ because the expression for the Laplacian in this dimension has the term $((n+1)/r)\partial \tilde \rho/\partial r$. + It is important to understand the asymptotic behavior of our functions at spatial infinity. -**Claim.** -For any fixed $t$ we have the following asymptotic expansions at + +**Claim.** For any fixed $t$ we have the following asymptotic expansions at spatial infinity. -\begin{enumerate} -\item[(1)] - $e^{\tilde{h}(r^2,t)}$ is asymptotic to $\frac{1}{(1-t)r}$. -\item[(2)] $\tilde{h}(r^2,t)$ is asymptotic to $-\log r$. -\item[(3)] $\frac{\partial \tilde{h}}{\partial w}(r^2,t)$ is asymptotic to + +- **(1)** $e^{\tilde{h}(r^2,t)}$ is asymptotic to $\frac{1}{(1-t)r}$. +- **(2)** $\tilde{h}(r^2,t)$ is asymptotic to $-\log r$. +- **(3)** $\frac{\partial \tilde{h}}{\partial w}(r^2,t)$ is asymptotic to $-\frac{1}{2r^2}$. -\item[(4)] $r^{-1} \frac{\partial r}{\partial t}$ is asymptotic to +- **(4)** $r^{-1} \frac{\partial r}{\partial t}$ is asymptotic to $\frac{C}{r}$. -\item[(5)] $\frac{\partial B(r^2,t)}{\partial r}$ is asymptotic to $C$. -\item[(6)] $|G( \tilde{\rho},r^2,t)| \leq C_*< \infty$ where +- **(5)** $\frac{\partial B(r^2,t)}{\partial r}$ is asymptotic to $C$. +- **(6)** $|G( \tilde{\rho},r^2,t)| \leq C_*< \infty$ where $C_* =C_*\left(\sup\{|\tilde{\rho}|,\tilde{h}\}\right)$ is a constant depending only on $\sup\{|\tilde{\rho}|,\tilde{h}\}$. -\end{enumerate} + -**Proof.** The first item is immediate from Proposition~. The +**Proof.** The first item is immediate from \entryref{77ca4669ffb3}. The second and third follow immediately from the first. The fourth is a -consequence of the fact that by Proposition~ $dr/dt$ is +consequence of the fact that by \entryref{77ca4669ffb3} $dr/dt$ is asymptotic to a constant at infinity on each time-slice. The fifth follows immediately from the fourth and the definition of $B(r^2,t)$. Given all these asymptotic expressions, the last is @@ -1510,12 +1615,14 @@ and $t$. The purpose of this subsection is to prove the following short-time existence theorem for the harmonic map flow equation. + **Proposition.** For any $t_0\ge 0$ for which there is a standard solution $g(t)$ defined on $[0,T_1]$ with $t_0t_0$ and a solution -to Equation~() with initial condition -$\tilde{\rho}(r,t_0)=0$ defined on the time-interval $[t_0,T]$. +to Equation (12.18) with initial condition +$\tilde{\rho}(r,t_0)=0$ defined on the time-interval $[t_0,T]$. + - At this point to simplify the notation we shift time by $-t_0$ so +At this point to simplify the notation we shift time by $-t_0$ so that our initial time is $0$, though our initial metric is not $g_0$ but rather is the time $t_0$-slice of the standard solution we are considering, so that now $t_0=0$ and our initial condition is @@ -1528,11 +1635,13 @@ $$ H(x,y,t)=\frac{1}{(4 \pi t)^{(n+2)/2}} e^{-\frac{|x-y|^2}{4t}} $$ -be the heat kernel. We solve () by successive approximation -. +be the heat kernel. We solve (12.18) by successive approximation +[LiTam]. Define -$$F(x, \tilde{\rho},\nabla\tilde{\rho},t) + +$$ +F(x, \tilde{\rho},\nabla\tilde{\rho},t) = \nabla \left[ (n-1)\tilde{h}-B \right] \cdot \nabla\tilde{\rho} + \left | \nabla \tilde{\rho} \right |^2 +G( \tilde{\rho},x,t) $$ @@ -1540,22 +1649,15 @@ $$ Let $\tilde{\rho}_0(x,t)=0$ and for $i \geq 1$ we define $\tilde{\rho}_i$ by -$$ - $$ \begin{aligned} \tilde{\rho}_i=\int_0^t \int_{\mathbb{R}^{n+2}} H(x,y,t-s)F(y,\tilde{\rho}_{i-1}, \nabla\tilde{\rho}_{i-1} ,t)dyds - \end{aligned} $$ -$$ - which solves -$$ - $$ \begin{aligned} \frac{\partial\tilde{\rho}_i}{\partial t} = \Delta \tilde{\rho}_i + @@ -1564,29 +1666,25 @@ F(x, \tilde{\rho}_{i-1},\nabla\tilde{\rho}_{i-1},t) \qquad \end{aligned} $$ -$$ - To show the existence of $\tilde{\rho}_i$ by induction, it suffices to prove the following statement: For any $i \geq 1$, if $|\tilde{\rho}_{i-1}| , |\nabla \tilde{\rho}_{i-1}| $ are bounded, then $\tilde{\rho}_i$ exists and $|\tilde{\rho}_{i}|, |\nabla -\tilde{\rho}_{i}|$ are bounded. Assume $|\tilde{\rho}_{i-1}| \leq +\tilde{\rho}_{i}|$ are bounded. Assume $|\tilde{\rho}_{i-1}| \leq C_1, |\nabla \tilde{\rho}_{i-1}| \leq C_2$ are bounded on $\mathbb{R}^{n+2} \times [0,T]$; then it follows from -Claim~ that $G(\tilde{\rho}_{i-1},**x**,t)$ is bounded +\entryref{ac4ac4fb06f8} that $G(\tilde{\rho}_{i-1},\mathbf{x},t)$ is bounded on $\mathbb{R}^{n+2} \times [0,T]$ $$ |G(\tilde{\rho}_{i-1},x,t)| \leq C_*(C_1,\tilde{h} ), $$ -and also because of Claim~ both $|\nabla B|$ and +and also because of \entryref{ac4ac4fb06f8} both $|\nabla B|$ and $|\nabla \tilde h|$ are bounded on all of $\Ar^{n+2}\times [0,T]$, it follows that $F(x, \tilde{\rho}_{i-1},\nabla\tilde{\rho}_{i-1},t)$ is bounded: -$$ - $$ \begin{aligned} & |F(x, \tilde{\rho}_{i-1},\nabla\tilde{\rho}_{i-1},t)| \\ @@ -1595,8 +1693,6 @@ $$ \end{aligned} $$ -$$ - Hence $\tilde{\rho}_i$ exists. The bounds on $|\tilde{\rho}_{i}|$ and $|\nabla \tilde{\rho}_{i}|$ @@ -1609,8 +1705,6 @@ $$ and -$$ - $$ \begin{aligned} & | \nabla \tilde{\rho}_i | = | \int_0^t \int_{\mathbb{R}^{n+2}} @@ -1625,30 +1719,22 @@ e^{-\frac{|x-y|^2}{4{(t-s)}} } \frac{|x-y|}{2{(t-s)}} C_3 dyds \\ \end{aligned} $$ -$$ - Assuming, as we shall, that $T\le \min \{\frac{C_3}{C_1},\frac{\pi C_2^2}{ 4(n+2)^2C_3^2} \}$, then for $0 \leq t \leq T$ we have for all $i$, -$$ - $$ \begin{aligned} -|\tilde{\rho}_{i}| \leq C_1 \qquad and \qquad |\nabla +|\tilde{\rho}_{i}| \leq C_1 \qquad \mathit{and} \qquad |\nabla \tilde{\rho}_{i}| \leq C_2. \end{aligned} $$ -$$ - We prove the convergence of $\tilde{\rho}_i$ to a solution of -() via proving that it is a Cauchy sequence in -$C^1$-norm. Note that $\tilde{\rho}_i- \tilde{\rho}_{i-1}$ +(12.18) via proving that it is a Cauchy sequence in +$C^1$-norm. Note that $\tilde{\rho}_i- \tilde{\rho}_{i-1}$ satisfies -$$ - $$ \begin{aligned} & \frac{\partial (\tilde{\rho}_i- \tilde{\rho}_{i-1})}{\partial t} = @@ -1659,12 +1745,8 @@ $$ \end{aligned} $$ -$$ - where -$$ - $$ \begin{aligned} & F(x, @@ -1673,24 +1755,20 @@ $$ = & [(n-1) \nabla \tilde{h}- \nabla B +\nabla (\tilde{\rho}_{i-1}+ \tilde{\rho}_{i-2})] \cdot \nabla(\tilde{\rho}_{i-1} - \tilde{\rho}_{i-2}) \\ -& + G( \tilde{\rho}_{i-1},**x**,t)-G(\tilde{\rho}_{i-2},**x**,t) +& + G( \tilde{\rho}_{i-1},\mathbf{x},t)-G(\tilde{\rho}_{i-2},\mathbf{x},t) \end{aligned} $$ -$$ - By lengthy but straightforward calculations one can verify the -Lipschitz property of $G(\tilde{\rho},**x**,t)$ +Lipschitz property of $G(\tilde{\rho},\mathbf{x},t)$ $$ -|G(\tilde{\rho}_{i-1},**x**,t)-G(\tilde{\rho}_{i-2},**x**,t)| +|G(\tilde{\rho}_{i-1},\mathbf{x},t)-G(\tilde{\rho}_{i-2},\mathbf{x},t)| \leq C_\&(C_1,C _2, \tilde{f},\tilde{f}_0) \cdot | \tilde{\rho}_{i-1}-\tilde{\rho}_{i-2}|. $$ -This and () implies - -$$ +This and (12.21) implies $$ \begin{aligned} @@ -1700,19 +1778,14 @@ $$ \nonumber \\ \leq & C_4 \cdot | \tilde{\rho}_{i-1}-\tilde{\rho}_{i-2}| +C_5 \cdot |\nabla \tilde{\rho}_{i-1}- \nabla \tilde{\rho}_{i-2} | - \end{aligned} $$ -$$ - where $C_4 = C_\&(C_1,C_2, \tilde{f},\tilde{f}_0)$ and $C_5 = [(n-1) \sup |\nabla \tilde{f}| +\sup |\nabla B|+2C_2]$. Let -$$ - $$ \begin{aligned} & A_i(t)= \sup_{0 \leq s \leq t, x \in \mathbb{R}^{n+2}} @@ -1724,15 +1797,11 @@ $$ \end{aligned} $$ -$$ - -From Equations~() and~() we can estimate +From Equations (12.22) and (12.23) we can estimate $|\tilde{\rho}_i- \tilde{\rho}_{i-1}|$ and $ |\nabla(\tilde{\rho}_i- \tilde{\rho}_{i-1})|$ in the same way as we estimate $|\tilde{\rho}_{i}|$ and $|\nabla\tilde{\rho}_i|$ above; we conclude -$$ - $$ \begin{aligned} & A_i(t)\leq [ C_4 A_{i-1}(t) +C_5 B_{i-1}(t)] \cdot t \\ @@ -1741,8 +1810,6 @@ B_{i-1}(t)]}{\sqrt{\pi}} \cdot \sqrt{t}. \end{aligned} $$ -$$ - Let $C_6 = \max \{C_4,C_5 \}$; then we get $$ @@ -1751,7 +1818,7 @@ A_i(t)+B_i(t) \leq \left( C_6t+ \frac{2(n+2)C_6 \right). $$ -Now suppose that $T\le T_2$ where $T_2$ satisfies $C_6T_2+ +Now suppose that $T\le T_2$ where $T_2$ satisfies $C_6T_2+ \frac{2(n+2)C_6 \sqrt{T_2}}{\sqrt{\pi}} =\frac{1}{2}$; then for all $t\le T$ we have @@ -1766,9 +1833,7 @@ $C^1(\mathbb{R}^{n+2})$. Let $\lim_{i \rightarrow +\infty} \tilde{\rho}_{i} \rightarrow \nabla \tilde{\rho}_{\infty}$ and $F(x,\tilde{\rho}_{i-1},\nabla \tilde{\rho}_{i-1},t) \rightarrow F(x, \tilde{\rho}_{\infty}, \nabla \tilde{\rho}_{\infty},t)$ -uniformly. Hence we get from (), - -$$ +uniformly. Hence we get from (12.19), $$ \begin{aligned} @@ -1777,15 +1842,13 @@ $$ \end{aligned} $$ -$$ - -The next argument is similar to the argument in , p.21. +The next argument is similar to the argument in [LiTam], p.21. The function $\tilde{\rho}_{i}$ is a smooth solution of -() with $\tilde{\rho}_i(x,0)=0$. Also, both +(12.20) with $\tilde{\rho}_i(x,0)=0$. Also, both $\tilde{\rho}_{i}$ and $F(x,\tilde{\rho}_{i_1},\nabla \tilde{\rho}_{i-1},t)$ are uniformly bounded on $\mathbb{R}^{n+2} \times[0,T]$. Thus, by -Theorem 1.11 , p.211 and Theorem 12.1 , p.223, +Theorem 1.11 [LSU], p.211 and Theorem 12.1 [LSU], p.223, for any compact $K \subset \mathbb{R}^{n+2}$ and any $0 +**Proposition.** For a standard solution $(\mathbb{R}^n,g(t)),\ 0\le t ### The uniqueness for the solutions of Ricci-DeTurck flow We prove the following general uniqueness result for Ricci-DeTurck flow on open manifolds. -**Proposition.** -Let $\hat g_1( t) $ and $\hat g_2( t),\ 0 \leq t\leq T $, be two + +**Proposition.** Let $\hat g_1( t) $ and $\hat g_2( t),\ 0 \leq t\leq T $, be two bounded solutions of the Ricci-DeTurck flow on complete and noncompact manifold $M^{n}$ with initial metric $g_1( t_0) =g_2(t_ 0) =g$. Suppose that for some $10,$ there is a $k_0$ arbitrarily large with @@ -2039,14 +2062,13 @@ _{C^{1}\left(\partial\Omega_{k_0}\right) ,g}\leq\epsilon, $$ Then $ \hat g_1( t) =\hat g_2( t)$. + **Proof.** Letting $\tilde \nabla$ be the covariant derivative determined by -$g$, then, using the Ricci-DeTurck flow () for -$\hat g_1$ and $\hat g_2$, we can make the following estimate for +$g$, then, using the Ricci-DeTurck flow (12.10) for +$\hat g_1$ and $\hat g_2$, we can make the following estimate for an appropriate constant $C_{14}$ depending on $g$. -$$ - $$ \begin{aligned} & \frac{\partial}{\partial t}\left\vert \hat g_1( t) -\hat g_2( t) @@ -2085,8 +2107,6 @@ g_2(t) \right) \right\vert _{g}^{2}+\frac{C_{14}^{2}}{4C^{-1} \end{aligned} $$ -$$ - where the last inequality comes from completing the square to replace the last term in the previous expression. Thus, we have proved @@ -2098,7 +2118,6 @@ g_1^{\alpha\beta}\tilde{\nabla}_{\alpha} \tilde{\nabla}_{\beta}\left\vert \hat g_1\left( t\right) -\hat g_2\left( t\right) \right\vert _{g}^{2}+C_{15}\left\vert \hat g_1\left( t\right) -\hat g_2\left( t\right) \right\vert _{g}^{2} - $$ pointwise on $\Omega_{k}$ with $C_{15}$ a constant that depends only @@ -2107,7 +2126,7 @@ on $n$, $C$ and $g$. Suppose that $\hat g_1\left( t\right) \neq \hat g_2 \left( t\right) $ for some $t$. Then there is a point $x_{0}$ such that $\left\vert \hat g_1(x_0, t) - \hat g_2(x_0, t) \right\vert _{g}^{2} ->\epsilon_{0}$ for some $\epsilon_{0}>0$. +>\epsilon_{0}$ for some $\epsilon_{0}>0$. We choose a $k_0$ sufficiently large that $x_{0}\in \Omega_{k_0}$ and for all $t'\in [t_0,T]$ we have @@ -2120,7 +2139,7 @@ $$ where $\epsilon >0$ is a constant to be chosen later. Recall we have the initial condition $\left\vert \hat g_1( 0) -\hat -g_2(0) \right\vert _{g}^{2}=0$. Using Equation~() and applying the maximum principle to $\left\vert \hat g_1( +g_2(0) \right\vert _{g}^{2}=0$. Using Equation (12.28) and applying the maximum principle to $\left\vert \hat g_1( t) -\hat g_2( t) \right\vert _{g}^{2}$ ) on the domain $\Omega_{k_0}$, we get @@ -2135,29 +2154,30 @@ proposition. Let $g_1(t),\ 0\le tt_0$ for the Ricci flows $g_1(t)$ and $g_2(t)$. Let $\hat{g}_1(t)= (\psi^{-1}(t))^*g_1(t)$ and $\hat{g}_2(t) = (\psi^{-1}(t))^*g_2(t)$. Then $\hat{g}_1(t)$ and $\hat{g}_2(t)$ are two solutions of the Ricci-DeTurck flow with $\hat{g}_1(t_0)=\hat{g}_2(t_0)$. Choose $T' \in (t_0,T]$ such that $\hat{g}_1(t)$ and $\hat{g}_2(t)$ are $\delta$-close to -$\hat{g}_1(t_0)$ as required in Proposition~. It follows -from Lemma~ and the decay estimate in -Proposition~ that $\hat{g}_1(t)$ and $\hat{g}_2(t)$ +$\hat{g}_1(t_0)$ as required in \entryref{c5e225b067f1}. It follows +from \entryref{77ca4669ffb3} and the decay estimate in +\entryref{5fbeadbba61c} that $\hat{g}_1(t)$ and $\hat{g}_2(t)$ are bounded solutions and that they have same sequential asymptotic -behavior at infinity. We can apply Proposition~ to -conclude $\hat{g}_1(t)=\hat{g}_2(t)$ on $t_0 \leq t \leq T'$. We +behavior at infinity. We can apply \entryref{c5e225b067f1} to +conclude $\hat{g}_1(t)=\hat{g}_2(t)$ on $t_0 \leq t \leq T'$. We have proved: -**Corollary.** -Let $g_1(t)$ and $g_2(t)$ be standard solutions. Suppose that + +**Corollary.** Let $g_1(t)$ and $g_2(t)$ be standard solutions. Suppose that $g_1(t)=g_2(t)$ for all $t\in [0,t_0]$ for some $t_0\ge 0$. The Ricci-DeTurck solutions $\hat{g}_1(t)$ and $\hat{g}_2(t)$ constructed from standard solutions $g_1(t)$ and $g_2(t)$ with $g_1(t_0)=g_2(t_0)$ exist and satisfy $\hat{g}_1(t)=\hat{g}_1(t)$ for $t \in [t_0,T']$ for some $T'>t_0$. + ## Completion of the proof of uniqueness @@ -2182,17 +2202,17 @@ There is one more case to rule out. **Case 3: $I$ is a closed interval $I=[0,t_0]$.** -In this case, of course, $t_0t_0$. Using these harmonic map flows we construct solutions $\hat g_1(t)$ and $\hat g_2(t)$ to the Ricci-DeTurck flow defined on -the interval $[t_0,T]$. According to Corollary~, there is +the interval $[t_0,T]$. According to \entryref{a501760fb0e6}, there is a uniqueness theorem for these Ricci-DeTurck flows, which implies that $\hat g_1(t)=\hat g_2(t)$ for all $t\in [t_0,T']$ for some -$T'>t_0$. Invoking Corollary~ we conclude that +$T'>t_0$. Invoking \entryref{5b79e248c5cb} we conclude that $g_1(t)=g_2(t)$ for all $t\in [0,T']$, contradicting the maximality of the interval $I$. @@ -2204,7 +2224,7 @@ standard flow. ### $T=1$ and existence of canonical neighborhoods At this point we have established all the properties claimed in -Theorem~ for the standard flow except for the fact that +\entryref{f44d182ef27e} for the standard flow except for the fact that $T$, the endpoint of the time-interval of definition, is equal to $1$. We have shown that $T\le 1$. In order to establish the opposite inequality, we must show the existence of canonical neighborhoods @@ -2213,18 +2233,18 @@ for the standard solution. Here is the result about the existence of canonical neighborhoods for the standard solution. -**Theorem.** -Fix $0<\epsilon<1$. Then there is $r>0$ such that for any point + +**Theorem.** Fix $0<\epsilon<1$. Then there is $r>0$ such that for any point $(x_0,t_0)$ in the standard flow with $R(x_0,t_0)\ge r^{-2}$ the following hold. -\begin{enumerate} -\item[(1)] $t_0>r^2$. -\item[(2)] $(x_0,t_0)$ has a strong canonical + +- **(1)** $t_0>r^2$. +- **(2)** $(x_0,t_0)$ has a strong canonical $(C(\epsilon),\epsilon)$-neighborhood. If this canonical neighborhood is a strong $\epsilon$-neck centered at $(x_0,t_0)$, then the strong neck extends to an evolving neck defined for backward rescaled time $(1+\epsilon)$. -\end{enumerate} + **Proof.** Take an increasing sequence of times $t'_n$ converging to $T$. Since the curvature of $(\Ar^3,g(t))$ is locally bounded in time, @@ -2235,66 +2255,67 @@ conclusion of the theorem does not hold. (There clearly are such points since the conclusion of the theorem fails for all $(x,0)$.) Pick $(x_n,t_n)$ with $t_n\le t'_n$, with $R(x_n,t_n)\ge R_n/2$ and such that the conclusion of the theorem does not hold for -$(x_n,t_n)$. To prove the theorem we must show that $\overlinelim_{n\rightarrow\infty}R(x_n,t_n)<\infty$. Suppose the contrary. -By passing to a subsequence we can suppose that $lim_{n\rightarrow \infty}R(x_n,t_n)=\infty$. We set +$(x_n,t_n)$. To prove the theorem we must show that $\overline\mathit{lim}_{n\rightarrow\infty}R(x_n,t_n)<\infty$. Suppose the contrary. +By passing to a subsequence we can suppose that $\mathit{lim}_{n\rightarrow \infty}R(x_n,t_n)=\infty$. We set $Q_n=R(x_n,t_n)$. We claim that all the hypotheses of -Theorem~ apply to the sequence +\entryref{a9e023d082dc} apply to the sequence $(\Ar^3,g(t),(x_n,t_n))$. First, we show that all the hypotheses of -Theorem~ (except the last) hold. Since $(\Ar^3,g(t))$ +\entryref{d51aab8a8065} (except the last) hold. Since $(\Ar^3,g(t))$ has non-negative curvature all these flows have curvature pinched -toward positive. By Theorem~ there are $r>0$ and +toward positive. By \entryref{ed069175cdf9} there are $r>0$ and $\kappa>0$ so that all these flows are $\kappa$-non-collapsed on scales $\le r$. By construction if $t\le t_n$ and $R(y,t)>2Q_n\ge -R_n$ then the point $(y,t)$ has a strong canonical +R_n$ then the point $(y,t)$ has a strong canonical $(C(\epsilon),\epsilon)$-neighborhood. We are assuming that $Q_n\rightarrow \infty$ as $n\rightarrow \infty$ in order to achieve the contradiction. Since all time-slices are complete, all balls of finite radius have compact closure. Lastly, we need to show that the extra hypothesis of -Theorem~ (which includes the last hypothesis of -Theorem~) is satisfied. This is clear since +\entryref{a9e023d082dc} (which includes the last hypothesis of +\entryref{d51aab8a8065}) is satisfied. This is clear since $t_n\rightarrow T$ as $n\rightarrow \infty$ and $Q_n\rightarrow -\infty$ as $n\rightarrow \infty$. Applying Theorem~ we +\infty$ as $n\rightarrow \infty$. Applying \entryref{a9e023d082dc} we conclude that after passing to a subsequence there is a limiting flow which is a $\kappa$-solution. Clearly, this and -Corollary~ imply that for all sufficiently large -$n$ (in the subsequence) the neighborhood as required by the +\entryref{e724298e5bc3} imply that for all sufficiently large +$n$ (in the subsequence) the neighborhood as required by the theorem exists. This contradicts our assumption that none of the points $(x_n,t_n)$ have these neighborhoods. This contradiction proves the result. -### Completion of the proof of -Theorem~\protect{} +### Completion of the proof of Theorem \entryref{f44d182ef27e} The next proposition establishes the last of the conditions claimed -in Theorem~. +in \entryref{f44d182ef27e}. + **Theorem.** For the standard flow $T=1$. + -**Proof.** We have already seen in Corollary~ that $T\le 1$. Suppose now +**Proof.** We have already seen in \entryref{2db418fba1c1} that $T\le 1$. Suppose now that $T<1$. Take $T_00$ is the -constant given in Theorem~, we have $\partial +constant given in \entryref{85e3c925177e}, we have $\partial R/\partial t(x,t)\le C(\epsilon)R^2(x,t)$. Thus, provided that $T-T_0$ is sufficiently small, there is a uniform bound to $R(x,t)$ for all $x\in \Ar^3\setminus X$ and all $t\in [T_0,T)$. Using -Theorem~ and the fact that the standard flow is +\entryref{e0b6a1f646a7} and the fact that the standard flow is $\kappa$-non-collapsed implies that the restrictions of the metrics $g(t)$ to $\Ar^3\setminus X$ converge smoothly to a limiting Riemannian metric $g(T)$ on $\Ar^3\setminus X$. Fix a non-empty open subset $\Omega\subset \Ar^3\setminus X$ with compact closure. For each $t\in [0,T)$ let $V(t)$ be the volume of -$(\Omega,g(t)|_{\Omega})$. Of course, $lim_{t\rightarrow -T}V(t)=Vol_{g(T)}(\Omega)>0$. +$(\Omega,g(t)|_{\Omega})$. Of course, $\mathit{lim}_{t\rightarrow +T}V(t)=\mathit{Vol}_{g(T)}(\Omega)>0$. Since the metric $g(T)$ exists in a neighborhood of infinity and has bounded curvature there, if the limit metric $g(T)$ exists on all of @@ -2302,56 +2323,82 @@ $\Ar^3$, then we can extend the flow keeping the curvature bounded. This contradicts the maximality of our flow subject to the condition that the curvature be locally bounded in time. Consequently, there is a point $x\in \Ar^3$ for which the limit metric $g(T)$ does not -exist. This means that $\overlinelim_{t\rightarrow -T}R(x,t)=\infty$. That is to say, there is a sequence of +exist. This means that $\overline\mathit{lim}_{t\rightarrow +T}R(x,t)=\infty$. That is to say, there is a sequence of $t_n\rightarrow T$ such that setting $Q_n=R(x,t_n)$, we have $Q_n\rightarrow \infty$ as $n$ tends to infinity. By -Theorem~ the second hypothesis in the statement of -Theorem~ holds for the sequence +\entryref{85e3c925177e} the second hypothesis in the statement of +\entryref{d51aab8a8065} holds for the sequence $(\Ar^3,g(t),(x,t_n))$. All the other hypotheses of this theorem as -well as the extra hypothesis in Theorem~ obviously -hold for this sequence. Thus, according to Theorem~ +well as the extra hypothesis in \entryref{a9e023d082dc} obviously +hold for this sequence. Thus, according to \entryref{a9e023d082dc} the based flows $(\Ar^3,Q_ng(Q_n^{-1}t'+t_n),(x,0))$ converge smoothly to a $\kappa$-solution. Since the asymptotic volume of any -$\kappa$-solution is zero (see Theorem~), we see that +$\kappa$-solution is zero (see \entryref{7468282bd017}), we see that for all $n$ sufficiently large, the following holds: + **Claim.** For any $\epsilon>0$, there is $A<\infty$ such that for all $n$ sufficiently large we have -$$Vol(B_{Q_ng}(x,t_n,A))<\epsilon A^3.$$ + +$$ +\mathit{Vol}(B_{Q_ng}(x,t_n,A))<\epsilon A^3. +$$ + Rescaling, we see that for all $n$ sufficiently large we have -$$Vol\,B_g(x,t_n,A/\sqrt{Q_n})< \epsilon(A/\sqrt{Q_n})^3.$$ + +$$ +\mathit{Vol} B_g(x,t_n,A/\sqrt{Q_n})< \epsilon(A/\sqrt{Q_n})^3. +$$ + Since the curvature of $g(t_n)$ is non-negative and since the $Q_n$ tend to $\infty$, it follows from the Bishop-Gromov Inequality -(Proposition~) that for any $00$, for all $n$ sufficiently large we have -$$Vol\,B_g(x,t_n,A)<\epsilon A^3.$$ + +$$ +\mathit{Vol} B_g(x,t_n,A)<\epsilon A^3. +$$ On the other hand, since $\Omega$ has compact closure, there is an $A_1<\infty$ with $\Omega\subset B(x,0,A_1)$. Since the curvature of $g(t)$ is non-negative for all $t\in [0,T)$, it follows from -Lemma~ that the distance is a non-increasing +\entryref{51a2afee4465} that the distance is a non-increasing function of $t$, so that for all $t\in [0,T)$ we have $\Omega\subset B(x,t,A_1)$. Applying the above, for any $\epsilon>0$ for all $n$ sufficiently large we have -$$Vol\,(\Omega,g(t_n))\le Vol_g\,B(x,t_n,A_1)<\epsilon -(A_1)^3.$$ But this contradicts the fact that -$$lim_{n\rightarrow -\infty}Vol\,\Omega,g(t_n)=Vol\,(\Omega,g(T))>0.$$ This + +$$ +\mathit{Vol} (\Omega,g(t_n))\le \mathit{Vol}_g B(x,t_n,A_1)<\epsilon +(A_1)^3. +$$ + + But this contradicts the fact that + +$$ +\mathit{lim}_{n\rightarrow +\infty}\mathit{Vol} \Omega,g(t_n)=\mathit{Vol} (\Omega,g(T))>0. +$$ + + This contradiction proves that $T=1$. -This completes the proof of Theorem~. +This completes the proof of \entryref{f44d182ef27e}. ## Some corollaries Now let us derive extra properties of the standard solution that will be important in our applications. -**Proposition.** -There is a constant $c>0$ such that for all $(p,t)$ in the standard + +**Proposition.** There is a constant $c>0$ such that for all $(p,t)$ in the standard solution we have -$$R(p,t)\ge \frac{c}{1-t}.$$ + +$$ +R(p,t)\ge \frac{c}{1-t}. +$$ + **Proof.** First, let us show that there is not a limiting metric $g(1)$ defined on all of $\Ar^3$. This does not immediately contradict the @@ -2360,51 +2407,58 @@ maximal subject to having curvature locally bounded in time. Assume that a limiting metric $(\Ar^3,g(1))$ exists. First, notice that from the canonical neighborhood assumption and -Lemma~ we see that the curvature of $g(T)$ must be +\entryref{0ddfe578961e} we see that the curvature of $g(T)$ must be unbounded at spatial infinity. On the other hand, by -Proposition~ every point of $(\Ar^3,g(1))$ of +\entryref{7c2758f0e2b3} every point of $(\Ar^3,g(1))$ of curvature greater than $R_0$ has a $(2C,2\epsilon)$-canonical neighborhood. Hence, since $(\Ar^3,g(1))$ has unbounded curvature, it then has $2\epsilon$-necks of arbitrarily small scale. This -contradicts Proposition~. (One can also rule this +contradicts \entryref{cb9ca6d6db9f}. (One can also rule this possibility out by direct computation using the spherical symmetry of the metric.) This means that there is no limiting metric $g(1)$. -The next step is to see that for any $p\in \Ar^3$ we have $lim_{t\rightarrow 1}R(p,t)=\infty$. Let $\Omega\subset \Ar^3$ be -the subset of $x\in \Ar^3$ for which $liminf_{t\rightarrow +The next step is to see that for any $p\in \Ar^3$ we have $\mathit{lim}_{t\rightarrow 1}R(p,t)=\infty$. Let $\Omega\subset \Ar^3$ be +the subset of $x\in \Ar^3$ for which $\mathit{liminf}_{t\rightarrow 1}R(x,t)<\infty$. We suppose that $\Omega\not=\emptyset$. According -to Theorem~ the subset $\Omega$ is open and the metrics +to \entryref{af00774dbaa9} the subset $\Omega$ is open and the metrics $g(t)|_\Omega$ converge smoothly to a limiting metric $g(1)|_\Omega$. On the other hand, we have just seen that there is not a limit metric $g(1)$ defined everywhere. This means that there -is $p\in \Ar^3$ with $lim_{t\rightarrow 1}R(p,t)=\infty$. +is $p\in \Ar^3$ with $\mathit{lim}_{t\rightarrow 1}R(p,t)=\infty$. Take a sequence $t_n$ converging to $1$ and set $Q_n=R(p,t_n)$. By -Theorem~ we see that, possibly after passing to a +\entryref{a9e023d082dc} we see that, possibly after passing to a subsequence, the based flows $(\Ar^3,Q_ng(t'-t_n),(p,0))$ converge -to a $\kappa$-solution. Then by Proposition~ for any -$\epsilon>0$ there is $A<\infty$ such that $Vol\,B_{Q_ng}(p,t_n,A)<\epsilon A^3$, and hence after rescaling we -have $Vol\,B_g(p,t_n,A/\sqrt{Q_n})<\epsilon (A/\sqrt{Q_n})^3$. -By the Bishop-Gromov inequality (Proposition~) it +to a $\kappa$-solution. Then by \entryref{7468282bd017} for any +$\epsilon>0$ there is $A<\infty$ such that $\mathit{Vol} B_{Q_ng}(p,t_n,A)<\epsilon A^3$, and hence after rescaling we +have $\mathit{Vol} B_g(p,t_n,A/\sqrt{Q_n})<\epsilon (A/\sqrt{Q_n})^3$. +By the Bishop-Gromov inequality (\entryref{1a1b2c4fdc9b}) it follows that for any $00$ and for all $n$ -sufficiently large, we have $Vol\,B_g(p,t_n,A)<\epsilon A^3$. +sufficiently large, we have $\mathit{Vol} B_g(p,t_n,A)<\epsilon A^3$. Take a non-empty subset $\Omega'\subset \Omega$ with compact -closure. Of course, $Vol\,(\Omega',g(t))$ converges to $Vol\,(\Omega',g(T))>0$ as $t\rightarrow T$. Then there is +closure. Of course, $\mathit{Vol} (\Omega',g(t))$ converges to $\mathit{Vol} (\Omega',g(T))>0$ as $t\rightarrow T$. Then there is $A<\infty$ such that for each $n$, the subset $\Omega'$ is contained in the ball $B(p_0,t_n,A)$. This is a contradiction since it implies that for any $\epsilon>0$ for all $n$ sufficiently large we have -$Vol\,(\Omega',g(t))<\epsilon A^3$. This completes the proof -that for every $p\in \Ar^3$ we have $lim_{t\rightarrow +$\mathit{Vol} (\Omega',g(t))<\epsilon A^3$. This completes the proof +that for every $p\in \Ar^3$ we have $\mathit{lim}_{t\rightarrow 1}R(p,t)=\infty$. Fix $\epsilon>0$ sufficiently small and set $C=C(\epsilon)$. Then for every $(p,t)$ with $R(p,t)\ge r^{-2}$ we have -$$\left|\frac{dR}{dt}(p,t)\right|\le CR^2(p,t).$$ + +$$ +\left|\frac{dR}{dt}(p,t)\right|\le CR^2(p,t). +$$ + Fix $t_0=1-1/2r^2C$. Since the flow has curvature locally bounded in time, there is $2C\le C'<\infty$ such that $R(p,t_0)\le 1/(C'(1-t_0)$ for all $p\in \Ar ^3$. Since $R(p,t_0)=1/C'(1-t_0)$, for all $t\in [t_0,1)$ we have -$$R(p,t)0$ there is $C'(\epsilon)<\infty$ such that for + +**Theorem.** For any $\epsilon>0$ there is $C'(\epsilon)<\infty$ such that for any point $x$ in the standard solution one of the following holds -(see Fig.~). -\begin{enumerate} -\item[(1)] $(x,t)$ is contained in the core of a $(C'(\epsilon),\epsilon)$-cap. -\item[(2)] $(x,t)$ is the center of an evolving $\epsilon$-neck $N$ whose initial time-slice is +(see Fig. 12.2). + +- **(1)** $(x,t)$ is contained in the core of a $(C'(\epsilon),\epsilon)$-cap. +- **(2)** $(x,t)$ is the center of an evolving $\epsilon$-neck $N$ whose initial time-slice is $t=0$ and whose initial time-slice is disjoint from the surgery cap. -\item[(3)] $(x,t)$ is the center of an evolving $\epsilon$-neck defined for rescaled time +- **(3)** $(x,t)$ is the center of an evolving $\epsilon$-neck defined for rescaled time $1+\epsilon$. -\end{enumerate} + + **Remark.** At first glance it may seem impossible for a point $(x,t)$ in the standard solution to be the center of an evolving $\epsilon$-neck defined for rescaled time $1+\epsilon$ since the standard solution @@ -2433,14 +2488,15 @@ centered at $(x,t)$ is $R(x,t)^{-1/2}$. As $t$ approaches one, $R(x,t)$ goes to infinity, so that rescaled time $1$ at $(x,t)$ is an arbitrarily small time interval measured in the scale of the standard solution. + -**Proof.** By Theorem~, there is $r_0$ such that if $R(x,t)\ge -r_0^{-2}$, then $(x,t)$ has a $(C,\epsilon)$-canonical neighborhood +**Proof.** By \entryref{85e3c925177e}, there is $r_0$ such that if $R(x,t)\ge +r_0^{-2}$, then $(x,t)$ has a $(C,\epsilon)$-canonical neighborhood and if this canonical neighborhood is a strong $\epsilon$-neck centered at $x$, then that neck extends to an evolving neck defined -for rescaled time $(1+\epsilon)$. By Proposition~, there +for rescaled time $(1+\epsilon)$. By \entryref{ffae4504318a}, there is $\theta<1$ such that if $R(x,t)\le r_0^{-2}$ then $t\le \theta$. -By Proposition~, there is a compact subset $X\subset \Ar +By \entryref{77ca4669ffb3}, there is a compact subset $X\subset \Ar ^3$ such that if $t\le \theta$ and $x\notin X$, then there is an evolving $\epsilon$-neck centered at $x$ whose initial time is zero and whose initial time-slice is at distance at least one from the @@ -2448,20 +2504,23 @@ surgery cap. Lastly, by compactness there is $C'<\infty$ such that every $(x,t)$ for $x\in X$ and every $t\le \theta$ is contained in the core of a $(C',\epsilon)$-cap. -**Corollary.** -Fix $\epsilon>0$. Suppose that $(q,t)$ is a point in the standard + +**Corollary.** Fix $\epsilon>0$. Suppose that $(q,t)$ is a point in the standard solution with $t\le R(q,t)^{-1}(1+\epsilon))$ and with $(q,0)\in B(p_0,0,(\epsilon^{-1}/2)+A_0+5)$. Then $(q,t)$ is contained in an $(C'(\epsilon),\epsilon)$-cap. + + **Remark.** Recall that $p_0$ is the origin in $\Ar^3$ and hence is the tip of -the surgery cap. Also, $A_0$ is defined in Lemma~. +the surgery cap. Also, $A_0$ is defined in \entryref{0b07beec84c0}. + -**Corollary.** -For any $\epsilon>0$ let $C'=C'(\epsilon)$ be as in -Theorem~. Suppose that we have a sequence of + +**Corollary.** For any $\epsilon>0$ let $C'=C'(\epsilon)$ be as in +\entryref{6ff28dba4e56}. Suppose that we have a sequence of generalized Ricci flows $({\mathcal M}_n,G_n)$, points $x_n\in -{\mathcal M}_n$ with $**t**(x_n)=0$, neighborhoods $U_n$ of $x_n$ +{\mathcal M}_n$ with $\mathbf{t}(x_n)=0$, neighborhoods $U_n$ of $x_n$ in the zero time-slice of ${\mathcal M}_n$, and a constant $0<\theta<1$. Suppose that there are embeddings $\rho_n\colon U_n\times [0,\theta)\to {\mathcal M}_n$ compatible with time and the @@ -2469,16 +2528,16 @@ vector field so that the Ricci flows $\rho_n^*G_n$ on $U_n$ based at $x_n$ converge geometrically to the restriction of the standard solution to $[0,\theta)$. Then for all $n$ sufficiently large, and any point $y_n$ in the image of $\rho_n$ one of the following holds: -\begin{enumerate} -\item[(1)] $y_n$ is contained in the core of a + +- **(1)** $y_n$ is contained in the core of a $(C'(\epsilon),\epsilon)$-cap -\item[(2)] $y_n$ is the center of a strong $\epsilon$-neck -\item[(3)] $y_n$ is the center of an evolving $\epsilon$-neck whose +- **(2)** $y_n$ is the center of a strong $\epsilon$-neck +- **(3)** $y_n$ is the center of an evolving $\epsilon$-neck whose initial time-slice is at time $0$. -\end{enumerate} + -**Proof.** This follows immediately from Theorem~ and -Proposition~. +**Proof.** This follows immediately from \entryref{6ff28dba4e56} and +\entryref{7c2758f0e2b3}. There is one property that we shall use later in proving the finite-time extinction of Ricci flows with surgery for manifolds @@ -2486,16 +2545,17 @@ with finite fundamental group (among others). This is a distance decreasing property which we record here. Notice that for the standard initial metric constructed in -Lemma~ we have the following: +\entryref{f1d3021d32b8} we have the following: -**Lemma.** -Let $S^2$ be the unit sphere in $T_0\Ar^3$. Equip it with the metric + +**Lemma.** Let $S^2$ be the unit sphere in $T_0\Ar^3$. Equip it with the metric $h_0$ that is twice the usual metric (so that the scalar curvature of $h_0$ is $1$). We define a map $\rho\colon S^2\times [0,\infty)\to \Ar^3$ by sending the point $(x,s)$ to the point at distance $s$ from the origin in the radial direction from $0$ given by $x$ (all this measured in the metric $g_0$). Then $\rho^*g_0\le h_0\times ds^2$. + **Proof.** Clearly, the metric $\rho^*g_0$ is rotationally symmetric and its component in the $s$-direction is $ds^2$. On the other hand, since diff --git a/projects/poincare-conjecture/.astrolabe/docs-src/14-surgery-on-a-delta-neck.mdx b/projects/poincare-conjecture/.astrolabe/docs-src/14-surgery-on-a-delta-neck.mdx new file mode 100644 index 00000000..7fc5e7bf --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/docs-src/14-surgery-on-a-delta-neck.mdx @@ -0,0 +1,692 @@ + + +# Chapter 13 -- Surgery on a $\delta$-neck + +## Notation and the Statement of the Result + +In this chapter we describe the surgery process. For this chapter we +fix: + +- **(1)** A $\delta$-neck $(N,g)$ centered at a point $x_0$. +We denote by $\rho\colon S^2\times (-\delta^{-1},\delta^{-1})\to N$ +the diffeomorphism that gives the $\delta$-neck structure. +- **(2)** Standard initial conditions $(\Ar^3,g_0)$. + +We denote by $h_0\times ds^2$ the metric on $S^2\times \Ar$ which is +the product of the round metric $h_0$ on $S^2$ of scalar curvature +$1$ and the Euclidean metric $ds^2$ on $\Ar$. We denote by +$N^-\subset N$ the image $\rho((-\delta^{-1},0]\times S^2)$ and we +denote by $s\colon N^-\to (-\delta^{-1},0]$ the composition +$\rho^{-1}$ followed by the projection to the second factor. + +Recall that the standard initial metric $(\Ar^3,g_0)$ is invariant +under the standard $SO(3)$-action on $\Ar^3$. We let $p_0$ denote +the origin in $\Ar^3$. It is the fixed point of this action and is +called the *tip* +of the standard initial metric. Recall from \entryref{0b07beec84c0} that +there are $A_0>0$ and an isometry + +$$ +\psi\colon (S^2\times +(-\infty,4],h_0\times ds^2)\to (\Ar^3\setminus B(p_0,A_0),g_0). +$$ + + The +composition of $\psi^{-1}$ followed by projection onto the second +factor defines a map $s_1\colon \Ar^3\setminus B(p_0,A_0)\to +(-\infty,4]$. Lastly, there is $0 +**Claim.** Let $\xi\colon N\to \Ar^3 $ be the map that sends $\rho(S^2\times +[A_0+4,\delta^{-1}))$ to the origin $0\in \Ar^3$ (i.e., to the tip of +the surgery cap) and for every $s + +**Proof.** Since $R(x_0)g$ is within $\delta$ of $h_0\times ds^2$, it follows +that $R(x_0)g\ge \eta (h_0\times ds^2)$. But according to +\entryref{545ca0a10566} the map $\xi$ given in the statement of the +claim is a distance non-increasing map from $h_0\times ds^2$ to +$g_0$. The claim follows immediately. + + The last choices we need to make are of +constants $C_0<\infty$ and $q<\infty$, with $C_0 \gg q$, but both of these are +independent of $\delta$. These choices will be made later. Given all these +choices, we define a function + +$$ +f(s)=\begin{cases} 0 & s\le 0 \\ C_0\delta e^{-q/s} & s>0,\end{cases} +$$ + +and then we define the metric $\tilde g$ on ${\mathcal S}$ by first +defining a metric: + +$$ +\hat g=\begin{cases} \mathit{exp}(-2f(s))R(x_0)\rho^*g & \ \ \mathit{on} \ \ +s^{-1}(-\infty,1] \\ +\mathit{exp}(-2f(s))\left(\alpha(s)R(x_0)\rho^*g+(1-\alpha(s))\eta +g_0\right) & \ \ \mathit{on}\ \ s^{-1}([1,2]) \\ +\mathit{exp}(-2f(s))\eta g_0 & \ \ \mathit{on}\ \ s^{-1}([2,A_{r_0}]\\ +\left[\beta(s)\mathit{exp}(-2f(s))+(1-\beta(s))\mathit{exp}(-2f(4+A_0))\right]\eta +g_0 & \ \ \mathit{on} \ \ s^{-1}([A_{r_0},A']),\end{cases} +$$ + + where +$A_{r_0}=4+A_0-r_0$ and $A'=A_0+4$. + Then we define + +$$ +\widetilde g=R(x_0)^{-1}\hat g. +$$ + +See Fig. 13.1. + + +**Theorem.** There are constants $C_0,q,R_0<\infty$ and $\delta'_0>0$ such that the +following hold for the result $({\mathcal S},\widetilde g)$ of surgery on +$(N,g)$ provided that $R(x_0)\ge R_0$, $0<\delta\le \delta'_0$. Define $f(s)$ +as above with the constants $C_0,\delta$ and then use $f$ to define surgery on +a $\delta$-neck $N$ to produce $({\mathcal S},\tilde g)$. Then the following +hold. + +- Fix $t\ge 0$. For any $p\in N$, +let $X(p)=\mathit{max}(0,-\nu_{g}(p))$, where $\nu_ g(p)$ is the +smallest eigenvalue of $Rm_{ g}(p)$. Suppose that for all $p\in N$ +we have: + +- **(1)** $R(p)\ge \frac{-6}{1+4t}$, and +- **(2)** $R(p)\geq 2X(p)\left(\mathit{log}X(p)+\mathit{log}(1+t)-3\right)$, whenever $00$ there is $\delta'_1=\delta'_1(\delta'')>0$ such that if +$\delta\le \mathit{min}(\delta'_1,\delta'_0)$, then the restriction of +$\hat g$ to $B_{\hat g}(p_0,(\delta'')^{-1})$ in $({\mathcal S},\hat +g)$ is $\delta''$-close in the $C^{[1/\delta'']}$-topology to the +restriction of the standard initial metric $g_0$ to +$B_{g_0}(p_0,(\delta'')^{-1})$. + + +The rest of this chapter is devoted to the proof of this theorem. + +Before starting the curvature computations let us make a remark about the surgery cap. + + +**Definition.** The image in ${\mathcal S}$ of $B_{g_0}(p_0,0,A_0+4)$ is called the *surgery cap*. + + +The following is immediate from the definitions provided that +$\delta>0$ is sufficiently small. + + +**Lemma.** The surgery cap in $({\mathcal S},\tilde g)$ has a metric that +differs from the one coming from a rescaled version of the standard +solution. Thus, the image of this cap is not necessarily a metric +ball. Nevertheless for $\epsilon<1/200$ the image of this cap will +be contained in the metric ball in ${\mathcal S}$ centered at $p_0$ +of radius $R(x_0)^{-1/2}(A_0+5)$ and will contain the metric ball +centered at $p_0$ of radius $R(x_0)^{-1/2}(A_0+3)$. Notice also that +the complement of the closure of the surgery cap in ${\mathcal S}$ +is isometrically identified with $N^-$. + + +## Preliminary computations + +We shall compute in a slightly more general setup. + Let $I$ be an open interval contained in $(-\delta^{-1},4+A_0)$ and let +$h$ be a metric on $S^2\times I$ within $\delta$ in the +$C^{[1/\delta]}$-topology of the restriction to this open +submanifold of the standard metric $h_0\times ds^2$. We let $\hat +h=e^{-2f}h$. Fix local coordinates near a point $y\in S^2\times I$. +We denote by $\nabla$ the covariant derivative for $h$ and by +$\widehat\nabla$ the covariant derivative for $\hat h$. + We also denote by +$(R_{ijkl})$ the matrix of the Riemann curvature operator of $h$ +in the associated basis of $\wedge^2T(S^2\times I)$ and by $(\hat +R_{ijkl})$ the matrix of the Riemann curvature operator of $\hat +h$ with respect to the same basis. Recall the formula for the +curvature of a conformal change of metric (see, (3.34) on p.51 of +[Sakai]): + +$$ +\begin{aligned} +\hat R_{ijkl} & = & +e^{-2f}\left(R_{ijkl}-f_jf_kh_{il}+f_jf_lh_{ik}+f_if_kh_{jl}-f_if_lh_{jk}\right. +\\ & & \left.- +(\wedge^2h)_{ijkl}|\nabla +f|^2-f_{jk}h_{il}+f_{ik}h_{jl}+f_{jl}h_{ik}-f_{il}h_{jk}\right).\nonumber +\end{aligned} +$$ + +Here, $f_i$ means $\partial_if$, + +$$ +f_{ij}=\mathit{Hess}_{ij}(f)=\partial_if_j-f_l\Gamma_{ij}^l, +$$ + +and $\wedge^2h$ is the metric induced by $h$ on $\wedge^2TN$, so +that + +$$ +\wedge^2h_{ijkl}=h_{ik}h_{jl}-h_{il}h_{jk}. +$$ + +Now we introduce the notation $O(\delta)$. When we say that a +quantity is $O(\delta)$ we mean that there is some universal +constant $C$ such that, provided that $\delta>0$ is sufficiently +small, the absolute value of the quantity is $\le C\delta$. The +universal constant is allowed to change from inequality to +inequality. + +In our case we take local coordinates adapted to the $\delta$-neck: +$(x^0,x^1,x^2)$ where $x^0$ agrees with the $s$-coordinate and +$(x^1,x^2)$ are Gaussian local coordinates on the $S^2$ such that +$dx^1$ and $dx^2$ are orthonormal at the point in question in the +round metric $h_0$. The function $f$ is a function only of $x^0$. +Hence $f_i=0$ for $i=1,2$. Also, $f_0=\frac{q}{s^2}f$. It follows +that + +$$ +|\nabla f|_h=\frac{q}{s^2}f\cdot(1+O(\delta)), +$$ + + so that + +$$ +|\nabla +f|_h^2=\frac{q^2}{s^4}f^2\cdot (1+O(\delta)). +$$ + +Because the metric $h$ is $\delta$-close to the product $h_0\times +ds^2$, we see that $h_{ij}(y)=(h_0)_{ij}(y)+O(\delta)$ and the +Christoffel symbols $\Gamma_{ij}^k(y)$ of $h$ are within $\delta$ in +the $C^{([1/\delta]-1)}$-topology of those of the product metric +$h_0\times ds^2$. In particular, $\Gamma_{ij}^0=O(\delta)$ for all +$ij$. + The components $f_{ij}$ of the Hessian with +respect to $h$ are given by + +$$ +f_{00}=\left(\frac{q^2}{s^4}-\frac{2q}{s^3}\right)f+\frac{q}{s^2}fO(\delta), +$$ + +$$ +f_{i0}=\frac{q}{s^2}fO(\delta)\ \ \ \mathit{for}\ \ 1\le i\le 2, +$$ + +$$ +f_{ij}=\frac{q}{s^2}fO(\delta)\ \ \ \mathit{for}\ \ 1\le i,j\le 2. +$$ + +In the following $a,b,c,d$ are indices taking values $1$ and $2$. +Substituting in Equation (13.1) yields + +$$ +\begin{aligned} +\hat +R_{0a0b} & = & +e^{-2f}\left(R_{0a0b}+\frac{q^2}{s^4}f^2h_{ab}-h_{ab}(\frac{q^2}{s^4})f^2(1+O(\delta)) ++\left(\frac{q^2}{s^4}-\frac{2q}{s^3}\right)fh_{ab} \right. \\ +& & \left. +\frac{q}{s^2}fO(\delta)\right)\\ +& = & +e^{-2f}\left(R_{0a0b}+\left(\frac{q^2}{s^4}-\frac{2q}{s^3}\right)fh_{ab}+ +h_{ab}(\frac{q^2}{s^4})f^2O(\delta) ++\frac{q}{s^2}fO(\delta)\right) +\end{aligned} +$$ + + Also, we have + +$$ +\begin{aligned} +\hat +R_{ab0c} & = & +e^{-2f}\left(R_{ab0c}-(\wedge^2h)_{ab0c}(\frac{q^2}{s^4})f^2(1+O(\delta)) ++\frac{q}{s^2}fO(\delta)\right) \\ +& = & e^{-2f}\left(R_{ab0c}+(\frac{q^2}{s^4})f^2O(\delta) ++\frac{q}{s^2}fO(\delta)\right). +\end{aligned} +$$ + + Lastly, + +$$ +\begin{aligned} +\hat R_{1212} & = & +e^{-2f}\left(R_{1212}-(\wedge^2h)_{1212}\frac{q^2}{s^4}f^2(1+O(\delta))+\frac{q}{s^2}fO(\delta)\right) +\\ +& = & +e^{-2f}\left(R_{1212}-\frac{q^2}{s^4}f^2(1+O(\delta))+\frac{q}{s^2}fO(\delta)\right). +\end{aligned} +$$ + +Now we are ready to fix the constant $q$. We fix it so that for all $s\in +[0,4+A_0]$ we have + +$$ +q \gg (4+A_0)^2\ \ \ \mathit{and}\ \ \ \frac{q^2}{s^4}e^{-q/s} \ll 1. +$$ + +It follows immediately that $q^2/s^4\gg q/s^3$ for all $s\in [0,4+A_0]$. + We are not yet ready +to fix the constant $C_0$, but once we do we shall always require $\delta$ to +satisfy +$\delta\ll C_0^{-1}$ so that for all $s\in [0,4+A_0]$ we have + +$$ +\frac{q}{s^2}f^2\ll \frac{q^2}{s^4}f^2\ll \frac{q}{s^2}f\ll 1. +$$ + +(These requirements are not circular, since $C_0$ and $q$ are chosen +independent of $\delta$.) + + Using +these inequalities and putting our computations in matrix form show +the following. + + +**Corollary.** There is $\delta'_2>0$, depending on $C_0$ and $q$, +such that if $\delta\le \delta'_2$ then we have + +$$ +\left(\hat +R_{ijkl}\right)=e^{-2f}\left[\left(R_{ijkl}\right)+\begin{pmatrix} +-\frac{q^2}{s^4}f^2 & 0 \\ 0 & +\left(\frac{q^2}{s^4}-\frac{2q}{s^3}\right)f\begin{pmatrix} 1 & 0 +\\ 0 & 1\end{pmatrix} +\end{pmatrix}+\left(\frac{q}{s^2}fO(\delta)\right)\right]. +$$ + + +Similarly, we have the equation relating scalar curvatures + +$$ +\hat R=e^{2f}\left(R+4\triangle f-2|\nabla f|^2\right), +$$ + +and hence + +$$ +\hat R=e^{2f}\left(R+4\left(\frac{q^2}{s^4}-\frac{2q}{s^3}\right)f-2\frac{q^2}{s^4}f^2+ +\frac{q}{s^2}fO(\delta)\right). +$$ + + +**Corollary.** For any constant $C_0<\infty$ and any $\delta<\mathit{min}(\delta_2',C_0^{-1})$ we +have $\hat R\ge R$. + + +**Proof.** By our choice of $q$, since $C_0\delta<1$, then $f^2\ll f$ and $q^2/s^4\gg \mathit{max}(q/s^3,q/s^2)$ so that the result follows immediately from the above +formula. + +Now let us compute the eigenvalues of the curvature $ R_{ijkl}(y)$ +for any $y\in S^2\times I$. + + +**Lemma.** There is a $\delta_3'>0$ such that the following hold if $\delta\le +\delta_3'$. Let $\{e_0,e_1,e_2\}$ be an orthonormal basis for the +tangent space at a point $y\in S^2\times I$ for the metric +$h_0\times ds^2$ with the property that $e_0$ points in the +$I$-direction. Then there is a basis $\{f_0,f_1,f_2\}$ for this +tangent space so that the following hold: + +- **(1)** The basis is orthonormal in the metric $h$. +- **(2)** The change of basis matrix expressing the $\{f_0,f_1,f_2\}$ +in terms of $\{e_0,e_1,e_2\}$ is of the form $\mathit{Id}+O(\delta)$. +- **(3)** The Riemann curvature of $h$ in the basis $\{f_0\wedge f_1,f_1\wedge f_2,f_2\wedge f_0\}$ +of $\wedge^2T_y(S^2\times I)$ is + +$$ +\begin{pmatrix} 1/2 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 +\end{pmatrix}+O(\delta). +$$ + + +**Proof.** Since $h$ is within $\delta$ of $h_0\times ds^2$ in the +$C^{[1/\delta]}$-topology, it follows that the matrix for $h(y)$ in +$\{e_0,e_1,e_2\}$ is within $O(\delta)$ of the identity matrix, and the matrix +for the curvature of $h$ in the associated basis of $\wedge^2T_y(S^2\times I)$ +is within $O(\delta)$ of the curvature matrix for $h_0\times ds^2$, the latter +being the diagonal matrix with diagonal entries $\{1/2,0,0\}$. Thus, the usual +Gram-Schmidt orthonormalization process constructs the basis $\{f_0,f_1,f_2\}$ +satisfying the first two items. Let $A=(A^{ab})$ be the change of basis matrix +expressing the $\{f_a\}$ in terms of the $\{e_b\}$, so that $A=\mathit{Id}+O(\delta)$. The curvature of $h$ in this basis is then given by +$B^\mathit{tr}(R_{ijkl})B$ where $B=\wedge^2A$ is the induced change of basis matrix +expressing the basis $\{f_0\wedge f_1,f_1\wedge f_2,f_2\wedge f_0\}$ in terms +of $\{e_0\wedge e_1,e_1\wedge e_2,e_2\wedge e_0\}$. Hence, in the basis +$\{f_0\wedge f_1,f_1\wedge f_2,,f_2\wedge f_0\}$ the curvature matrix for $h$ +is within $O(\delta)$ of the same diagonal matrix. For $\delta$ sufficiently +small then the eigenvalues of the curvature matrix for $h$ are within +$O(\delta)$ of $(1/2,0,0)$. + + +**Corollary.** The following holds provided that $\delta\le \delta_3'$. It is +possible to choose the basis $\{f_0,f_1,f_2\}$ satisfying the +conclusions of \entryref{52d3cb562bdc} so that in addition the +curvature matrix for $(R_{ijkl}(y))$ is of the form + +$$ +\begin{pmatrix} \lambda & 0 & 0 \\ 0 & \alpha & \beta \\ 0 & \beta & \gamma\end{pmatrix} +$$ + +with $|\lambda-\frac{1}{2}|\le O(\delta)$ and +$|\alpha|,|\beta|,|\gamma|\le O(\delta)$. + + +**Proof.** We have an $h$-orthonormal basis $\{f_0\wedge f_1,f_1\wedge +f_2,f_2\wedge f_0\}$ for $\wedge^2T_y(S^2\times \Ar)$ in which the +quadratic form $(R_{ijkl}(y)$ is + +$$ +\begin{pmatrix} 1/2 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{pmatrix} ++O(\delta). +$$ + + It follows that the restriction to the $h$-unit sphere +in $\wedge^2T_y(S^2\times \Ar)$ of this quadratic form achieves its +maximum value at some vector $v$, which, when written out in this +basis, is given by $(x,y,z)$ with $|y|,|z|\le O(\delta)$ and +$|x-1|\le O(\delta)$. Of course, this maximum value is within +$O(\delta)$ of $1/2$. Clearly, on the $h$-orthogonal subspace to +$v$, the quadratic form is given by a matrix all of whose entries +are $O(\delta)$ in absolute value. This gives us a new basis of +$\wedge^2T_y(S^2\times I)$ within $O(\delta)$ of the given basis in +which $(R_{ijkl}(y))$ is diagonal. The corresponding basis for +$T_y(S^2\times\Ar)$ is as required. + +Now we consider the expression $(\hat R_{ijkl}(y))$ in this basis. + + +**Lemma.** Set $\delta'_4=\mathit{min}(\delta_2',\delta_3')$. Suppose +that $\delta\le \mathit{min}(\delta'_4,C_0^{-1})$. Then in the basis +$\{f_0,f_1,f_2\}$ for $T_y(S^2\times I)$ as in \entryref{ad2042196c95} we +have + +$$ +(\hat R_{ijkl}(y)) =e^{-2f}\left[\begin{pmatrix} \lambda & 0 & 0 +\\ 0 & \alpha & \beta \\ 0 & \beta & \gamma\end{pmatrix} ++\begin{pmatrix} -\frac{q^2}{s^4}f^2 & 0 \\ 0 & +\left(\frac{q^2}{s^4}-\frac{2q}{s^3}\right)f\begin{pmatrix} 1 & 0 +\\ 0 & 1\end{pmatrix} +\end{pmatrix}+\left(\frac{q^2}{s^4}fO(\delta)\right)\right] +$$ + +where $\lambda,\alpha,\beta,\gamma$ are the constants in +\entryref{ad2042196c95} and the first matrix is the expression for +$(R_{ijkl}(y))$ in this basis. + + +**Proof.** We simply conjugate the expression in Equation (13.3) by the change of +basis matrix and use the fact that by our choice of $q$ and the fact that +$C_0\delta<1$, we have $f\gg f^2$ and $q/s^3\ll q^2/s^4$. + + +**Corollary.** Assuming that $\delta\le \mathit{min}(\delta'_4,C_0^{-1})$, there is an +$h$-orthonormal basis $\{f_0,f_1,f_2\}$ so that in the associated basis for +$\wedge^2T_y(S^2\times I)$ the matrix $(R_{ijkl}(y))$ is diagonal and given by + +$$ +\begin{pmatrix} \lambda & 0 & 0 \\ 0 & \mu & 0 \\ 0 & 0 & +\nu\end{pmatrix} +$$ + + with $|\lambda-1/2|\le O(\delta)$ and +$|\mu|,|\nu|\le O(\delta)$. Furthermore, in this same basis the +matrix $(\hat R_{ijkl}(y))$ is + +$$ +e^{-2f}\left[\begin{pmatrix}\lambda & 0 & 0 \\ 0 & \mu & 0 \\ 0 & 0 & +\nu\end{pmatrix}+\begin{pmatrix} -\frac{q^2}{s^4}f^2 & 0 \\ 0 & +\left(\frac{q^2}{s^4}-\frac{2q}{s^3}\right)f\begin{pmatrix} 1 & 0 \\ +0 & 1\end{pmatrix}\end{pmatrix}+\frac{q^2}{s^4}fO(\delta)\right]. +$$ + + +**Proof.** To diagonalize $(R_{ijkl}(y))$ we need only rotate in the +$\{f_1\wedge f_2,f_2\wedge f_3\}$-plane. Applying this rotation to +the expression in \entryref{52d3cb562bdc} gives the result. + + +**Corollary.** There is a constant $A<\infty$ such that the following holds for the given value of $q$ +and any $C_0$ provided that $\delta$ is sufficiently small. +Suppose that the eigenvalues for the curvature matrix of $h$ at $y$ +are $\lambda\ge \mu\ge \nu$. +Then the eigenvalues for the curvature of $\hat h$ at the point $y$ +are given by $\lambda',\mu',\nu'$, where + +$$ +\left|\lambda'-e^{2f}\left(\lambda-\frac{q^2}{s^4}f^2\right)\right| +\le \frac{q^2}{s^4}fA\delta +$$ + +$$ +\left|\mu'-e^{2f}\left(\mu+\left(\frac{q^2}{s^4}-\frac{2q}{s^3}\right)f\right)\right|\le +\frac{q^2}{s^4}fA\delta +$$ + +$$ +\left|\nu'-e^{2f}\left(\nu+\left(\frac{q^2}{s^4}-\frac{2q}{s^3}\right)f\right)\right|\le +\frac{q^2}{s^4}fA\delta. +$$ + +In particular, we have + +$$ +\nu'\ge e^{2f}\left(\nu+\frac{q^2}{2s^4}f\right) +$$ + +$$ +\mu'\ge e^{2f}\left(\mu+\frac{q^2}{2s^4}f\right). +$$ + + +**Proof.** Let $\{f_0,f_1,f_2\}$ be the $h$-orthonormal basis given in +\entryref{347896f0385b}. Then $\{e^ff_0,e^ff_1,e^ff_2\}$ is orthonormal for $\hat +h=e^{-2f}h$. This change multiplies the curvature matrix by $e^{4f}$. Since +$f\ll 1$, $e^{4f}<2$ so that the expression for $(\hat R_{ijkl}(y))$ in this +basis is exactly the same as in \entryref{0a41b5c61dd2} except that the factor +in front is $e^{2f}$ instead of $e^{-2f}$. Now, it is easy to see that since +$((q^2/s^4)fA\delta)^2\ll (q^2/s^4)fA\delta$ the eigenvalues will differ from +the diagonal entries by at most a constant multiple of $(q^2/s^4)fA\delta$. + + The first three inequalities are immediate +from the previous corollary. The last two follow since $q^2/s^4\gg q/s^3$ and +$\delta\ll 1$. + +One important consequence of this computation is the following: + + +**Corollary.** For the given value of $q$ and for any $C_0$, +assuming that $\delta>0$ is sufficiently small, the smallest +eigenvalue of $\mathit{Rm}_{\hat h}$ is greater than the smallest eigenvalue +of $\mathit{Rm}_h$ at the same point. Consequently, at any point where $h$ +has non-negative curvature so does $\hat h$. + + +**Proof.** Since $|\lambda-1/2|, |\mu|, |\nu|$ are all $O(\delta)$ and since +$\frac{q^2}{s^4}f\ll 1$, it follows that the smallest eigenvalue of $(\hat +R_{ijkl}(y))$ is either $\mu'$ or $\nu'$. But it is immediate from the above +expressions that $\mu'>\mu$ and $\nu'>\nu$. This completes the proof. + +Now we are ready to fix $C_0$. +There is a universal constant $K$ such that for all $\delta>0$ sufficiently small +and for any $\delta$-neck $(N,h)$ of +scale one, every eigenvalue of $\mathit{Rm}_h$ is at least $-K\delta$. +We set + +$$ +C_0=2Ke^{q}. +$$ + + +**Lemma.** With these choices of $q$ and $C_0$ for any $\delta>0$ sufficiently small we have +$\nu'>0$ and $\mu'>0$ for $s\in [1,4+A_0]$ and $\lambda'>1/4$. + + +**Proof.** Then by the previous result we have + +$$ +\nu'\ge e^{2f}\left(\nu+\frac{q^2}{2s^4}f\right). +$$ + +It is easy to see that since $q\gg (4+A_0)$ the function $(q^2/2s^4)f$ is an +increasing function on $[1,4+A_0]$. Its value at $s=1$ is +$(q^2/2)e^{-q}C_0\delta>K\delta$. Hence $\nu+\frac{q^2}{2s^4}f>0$ for all $s\in +[1,4+A_0]$ and consequently $\nu'>0$ on this submanifold. The same argument +shows $\mu'>0$. Since $q^2/s^4 f^2\ll 1$ and $0 +**Proposition.** Fix $\delta>0$ sufficiently small. Suppose that for some $t\ge 0$ +and every point $p\in N$ the curvature of $h$ satisfies: + +- **(1)** $R(p)\ge \frac{-6}{1+4t}$, and +- **(2)** $R(p)\geq 2X(p)\left(\mathit{log}X(p)+\mathit{log}(1+t)-3\right)$ whenever $0 + +**Proof.** According to \entryref{e70bacf8aab0}, the smallest eigenvalue of $\hat +h$ at any point $p$ is greater than or equal to the smallest +eigenvalue of $h$ at the corresponding point. According to +\entryref{7f0b4a692c5e}, $\hat R(p)\ge R(p)$ for every $p\in +{\mathcal S}$. Hence, $X_{\hat h}(p)\le X_h(p)$. If $X_h(p)\ge +e^3/(1+t)$, then we have + +$$ +\hat R(p)\ge R(p)\ge 2X_h(p)(\mathit{log}(X_h(p)+\mathit{log}(1+t)-3)\ge 2X_{\hat h}(p) +(\mathit{log}(X_{\hat h}(p)+\mathit{log}(1+t)-3). +$$ + +If $X_h(p)2X_{\hat p}(p)(\mathit{log}(X_{\hat h}(p)+\mathit{log}(1+t)-3). +$$ + +This completes the proof in both cases. + +This establishes the first item in the conclusion of +\entryref{cf61b731c6f2} for $\delta>0$ sufficiently small on +$s^{-1}(-\delta^{-1},4)$. As we have seen in \entryref{cc9110b4d2fe}, +the curvature is positive on $s^{-1}[1,4)$. + +### Proof of the first two items for $s\ge 4$ + +Now let us show that the curvature on $\tilde g$ is positive in the region +$s^{-1}([4,4+A_0])$. First of all in the preimage of the +interval $[4,4+A_0,-r_0]$ this follows +from \entryref{e70bacf8aab0} and the fact that $\eta g_0$ has non-negative +curvature. As for the region $s^{-1}([4+A_0-r_0,4+A_0])$, as $\delta$ tends to +zero, the metric here tends smoothly to the restriction of the metric $g_0$ to +that subset. The metric $g_0$ has positive curvature on +$s^{-1}([4+A_0-r_0,4+A_0])$. Thus, for all $\delta>0$ sufficiently small the +metric $\tilde g$ has positive curvature on all of $s^{-1}([4+A_0-r_0,4+A_0])$. +This completes the proof of the first two items. + +### Proof of the third item + + By +construction the restriction of the metric $\widetilde g$ to +$s^{-1}((-\delta^{-1},0])$ is equal to the metric $\rho^*g$. Hence, +in this region the mapping is an isometry. In the region +$s^{-1}([0,4])$ we have $R(x_0)\rho^*g\ge \eta g_0$ so that by +construction in this region $\rho^*g\ge \tilde g$. Lastly, in the +region $s^{-1}([4,A_0+4])$ we have $R(x_0)^{-1}\eta g_0\ge \tilde +g$. On the other hand, it follows from \entryref{6168ffcb3bed} that +the map from $([0,\delta^{-1}]\times S^2,R(x_0)\rho^* g)$ to +$(B(p_0,4+A_0),\eta g)$ is distance decreasing. This completes the +proof of the third item. + +### Completion of the proof + +As $\delta$ goes to zero, $f$ tends to zero in the +$C^\infty$-topology and $\eta$ limits to $1$. From this the fourth +item is clear. + +This completes the proof of \entryref{cf61b731c6f2}. + +## Other properties of the result of surgery + + +**Lemma.** Provided that $\delta>0$ is sufficiently small the following holds. Let $(N,g)$ +be a $\delta$-neck and let $({\mathcal S},\tilde g)$ be the result of surgery +along the cental $2$-sphere of this neck. Then for any $0 + +**Proof.** The Riemannian manifold $({\mathcal S},\tilde g)$ is identified by a +diffeomorphism with the union of $s_{N}^{-1}(-\delta^{-1},0]$ to +$B_{g_0}(p_0,A_0+4)$ glued along their boundaries. Thus, we have a natural +identification of ${\mathcal S}$ with the ball $B_{g_0}(p,A_0+4+\delta^{-1})$ +in the standard solution. This identification pulls back the metric $\tilde g$ +to be within $2\delta$ of the standard initial metric. The result then follows +immediately for $\delta$ sufficiently small. diff --git a/projects/poincare-conjecture/.astrolabe/docs-src/15-ricci-flow-with-surgery-the-definition.mdx b/projects/poincare-conjecture/.astrolabe/docs-src/15-ricci-flow-with-surgery-the-definition.mdx new file mode 100644 index 00000000..885a2fb1 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/docs-src/15-ricci-flow-with-surgery-the-definition.mdx @@ -0,0 +1,594 @@ + + +# Chapter 14 -- Ricci Flow with surgery: the definition + +In this chapter we introduce Ricci flows with surgery. These objects +are closely related to generalized Ricci flows but they differ +slightly. The space-time of a Ricci flow with surgery has an open +dense subset that is a manifold, and the restriction of the Ricci +flow with surgery to this open subset is a generalized Ricci flow. +Still there are other, more singular points allowed in a Ricci flow +with surgery. + +## Surgery space-time + + +**Definition.** By a *space-time* we mean a paracompact Hausdorff space +${\mathcal M}$ with a continuous function $\mathbf{t}\colon {\mathcal +M}\to \Ar$, called *time*. We require that the image of $\mathbf{t}$ be an interval $I$, finite or infinite with or without +endpoints, in $\Ar$. The interval $I$ is called the *time-interval of +definition of space-time*. The initial point of $I$, if there is +one, is the *initial time* and the final point of $I$, if there +is one, is the *final time*. The level sets of $\mathbf{t}$ are +called the *time-slices* of space-time, and the preimage of the +initial (resp., final) point of $I$ is the *initial* (resp., +*final*) time-slice. + + +We are interested in a certain class of space-times, which we call +*surgery space-times*. These objects have a 'smooth structure' +(even though they are not smooth manifolds). As in the case of a +smooth manifold, this smooth structure is given by local coordinate +charts with appropriate overlap functions. + +### An exotic chart + +There is one exotic chart, and we begin with its description. To +define this chart we consider the open unit square $(-1,1)\times +(-1,1)$. We shall define a new topology, denoted by ${\mathcal P}$, +on this square. The open subsets of ${\mathcal P}$ are the open +subsets of the usual topology on the open square together with open +subsets of $(0,1)\times [0,1)$. Of course, with this topology the +'identity' map $\iota\colon {\mathcal P}\to (-1,1)\times (-1,1)$ is +a continuous map. Notice that the restriction of the topology of +${\mathcal P}$ to the complement of the closed subset $[0,1)\times +\{0\}$ is a homeomorphism onto the corresponding subset of the open +unit square. Notice that the complement of $(0,0)$ in ${\mathcal P}$ +is a manifold with boundary, the boundary being $(0,1)\times\{0\}$. +(See Fig. 0.5 in the Introduction.) + +Next, we define a 'smooth structure' on ${\mathcal P}$ by defining a +sheaf of germs of 'smooth' functions. The restriction of this sheaf +of germs of 'smooth functions' to the complement of $(0,1)\times +\{0\}$ in ${\mathcal P}$ is the usual sheaf of germs of smooth +functions on the corresponding subset of the open unit square. +In particular, a function is smooth near $(0,0)$ if and only if its restriction +to some neighborhood of $(0,0)$ is the pullback under $\iota$ +of a usual smooth function on a +neighborhood of the origin in the square. + Now +let us consider the situation near a point of the form $x=(a,0)$ for +some $0 +**Definition.** A *surgery space-time* is a +space-time ${\mathcal M}$ equipped with a maximal atlas of charts +covering ${\mathcal M}$, each chart being of one of the three types +listed above, with the overlap functions being diffeomorphisms +preserving the functions $\mathbf{t}$ and the vector fields $\chi$. The +points with neighborhoods of the first type are called *smooth +points*, those with neighborhoods of the second type but not the +first type are called *exposed points*, +and all the other points are called *singular +points*. Notice that the union of +the set of smooth points and the set of exposed points forms a +smooth manifold with boundary (possibly disconnected). Each +component of the boundary of this manifold is contained in a single +time-slice. The union of those components contained in a time +distinct from the initial time and the final time is called the *exposed region*. and the boundary points of +the closure of the exposed region form the set of the singular +points of ${\mathcal M}$. (Technically, the exposed points are +singular, but we reserve this word for the most singular points.) An +$(n+1)$-dimensional surgery space-time is by definition of +homogeneous dimension $n+1$. + +By construction, the local smooth functions $\mathbf{t}$ are compatible +on the overlaps and hence fit together to define a global smooth +function $\mathbf{t}\colon {\mathcal M}\to \Ar$, called the *time* +function. The level sets of this function are called the *time-slices* of the space-time, and $\mathbf{t}^{-1}(t)$ is denoted +$M_t$. Similarly, the tangent bundles of the various charts are +compatible under the overlap diffeomorphisms and hence glue together +to give a global smooth tangent bundle on space-time. The smooth +sections of this vector bundle, the smooth vector fields on space +time, act as derivations on the smooth functions on space-time. The +tangent bundle of an $(n+1)$-dimensional surgery space-time is a +vector bundle of dimension $(n+1)$. Also, by construction the local +vector fields $\chi$ are compatible and hence glue together to +define a global vector field, denoted $\chi$. The vector field and +time function satisfy + +$$ +\chi(\mathbf{t})=1. +$$ + +At the manifold points (including the exposed points) it is a usual +vector field. Along the exposed region and the initial time-slice +the vector field points into the manifold; along the final +time-slice it points out of the manifold. + + + +**Definition.** Let ${\mathcal M}$ be a surgery space-time. Given a space $K$ and an +interval $J\subset \Ar$ we say that an embedding $K\times J\to +{\mathcal M}$ is *compatible with time and the vector field* +if: (i) the restriction of $\mathbf{t}$ to the image agrees with the +projection onto the second factor and (ii) for each $x\in X$ the +image of $\{x\}\times J$ is the integral curve for the vector field +$\chi$. + If in addition $K$ is a subset of $M_{t}$ +we require that $t\in J$ and that the map $K\times\{t\}\to M_{t}$ be +the identity. Clearly, by the uniqueness of integral curves for vector fields, two such +embeddings agree on their common interval of definition, so that, +given $K\subset M_{t}$ there is a maximal interval $J_K$ containing +$t$ such that such an embedding is defined on $K\times J_K$. In the +special case when $K=\{x\}$ for a point $x\in M_{t}$ we say that +such an embedding is *the maximal flow line* through $x$. The +embedding of the maximal interval through $x$ compatible with time +and the vector field $\chi$ is called *the domain of definition* +of the flow line through $x$. For a more general subset $K\subset +M_{t}$ there is an embedding $K\times J$ compatible with time and +the vector field $\chi$ if and only if, for every $x\in K$, the +interval $J$ is contained in the domain of definition of the flow +line through $x$. + + + +**Definition.** Let ${\mathcal M}$ be a surgery space-time with $I$ as its time +interval of definition. We say that $t\in I$ is a *regular* time +if there is an interval $J\subset I$ which is an open neighborhood +in $I$ of $t$, and a diffeomorphism $M_{t}\times J\to \mathbf{t}^{-1}(J)\subset{\mathcal M}$ compatible with time and the vector +field. A time is *singular* if it is not regular. Notice that if +all times are regular, then space-time is a product $M_t\times I$ +with $\mathbf{t}$ and $\chi$ coming from the second factor. + + + +**Lemma.** Let ${\mathcal M}$ be an $(n+1)$-dimensional surgery space-time, and fix $t$. +The restriction of the smooth structure on ${\mathcal M}$ to the time-slice +$M_{t}$ induces the structure of a smooth $n$-manifold on this time-slice. That +is to say, we have a smooth embedding of $M_{t}\to {\mathcal M}$. This smooth +embedding identifies the tangent bundle of $M_{t}$ with a codimension-one +subbundle of the restriction of tangent bundle of ${\mathcal M}$ to $M_{t}$. +This subbundle is complementary to the line field spanned by $\chi$. These +codimension-one subbundles along the various time-slices fit together to form +a smooth, codimension-one subbundle of the tangent bundle of space-time. + + +**Proof.** These statements are immediate for any coordinate patch, and hence +are true globally. + + +**Definition.** We call the codimension-one subbundle of the tangent bundle of +${\mathcal M}$ described in the previous lemma the *horizontal +subbundle*, and we denote it +${\mathcal HT}({\mathcal M})$. + + +## The generalized Ricci flow equation + +In this section we introduce the Ricci flow equation for surgery +space-times, resulting in an object that we call Ricci flow with +surgery. + +### Horizontal metrics + + +**Definition.** By a *horizontal metric $G$ on a surgery space-time ${\mathcal +M}$* we mean a $C^\infty$ metric on ${\mathcal HT}{\mathcal M}$. For +each $t$, the horizontal metric $G$ induces a Riemannian metric, +denoted $G(t)$, on the time-slice $M_t$. Associated to a horizontal +metric $G$ we have the *horizontal covariant derivative*, +denoted $\nabla$. This is a pairing between horizontal vector fields + +$$ +X\otimes Y\mapsto \nabla_XY. +$$ + +On each time slice $M_t$ it is the usual Levi-Civita connection associated to +the Riemannian metric $G(t)$. Given a function $F$ on space-time, by its +gradient $\nabla F$ we mean its horizontal gradient. The value of this gradient +at a point $q\in M_t$ is the usual $G(t)$-gradient of $F|_{M_t}$. In +particular, $\nabla F$ is a smooth horizontal vector field on space-time. The +horizontal metric $G$ on space-time has its (horizontal) curvatures $\mathit{Rm}_G$. +These are smooth symmetric endomorphisms of the second exterior power of +${\mathcal HT}{\mathcal M}$. The value of $\mathit{Rm}_G$ at a point $q\in M_t$ is +simply the usual Riemann curvature operator of $G(t)$ at the point $q$. +Similarly, we have the (horizontal) Ricci curvature $\mathit{Ric}=\mathit{Ric}_G$, a +section of the symmetric square of the horizontal cotangent bundle, and the +(horizontal) scalar curvature denoted $R=R_G$. + + +The only reason for working in ${\mathcal HT}{\mathcal M}$ rather +than individually in each slice is to emphasize the fact that all +these horizontal quantities vary smoothly over the surgery +space-time. + +Suppose that $t\in I$ is not the final time and suppose that +$U\subset M_{t}$ is an open subset with compact closure. Then there +is $\epsilon>0$ and an embedding $i_U\colon +U\times[t,t+\epsilon)\subset {\mathcal M}$ compatible with time and +the vector field. Of course, two such embeddings agree on their +common domain of definition. Notice also that for each $t'\in +[t,t+\epsilon)$ the restriction of the map $i_U$ to $U\times \{t'\}$ +induces an diffeomorphism from $U$ to an open subset $U_{t'}$ of +$M_{t'}$. It follows that the local flow generated by the vector +field $\chi$ preserves the horizontal subbundle. Hence, the vector +field $\chi$ acts by Lie derivative on the sections of ${\mathcal +HT}({\mathcal M})$ and on all associated bundles (for example the +symmetric square of the dual bundle). + +### The equation + + +**Definition.** A *Ricci flow with surgery* is a pair $({\mathcal M},G)$ +consisting of a surgery space-time ${\mathcal M}$ and a horizontal +metric $G$ on ${\mathcal M}$ such that for every $x\in {\mathcal M}$ + we have + +$$ +{\mathcal L}_\chi(G)(x)=-2\mathit{Ric}_G(x)) +$$ + + as sections +of the symmetric square of the dual to ${\mathcal HT}({\mathcal +M})$. If space-time is $(n+1)$-dimensional, then we say that the +Ricci flow with surgery is $n$-dimensional (meaning of course that +each time-slice is an $n$-dimensional manifold). + + + +**Remark.** Notice that at an exposed point and at points at the initial and the final time + the Lie derivative is a one-sided derivative. + + +### Examples of Ricci flows with surgery + + +**Example.** One example of a Ricci flow with surgery is ${\mathcal M}=M_0\times +[0,T)$ with time function $\mathbf{t}$ and the vector field $\chi$ +coming from the second factor. In this case the Lie derivative +${\mathcal L}_\chi$ agrees with the usual partial derivative in the +time direction, and hence our generalized Ricci flow equation is the +usual Ricci flow equation. This shows that an ordinary Ricci flow is indeed +a Ricci flow with surgery. + + +The next lemma gives an example of a Ricci flow with surgery where +the topology of the time-slices changes. + + +**Lemma.** Suppose that we have manifolds $M_1\times (a,b]$ and $M_2\times +[b,c)$ and compact, smooth codimension-$0$ submanifolds +$\Omega_1\subset M_1$ and $\Omega_2\subset M_2$ with open +neighborhoods $U_1\subset M_1$ and $U_2\subset M_2$ respectively. +Suppose we have a diffeomorphism $\psi\colon U_1\to U_2$ carrying +$\Omega_1$ onto $\Omega_2$. Let $(M_1\times (a,b])_0$ be the subset +obtained by removing $(M_1\setminus \Omega_1)\times \{b\}$ from +$M_1\times (a,b]$. Form the topological space + +$$ +{\mathcal M}=(M_1\times (a,b])_0\cup M_2\times [b,c) +$$ + +where $\Omega_1\times\{b\}$ in $(M_1\times (a,b])_0$ is identified +with $\Omega_2\times \{b\}$ using the restriction of $\psi$ to +$\Omega_1$. Then ${\mathcal M}$ naturally inherits the structure of +a surgery space-time where the time function restricts to +$(M_1\times (a,b])_0$ and to $M_2\times [b,c)$ to be the projection +onto the second factor and the vector field $\chi$ agrees with the +vector fields coming from the second factor on each of $(M_1\times +(a,b])_0$ and $M_2\times [b,c)$. + + Lastly, given Ricci +flows $(M_1,g_1(t)),\ a + +**Proof.** As the union of Hausdorff spaces along closed subsets, ${\mathcal +M}$ is a Hausdorff topological space. The time function is the one +induced from the projections onto the second factor. For any point +outside the $b$ time-slice there is the usual smooth coordinate +coming from the smooth manifold $M_1\times (a,b)$ (if $tb$). At any point of +$(M_2\setminus\Omega_2)\times \{b\}$ there is the smooth manifold +with boundary coordinate patch coming from $M_2\times[b,c)$. For any +point in $\mathit{int}(\Omega_1)\times\{b\}$ we have the smooth +manifold structure obtained from gluing $(\mathit{int}(\Omega_1))\times +(a,b]$ to $\mathit{int}(\Omega_2)\times [b,c)$ along the $b$ time-slice +by $\psi$. Thus, at all these points we have neighborhoods on which +our data determine a smooth manifold structure. Lastly, let us +consider a point $x\in +\partial \Omega_1\times \{b\}$. +Choose local coordinates $(x^1,\ldots,x^n)$ for a neighborhood $V_1$ +of $x$ such that $\Omega_1\cap V_1=\{x^n\le 0\}$. We can assume that +$\psi$ is defined on all of $V_1$. Let $V_2=\psi(V_1)$ and take the + local coordinates on $V_2$ induced from the $x^i$ on $V_1$. + Were we to identify $V_1\times (a,b]$ +with $V_2\times [b,c)$ along the $b$ time-slice using this map, then +this union would be a smooth manifold. There is a +neighborhood of the point $(x,b)\in {\mathcal M}$ which is obtained +from the smooth manifold $V_1\times (a,b]\cup_\psi V_2\times [b,c)$ +by inducing a new topology where the open subsets are, in addition +to the usual ones, any open subset of the form $\{x^n>0\}\times +[b,b')$ where $b +**Proposition.** Suppose that $({\mathcal M},G)$ is a Ricci flow with surgery. Let +$\mathit{int}{\mathcal M}$ be the open subset consisting of all +smooth $(n+1)$-manifold points, plus all manifold-with-boundary +points at the initial time and the final time. This space-time +inherits the structure of a smooth manifold with boundary. This +structure together with the restrictions to it of $\mathbf{t}$ and the +vector field $\chi$ and the restriction of the horizontal metric $G$ +form a generalized Ricci flow whose underlying smooth manifold is $\mathit{int}{\mathcal M}$. + + +### Scaling and translating Ricci flows with surgery + +Suppose that $({\mathcal M},G)$ is a Ricci flow with surgery. Let +$Q$ be a positive constant. Then we can define a new Ricci flow +with surgery by setting $G'=QG$, $\mathbf{t'}=Q\mathbf{t}$ and +$\chi'=Q^{-1}\chi$. It is easy to see that the resulting data still +satisfies the generalized Ricci flow equation, +Equation (14.1). We denote this new Ricci flow with surgery by +$(Q{\mathcal M},QG)$ where the changes in $\mathbf{t}$ and $\chi$ are +indicated by the factor $Q$ in front of the space-time. + +It is also possible to translate a Ricci flow with surgery $({\mathcal +M},G)$ by replacing the time function $\mathbf{t}$ by $\mathbf{t'}=\mathbf{t}+a$ for any constant $a$, and leaving $\chi$ and $G$ unchanged. + +### More basic definitions + + +**Definition.** Let $({\mathcal M},G)$ be a Ricci flow with surgery, and let $x$ be +a point of space-time. Set $t=\mathbf{t}(x)$. For any $r>0$ we define +$B(x,t,r)\subset M_t$ to be the metric ball of radius $r$ centered +at $x$ in the Riemannian manifold $(M_t,G(t))$. + + + +**Definition.** Let $({\mathcal M},G)$ be a Ricci flow with surgery, and let $x$ be +a point of space-time. Set $t=\mathbf{t}(x)$. For any $r>0$ and $\Delta +t>0$ we say that the *backward parabolic neighborhood* +$P(x,t,r,-\Delta t)$ *exists* in ${\mathcal M}$ if there is an +embedding $B(x,t,r)\times (t-\Delta t,t]\to {\mathcal M}$ compatible +with time and the vector field. Similarly, we say +that the *forward parabolic neighborhood* $P(x,t,r,\Delta t)$ +*exists* in ${\mathcal M}$ if there is an embedding +$B(x,t,r)\times [t,t+\Delta t)\to {\mathcal M}$ compatible with time +and the vector field. A *parabolic neighborhood* is either a +forward or backward parabolic neighborhood. + + + +**Definition.** Fix $\kappa>0$ and $r_0>0$. We say that a Ricci flow with surgery +$({\mathcal M},G)$ is *$\kappa$-noncollapsed on scales $\le +r_0$* if the following holds for every point $x\in {\mathcal M}$ and +for every $r\le r_0$. Denote $\mathbf{t}(x)$ by $t$. If the parabolic +neighborhood $P(x,t,r,-r^2)$ exists in ${\mathcal M}$ and if +$|\mathit{Rm}_G|\le r^{-2}$ on $P(x,t,r,-r^2)$, then $\mathit{Vol} B(x,t,r)\ge \kappa r^3$. + + + +**Remark.** For $\epsilon>0$ sufficiently small, +an $\epsilon$-round component satisfies the first condition in the above definition + for some $\kappa>0$ depending only on +the order of the fundamental group of the underlying manifold, but there is no universal +$\kappa>0$ that works for all $\epsilon$-round manifolds. Fixing an integer $N$ let ${\mathcal C}_N$ +be the class of closed $3$-manifolds + with the property that any finite free factor of $\pi_1(M)$ +has order at most $N$. Then any $\epsilon$-round component of any +time-slice of any Ricci flow $({\mathcal M},G)$ whose initial +conditions consist of a manifold in ${\mathcal C}_N$ will have +fundamental group of order at most $N$ and hence will satisfy the first condition +in the above definition for some $\kappa>0$ depending only on $N$. + + +We also have the notion of the curvature being pinched toward +positive, analogous to the notions for Ricci flows and generalized Ricci flows. + + +**Definition.** Let $({\mathcal M},G)$ be a $3$-dimensional Ricci flow with surgery, +whose time domain of definition is contained in $[0,\infty)$. For +any $x\in {\mathcal M}$ we denote the eigenvalues of $\mathit{Rm}(x)$ by +$\lambda(x)\ge \mu(x)\ge \nu(x)$ and we set $X(x)=\mathit{max}(0,-\nu(x))$. + We say +that its *curvature is pinched toward positive* if the following +hold for every $x\in {\mathcal M}$: + +- **(1)** $R(x)\ge \frac{-6}{1+4\mathbf{t}(x)}$. +- **(2)** $R(x)\geq 2X(x)\left(\mathit{log}X(x)+\mathit{log}(1+\mathbf{t}(x))-3\right)$, whenever $0 + +Lastly, there is the definition of canonical neighborhoods for a +Ricci flow with surgery, there is the following extension of the +notion for a generalized Ricci flow. + + +**Definition.** Fix constants $(C,\epsilon)$ and a constant $r$. We say that a Ricci +flow with surgery $({\mathcal M},G)$ *satisfies the strong +$(C,\epsilon)$-canonical neighborhood assumption with parameter $r$* +if every point $x\in {\mathcal M}$ with $R(x)\ge r^{-2}$ has a strong +$(C,\epsilon)$-canonical neighborhood in ${\mathcal M}$. In all +cases except that of the strong $\epsilon$-neck, the strong +canonical neighborhood of $x$ is a subset of the time-slice +containing $x$, and the notion of a $(C,\epsilon)$-canonical +neighborhood has exactly the same meaning as in the case of an +ordinary Ricci flow. In the case of a strong $\epsilon$-neck +centered at $x$ this means that there is an embedding +$\left(S^2\times (-\epsilon^{-1},\epsilon^{-1})\right)\times (\mathbf{t}(x)-R(x)^{-1},\mathbf{t}(x)]\to {\mathcal M}$, mapping $(q_0,0)$ to $x$, + where $q_0$ is the basepoint of $S^2$, an embedding +compatible with time and the vector field, +such that the pullback +of $G$ is a Ricci flow on $S^2\times (-\epsilon^{-1},\epsilon^{-1})$ +which, when the time is shifted by $-\mathbf{t}(x)$ and then the flow +is rescaled by $R(x)$, is within $\epsilon$ in the +$C^{[1/\epsilon]}$-topology of the standard evolving round cylinder +$\left(S^2\times(-\epsilon^{-1},\epsilon^{-1}),h_0(t)\times +ds^2\right),\ -1 + +Notice that $x$ is an exposed point or sufficiently close to an exposed point +then $x$ cannot be the center of a strong $\epsilon$-neck. diff --git a/projects/poincare-conjecture/.astrolabe/docs-src/16-controlled-ricci-flows-with-surgery.mdx b/projects/poincare-conjecture/.astrolabe/docs-src/16-controlled-ricci-flows-with-surgery.mdx new file mode 100644 index 00000000..4f77f18c --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/docs-src/16-controlled-ricci-flows-with-surgery.mdx @@ -0,0 +1,785 @@ + + +# Chapter 15 -- Controlled Ricci flows with surgery + + We do not +wish to consider all Ricci flows with surgery. Rather we shall +concentrate on $3$-dimensional flows (that is to say $4$-dimensional +space-times) whose singularities are closely controlled both +topologically and geometrically. We introduce the hypotheses that we +require these evolutions to satisfy. Then main result, which is +stated in this chapter and proved in the next two, is that these +controlled $3$-dimensional Ricci flows with surgery always exist for +all time with any compact $3$-manifold as initial metric. + +### Normalized initial conditions + +Consider a compact connected Riemannian $3$-manifold $(M,g(0))$ +satisfying + +- **(1)** $|\mathit{Rm}(x,0)|\le 1$ for all $x\in M$ and +- **(2)** for every $x\in M$ we have $\mathit{Vol} B(x,0,1)\ge \omega/2$ where +$\omega$ is the volume of the unit ball in $\Ar^3$. + +Under these conditions we say that $(M,g(0))$ is *normalized*. +Also, if $(M,g(0))$ is the initial manifold of a Ricci flow with +surgery then we say that it is a *normalized initial metric*. Of +course, given any compact Riemannian $3$-manifold $(M,g(0))$ there +is a positive constant $Q<\infty$ such that $(M,Qg(0))$ is +normalized. + +Starting with a normalized initial metric implies that the flow exists +and has uniformly bounded curvature for a fixed amount of time. This is the +content of the following claim which is an immediate corollary of +\entryref{e6b1bd9fa0d7}, \entryref{b6bc786f7f44}, \entryref{ed4b72caf9f9}, +and \entryref{c1330c5a8be3}. + + +**Claim.** There is $\kappa_0$ such that the following holds. Let $(M,g(0))$ be a normalized initial +metric. Then the solution to the Ricci flow equation with these +initial conditions exists for $t\in [0,2^{-4}]$, and $|R(x,t)|\le +2$ for all $x\in M$ and all $t\in [0,2^{-4}]$. Furthermore, for any +$t\in [0,2^{-4}]$ and any $x\in M$ and any $r\le \epsilon$ we have +$\mathit{Vol} B(x,t,r)\ge \kappa_0r^3$. + + +## Gluing together evolving necks + + +**Proposition.** There is $0<\beta<1/2$ such that the following holds for any +$\epsilon<1$. Let $(N\times [-t_0,0],g_1(t))$ be an evolving +$\beta\epsilon$-neck centered at $x$ with $R(x,0)=1$. Let $ +(N'\times (-t_1,-t_0],g_2(t))$ be a strong $\beta\epsilon/2$-neck. +Suppose we have an isometric embedding of $N\times \{-t_0\}$ with +$N'\times \{-t_0\}$ and the strong $\beta\epsilon/2$-neck structure on $N'\times +(-t_1,-t_0]$ is centered at the image of $(x,-t_0]$. Then the union + +$$ +N\times [-t_0,0]\cup N'\times (-t_1,-t_0] +$$ + + with the induced +one-parameter family of metrics contains a strong $\epsilon$-neck +centered at $(x,0)$. + + +**Proof.** Suppose that the result does not hold. Take a sequence of $\beta_n$ +tending to zero and counterexamples $(N_n\times +[-t_{0,n},0],g_{1,n}(t));\ (N'_n\times +(-t_{1,n},-t_{0,n}],g_{2,n}(t))$. Pass to a subsequence so that the +$t_{0,n}$ tend to a limit $t_{0,\infty}\ge 0$. Since $\beta_n$ tends +to zero, we can take a smooth limit of a subsequence and this limit +is an evolving cylinder $(S^2\times \Ar, h(t)\times ds^2)$, where +$h(t)$ is the round metric of scalar curvature $1/(1-t)$ defined for +some amount of backward time. Notice that, for all $\beta$ +sufficiently small, on a $\beta\epsilon$-neck the derivative of the +scalar curvature is positive. Thus, $R_{g_{1,n}}(x,-t_{0,n})<1$. +Since we have a strong neck structure on $N'_n$ centered at +$(x,-t_{0,n})$, this implies that $t_{1,n}>1$ so that the limit is +defined for at least time $t\in [0, 1+t_{0,\infty})$. If +$t_{0,\infty}>0$, then, restricting to the appropriate subset of +this limit, a subset with compact closure in space-time, it follows +immediately that for all $n$ sufficiently large there is a strong +$\epsilon$-neck centered at $(x,0)$. This contradicts the assumption +that we began with a sequence of counterexamples to the +proposition. + +Let us consider the case when $t_{0,\infty}=0$. In this case the +smooth limit is an evolving round cylinder defined for time +$(-1,0]$. + Since $t_{1,n}>1$ we +see that for any $A<\infty$ for all $n$ sufficiently large the ball +$B(x_n,0,A)$ has compact closure in every time-slice and there are +uniform bounds to the curvature on $B(x_n,0,A)\times (-1,0]$. This +means that the limit is uniform for time $(-1,0]$ on all these +balls. Thus, once again for all $n$ sufficiently large we see that +$(x,0)$ is the center of a strong $\epsilon$-neck in the union. In +either case we have obtained a contradiction, and hence we have +proved the result. See Fig. 15.1. + +### First assumptions + +**Choice of $C$ and $\epsilon$:** The first thing we need +to do is fix for the rest of the argument $C<\infty$ and +$\epsilon>0$. We do this in the following way. We fix $0<\epsilon\le +\mathit{min}(1/200,\left(\sqrt{D}(A_0+5)\right)^{-1},\bar\epsilon_1/2,\bar +\epsilon'/2,\epsilon_0)$ where $\bar\epsilon_1$ is the constant from +\entryref{cb9ca6d6db9f}, $\bar\epsilon'$ is the constant from +\entryref{c2be59f6090c}, $\epsilon_0$ is the constant from Section 10.1, +and $A_0$ and $D$ are the constants from +\entryref{0b07beec84c0}. + We fix $\beta<1/2$, the constant from +\entryref{a992d6a89479}. Then we let $C$ be the maximum of the +constant $C(\epsilon)$ as in \entryref{fc2d2036a96c} and +$C'(\beta\epsilon/3)+1$ as in \entryref{6ff28dba4e56}. + +For all such $\epsilon$, \entryref{ecd62fc2ea2b} holds for $\epsilon$ and +\entryref{cb9ca6d6db9f}, \entryref{7c2758f0e2b3} and +Corollaries \entryref{fc2d2036a96c} and \entryref{e724298e5bc3} and Theorems \entryref{d51aab8a8065} and \entryref{a9e023d082dc} + hold for $2\epsilon$. Also, all the topological results of the Appendix hold +for $2\epsilon$ and $\alpha=10^{-2}$. + +Now let us turn to the assumptions we shall make on the Ricci flows +with surgery that we shall consider. Let ${\mathcal M}$ be a +space-time. Our first set of assumptions are basically topological +in nature. They are: + +i̱gskip + + **Assumption (1). Compactness and dimension:** *Each +time-slice $M_t$ of space-time is a compact $3$-manifold +containing no embedded $\Ar P^2$ with trivial normal bundle.* + +i̱gskip + +**Assumption (2). Discrete singularities:** *The set of +singular times is a discrete subset of $\Ar$.* + +i̱gskip + + **Assumption (3). Normalized initial conditions:** *$0$ is the initial time of the Ricci flow with surgery and the +initial metric $(M_0,G(0))$ is normalized.* + +i̱gskip + + It follows +from Assumption (2) that for any time $t$ in the time-interval of +definition of a Ricci flow with surgery, with $t$ being distinct +from the initial and final times (if these exist), for all +$\delta>0$ sufficiently small, the only possible singular time in +$[t-\delta,t+\delta]$ is $t$. Suppose that $t$ is a singular time. +The singular locus at time $t$ is a closed, smooth subsurface +$\Sigma_{t}\subset M_{t}$. From the local +model, near every point of $x\in \Sigma_{t}$ we see that this +surface separates $M_{t}$ into two pieces: + +$$ +M_{t}=C_{t}\cup_{\Sigma_{t}}E_{t}, +$$ + +where $E_{t}$ is the exposed region at time $t$ and $C_{t}$ is the +complement of the interior of $E_{t}$ in $M_{t}$. We call $C_t$ the +*continuing region*. $C_t\subset M_t$ is the maximal subset of +$M_t$ for which there is $\delta>0$ and an embedding $C_t\times +(t-\delta,t]\to {\mathcal M}$ compatible with time and the vector +field. + + **Assumption (4). Topology of the exposed regions:** *At all singular times $t$ we require that $E_{t}$ be a finite +disjoint union of $3$-balls. In particular, $\Sigma_{t}$ is a +finite disjoint union of $2$-spheres.* + +i̱gskip + +The next assumptions are geometric in nature. +Suppose that $t$ is a surgery time. Let ${\mathcal M}_{(-\infty,t)}$ be $\mathbf{t}^{-1}((-\infty,t))$ +and let $(\widehat {\mathcal M}_{(-\infty,t)},\widehat G)$ be the maximal extension of +$({\mathcal M}_{(-\infty,t)},G)$ to time $t$, as given in \entryref{8651cf20bd2d}. + +**Assumption (5). Boundary components of the exposed +regions:** *There is a surgery control parameter +function, +$\delta(t)>0$, a non-increasing function of +$t$, such that each component of $\Sigma_{t}\subset M_{t}$ is the +central $2$-sphere of a strong $\delta(t)$-neck in $(\widehat{\mathcal +M}_{(-\infty,t)},\widehat G)$.* + +i̱gskip + +Suppose that $t$ is a singular time. Then for all $t^- +**Proposition.** Suppose that $({\mathcal M},G)$ is a generalized Ricci flow satisfying Assumptions +(1) -- (7). Let $t$ be a singular time. Then the following holds +for any $t^- + +**Proof.** Fix $t' +**Corollary.** Let $({\mathcal M},G)$ be a generalized Ricci flow satisfying Assumptions +(1) -- (7) with initial conditions $(M,g(0))$. Suppose +that for some $T$ the time-slice $M_T$ of this generalized flow +satisfies Thurston's Geometrization Conjecture. Then the same is true for the manifold +$M_t$ for any $t\le T$, and in particular $M$ satisfies Thurston's +Geometrization Conjecture. In addition, + +- **(1)** If for some $T>0$ the manifold $M_T$ is empty, then $M$ is a +connected sum of manifolds diffeomorphic to $2$-sphere bundles over +$S^1$ and $3$-dimensional space-forms, i.e., compact +$3$-manifolds that admit a metric of constant positive curvature. +- **(2)** If for some $T>0$ the manifold $M_T$ is empty and if $M$ is connected and simply connected, +then $M$ is diffeomorphic to $S^3$. +- **(3)** If for some $T>0$ the manifold $M_T$ is empty and if $M$ has finite fundamental group, then $M$ +is a $3$-dimensional space-form. + + +**Proof.** Suppose that $M_T$ satisfies the Thurston Geometrization Conjecture and that $t_0$ is +the largest surgery time $\le T$. (If there is no such surgery time +then $M_T$ is diffeomorphic to $M$ and the result is established.) +Let $T'0$ also decays to zero rapidly as time goes to infinity. +Here then are the functions that will play the crucial role in defining +the surgery process. + + +**Definition.** We have: (i) *a canonical neighborhood parameter*, + $r(t)>0$, and +(ii) *a surgery control parameter* $\overline\delta(t)>0$. We use these +to define the surgery scale function $h(t)$. Set +$\rho(t)=\overline\delta(t)r(t)$. Let $h(t)= +h(\rho(t),\overline\delta(t))\le +\rho(t)\cdot\overline\delta(t)=\overline\delta^2(t)r(t)$ be the +function given by \entryref{074e02b067b8}. We require that $h(0)\le +R_0^{-1/2}$ where $R_0$ is the constant from +\entryref{cf61b731c6f2}. + + In addition, there is a +function $\kappa(t)>0$ called the *non-collapsing parameter*. +All three functions $r(t)$, $\overline\delta(t)$ and $\kappa(t)$ are +required to be positive, non-increasing functions of $t$. + + +We shall consider Ricci flows with surgery $({\mathcal M},G)$ that +satisfy Assumptions (1) -- (7) and also satisfy: + + **For any singular time $t$ the surgery at time $t$ is +performed with control $\overline\delta(t)$ and at scale +$h(t)=h(\rho(t),\overline\delta(t))$, where +$\rho(t)=\overline\delta(t)r(t)$, in the sense that each boundary +component of $C_t$ is the central $2$-sphere of a strong +$\overline\delta(t)$-neck centered at a point $y$ with +$R(y)=h(t)^{-2}$.** + +There is quite a bit of freedom in the choice of these parameters. +But it is not complete freedom. They must decay rapidly enough as +functions of $t$. We choose to make $r(t)$ and $\kappa(t)$ step +functions, and we require $\overline\delta(t)$ to be bounded above +by a step function of $t$. Let us fix the step sizes. + + +**Definition.** We set $t_0=2^{-5}$, and for any $i\ge 0$ we define $T_i=2^it_0$. + + +The steps we consider are $[0,T_0]$ and then $[T_i,T_{i+1}]$ for +every $i\ge 0$. The first step is somewhat special. Suppose that +$({\mathcal M},G)$ is a Ricci flow with surgery with normalized +initial conditions. Then + according to +\entryref{f5e180ecfd18} the flow exists on $[0,T_1]$ and the norm +of the Riemann curvature is bounded by $2$ on $[0,T_1]$, so that +by Assumption (7) there are no surgeries in this time interval. Also, +by \entryref{f5e180ecfd18} there is a $\kappa_0>0$ so +that $\mathit{Vol} B(x,t,r)\le \kappa_0r^3$ for every $t\le T_1$ and +$x\in M_t$ and every $r\le \epsilon$. + + +**Definition.** *Surgery parameter sequences* are sequences + +- **(i)** $\mathbf{r}=r_0\ge r_1\ge r_2\ge \cdots >0$, with $r_0=\epsilon$, +- **(ii)** $\mathbf{K}=\kappa_0\ge \kappa_1\ge \kappa_2\ge \cdots >0$ +with $\kappa_0$ as in \entryref{f5e180ecfd18}, and +- **(iii)** $ +\Delta=\delta_0\ge \delta_1\ge \delta_2\ge \cdots +>0$ with $\delta_0=\mathit{min}(\beta\epsilon/3,\delta_0',K^{-1},D^{-1})$ + where $\delta_0'$ is the constant +from \entryref{cf61b731c6f2} and $\beta<1/2$ is the constant from +\entryref{a992d6a89479}, $\epsilon$ is the constant that we have +already fixed, and $K$ and $D$ are the constants from +\entryref{0b07beec84c0}. + +We shall also refer to partial sequences defined for indices $0,\ldots, i$ for some $i>0$ +as surgery parameter sequences if they are positive, non-increasing and if their initial +terms satisfy the conditions given above. + + +We let $r(t)$ be the step function whose value on $[T_i,T_{i+1})$ is $r_{i+1}$ +and whose value on $[0,T_0)$ is $r_0$. We say that a Ricci flow with surgery +satisfies the strong $(C,\epsilon)$-canonical neighborhood assumption with parameter $\mathbf{r}$ if it satisfies this condition with respect to the step function $r(t)$ +associated with $\mathbf{r}$. This means that any $x\in {\mathcal M}$ with +$R(x)\ge r^{-2}(\mathbf{t}(x))$ has a strong $(C,\epsilon)$-canonical neighborhood in +${\mathcal M}$. Let $\kappa(t)$ be the step function whose value on +$[T_i,T_{i+1})$ is $\kappa_{i+1}$ and whose value on $[0,T_0)$ is $\kappa_0$. +Given $\kappa>0$, we say that a Ricci flow defined on $[0,t]$ is $\kappa$-non-collapsed +on scales $\le \epsilon$ provided that for every point $x$ not contained in a +component of its time-slice with positive sectional curvature, +if for some $r\le \epsilon$, the parabolic neighborhood $P(x,\mathbf{t}(x),r,-r^2)$ exists in ${\mathcal M}$ +and the norm of the Riemann curvature is bounded on this backward +parabolic neighborhood by $r^{-2}$, then $\mathit{Vol} B(x,\mathbf{t}(x),r)\ge +\kappa r^3$. +We say that a Ricci flow with surgery is $\mathbf{K}$-non-collapsed on scales +$\epsilon$ if for every $t\in [0,\infty)$ the restriction of the flow to +$[0,t]$ is $\kappa(t)$-non-collapsed on scales $\le \epsilon$. + Lastly, we fix a non-increasing function $\overline\delta(t)>0$ +with $\overline\delta(t)\le \delta_{i+1}$ if $t\in [T_i,T_{i+1})$ for all $i\ge +0$ and $\overline\delta(t)\le \delta_0$ for $t\in [0,T_0)$. We denote the fact +that such inequalities hold for all $t$ by saying $\overline\delta(t)\le +\Delta$. + +Having fixed surgery parameter sequences $\mathbf{K}$, $\mathbf{r}$ and +$\Delta$, defined step functions $r(t)$ and $\kappa(t)$, and fixed +$\overline\delta(t)\le \Delta$ as above, we shall consider only +Ricci flows with surgery where the surgery at time $t$ is defined +using the surgery parameter functions $r(t)$ and +$\overline\delta(t)$. In addition, we require that these Ricci +flows with surgery satisfy Assumptions (1) -- (7). + +What we shall show is that there are surgery parameter +sequences $\mathbf{r}$, $\mathbf{K}$ and $\Delta$ with the property that +for any normalized initial metric and any positive, +non-increasing function $\overline\delta(t)\le \Delta$, it is +possible to construct a Ricci flow with surgery using the surgery +parameters $r(t)$ and $\overline\delta(t)$ with the given initial +conditions and furthermore that this Ricci flow with surgery +satisfies the Assumptions (1) -- (7), has curvature pinched toward +positive, satisfies the canonical neighborhood assumption, and satisfies the +non-collapsing assumption using these parameters. + +In fact we shall prove this inductively, constructing the step +functions inductively one step at a time. Thus, given surgery +parameter sequences indexed by $0,\ldots,i$ we show that there are appropriate choices of +$r_{i+1}, \kappa_{i+1}$ and $\delta_{i+1}$ such that the following is true. +Given a Ricci flow with +surgery defined on time $[0,T_i)$ satisfying all the properties with +respect to the first set of data, that Ricci +flow with surgery extends to one defined for time $[0,T_{i+1})$ and satisfies +Assumptions (1) -- (7), the canonical neighborhood assumption and the +non-collapsing assumption with respect to the extended surgery +parameter sequences, and has curvature pinched toward positive. +As stated this is not quite true; there is a slight twist: we must also assume +that $\overline\delta(t)\le \delta_{i+1}$ for all $t\in [T_{i-1},T_{i+1})$. +It is for this reason that we consider +pairs consisting of sequences $\Delta$ and a surgery control +parameter $\overline\delta(t)$ bounded above by $\Delta$. + +## The process of surgery + +Suppose given surgery parameter sequences $\{r_0,\ldots,r_{i+1}\}$, +$\{\kappa_0,\ldots,\kappa_{i+1}\}$ and +$\Delta_i=\{\delta_0,\ldots,\delta_i\}$ and also given a positive, +decreasing function $\overline\delta(t)\le \Delta_i$, defined for +$t\le T_{i+1}$ with $\delta_0=\mathit{min}(\alpha\epsilon/3,\delta_0',K^{-1},D^{-1})$ as above. Suppose +that $({\mathcal M},G)$ is a Ricci flow with surgery defined for +$t\in [0,T)$ that goes singular at time $T\in (T_i,T_{i+1}]$. We +suppose that it satisfies Assumptions (1) -- (7). Since the flow has +normalized initial conditions and goes singular at time $T$, it +follows that $i\ge 1$. We suppose that $({\mathcal M},G)$ satisfies +the $(C,\epsilon)$-canonical neighborhood assumption with parameter +$r_{i+1}$ and that its curvature is pinched toward positive. By +\entryref{af00774dbaa9} we know that there is a maximal extension +$(\widehat{\mathcal M},\widehat G)$ of this generalized flow to time +$T$ with the $T$ time-slice being $(\Omega(T),G(T))$. Set +$\rho=\overline\delta(T)r_{i+1}$, and set +$h(T)=h(\rho(T),\overline\delta(T))$ as in \entryref{074e02b067b8}. +Since $\overline\delta(T)\le \delta_0<1$, we see that +$\rho +**Definition.** The operation described in the previous paragraph is the *surgery +operation at time $T$* using the surgery parameters $\overline\delta(T)$ and $r_{i+1}$. + + +## Statements about the existence of Ricci flow with surgery + +What we shall establish is the existence of surgery satisfying Assumptions (1) -- (7) +above and also satisfying the curvature pinched toward positive assumption, the +strong canonical neighborhood assumption, and the non-collapsing assumption. +This requires first of all that we begin with a compact, Riemannian +$3$-manifold $(M,g(0))$ that is normalized, which we are assuming. It also +requires careful choice of upper bounds $\Delta=\{\delta_i\}$ for the surgery +control parameter $\overline\delta(t)$ and careful choice of the canonical neighborhood +parameter $\mathbf{r}=\{r_i\}$ and of the non-collapsing step function $\mathbf{K}=\{\kappa_i\}$. + +Here is the statement that we shall establish. + + +**Theorem.** There are surgery parameter sequences + +$$ +\mathbf{K}=\{\kappa_i\}_{i=1}^\infty,\Delta=\{\delta_i\}_{i=1}^\infty,\mathbf{r}=\{r_i\}_{i=1}^\infty +$$ + + such that the following holds. Let $r(t)$ +be the step function whose value on $[T_{i-1},T_i)$ is $r_i$. +Suppose that $\overline\delta\colon [0,\infty)\to \Ar^+$ is any +non-increasing function with $\overline\delta(t)\le \delta_i$ +whenever $t\in [T_{i-1},T_i)$. Then the following holds: Suppose +that $({\mathcal M},G)$ is a Ricci flow with surgery defined for +$0\le t + +This of course leads immediately to the existence result for Ricci +flows with surgery defined for all time with any normalized initial +conditions. + + +**Corollary.** Let $\mathbf{K}$, $\mathbf{r}$ and $\Delta$ be surgery parameter sequences +provided by the previous theorem. Let $\overline\delta(t)$ be a +non-increasing positive function with $\overline\delta(t)\le +\Delta$. Let $M$ be a compact $3$-manifold containing no $\Ar P^2$ +with trivial normal bundle. Then there is a Riemannian metric $g(0)$ +on $M$ and a Ricci flow with surgery defined for $0\le t<\infty$ +with initial metric $(M,g(0))$. This Ricci flow with surgery +satisfies the seven assumptions and is $\mathbf{K}$-non-collapsed on +scales $\le \epsilon$. It also satisfies the strong +$(C,\epsilon)$-canonical neighborhood assumption with parameter +$\mathbf{r}$ and has curvature pinched toward positive. Furthermore, +any surgery at a time $t\in [T_i,T_{i+1})$ is done using +$\overline\delta(t)$ and $r_{i+1}$. + + +**Proof.** (Assuming \entryref{703e4cfeece7}) +Choose a metric $g(0)$ so that $(M,g_0)$ is normalized. This is +possible by beginning with any Riemannian metric on $M$ and scaling +it by a sufficiently large positive constant to make it normalized. +According to \entryref{c1330c5a8be3} and the definitions of +$T_i$ and $\kappa_0$ there is a Ricci flow $(M,g(t))$ with these +initial conditions defined for $0\le t\le T_2$ satisfying Assumptions (1) -- (7) +and the three conditions of the previous theorem. The assumption that +$M$ has no embedded $\Ar P^2$ with trivial normal bundle is needed so that Assumption (1) holds +for this Ricci flow. + Hence, by the previous theorem we can extend this +Ricci flow to a Ricci flow with surgery defined for all $0\le +t<\infty$ satisfying the same conditions. + +Showing that after surgery Assumptions (1) -- (7) continue to hold and +that the curvature is pinched toward positive is direct and only requires that +$\overline\delta(t)$ be smaller than some universal positive constant. + + +**Lemma.** Suppose that $({\mathcal M},G)$ is a Ricci flow with surgery going +singular at time $T\in [T_{i-1},T_i)$. We suppose that $({\mathcal +M},G)$ satisfies Assumptions (1) - (7), has curvature pinched toward +positive, satisfies the strong $(C,\epsilon)$-canonical neighborhood +assumption with parameter $\mathbf{r}$ and is $\mathbf{K}$ non-collapsed. +Then the result of the surgery operation at time $T$ on $({\mathcal M},G)$ is a + Ricci flow with surgery defined on $[0,T')$ for some $T'>T$. + The resulting Ricci flow with +surgery satisfies Assumptions (1) -- (7). It also has curvature pinched +toward positive. + + +**Proof.** It is immediate from the construction and \entryref{b186908cc3b4} that +the result of performing the surgery operation at time $T$ on a Ricci flow with +surgery produces a new Ricci flow with surgery. Assumptions (1) -- (3) +clearly hold for the result. and Assumptions (4) and (5) hold because of +the way that we do surgery. Let us consider Assumption (6). Fix $t'\rho^{-2}>r_i^{-2}. +$$ + +Since $M_{t'}\setminus \psi(\mathit{int} C_T\times \{t'\})$ is compact +for every $t'$, there is $T_1r_i^{-2}$ for +all $p\in M_{t'}\setminus \psi(\mathit{int}C_T\times \{t'\})$ and all +$t\in [T_1,T)$. Hence, by our assumptions all these points have +strong $(C,\epsilon)$-canonical neighborhoods. This establishes that +Assumption (6) holds at the singular time $T$. By hypothesis +Assumption (6) holds at all earlier singular times. Clearly, from the +construction the Ricci flow on $[T,T')$ is maximal. Hence, +Assumption (7) holds for the new Ricci flow with surgery. + + From \entryref{cf61b731c6f2} the +fact that $\delta(T)\le \delta_i\le \delta_0\le\delta'_0$ and +$h(T)\le R_0^{-1/2}$ imply that the Riemannian manifold $(M_T,G(T))$ +has curvature pinched toward positive for time $T$. It then follows +from \entryref{fc44d8653f35} that the Ricci flow defined on $[T,T')$ +with $(M_T,G(T))$ as initial conditions has curvature pinched toward +positive. The inductive hypothesis is that on the time-interval +$[0,T)$ the Ricci flow with surgery has curvature pinched toward +positive. This completes the proof of the lemma. + + +**Proposition.** Suppose that $({\mathcal M},G)$ is a Ricci flow with surgery +satisfying Assumptions (1) -- (7) in Section 15.1.1. +Suppose that $T$ is a surgery time, suppose that the surgery +control parameter $\delta(T)$ is less than $\delta_0$ in +\entryref{22c40a5664ac}, and suppose that the scale of the +surgery $h(T)$ is less than $R_0^{-1/2}$ where $R_0$ is the constant +from \entryref{cf61b731c6f2}. Fix $t' + +**Proof.** This is immediate from the third item in \entryref{cf61b731c6f2}. + + +**Remark.** If we have a non-separating surgery $2$-sphere then there will a +component $X(T)$ with surgery caps on both sides of the surgery +$2$-sphere and hence we cannot extend the map even continuously +over all of $X(t')$. + + +The other two inductive properties in \entryref{703e4cfeece7} -- that the result is +$\mathbf{K}$-non-collapsed and also that it satisfies the strong +$(C,\epsilon)$-canonical neighborhood assumption with parameter $\mathbf{r}$ - +require appropriate inductive choices of the sequences. The arguments +establishing these are quite delicate and intricate. They are given in the next +two sections. + +## Outline of the proof of Theorem \entryref{703e4cfeece7} + +Before giving the proof proper of \entryref{703e4cfeece7} let us outline +how the argument goes. We shall construct the surgery parameter +sequences $\Delta$, $\mathbf{r}$, and $\mathbf{K}$ inductively. Because of +\entryref{c1330c5a8be3} we have the beginning of the inductive +process. We suppose that we have defined sequences as required up to +index $i$ for some $i\ge 1$. Then we shall extend them one more step +to sequences defined up to $(i+1)$, though there is a twist: to do +this we must redefine $\delta_i$ in order to make sure that the +extension is possible. In Chapter 16 we establish the +non-collapsing result assuming the strong canonical neighborhood +result. More precisely, suppose that we have a Ricci flow with +surgery $({\mathcal M},G)$ defined for time $0\le t0$. +Then there is $\delta(r_{i+1})>0$ and $\kappa_{i+1}>0$ such that, +provided that $\overline\delta(t)\le \delta(r_{i+1})$ for all $t\in +[T_{i-1},T)$, the Ricci flow with surgery $({\mathcal M},G)$ is +$\kappa_{i+1}$ non-collapsed on scales $\le \epsilon$. + +In Section 17.1 we show that the strong $(C,\epsilon)$-canonical neighborhood +assumption extends for some parameter $r_{i+1}$, assuming again that +$\overline\delta(t)\le \delta(r_{i+1})$ for all $t\in [T_{i-1},T)$. + +Lastly, in Section 17.2 we complete the proof by +showing that the number of surgeries possible in $[0,T_{i+1})$ is +bounded in terms of the initial conditions and $\overline\delta(T)$. +The argument for this is a simple volume comparison argument -- +under Ricci flow with normalized initial conditions, the volume +grows at most at a fixed exponential rate and under each surgery an +amount of volume, bounded below by a positive constant depending +only on $\overline \delta(T_{i+1})$, is removed. diff --git a/projects/poincare-conjecture/.astrolabe/docs-src/17-proof-of-the-non-collapsing.mdx b/projects/poincare-conjecture/.astrolabe/docs-src/17-proof-of-the-non-collapsing.mdx new file mode 100644 index 00000000..2a0a9ff0 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/docs-src/17-proof-of-the-non-collapsing.mdx @@ -0,0 +1,1834 @@ + + +# Chapter 16 -- Proof of the non-collapsing + +The precise statement of the non-collapsing result is given in the +next section. Essentially, the proof of non-collapsing in the +context of Ricci flow with surgery is the same as the proof in the +case of ordinary Ricci flows. Given a point $x\in {\mathcal M}$, one +finds a parabolic neighborhood whose size, $r'$, is determined by +the constants $r_i$, $C$ and $\epsilon$, contained in $\mathbf{t}^{-1}([T_{i-1},T_i))$ and on which the curvature is bounded by +$(r')^{-2}$. Hence, by the inductive hypothesis, the final +time-slice of this neighborhood is $\kappa_i$-non-collapsed. +Furthermore, we can choose this neighborhood so that the reduced +${\mathcal L}$-length of its central point from $x$ is bounded by +$3/2$. This allows us to produce an open subset at an earlier time +whose reduced volume is bounded away from zero. Then using +\entryref{3defa9b2209c} we transfer this conclusion to a non-collapsing +conclusion for $x$. The main issue in this argument is to show that +there is a point in each earlier time-slice whose reduced length +from $x$ is at most $3/2$. We can argue as in the case of a Ricci +flow if we can show that any curve parameterized by backward time +starting at $x$ (a point where the hypothesis of +$\kappa$-non-collapsing holds) that comes close to a surgery cap +either from above or below must have large ${\mathcal L}$-length. In +establishing the relevant estimates we are forced to require that +$\delta_i$ be sufficiently small. + +## The statement of the non-collapsing result + +Here, we shall assume that after surgery the strong canonical +neighborhood assumption holds, and we shall establish the +non-collapsing result. + + +**Proposition.** Suppose that for some $i\ge 0$ we have surgery parameter sequences $\delta_0\ge +\delta_1\ge\cdots\ge \delta_i>0$, $\epsilon=r_0\ge r_1\ge \cdots\ge +r_i>0$ and $\kappa_0\ge \kappa_1\ge \cdots\ge \kappa_i>0$. + Then there is $0<\kappa\le \kappa_i$ and for any $0 + + +**Remark.** Implicitly, +$\kappa$ and $\delta(r_{i+1})$ are also allowed to depend on $t_0, \epsilon$, +and $C$, which are fixed, and also $i+1$. +Also recall that the non-collapsing condition allows for two outcomes: if $x$ is a point at which the +hypothesis of the non-collapsing hold, then +there is a lower bound on the volume of a ball centered at $x$, or $x$ +is contained in a component of its time-slice that has positive sectional curvature. + + +## The proof of non-collapsing when $R(x)=r^{-2}$ with $r\le r_{i+1}$ + +Let us begin with an easy case of the non-collapsing result, where +non-collapsing follows easily from the strong canonical neighborhood +assumption, rather than from using ${\mathcal L}$-length and +monotonicity along ${\mathcal L}$-geodesics. We suppose that we have +a Ricci flow with surgery $({\mathcal M},G)$ defined for $0\le t +**Proposition.** Let $x\in {\mathcal M}$ with $\mathbf{t}(x)=t$ and with $R(x)=r^{-2}\ge r_{i+1}^{-2}$. Then there is $\kappa>0$ depending only +on $C$ such that ${\mathcal M}$ is $\kappa$-non-collapsed at $x$; that is to say, +if $R(x)=r^{-2}$ with $r\le r_{i+1}$, then $\mathit{Vol} B(x,t,r)\ge \kappa r^3$, or $x$ is contained in a component of $M_t$ with positive sectional curvature. + + +**Proof.** Since +$R(x)\ge r_{i+1}^{-2}$, by assumption any such $x$ has a strong +$(C,\epsilon)$-canonical neighborhood. If this neighborhood is a +strong $\epsilon$-neck centered at $x$ then the result is clear for +a non-collapsing constant $\kappa$ which is universal. If the +neighborhood is an $\epsilon$-round component containing $x$, then +$x$ is contained in a component of positive sectional curvature. Likewise, +if $x$ is contained in a $C$-component then by definition it is contained in a component of its +time-slice with positive sectional curvature. + +Lastly, we suppose that $x$ is contained in the core $Y$ of a +$(C,\epsilon)$-cap ${\mathcal C}$. Let $r'>0$ be such that the +supremum of $|\mathit{Rm}|$ on $B(x,t,r')$ is $(r')^{-2}$. +Then, by the +definition + of a $(C,\epsilon)$-cap, +$\mathit{vol} B(x,t,r')\ge C^{-1}(r')^3$. +Clearly, $r'\le r$ and there is a point $y\in \overline{B(x,t,r')}$ +with $R(y)=(r')^{-2}$. On the other hand, by the +definition of a $(C,\epsilon)$-cap, we have $R(y)/R(x)\le C$, so that $r'/r\ge C^{-1/2}$. +Thus, the +volume of $B(x,t,r)$ is at least $C^{-5/2}r^3$. + +This completes an examination of all cases and establishes the +proposition. + +## Minimizing ${\mathcal L}$-geodesics exist when $R(x)\le r^{-2}_{i+1}$: the statement + +The proof of the non-collapsing result when $R(x)=r^{-2}$ with +$r_{i+1} +**Proposition.** For each $r_{i+1}$ with $00$ (depending implicitly on $t_0$, $C$, +$\epsilon$, and $i$) such that if $\overline\delta(t)\le \delta$ for +all $t\in [T_{i-1},T_{i+1}]$ then the following holds. Let +$({\mathcal M},G)$ be a Ricci flow with surgery satisfying the +hypothesis of \entryref{4ce8b5da9c58} with respect to the given +sequences and $r_{i+1}$, and let $x\in {\mathcal M}$ have $\mathbf{t}(x)=T$ with $T\in [T_i,T_{i+1})$. Suppose that for some $r\ge +r_{i+1}$ the parabolic neighborhood $P(x,r,T,-r^2)$ exists in +${\mathcal M}$ and $|\mathit{Rm}|\le r^{-2}$ on this neighborhood. Then +there is an open subset $U$ of $\mathbf{t}^{-1}[T_{i-1},T)$ contained +in the open subset of smooth manifold points of ${\mathcal M}$ with +the following properties: + +- **(1)** For every $y$ in $U$ there is a minimizing ${\mathcal L}$-geodesic +connecting $x$ to $y$. +- **(2)** $U_t=U\cap \mathbf{t}^{-1}(t)$ is non-empty for every $t\in [T_{i-1},T)$ +- **(3)** For each $t\in [T_{i-1},T)$ the restriction of ${\mathcal +L}$ to $U_t$ achieves its minimum and that minimum is at most +$3\sqrt{(T-t)}$. +- **(4)** The subset of $U$ consisting of all $y$ with the property that +${\mathcal L}(y)\le {\mathcal L}(y')$ for all $y'\in \mathbf{t}^{-1}(\mathbf{t}(y))$ has the property that its intersection with +$\mathbf{t}^{-1}(I)$ is compact for every compact interval $I\subset +[T_{i-1},T)$. + + +The basic idea in proving this result is to show that all paths +beginning at $x$ and parameterized by backward time that come close +to the exposed regions have large ${\mathcal L}$-length. If we can +establish this, then the existence of such paths will not be an +impediment to using the analytic estimates from +Chapter 6 to show that for each $t\in [T_{i-1},T)$ +there is a point whose ${\mathcal L}$-length from $x$ is at most +$3\sqrt{T-t}$, and that the set of points that minimize the +${\mathcal L}$-length from $x$ in a given time-slice form a compact +set. + +Given \entryref{89a60eb69c3f}, arguments from +Chapter 8 will be applied to complete the proof of +\entryref{4ce8b5da9c58}. + +## Evolution of neighborhoods of surgery caps + + We begin this analysis required to prove \entryref{89a60eb69c3f} + by studying the +evolution of surgery caps. +\entryref{ad4f67ea75bc} below is the main result along these lines. +Qualitatively, it says that if the surgery control parameter +$\delta$ is sufficiently small, then as a surgery cap evolves in a +Ricci flow with surgery it stays near the rescaled version of the +standard flow for any rescaled time less than one unless the entire +cap is removed (all at once) by some later surgery. In that case, +the evolution of the cap is close to the rescaled version of the +standard flow until it is removed. Using this result we will show +that if a path parameterized by backward time has final point near +a surgery cap and has initial point with scalar curvature not too +large, then this path must enter this evolving neighborhood either +from the 'top' or 'size' and because of the estimates that we derive +in this chapter such a path must have large ${\mathcal L}$-length. + + +**Proposition.** Given $A<\infty$, $\delta''>0$ and $0<\theta<1$, there is +$\delta''_0=\delta''_0(A,\theta,\delta'')$ ($\delta_0''$ also +depends on $r_{i+1}$, $C$, and $\epsilon$, which are all now fixed) +such that the following holds. Suppose that $({\mathcal M},G)$ is a +Ricci flow with surgery defined for $0\le t + +**Proof.** The method of proof is to assume that the result is false and take +a sequence of counterexamples with surgery control parameters +$\delta_n$ tending to zero. In order to derive a contradiction we +need to be able to take smooth limits of rescaled versions of these +Ricci flows with surgery, where the base points are the tips of the +surgery caps. This is somewhat delicate since the surgery cap is not +the result of moving forward for a fixed amount of time under Ricci +flow, and consequently Shi's theorem does not apply. Fortunately, +the metrics on the cap are bounded in the $C^\infty$-topology so +that Shi's theorem with derivatives does apply. Let us start +by examining limits of the sort we need to take. + + +**Claim.** Let $(N,g_N)$ be a strong $\delta'$-neck with $N_0$ its middle half. +Suppose that $({\mathcal S},g)$ is the result of doing surgery on +(the central $2$-sphere) of $N$, adding a surgery cap ${\mathcal C}$ +to $N^-$. Let $h$ be the scale of $N$. Let $({\mathcal S}_0(N),g')$ +be the union of $N^-_0\cup {\mathcal C}$ with its induced metric as +given in Section 13.1, and let $({\mathcal +S}_0(N),\widehat g_0)$ be the result of rescaling $g_0$ by $h^{-2}$. +Then for every $\ell<\infty$ there is a uniform bound to +$|\nabla^\ell \mathit{Rm}_{\widehat g_0}(x)|$ for all $x\in {\mathcal +S}_0(N)$. + + +**Proof.** Since $(N,g_N)$ is a strong $\delta'$-neck of scale $h$, there is a +Ricci flow on $N$ defined for backward time $h^2$. After rescaling +by $h^{-2}$ we have a flow defined for backward time $1$. +Furthermore, the curvature of the rescaled flow is bounded on the +interval $(-1,0]$. Since the closure of $N_0$ in $N$ is compact, the +restriction of $h^{-2}g_N$ to $N_0\subset N$ at time $0$ is +uniformly bounded in the $C^\infty$-topology by Shi's theorem +(\entryref{ed4b72caf9f9}). The bound on the $k^{th}$-derivatives of the +curvature depends only on the curvature bound and hence can be taken +to be independent of $\delta'>0$ sufficiently small and also +independent of the strong $\delta'$-neck $N$. Gluing in the cap with +a $C^\infty$-metric that converges smoothly to the standard initial +metric $g_0$ as $\delta'$ tends to zero using a fixed +$C^\infty$-partition of unity produces a family of manifolds +uniformly bounded in the $C^\infty$-topology. + +This leads immediately to: + + +**Corollary.** Given a sequence of $\delta'_n\rightarrow 0$ and strong +$\delta'_n$-necks $(N(n),g_{N(n)})$ of scales $h_n$ and results of +surgery $(S_0(N(n)),g(n))$ with tips $p_n$ as in the previous claim, +then after passing to a subsequence there is a smooth limit +$(S_\infty,g_\infty,p_\infty)$ of a subsequence of the +$(S_0(N(n)),h_n^{-2}g_0(n)),p_n)$. This limit is the metric from Section 13.1 +that gives the standard initial conditions for a surgery cap. + + +**Proof.** That there is a smooth limit of a subsequence is immediate from the +previous claim. Since the $\delta_n$ tend to zero, it is clear that the +limiting metric is the standard initial metric. + + +**Lemma.** Suppose that we have a sequence of $3$-dimensional Ricci flows +with surgeries $({\mathcal M}_n,G_n)$ that satisfy the strong +$(C,\epsilon)$-canonical neighborhood assumption with parameter +$r_{i+1}$, and have curvature pinched toward positive. Suppose that +there are surgeries in ${\mathcal M}_n$ at times $t_n$ with surgery +control parameters $\delta'_n$ and scales $h_n$. Let $p_n$ be the +tip of a surgery cap for the surgery at time $ t_n$. Also suppose +that there is $0\le\theta_n<1$ such that for every $A<\infty$, +for all $n$ sufficiently large +there are embeddings +$B(p_a, t_n,A h_n)\times [t_n, t_n+ h^2_n\theta_n)\to {\mathcal +M}_n$ compatible with time and the vector field. Suppose that +$\delta'_n\rightarrow 0$ and $\theta_n\rightarrow \theta<1$ as $n\rightarrow +\infty$. Let $({\mathcal M}'_n,G'_n,p_n)$ be the Ricci flow with +surgery obtained by shifting time by $-t_n$ so that surgery occurs +at $\mathbf{t}=0$ and rescaling by $h_n^{-2}$ so that the scale of the +surgery becomes one. Then, after passing to a subsequence, the +sequence converges smoothly to a limiting flow +$(M_\infty,g_\infty(t),(p_\infty,0)),\ 0\le t<\theta$. This limiting flow is isomorphic to +the restriction of the standard flow to time $0\le t<\theta$. + + +**Proof.** Let $Q<\infty$ be an upper bound for the scalar curvature of the +standard flow on the time interval $[0,\theta)$. Since +$\delta'_n\rightarrow 0$, according to the previous corollary, there is +a smooth limit at time $0$ for a subsequence, and this limit is the +standard initial metric. Suppose that, for some $0\le +\theta'<\theta$, we have established that there is a smooth limiting +flow on $[0,\theta']$. Since the initial conditions are the standard +solution, it follows from the uniqueness statement in +\entryref{f44d182ef27e} that in fact the limiting flow is +isomorphic to the restriction of the standard flow to this time +interval. Then the scalar curvature of the limiting flow is bounded +by $Q$. Hence, for any $A<\infty$, for all $n$ sufficiently large, +the scalar curvature of the restriction of $G'_n$ to the image of +$B_{G'_n}(p_n,0,2A)\times[0,\theta']$ is bounded by $2Q$. According +to \entryref{0ddfe578961e} there is an $\eta>0$ and a constant +$Q'<\infty$, each depending only on $Q$, $r_{i+1}$, $C$ and +$\epsilon$, such that for all $n$ sufficiently large, the scalar +curvature of the restriction of $G_n'$ to $B_{G_n'}(p_n,0,A)\times +[0,\mathit{min}(\theta'+\eta,\theta_n))$ is bounded by $Q'$. Because of +the fact that the curvature is pinched toward positive, this implies +that on the same set the sectional curvatures are uniformly bounded. +Hence, by Shi's theorem with derivatives +(\entryref{e0b6a1f646a7}), it follows that there are uniform bounds +for the curvature in the $C^\infty$-topology. Thus, passing to a +subsequence we can extend the smooth limit to the time interval +$[0,\theta'+\eta/2]$ unless $\theta'+\eta/2\ge \theta$. Since $\eta$ +depends on $\theta$ (through $Q$), but is independent of $\theta'$, +we can repeat this process extending the time-interval of definition +of the limiting flow by $\eta/2$ until $\theta'+\eta/2\ge \theta$. +Now suppose that $\theta'+\eta/2\ge \theta$. Then the argument shows +that by passing to a subsequence we can extend the limit to any +compact subinterval of $[0,\theta)$. Taking a diagonal sequence +allows us to extend it to all of $[0,\theta)$. By the uniqueness of +the standard flow, this limit is the standard flow. + + +**Corollary.** With the notation and assumptions of the previous lemma, for all +$A<\infty$, and any $\delta''>0$, then for all $n$ sufficiently +large, the restriction of $G_n'$ to the image +$B_{G_n'}(p_n,0,A)\times [0,\theta_n)$ is within $\delta''$ in the +$C^{[1/\delta'']}$-topology of the restriction of the standard +solution to the ball of radius $A$ about the tip for time $0\le +t<\theta_n$. + + +**Proof.** Let $\eta>0$ depending on $\theta$ (though $Q$) as well as +$r_{i+1}$, $C$ and $\epsilon$ be as in the proof of the previous +lemma, and take $0<\eta'<\eta$. For all $n$ sufficiently large +$\theta_n>\theta-\eta'$, and consequently for all $n$ sufficiently +large there is an embedding $B_{G_n}(p_n,t_n,Ah_n)\times +[t_n,t_n+h_n^2(\theta-\eta')]$ into ${\mathcal M}_n$ compatible with +time and with the vector field. For all $n$ sufficiently large, we +consider the restriction of $G_n'$ to $B_{G_n'}(p_n,0,A)\times +[0,\theta-\eta']$. These converge smoothly to the restriction of the +standard flow to the ball of radius $A$ on the time interval +$[0,\theta-\eta']$. In particular, for all $n$ sufficiently large, +the restrictions to these time intervals are within $\delta''$ in +the $C^{[1/\delta'']}$-topology of the standard flow. Also, for all $n$ sufficiently +large, $\theta_n-(\theta-\eta')<\eta$. Thus, by +\entryref{0ddfe578961e}, we see that the scalar curvature of $G_n'$ +is uniformly bounded (independent of $n$) on +$B_{G_n'}(p_n,0,A)\times [0,\theta_n)$. By the assumption that the +curvature is pinched toward positive, this means that the sectional +curvatures of the $G_n'$ are also uniformly bounded on these sets, +and hence so are the Ricci curvatures. (Notice that these bounds are +independent of $\eta'>0$.) By Shi's theorem with derivatives +(\entryref{e0b6a1f646a7}), we see that there are uniform bounds on +the curvatures in the $C^\infty$-topology on these subsets, and +hence bounds in the $C^\infty$-topology on the Ricci curvature. +These bounds are independent of both $n$ and $\eta'$. Thus, choosing +$\eta'$ sufficiently close to zero, so that $\theta_n-\eta'$ is also +close to $\theta$ for all $n$ sufficiently large, we see that for all such large $n$ +and all +$t\in [\theta-\eta',\theta)$, the +restriction of $G_n'$ to $B_{G_n'}(p_n,0,A)\times \{t\}$ is +arbitrarily close in the $C^{[1/\delta'']}$-topology to +$G_n'(\theta-\eta')$. The same is of course true of the standard +flow. This completes the proof of the corollary. + +Now we turn to the proof proper of \entryref{ad4f67ea75bc}. We fix +$A<\infty$, $\delta''>0$ and $\theta<1$. We are free to make $A$ +larger so we can assume by \entryref{77ca4669ffb3} that for the +standard flow the restriction of the flow to $B(p_0,0,A)\setminus +B(p_0,0,A/2)$ remains close to a standard evolving $S^2\times +[A/2,A]$ for time $[0,\theta]$. Let $K<\infty$ be a constant with +the property that $R(x,t)\le K$ for all $x\in B(p_0,0,A)$ in the +standard flow and all $t\in [0,\theta]$. If there is no +$\delta''_0>0$ as required, then we can find a sequence +$\delta'_n\rightarrow 0$ as $n\rightarrow\infty$ and Ricci flows with +surgery $({\mathcal M}_n,G_n)$ with surgeries at time $t_n$ with +surgery control parameter $\delta_n(t_n)\le \delta_n'$ and surgery +scale parameter $ h_n=h(r_{i+1}\delta_n( t_n),\delta_n(t_n))$ +satisfying the hypothesis of the lemma but not the conclusion. Let +$T'_n$ be the final time of $({\mathcal M}_n,G_n)$. Let $\theta_n\le +\theta$ be maximal subject to the condition that there is an +embedding $\rho_n\colon B_{G_n}(x, t_n,Ah_n)\times [ t_n, +t_n+h_n^2\theta_n)\to {\mathcal M}_n$ compatible with time and the +vector field. Let $G'_n$ be the result of shifting the time by +$-t_n$ and scaling the result by $h_n^{-2}$. According to +\entryref{712124d7ea02}, for all $n$ sufficiently large, the +restriction of $G_n'$ to the image of $\rho_n$ is within $\delta''$ +in the $C^{[1/\delta'']}$-topology of the standard flow restricted +to the ball of radius $A$ about the tip of the standard solution on +the time interval $[0,\theta_n)$. If $\theta_n=\mathit{min}(\theta,(T_n'-t_n)/h_n^2)$, then the first conclusion of +\entryref{ad4f67ea75bc} holds for $({\mathcal M}_n,G_n)$ for all +$n$ sufficiently large which contradicts our assumption that the +conclusion of this proposition holds for none of the $({\mathcal +M}_n,G_n)$. If on the other hand $\theta_n<\mathit{min}(\theta,(T_n'-t_n)/h_n^2)$, we need only show that all of +$B(x_n,t_n,Ah_n)$ disappears at time $t_n+h_n^2\theta_n$ in order to +show that the second conclusion of \entryref{ad4f67ea75bc} holds +provided that $n$ is sufficiently large. Again this would contradict +the fact that the conclusion of this proposition holds for none of +the $({\mathcal M}_n,G_n)$. + +So now let us suppose that $\theta_n<\mathit{min}(\theta,(T'_n-t_n)/h_n^2)$. Since there is no further extension +in forward time for $B(p_n,t_n,A h_n)$, it must be the case that $ +t_n+ h_n^2\theta_n$ is a surgery time and there is some flow line +starting at a point of $B(p_n,t_n,A h_n)$ that does not continue to +time $t_n+h_n^2\theta_n$. It remains to show that in this case that +for any $t< t_n+ h_n^2\theta_n$ sufficiently close to $t_n+ +h_n^2\theta_n$ we have $\rho_n\left(B_{G_n}(x, t_n,Ah_n)\times +\{t\}\right)\subset D_t$, the region in $M_t$ that disappears at +time $t_n+h_n^2\theta_n$. + + +**Claim.** Suppose that $\theta_n<\mathit{min}(\theta,(T'_n- t_n)/ h_n^2)$. Let +$\Sigma_{1},\ldots,\Sigma_{k}$ be the $2$-spheres along which +we do surgery at time $ t_n+h_n^2\theta_n$. Then for any $t< t_n+ +h_n^2\theta_n$ sufficiently close to $ t_n+ h_n^2\theta_n$ the +following holds provided that $\delta_n'$ is sufficiently small. The image + +$$ +\rho_n\left(B_{g_n}(x,t_n,A h_n)\times +\{t\}\right) +$$ + + is disjoint from the images $\{\Sigma_{i}(t)\}$ of the $\{\Sigma_{i}\}$ +under the +backward flow to time $t$ of the spheres $\Sigma_{i}$ along which +we do surgery at time $t_n+h_n^2\theta_n$. + + +**Proof.** There is a constant $K'<\infty$ depending on $\theta$ such that for +the standard flow we have $R(x,t)\le K'$ for all $x\in B(p_0,0,A)$ +and all $t\in [0,\theta)$ for the standard solution. Consider the +embedding $\rho_n\left(B(p_n,t_n,Ah_n)\times [t_n,t_n+h_n^2 +\theta_n)\right)$. After time shifting by $-t_n$ and rescaling by +$h_n^{-2}$, the flow $G_n'$ on the image of $\rho_n$ is within +$\delta''$ of the standard flow. Thus, we see that for all $n$ sufficiently +large and for every point $x$ in the image of $\rho_n$ we have $ +R_{G_n'}(x)\le 2K'$ and hence $R_{G_n}(x)\le 2K'h_n^{-2}$. + +Let $h_n'$ be the scale of the surgery at time $t_n+ h_n^2\theta_n$. +(Recall that $h_n$ is the scale of the surgery at time $t_n$.) +Suppose that $\rho_n(B(p_n,t_n,Ah_n)\times \{t'\})$ meets one of the +surgery $2$-spheres $\Sigma_{i}(t')$ at time $t'$ at a point +$y(t')$. Then, for all $t\in [t',t_n+h_n^2\theta_n)$ we have + the image $y(t)$ of $y(t')$ under the flow. All these points +$y(t)$ are points of intersection of $\rho_n(B(p,t_n,Ah_n)\times +\{t\})$ with $\Sigma_{i}(t)$. Since $y(t)\in \rho_n(B(p,t_n,A +h_n)\times\{t\})$, we have $ R(y(t))\le 2K' h_n^{-2}$. On the other +hand $R(y(t)) (h'_n)^2$ is within $O(\delta)$ of $1$ as $t$ tends +to $t_n+h_n^2\theta_n$. This means that $h_n/h_n'\le \sqrt{3K'}$ for +all $n$ sufficiently large. Since the standard solution has +non-negative curvature, the metric is a decreasing function of $t$, +and hence the diameter of $B(p_0,t,A)$ is at most $ 2A$ in the standard +solution. Using \entryref{712124d7ea02} we see that for all $n$ +sufficiently large, the diameter of $\rho_n\left(B(p,t_n,Ah_n)\times +\{t\}\right)$ is at most $Ah_n\le 4\sqrt{K'} Ah_n'$. + This +means that for $\delta'_n$ sufficiently small the distance at time +$t$ from $\Sigma_{i}(t)$ to the complement of the $t$ time-slice +of the strong $\delta_n( t_n+ h_n^2\theta_n)$-neck $N_{i}(t)$ +centered at $\Sigma_{i}(t)$ (which is at least +$(\delta'_n)^{-1}h_n'/2$) is much larger than the diameter of + +$$ +\rho_n(B(p_n,t_n,Ah_n)\times \{t\}). +$$ + + Consequently, for all $n$ +sufficiently large, the image $\rho_n(B(p_n,t_n,Ah_n)\times \{t\})$ +is contained in $N_{i}(t)$. But by our choice of $A$, and +\entryref{712124d7ea02} there is an $\epsilon$-neck of rescaled +diameter approximately $Ah_n/2$ contained in $\rho_n(B(p_n,t_n,A +h_n)\times \{t\})$. By \entryref{98c6a95f8556} the spheres coming +from the neck structure in + +$$ +\rho_n(B(p_n,t_n,Ah_n)\times \{t\}) +$$ + + are + isotopic in $N_{i}(t)$ to the central $2$-sphere of this neck. +This is a contradiction because in $N_{i}(t)$ the central +$2$-sphere is homotopically non-trivial whereas the spheres in +$\rho_n(B(p_n,t_n,A\bar h_n)\times \{t\})$ clearly bound +$3$-disks. + +Since $\rho_n(B(p_n,t_n,Ah_n)\times\{t\})$ is disjoint from the +backward flow to time $t$ of all the surgery $2$-spheres +$\Sigma_{i}(t)$ and since $\rho_n(B(p_n,t_n,Ah_n)\times \{t\})$ is +connected, + if there is a flow line starting at some point $z\in +B(p,t_n,Ah_n)$ that disappears at time $t_n+ h_n^2\theta_n$, then +the flow from every point of $B(p,t_n,Ah_n)$ disappears at time $ +t_n+ h_n^2\theta_n$. This shows that if $\theta_n<\mathit{min}(\theta,T'_n- t_n/h_n^{2})$, and if there is no extension of +$\rho_n$ to an embedding defined at time $t_n+h^2_n\theta_n$, then +all forward flow lines beginning at points of $B(p,t_n,A h_n)$ +disappear at time $ t_n+h_n^2\theta_n$, which of course means that +for all $t +**Remark.** Notice that it is indeed possible that $B_{G}(x,t,Ah)$ is removed at +some later time, for example as part of a capped $\epsilon$-horn +associated to some later surgery. + + +## A length estimate + +We use the result in the previous section about the evolution of +surgery caps to establish the length estimate on paths parameterized +by backward time approaching a surgery cap from above. + + +**Definition.** Let $c>0$ be the constant from \entryref{ffae4504318a}. Fix +$0<\overline\delta_0<1/4$ such that if $g$ is within +$\overline\delta_0$ of $g_0$ in the $C^{[1/\overline +\delta]}$-topology then $|R_{g'}(x)-R_{g_0}(x)| + +Here is the length estimate. + + +**Proposition.** For any $\ell<\infty$ there is $A_0=A_0(\ell)<\infty$, +$0<\theta_0=\theta_0(\ell)<1$, and for any $A\ge A_0$ for the +constant +$\delta''=\delta''(A)=\delta''_0(A,\theta_0,\overline\delta_0)>0$ +from \entryref{ad4f67ea75bc} the following holds. Suppose that +$({\mathcal M},G)$ is a Ricci flow with surgery defined for $0\le +t< T<\infty$. Suppose that it satisfies the strong +$(C,\epsilon)$-canonical neighborhood assumption at all points $x$ +with $R(x)\ge r_{i+1}^{-2}$. Suppose also that the solution has +curvature pinched toward positive. Suppose that there is a surgery +at some time $\bar t$ with $T_{i-1}\le \bar t\ell. +$$ + +See Fig. 16.2. + + +**Proof.** The logic of the proof is as follows. We +fix $\ell<\infty$. We shall determine the relevant value of +$\theta_0$ and then of $A_0$ in the course of the argument. Then for any $A\ge A_0$ +we define + $\delta''(A)= \delta''_0(A,\theta_0,\overline\delta_0)$, as in +\entryref{ad4f67ea75bc}. + +The integral expression is invariant under time translation and also +under rescaling. Thus, we can (and do) assume that $\bar t=0$ and +that the scale $h$ of the surgery is $1$. We use the embedding of +$P(p,0,A,T')\to {\mathcal M}$ and write the restriction of the flow +to this subset as a one-parameter family of metrics $g(t),\ 0\le +t\le T'$, on $B(p,0,A)$. With this renormalization, $0\le t'\le +1/2$, also $T'\le \theta_0$, and $\tau=T'-t$. + +Let us first consider the case when $T'=\theta_0\le T$. Consider the +standard flow $(\Ar^3,g_0(t))$, and let $p_0$ be its tip. According +to \entryref{ffae4504318a}, for all $x\in \Ar^3$ and all $t\in +[0,1)$ we have $R_{g_0}(x,t)\ge c/(1-t)$. By \entryref{ad4f67ea75bc} and +since we are assuming that $\overline\delta(\bar t)\le \delta''= +\delta''_0(A,\theta_0,\overline\delta_0)$, we have that $R(a,t)\ge +c/2(1-t)$ for all $a\in B(p,0,A)$ and all $t\in [0,\theta]$. Thus, + we have + +$$ +\begin{aligned} +\int_0^{\theta_0-t'}\left(R(\gamma(\tau))+|X_\gamma(\tau)|^2\right)d\tau +& \ge & \int_{t'}^{\theta_0} +\frac{c}{2(1-t)}dt \\ +& = & \frac{-c}{2}\left(\mathit{log}(1-\theta_0)-\mathit{log}(1-t')\right)dt \\ +& \ge & \frac{-c}{2}\left(\mathit{log}(1-\theta_0)+\mathit{log}(2)\right). +\end{aligned} +$$ + + Hence, if +$\theta_0<1$ sufficiently close to $1$, the integral will be $> \ell$. +This fixes the value of $\theta_0$. + + +**Claim.** There is $A'_0<\infty$ with the property that for any $A\ge A'_0$ +the restriction of the standard solution $g_0(t)$ to +$\left(B(p_0,0,A)\setminus B(p_0,0,A/2)\right)\times [0,\theta_0]$ +is close to an evolving family $(S^2\times [A/2,A],h_0(t)\times +ds^2)$. In particular, for any $t\in [0,\theta_0]$, the +$g_0$-distance at time $t$ from $B(p_0,0,A/2)$ to the complement of +$B(p_0,0,A)$ in the standard solution is more than $A/4$. + + +**Proof.** This is immediate from \entryref{77ca4669ffb3} and the fact that +$\theta_0<1$. + +Now fix $A_0=\mathit{max}(A_0', 10\sqrt{\ell})$ and let $A\ge A_0$. + +Since $\overline\delta_0<1/4$ and since $T'\le \theta_0$, for $\overline\delta(\bar t)\le +\delta''_0(A,\theta_0,\overline\delta_0)$ +by \entryref{ad4f67ea75bc} +the $g(T')$-distance between $B(p,0,A/2)$ and $\partial +B(p,0,A)$ is at least $A/5$. + + Since the +flow on $B(p,0,A)\times[0,T']$ is within $\overline\delta_0$ of +the standard solution, and since the curvature of the standard +solution is non-negative, for any horizontal tangent vector $X$ at +any point of $B(p,0,A)\times [0,T']$ we have that + +$$ +\mathit{Ric}_g(X,X)\ge -\frac{1}{4}|X|_{g_0}^2\ge -\frac{1}{2}|X|^2_g, +$$ + + and hence + +$$ +\frac{d}{dt}|X|_g^2\le |X|_g^2. +$$ + +Because $T'\le 1$, we see that + +$$ +|X|^2_{g(T')}\le e\cdot|X|^2_{g(t)}<3|X|^2_{g(t)} +$$ + +for any $t\in [0,T']$. + +Now suppose that $\gamma(0)\in \partial B(p,0,A)\times\{T'\}$. Since +the image of $\gamma$ is contained in the closure of $P(p,0,A,T')$ +for every $\tau\in [0,T']$ we have +$\sqrt{3}|X_\gamma(\tau)|_{g(T'-\tau)}\ge |X_\gamma(\tau)|_{g(T')}$. +Since the flow $g(t)$ on $P(p,0,A, T')$ is within +$\overline\delta_0$ in the $C^{[1/\overline\delta_0]}$-topology of +the standard flow on the corresponding parabolic neighborhood, +$R(\gamma(t))\ge 0$ for all $t\in [0,T']$. Thus, because of these +two estimates we have + +$$ +\int_0^{T'-t'}\left(R(\gamma(\tau))+|X_\gamma(\tau)|^2\right)d\tau\ge +\int_0^{T'-t'} +\frac{1}{3}|X_\gamma(\tau)|_{g(T')}^2d\tau. +$$ + + Since $\gamma(0)\in +\partial B(p,0,A)\times \{T'\}$ and $\gamma(T')\in B(p,0,A/2)$, it follows from +Cauchy-Schwarz that + +$$ +\begin{aligned} +(T'-t')^2\int_0^{T'}|X_\gamma(\tau)|_{g(T')}^2d\tau & +\ge & \left(\int_0^{T'-t'}|X_\gamma(\tau)|_{g(T')}d\tau\right)^2 +\\ +& \ge & \left(d_{g(T')}(B(p,0,A/2),\partial +B(p,0,A))\right)^2\ge \frac{A^2}{25}. +\end{aligned} +$$ + + Since $T'-t'<1$, it +immediately follows from this and Equation (16.1) that + +$$ +\int_0^{T'-t'}\left(R(\gamma(\tau))+|X_\gamma(\tau)|^2\right)d\tau\ge +\frac{A^2}{75}. +$$ + + Since $A\ge A_0\ge 10\sqrt{ \ell}$, this +expression is $>\ell$. This completes the proof of +\entryref{0a4d6f17810d} + +### Paths with short ${\mathcal L}_+$-length avoid the surgery caps + +Here we show that a path parameterized by backward time that ends in +a surgery cap (or comes close to it) must have +long ${\mathcal L}$-length. Let $({\mathcal M},G)$ be a Ricci flow +with surgery, and let $x\in {\mathcal M}$ be a point with $\mathbf{t}(x)=T\in (T_i,T_{i+1}]$. We suppose that these data satisfy the +hypothesis of \entryref{89a60eb69c3f} with respect to the given +sequences and $r\ge r_{i+1}>0$. In particular, the parabolic +neighborhood $P(x,T,r,-r^2)$ exists in ${\mathcal M}$ and $|\mathit{Rm}|$ is +bounded on this parabolic neighborhood by $r^{-2}$. + +Actually, here we do not work directly with the length function +${\mathcal L}$ defined from $x$, but rather with a closely related +function. We set $R_+(y)=\mathit{max}(R(y),0)$. + + +**Lemma.** Given $L_0<\infty$, there is +$\overline\delta_1=\overline\delta_1(L_0,r_{i+1})>0$, independent of +$({\mathcal M},G)$ and $x$, such that if $\overline\delta(t)\le +\overline\delta_1$ for all $t\in [T_{i-1},T)$, then for any curve +$\gamma(\tau),\ 0\le \tau\le \tau_0$, with $\tau_0\le T-T_{i-1}$, +parameterized by backward time with $\gamma(0)=x$ and with + +$$ +{\mathcal L}_+(\gamma)=\int_0^{\tau_0}\sqrt{\tau} +\left(R_+(\gamma(\tau))+|X_\gamma|^2\right)d\tau< L_0 +$$ + + the +following two statements hold: + +- **(2)** Set + +$$ +\tau'=\mathit{min}\left(\frac{r_{i+1}^4}{(256)L_0^2},\mathit{ln}(\sqrt[3]{2})r_{i+1}^2\right). +$$ + +Then for all $\tau\le \mathit{min}(\tau',\tau_0)$ we have $\gamma(\tau)\in P(x,T,r/2,-r^2)$. +- **(2)** Suppose that $\bar t\in [T-\tau_0,T)$ is a surgery time with $p$ +being the tip of the surgery cap at time $\bar t$ and with the scale +of the surgery being $\bar h$. Suppose $t'\in [\bar t,\bar t+\bar +h^2/2]$ is such that there is an embedding + +$$ +\rho\colon B(p,\bar t,(50+A_0)\bar h)\times [\bar t,t']\to {\mathcal M} +$$ + +compatible with time and the vector field. Then the image of $\rho$ +is disjoint from the image of $\gamma$. See Fig. 16.3. + + + +**Remark.** Recall that $(A_0+4)\bar h$ is the radius of the surgery cap (measured in the +rescaled version of the standard initial metric) that is glued in when +performing surgery with scale $\bar h$. + + +**Proof.** We define $\ell=L_0/\sqrt{\tau'}$, then define $A=\mathit{max}(A_0(\ell),2(50+A_0))$ and $\theta=\theta_0(\ell)$. Here, +$A_0(\ell)$ and $\theta_0(\ell)$ are the constants in +\entryref{0a4d6f17810d}. + Lastly, we require $\overline\delta_1\le \delta''(A)$ from +\entryref{0a4d6f17810d}. Notice that, by construction, + $\delta''(A)=\delta''_0(A,\theta,\overline\delta_0)$ from +\entryref{ad4f67ea75bc}. Thus, if $p$ is the tip of a surgery cap at time $\bar t$ +with the scale of the surgery being $\bar h$, then it follows that +for any $\Delta t\le \theta$, if there is an embedding + +$$ +\rho\colon B(p,\bar t,A\bar h)\times [\bar t,\bar t+\bar h^2\Delta t)\to {\mathcal M} +$$ + + compatible with time and the vector field, then the induced flow (after time shifting by $-\bar t$ + and scaling by $(\bar h)^{-2}$ is within $\overline\delta_0$ in the + $C^{[1/\overline\delta_0]}$-topology of the standard solution. + In particular, the scalar curvature +at any point of the image of $\rho$ is positive and is within a +multiplicative factor of two of the scalar curvature at the +corresponding point of the standard flow. + + Recall that we have $r\ge r_{i+1}$ and +that $P(x,T,r,-r^2)$ exists in ${\mathcal M}$ and that $|\mathit{Rm}|\le +r^{-2}$ on this parabolic neighborhood. We begin by proving by +contradiction that there is no $\tau\le \tau'$ with the property +that $\gamma(\tau)\not\in P(x,T,r/2,-r^2)$. Suppose there is such a +$\tau\le \tau'$. Notice that by construction +$\tau' +**Claim.** $\int_0^{\tau''}|X_\gamma(\tau)|d\tau> r/2\sqrt{2}$. + + +**Proof.** Since $|\mathit{Rm}|\le r^{-2}$ on $P(x,T,r,-r^2)$, we have $|\mathit{Ric}|\le +2r^{-2}$ on $P(x,T,r,-\tau'')$. Thus, for any tangent vector $v$ at +a point of $B(x,T,r)$ we have + +$$ +\left|\frac{d(\langle v,v\rangle_{G(T-\tau)})}{d\tau}\right|\le 2r^{-2}\langle v,v\rangle_{G(T-\tau)} +$$ + +for all +$\tau\in [0,\tau'']$. Integrating gives that for any $\tau\le +\tau''$ we have + +$$ +\mathit{exp}(-2r^{-2}\tau'')\langle v,v\rangle_{G(T)}\le \langle +v,v\rangle_{G(T-\tau)} \le \mathit{exp}(2r^{-2}\tau'')\langle +v,v\rangle_{G(T)}. +$$ + + Since $\tau''\le \tau'$ and $r\ge r_{i+1}$ by +the assumption on $\tau'$ we have + +$$ +\mathit{exp}(2r^{-2}\tau'')\le \mathit{exp}(2\sqrt[3]{2})< 2. +$$ + + This +implies that for all $\tau\le \tau''$ we have + +$$ +\frac{1}{\sqrt{2}}| X_\gamma(\tau)|_{G(T)}< +|X_\gamma(\tau)|_{G(T-\tau)}<\sqrt{2}|X_\gamma(\tau)|_{G(T)}, +$$ + +and hence + +$$ +\int_0^{\tau''}|X_\gamma(\tau)|d\tau> +\frac{1}{\sqrt{2}}\int_0^{\tau''}|X_\gamma(\tau)|_{G(T)}\ge +\frac{r}{2\sqrt{2}}, +$$ + + where we use the fact that +$d_T(\gamma(0),\gamma(\tau''))=r/2$. + +Applying Cauchy-Schwarz to $\tau^{1/4}|X_\gamma|$ and $\tau^{-1/4}$ on the +interval $[0,\tau'']$ yields + +$$ +\begin{aligned} +\int_0^{\tau''}\sqrt{\tau}\left(R_+(\gamma(\tau))+|X_\gamma(\tau)|^2\right)d\tau & +\ge & \int_0^{\tau''}\sqrt{\tau}|X_\gamma(\tau)|^2d\tau +\\ +& \ge & +\frac{\left(\int_0^{\tau''}|X_\gamma(\tau)|d\tau\right)^2}{\int_0^{\tau''}\tau^{-1/2}d\tau} +\\ & > & +\frac{r^2}{16\sqrt{\tau''}}\ge L_0. +\end{aligned} +$$ + +Of course, the integral from $0$ to $\tau''$ is less than or equal +the entire integral from $0$ to $\tau_0$ since the integrand is +non-negative, contradicting the assumption that ${\mathcal +L}_+(\gamma)\le L_0$. This completes the proof of the first +numbered statement. + +We turn now to the second statement. We impose a further condition +on $\overline\delta_1$. Namely, require that +$\overline\delta_1^22r_{i+1}^{-2}$). Thus, if +$\tau'\ge \tau_0$ we have completed the proof. Suppose that +$\tau'<\tau_0$. It suffices to establish that for every $\tau_1\in +[\tau',\tau_0]$ the point $\gamma(\tau_1)$ is not contained in the +image of $\rho$ for any surgery cap and any $t'$ as in the +statement. Suppose that in fact there is $\tau_1\in [\tau', \tau_0]$ +with $\gamma(\tau_1)$ contained in the image of $\rho(B(p,\bar +t,(A_0+50)\bar h)\times [\bar t,t'])$ where $\bar t\le t'\le \bar +t+(\bar h)^2/2$ and where $p$ is the tip + of some surgery cap at time $\bar t$. We estimate + +$$ +\begin{aligned} +\nonumber +\int_0^{\tau_0}\sqrt{\tau}\left(R_+(\gamma(\tau))+|X_\gamma(\tau)|^2\right)d\tau & & \\ +& \ge & +\int_{\tau'}^{\tau_0}\sqrt{\tau}\left(R_+(\gamma(\tau))+|X_\gamma(\tau)|^2\right)d\tau +\nonumber +\\ & \ge & +\sqrt{\tau'}\int_{\tau'}^{\tau_1}\left(R_+(\gamma(\tau))+|X_\gamma(\tau)|^2\right)d\tau. +\end{aligned} +$$ + + Let $\Delta t\le +T-\bar t$ be the supremum of the set of $s$ for which there is a +parabolic neighborhood $P(p,\bar t,A\bar h,s)$ embedded in $\mathbf{t}^{-1}((-\infty,T])\subset {\mathcal M}$. Let $\Delta t_1=\mathit{min}(\theta \bar h^2,\Delta t)$. We consider $P(p,\bar t,A\bar +h,\Delta t_1)$. First, notice that since $\bar h\le +\overline\delta_1^2r_{i}r_{i+1}^{-2}\ge r^{-2}$. In particular, the parabolic +neighborhood $P(x,T,r,-r^2)$ is disjoint from $P(p,\bar t,A\bar +h,\Delta t_1)$. This means that there is some $\tau''\ge \tau'$ such +that $\gamma(\tau'')\in +\partial P(p,\bar t,A\bar h,\Delta t_1)$ and $\gamma|_{[\tau'',\tau_1]}\subset +P(p,\bar t,A\bar h,\Delta t_1)$. There are two cases to consider. +The first is when $\Delta t_1=\theta \bar h^2$, $\tau''=T-(\bar +t+\Delta t_1)$ and $\gamma(\tau'')\in B(p,\bar t,A\bar +h)\times\{\bar t+\Delta t_1\}$. Then, according to +\entryref{0a4d6f17810d}, + +$$ +\int_{\tau''}^{\tau_1}R_+(\gamma(\tau))d\tau> \ell. +$$ + +Now let us consider the other case. If $\Delta t_1<\theta \bar h^2$, +this means that either $\bar t+\Delta t_1=T$ or, according to +\entryref{ad4f67ea75bc}, at the time $\bar t+\Delta t_1$ there is a +surgery that removes all of $B(p,\bar t,A\bar h)$. Hence, under +either possibility it must be the case that $\gamma(\tau'')\in +\partial B(p,\bar t,A\bar h)\times \{T-\tau''\}$. Thus, the remaining +case to consider is when, whatever $\Delta t_1$ is, +$\gamma(\tau'')\subset +\partial B(p,\bar t,A\bar h)\times \{T-\tau''\}$. +\entryref{0a4d6f17810d} and the fact that $R\ge 0$ on $P(p,\bar t,A\bar +h,\Delta t_1)$ imply that + +$$ +\ell<\int_{\tau''}^{\tau_1}\left(R(\gamma(\tau))+|X_\gamma(\tau)|^2\right)d\tau = +\int_{\tau''}^{\tau_1}\left(R_+(\gamma(\tau))+|X_\gamma(\tau)|^2\right)d\tau. +$$ + +Since $\ell=L_0/\sqrt{\tau'}$ and $\tau''\ge \tau'$, it follows from +Equation (16.2) that in both +cases + +$$ +{\mathcal L}_+(\gamma)\ge +\int_{\tau''}^{\tau_1}\sqrt{\tau}\left(R_+(\gamma(\tau))+|X_\gamma(\tau)|^2\right)d\tau> +\ell\sqrt{\tau'}=L_0, +$$ + + which contradicts our hypothesis. This +completes the proof of \entryref{86043191a3c2}. + +### Paths with small energy avoid the disappearing regions + +At this point we have shown that paths of small energy do not approach +the surgery caps from above. We also need to rule out that they can be arbitrarily +close from below. That is to say, + we need to see that paths whose ${\mathcal L}$-length is not +too large avoid neighborhoods of the disappearing regions at all +times just before the surgery time at which they disappear. Unlike +the previous estimates which were universal for all $({\mathcal +M},G)$ satisfying the hypothesis of \entryref{89a60eb69c3f}, in +this case the estimates will depend on the Ricci flow with surgery. +First, let us fix some notation. + + +**Definition.** Suppose that $\bar t$ is a surgery +time, that $\tau_1>0$, and that there are no other surgery times in +the interval $(\bar t-\tau_1,\bar t]$. Let $\{\Sigma_i(\bar t)\}_i$ +be the $2$-spheres on which we do surgery at time $\bar t$. Each +$\Sigma_i$ is the central $2$-sphere of a strong $\delta$-neck +$N_i$. We can flow the cylinders $J_0(\bar +t)=\cup_is_{N_i}^{-1}(-25,0])$ backward to any time + $t\in (\bar t-\tau_1,\bar t]$. Let $J_0(t)$ be +the result. There is an induced function, denoted +$\coprod_is_{N_i}(t)$, on $J_0(t)$. It takes values in $ (-25,0]$. +We denote the boundary of $J_0(t)$ by $\coprod_i\Sigma_i(t)$. Of +course, this boundary is the result of flowing +$\coprod_i\Sigma_i(\bar t)$ backward to time $t$. (These backward +flows are possible since there are no surgery times in $(\bar +t-\tau_1,\bar t)$.) For each $t\in [\bar t-\tau_1,\bar t)$ we also +have the disappearing region $D_t$: -- the region that disappears at +time $\bar t$. It is an open submanifold whose boundary is +$\coprod_i\Sigma_i(t)$. Thus, for every $t\in (\bar t-\tau_1,\bar +t)$ the subset $J(t)=J_0(t)\cup D_t$ is an open subset of $M_t$. We +define + +$$ +J(\bar t-\tau_1,\bar t)=\cup_{t\in (\bar t-\tau_1,\bar t)}J(t). +$$ + +Then $J(\bar t-\tau_1,\bar t)$ is an open subset of ${\mathcal M}$. +See Fig. 16.4. + + + +**Lemma.** Fix a Ricci flow with surgery $({\mathcal M},G)$, a point $x\in +{\mathcal M}$ and constants $r\ge r_{i+1}>0$ as in the statement of +\entryref{89a60eb69c3f}. For any $1<\ell<\infty$ the following +holds. Suppose that $\bar t\in [T_{i-1},T)$ is a surgery time and +that $\gamma(\tau)$ is a path with $\gamma(\tau)\in M_{\bar +t-\tau}$. Let $\{p_1,\ldots,p_k\}$ be the tips of all the surgery +caps at time $\bar t$ and let $\bar h$ be the scale of surgery at +time $\bar t$. Suppose that for some $0<\tau_1\le \ell^{-1} \bar +h^2$ there are no surgery times in the interval $(\bar t-\tau_1,\bar +t)$. We identify all $M_t$ for $t\in [\bar t-\tau_0,\bar t)$ with +$M_{\bar t-\tau_1}$ using the flow. Suppose that $\gamma(0)\in +M_{\bar t}\setminus \cup_{i=1}^kB(p_i,\bar t,(50+A_0)\bar h)$, and +lastly, suppose that + +$$ +\int_{0}^{\tau_1}|X_\gamma(\tau)|^2d\tau\le \ell. +$$ + +Then $\gamma$ is disjoint from the open subset $J(\bar t-\tau_1,\bar +t))$ of ${\mathcal M}$. + + +**Proof.** Suppose that the lemma is false and let $\gamma\colon [0,\bar +\tau]\to {\mathcal M}$ be a path satisfying the hypothesis of the +lemma with $\gamma(\bar \tau)\in J(\bar t-\tau_1,\bar t)$. + Since + +$$ +\gamma(0)\in +M_{\bar t}\setminus \cup_iB(p_i,\bar t,(50+A_0)\bar h), +$$ + + if follows +that $\gamma(0)$ is separated from the boundary of +$s_{N_i}^{-1}(-25,0]$ by distance at least $20\bar h$. Since the $ +J_0(t)$ are contained in the disjoint union of strong $\delta$-necks +$N_i$ centered at the $2$-spheres along which we do surgery, and +since $\tau_1\le \bar h^2/\ell<\bar h^2$, it follows that, provided +that $\delta$ is sufficiently small, for every $t\in [\bar +t-\tau_1,\bar t)$, the metric on $J_0(t)$ is at least +$1/2$ the metric on $ J_0(\bar t)$. + It follows that, for $\delta$ sufficiently small, if there is a $\tau\in [0,\tau_1]$ +with $\gamma(\tau)\in J(t)$ then $\int_0^{\tau_1}|X_\gamma|d\tau> 10\bar h$. +Applying Cauchy-Schwarz we see that + +$$ +\int_0^{\tau_1}|X_\gamma(\tau)|^2d\tau\ge (10\bar h)^2/\tau_1. +$$ + +Since $\tau_1\le \ell^{-1}(\bar h)^2$, we see that + +$$ +\int_0^{\tau'}|X_\gamma(\tau)|^2d\tau>\ell, +$$ + +contradicting our hypothesis. + +### Limits of a sequence of paths with short ${\mathcal L}$-length + +Now using Lemmas \entryref{86043191a3c2} and \entryref{5ed7c639bd97} we show that it is possible to take limits +of certain types of sequences of paths parameterized by backward time to create +minimizing ${\mathcal L}$-geodesics. + +We shall work with a compact subset of $\mathbf{t}^{-1}([T_{i-1},T])$ that is +obtained by removing appropriate open neighborhoods of the exposed regions. + + +**Definition.** Fix $\ell<\infty$. Let $\theta_0=\theta_0(\ell)$ be as in +\entryref{0a4d6f17810d}. For each surgery time $\bar t\in +[T_{i-1},T]$, let $h(\bar t)$ be the scale of the surgery. + Let $p_1,\ldots,p_k$ be +the tips of the surgery caps at time $\bar t$. For each $1\le j\le k$, +we consider $B_j(\bar +t)=B(p_j,\bar t,(A_0+10)h(\bar t))$, and we let $\Delta t_j\le \mathit{min}(\theta_0,(T-\bar t)/h^2(\bar t))$ be maximal subject to the condition that +there is an embedding $\rho_j\colon B_j(\bar t)\times [\bar t,\bar t+h^2(\bar +t)\Delta t_j)$ into ${\mathcal M}$ compatible with time and the vector field. +Clearly, $B'_j=B(p_j,\bar t,(10+A_0)h)\cap C_{\bar t}$ is contained in $J(\bar +t)$. Let $\bar t'$ be the previous surgery time if there is one, otherwise set +$\bar t'=0$. Also for each $\bar t$ we set $\tau_1(\ell,\bar t)=\mathit{min}\left(h(\bar t)^2/\ell,\bar t-\bar t'\right)$. For each $t\in (\bar +t-\tau_1(\ell,\bar t),\bar t)$ let $\widetilde J(t)\subset J(t)$ be the union +of $D_t$, the disappearing region at time $t$, and $\coprod_iB_i'(t)$, the +result of flowing $\coprod_iB'_i$ backward to time $t$. Then we set +$\widetilde J(\bar t-\tau_1(\ell,\bar t),\bar t)\subset J(\bar +t-\tau_1(\ell,\bar t),\bar t)$ equal to the union over $t\in (\bar +t-\tau_1(\ell,\bar t),\bar t)$ of $\widetilde J(t)$. + + By construction, +for each surgery time $\bar t$, the union + +$$ +\nu_\mathit{sing}(\ell,\bar +t)=\widetilde J(\bar t-\tau_1(\ell,\bar t),\bar t)\cup\bigcup +_iB_i\times [\bar t,\bar t+h^2(\bar t)\Delta t_i) +$$ + + is an open +subset of ${\mathcal M}$ containing all the exposed regions and +singular points at time $\bar t$. + +We define $Y(\ell )\subset \mathbf{t}^{-1}([T_{i-1},T])$ to be the +complement of the $\cup_{\bar t}\nu_\mathit{sing}(\ell,\bar t)$ where +the union is over all surgery times $\bar t\in [T_{i-1},T]$. +Clearly, $Y(\ell )$ is a closed subset of $\mathbf{t}^{-1}([T_{i-1},T])$ and hence $Y(\ell )$ is a compact subset +contained in the open subset of smooth points of ${\mathcal M}$. +(Notice that $Y(\ell )$ depends on $\ell $ because $\tau_1(\ell,\bar +t)$ and $\theta_0$ depend on $\ell$.) + + + +**Proposition.** Fix $0 + +**Proof.** Given $L_0$, we set + +$$ +\tau'=\mathit{min}\left(\frac{r_{i+1}^4}{(256)L_0^2},\mathit{ln}(\sqrt[3]{2})r_{i+1}^2\right) +$$ + +as in \entryref{86043191a3c2} and then define $\ell=L_0/\sqrt{\tau'}$. We +also let $A=\mathit{min}(2(50+A_0),A_0(\ell))$ and +$\theta_0=\theta_0(\ell)$ as in \entryref{0a4d6f17810d}. Lastly, we +let +$\overline\delta_1(L_0,r_{i+1})=\delta''(A)=\delta''(A,\theta_0,\overline\delta_0)$ +from Propositions \entryref{0a4d6f17810d} and \entryref{ad4f67ea75bc}. We suppose that +$\delta(t)\le \overline\delta_1(L_0,r_{i+1})$ for all $t\in +[T_{i-1},T]$. + +Let $\bar t\in [T_{i-1},T]$ be a surgery time, and let $\bar h$ be +the scale of the surgery at this time. For each surgery cap +${\mathcal C}$ with tip $p$ at a time $\bar t\in [T_{i-1},T]$ let +$\Delta t({\mathcal C})$ be the supremum of those $s$ with $0\le s\le \theta_0\bar +h^2$ for which there is an embedding + +$$ +\rho_{\mathcal C}\colon +B(p,\bar t,2(A_0+50)\bar h)\times[\bar t,\bar t+s)\to{\mathcal M} +$$ + +compatible with time and the vector field. We set + +$$ +P_0({\mathcal C})=\rho_{\mathcal C}\left(B(p,\bar t,(A+50)\bar h)\times +[\bar t,\bar t+\mathit{min}(\bar h^2/2,\Delta t({\mathcal C})))\right). +$$ + + +**Claim.** Any path $\gamma$ beginning at $x$ and parameterized by backward +time misses $P_0({\mathcal C})$ if ${\mathcal L}(\gamma) + +**Proof.** Set $\tau_0=T-\bar t$. Of course, $\tau_0\le T-T_{i-1}\le T_{i+1}-T_{i-1}$. +Consider the restriction of $\gamma$ to $[0,\tau_0]$. +We have + +$$ +\begin{aligned} +\int_0^{\tau_0}\sqrt{\tau}\left(R_+(\gamma_n(\tau))+|X_{\gamma_n}(\tau)|^2\right)d\tau + \\ +& \le & +\int_0^{T-T_{i-1}}\sqrt{\tau}\left(R_+(\gamma_n(\tau))+|X_{\gamma_n}(\tau)|^2\right)d\tau \\ +& \le & \int_0^{T-T_{i-1}}\sqrt{\tau}\left(R(\gamma_n(\tau))+|X_{\gamma_n}(\tau)|^2\right)d\tau+ +\int_0^{T-T_{i-1}} +6\sqrt{\tau}d\tau \\ +& = & +\int_0^{T-T_{i-1}}\sqrt{\tau}\left(R(\gamma_n(\tau))+|X_{\gamma_n}(\tau)|^2\right)d\tau ++4(T-T_{i-1})^{3/2} +\\ +& \le & +\int_0^{\tau_0}\sqrt{\tau}\left(R(\gamma_n(\tau))+|X_{\gamma_n}(\tau)|^2\right)d\tau+4(T_{i+1})^{3/2} +\end{aligned} +$$ + +Thus, the hypothesis that ${\mathcal L}(\gamma_n)\le L$ implies that + +$$ +\int_0^{\tau_0}\sqrt{\tau}\left(R_+(\gamma_n(\tau))+|X_{\gamma_n}(\tau)|^2\right)d\tau\le +L_0. +$$ + +The claim now follows immediately from \entryref{86043191a3c2}. + +Now set $t'$ equal to the last surgery time before $\bar t$ or set $t'=0$ if $\bar t$ is +the first surgery time. We set $\tau_1(\bar t)$ equal to the minimum of $\bar t-t'$ and +$\bar h^2/\ell$. + +Assume that $\gamma(0)=x$ and that ${\mathcal L}(\gamma)\le L$. It follows from +\entryref{86043191a3c2} that the restriction of the path $\gamma$ to $[0,\tau']$ lies +in a region where the Riemann curvature is bounded above by $r^{-2}\le +r_{i+1}^{-2}$. Hence, since $\bar h<\bar\delta(t)^2r_{i+1}\ll r_{i+1}$, this +part of the path is disjoint from all strong $\delta$-necks (evolving +backward for rescaled time $(-1,0]$). That is to say, $\gamma|_{[0,\tau']}$ is +disjoint from $J_0(t)$ for every $t\in (\bar t-\tau_1(\bar t),\bar t)$ for any +surgery time $\bar t\le T$. It follows immediately that $\gamma|_{[0,\tau']}$ +is disjoint from $J(\bar t-\tau_1(\bar t),\bar t)$. + + +**Claim.** For every surgery time $\bar t\in [T_{i-1},T]$, the path $\gamma$ +starting at $x$ with ${\mathcal L}(\gamma)\le L$ is disjoint from +$J(\bar t,\bar t-\tau_1(\bar t))$. + + +**Proof.** By the remarks above, it suffices to consider surgery times $\bar +t\le T-\tau'$. It follows immediately from the previous claim that +for any surgery time $\bar t$, with the scale of the surgery being +$\bar h$ and with $p$ being the tip of a surgery cap at this time, +we have $\gamma$ is disjoint from $B(p,\bar t,(50+A_0)\bar h)$. +Also, + +$$ +\int_{T-\bar t}^{T-\bar t+\tau_1(\bar t)}\sqrt{\tau}|X_\gamma(\tau)|^2d\tau +\le {\mathcal L}_+(\gamma)\le L_0. +$$ + +Since we can assume $T-\bar t\ge \tau'$ this implies that + +$$ +\int_{T-\bar t}^{T-\bar t-\tau_1(\bar t)}|X_\gamma(\tau)|^2d\tau\le L_0/\sqrt{\tau'}=\ell. +$$ + +The claim is now immediate from \entryref{5ed7c639bd97}. + +From these two claims we see immediately that $\gamma$ is contained in the +compact subset $Y(\ell)$ which is contained in the open subset of smooth points +of ${\mathcal M}$. This proves the third item in the statement of the +proposition. Now let us turn to the limit statements. + +Take a sequence of paths $\gamma_n$ as in the statement of +\entryref{d98ad25d9bc3}. + By \entryref{86043191a3c2} the +restriction of each $\gamma_n$ to the interval $[0,\mathit{min}(\bar\tau,\tau')]$ is contained in $P(x,T,r/2,-r^2)$. The +arguments in the proof of \entryref{cd062526b91a} (which involve +changing variables to $s=\sqrt{\tau}$) show that, after passing to a +subsequence, the restrictions of the $\gamma_n$ to $[0,\mathit{min}(\bar\tau,\tau')]$ converge uniformly to a path $\gamma$ defined +on the same interval. Furthermore, + +$$ +\int_0^{\mathit{min}(\bar\tau,\tau')}\sqrt{\tau}|X_\gamma(\tau)|^2d\tau\le \mathit{liminf}_{n\rightarrow\infty}\int_0^{\mathit{min}(\bar\tau,\tau')}\sqrt{\tau}|X_{\gamma_n}(\tau)|^2d\tau, +$$ + +so that + +$$ +\begin{aligned} +\int_0^{\mathit{min}(\bar\tau,\tau')}\sqrt{\tau}\left(R(\gamma(\tau)+|X_\gamma(\tau)|^2\right)d\tau\le \\ +& & \mathit{liminf}_{n\rightarrow\infty}\int_0^{\mathit{min}(\bar\tau,\tau')}\sqrt{\tau}\left(R(\gamma_n(\tau))+|X_{\gamma_n}(\tau)|^2\right)d\tau.\nonumber +\end{aligned} +$$ + +If $\bar\tau\le \tau'$, then we have established the existence of a +limit as required. Suppose now that $\bar\tau>\tau'$. We turn our +attention to the paths $\gamma_n|_{[\tau',\bar\tau]}$. Let +$T_{i-1}<\bar t\le T-\tau'$ be either a surgery time or $T-\tau'$, and let $t'$ be the +maximum of the last surgery time before $\bar t$ and $T_{i-1}$. We consider the +restriction of the $\gamma_n$ to the interval $[T-\bar t,T-t']$. As +we have seen, these restrictions are disjoint from $J(\bar +t-\tau_1(\bar t),\bar t)$ and also from the exposed region at time +$\bar t$, which is denoted $E(\bar t)$, and from $J_0(\bar t)$. Let + +$$ +Y=\mathbf{t}^{-1}([T-t',T-\bar t])\setminus +\left(J(\bar t-\tau_1(\bar t),\bar t)\cup (E(\bar t)\cup J_0(\bar +t))\right). +$$ + + This is a compact subset with the property that any +point $y\in Y$ is connected by a backward flow line lying entirely +in $Y$ to a point $y(t')$ contained in $M_{t'}$. + + Since $Y$ is compact there is a finite upper bound on the +Ricci curvature on $Y$, and hence to ${\mathcal L}_\chi(G)$ at any +point of $Y$. Since all backward flow lines from points of $Y$ +extend all the way to $M_{t'}$, it follows that there is a constant +$C'$ such that + +$$ +|X_{\gamma_n}(\tau)|_{G(t')}\le C'|X_{\gamma_n}(\tau)|_{G(t)} +$$ + +for all $t\in [t',\bar t]$. Our hypothesis that the ${\mathcal +L}(\gamma_n)$ are uniformly bounded, the fact that the curvature is +pinched toward positive and the fact that there is a uniform bound +on the lengths of the $\tau$-intervals implies that the + +$$ +\int_{T-\bar t}^{T-t'} \sqrt{\tau}|X_{\gamma_n}(\tau)|^2d\tau +$$ + +are uniformly bounded. Because $T-\bar t$ is at least $\tau'>0$, it +follows that the $\int_{T-\bar t}^{T-t'}|X_{\gamma_n}|^2d\tau$ have +a uniform upper bound. This then implies that there is a constant +$C_1$ such that for all $n$ we have + +$$ +\int_{T-\bar t}^{T-t'}|X_{\gamma_n}(\tau)|_{G(t')}^2d\tau\le +C_1. +$$ + + Thus, after passing to a subsequence, the $\gamma_n$ converge +uniformly to a continuous $\gamma$ defined on $[T-\bar t,T-t']$. +Furthermore, we can arrange that the convergence is a weak +convergence in $W^{1,2}$. This means that $\gamma$ has a derivative +in $L^2$ and + +$$ +\int_{T-\bar t}^{T-t'}|X_{\gamma}(\tau)|^2d\tau\le \mathit{liminf}_{n\rightarrow +\infty}\int_{T-\bar t}^{T-t'}|X_{\gamma_n}(\tau)|^2d\tau. +$$ + +Now we do this simultaneously for all $\bar t=T-\tau'$ and for all the finite number of surgery times +in $[T_{i-1},T-\tau']$. This +gives a limiting path $\gamma\colon [\tau',\bar\tau]\to {\mathcal M}$. +Putting together the above inequalities we see that the limit +satisfies + +$$ +\int_{\tau'}^{\bar\tau}\sqrt{\tau}\left(R(\gamma(\tau))+|X_\gamma(\tau)|^2\right)d\tau\le +\mathit{liminf}_{n\rightarrow\infty}\int_{\tau'}^{\bar\tau}\sqrt{\tau}\left(R(\gamma_n(\tau))+ +|X_{\gamma_n}(\tau)|^2\right)d\tau. +$$ + + Since we have already +arranged that there is a limit on $[0,\tau']$, this produces a +limiting path $\gamma\colon [0,\tau_0]\to {\mathcal M}$. By +Inequalities 16.5 and 16.6 we see that + +$$ +{\mathcal L}(\gamma)\le \mathit{liminf}_{i\rightarrow \infty}{\mathcal +L}(\gamma_n). +$$ + +The limit lies in the compact subset $Y(\ell)$ and hence is contained +in the open subset of smooth points of ${\mathcal M}$. + This completes the proof of the first statement of the proposition. + +Now suppose, in addition to the above, that all the $\gamma_n$ +have the same endpoint $y\in M_{T-\tau_0}$ and that $\mathit{lim}_{n\rightarrow\infty}{\mathcal L}(\gamma_n)$ is less than or equal +to the ${\mathcal L}$-length of any path parameterized by backward +time connecting $x$ to $y$. Let $\gamma$ be the limit of a +subsequence as constructed in the proof of the first part of this +result. Clearly, by what we have just established, $\gamma$ is a path +parameterized by backward time from $x$ to $y$ and ${\mathcal +L}(\gamma)\le \mathit{lim}_{n\rightarrow\infty}{\mathcal L}(\gamma_n)$. +This means that $\gamma$ is a minimizing ${\mathcal L}$-geodesic +connecting $x$ to $y$, an ${\mathcal L}$-geodesic contained in the open subset of smooth +points of ${\mathcal M}$. + +This completes the proof of the proposition. + + +**Corollary.** Given $L<\infty$, let +$\overline{\delta}_1=\overline\delta_1(L+4(T_{i+1}^{3/2},r_{i+1})$ be as given in \entryref{86043191a3c2}. +If +$\overline\delta(t)\le \overline\delta_1$ for all $t\in [T_{i-1},T_{i+1}]$, +then for any $x\in \mathbf{t}^{-1}([T_i,T_{i+1}))$ and for any $y\in +M_{T_{i-1}}$, if there is a path $\gamma$ parameterized by +backward time connecting $x$ to $y$ with ${\mathcal L}(\gamma)\le L$, +then there is a minimizing ${\mathcal L}$-geodesic contained in the +open subset of smooth points of ${\mathcal M}$ connecting $x$ to +$y$. + + +**Proof.** Choose an ${\mathcal L}$-minimizing sequence of paths from $x$ to +$y$ and apply the previous proposition. + +### Completion of the proof of Proposition \entryref{89a60eb69c3f} + +Having found a compact subset of the open subset of smooth points of ${\mathcal M}$ +that contains all paths parameterized by backward time whose ${\mathcal L}$-length is not +too large, we are in a position to prove \entryref{89a60eb69c3f}, which states +the existence of a minimizing ${\mathcal L}$-geodesics in ${\mathcal M}$ from $x$ +and gives estimates on their ${\mathcal L}$-lengths. + +**Proof.** (of \entryref{89a60eb69c3f}). +Fix $r\ge r_{i+1}>0$. Let $({\mathcal M},G)$ and $x\in {\mathcal M}$ be +as in the statement of \entryref{89a60eb69c3f}. We set +$L=8\sqrt{T_{i+1}}(1+T_{i+1})$, and we set + +$$ +\delta=\mathit{min}\left(\delta_i,\overline \delta_1(L+4(T_{i+1})^{3/2},r_{i+1})\right), +$$ + +where $\overline\delta_1$ is as given in \entryref{86043191a3c2}. +Suppose that $\overline\delta(t)\le +\delta$ for all $t\in [T_{i-1},T_{i+1})$. We set $U$ equal to the +subset of $\mathbf{t}^{-1}([T_{i-1},T))$ consisting of all points $y$ +for which there is a path $\gamma$ from $x$ to $y$, parameterized by +backward time, with ${\mathcal L}(\gamma) +**Claim.** The subset $Z'=\{z\in Z|{\mathcal L}_x(z)\le L/2\}$ has the property +that for any compact interval $I\subset [T_{i-1},T)$ the +intersection $\mathbf{t}^{-1}(I)\cap Z'$ is compact. + + +**Proof.** Fix a compact interval $I\subset [T_{i-1},T)$. Let $\{z_n\}$ be a +sequence in $Z'\cap \mathbf{t}^{-1}(I)$. By passing to a subsequence we +can assume that the sequence $\mathbf{t}(z_n)=t_n$ converges to some +$t\in I$, and that ${\mathcal L}_x(z_n)$ converges to some $D\le +L/2$. Since the surgery times are discrete, there is a neighborhood +$J$ of $t$ in $I$ such that the only possible surgery time in $J$ is +$t$ itself. By passing to a further subsequence if necessary, we can +assume that $t_n\in J$ for all $n$. Fix $n$. First, let us consider +the case when $t_n\ge t$. Let $\gamma_n$ be a minimizing ${\mathcal +L}$-geodesic from $x$ to $z_n$. Then we form the path $\widehat +\gamma_n$ which is the union of $\gamma_n$ followed by the flow line +for the vector field $\chi$ from the endpoint of $\gamma_n$ to +$M_t$. (This flowline exists since there is no surgery time in the +open interval $(t,t_n]$.) If $t_n0$ sufficiently small. Also, +extending $\gamma$ by adding a backward vertical flow line from $z$ +shows that the minimum value of ${\mathcal L}_x$ on $U_{t-\mu}$ is +at most $(D'+D)/2$ for all $\mu>0$ sufficiently small. (Such a +vertical flow line backward in time exists since $z\in U$ and hence +$z$ is contained in the smooth part of ${\mathcal M}$.) This +contradicts the fact that limit of the minimum values of ${\mathcal +L}_x$ on $U_{t_n}$ converge to $D$ as $t_n$ converges to $t$. This +contradiction proves that the minimum value of ${\mathcal L}_x$ on +$U_t$ is $D$, and consequently the point $y\in Z'$. This proves that +$Z'\cap \mathbf{t}^{-1}(I)$ is compact, establishing the claim. + +At this point we have established that Properties (1),(2), and (4); +So it remains only to prove Property (3) of \entryref{89a60eb69c3f}. + To do this we define the reduced length function +$l_x\colon U\to \Ar$ by + +$$ +l_x(q)=\frac{{\mathcal L}_x(q)}{2\sqrt{T-\mathbf{t}(q)}} \ \ \mathit{and} \ \ +l_x^\mathit{min}(\tau)=\mathit{min}_{q\in M_t}l_x(q). +$$ + + We consider the +subset ${\mathcal S}$ of $\tau'\in (0,T-T_{i-1}]$ with $l_x^\mathit{min}(\tau)\le L/2$ for all $\tau\le \tau'$. Recall that by the +choice of $L$, we have $3\sqrt{T-T_{i-1}}< L/2$. Clearly, the +minimum value of $l_x$ on $U_{T-\tau}$ converges to $0$ +as $\tau\rightarrow 0$, implying that this set is non-empty. Also, from +its definition, ${\mathcal S}$ is an interval with $0$ being one +endpoint. + + +**Lemma.** Let $l_x^\mathit{min}(\tau')$ be the minimum value of $l_x$ on +$U_{T-\tau'}$. For any $\tau\in {\mathcal S}$ we have $l_x^\mathit{min}(\tau)\le 3/2$. + + +**Proof.** Given that we have already established Properties 1,2 and 4 of +\entryref{89a60eb69c3f}, this is immediate from +\entryref{71669e7fd5ec}. + +Now let us establish that ${\mathcal S}=(0,T-T_{i-1}]$. As we remarked above, +${\mathcal S}$ is a non-empty interval with $0$ as one endpoint. Suppose that +it is of the form $(0,\tau]$ for some $\tau\tau$ but sufficiently close to +$\tau$, there is a point $y(\tau')\in U_{T-\tau'}$ with ${\mathcal +L}_x(y(\tau')) +**Claim.** There is $\tau_0$ with $\mathit{max}(\epsilon^2,T-T_i)\le \tau_0\le +T-T_{i-1}-\epsilon^2$ such that $R(\gamma(\tau_0))< r_i^{-2}$. + + +**Proof.** Let $T'=\mathit{max}(\epsilon^{2},T-T_i)$ and let +$T''=T-T_{i-1}-\epsilon^2$, and suppose that $R(\gamma(\tau))\ge +r_i^{-2}$ for all $\tau\in [T',T'']$. Then we see that + +$$ +\int_{T'}^{T''}\sqrt{\tau}\left(R(\gamma(\tau)+|X_\gamma(\tau)|^2\right)d\tau +\ge \frac{2}{3}r_i^{-2}\left((T'')^{3/2}-(T')^{3/2}\right). +$$ + + Since +$R\ge -6$ because the curvature is pinched toward positive, we see that + +$$ +\begin{aligned} +{\mathcal L}(\gamma) & \ge & \frac{2}{3}r_i^{-2}\left((T'')^{3/2}-(T')^{3/2}\right)-\int_0^{T'} +6\sqrt{\tau}d\tau-\int_{T''}^{T-T_{i-1}}6\sqrt{\tau}d\tau \\ +& = & \frac{2}{3}r_i^{-2}\left((T'')^{3/2}-(T')^{3/2}\right)- +4(T')^{3/2}-4\left((T-T_{i-1})^{3/2}-(T'')^{3/2}\right). +\end{aligned} +$$ + + +**Claim.** We have the following estimates: + +$$ +\begin{aligned} +(T'')^{3/2}-(T')^{3/2} & \ge & +\frac{1}{4}(T-T_{i-1})^{3/2} \\ +4(T')^{3/2} & \le & 4(T-T_{i-1})^{3/2} \\ +4\left((T-T_{i-1})^{3/2}-(T'')^{3/2}\right) & \le & +\frac{2t_0}{25}\sqrt{(T-T_{i-1})}. +\end{aligned} +$$ + + +**Proof.** Since $T_i-T_{i-1}\ge t_0$ and $T\ge T_i$, we see that +$T''/(T-T_{i-1})\ge 0.9$. If $T'=T-T_i$, then since +$T0.85$ and +$(2/3)^{3/2}\le 0.6$, the first inequality follows. + +The second inequality is clear since $T'<(T-T_{i-1})$. + +The last inequality is clear from the fact that +$T''=T-T_{i-1}-\epsilon^2$ and $\epsilon\le \sqrt{t_0/50}$. + + Putting these together yields + +$$ +{\mathcal L}(\gamma)\ge + \left[\left(\frac{1}{6}r_i^{-2}-4\right)\left(T-T_{i-1}\right)-\frac{2t_0}{25}\right] + \sqrt{T-T_{i-1}}. +$$ + +Since + +$$ +r_i^{-2}\ge r_0^{-2}\ge \epsilon^{-2}\ge 50/t_0, +$$ + +and $T-T_{i-1}\ge t_0$ we see that + +$$ +\begin{aligned} +{\mathcal +L}(\gamma) & \ge & +\left[\left(\frac{50}{6t_0}-4\right)t_0-\frac{2t_0}{25}\right]\sqrt{T-T_{i-1}} +\\ +& \ge & (8-5t_0)\sqrt{T-T_{i-1}} \\ +& \ge & 4\sqrt{T-T_{i-1}}. +\end{aligned} +$$ + +(The last inequality uses the fact that $t_0=2^{-5}$.) But this +contradicts the fact that ${\mathcal L}(\gamma)\le +3\sqrt{T-T_{i-1}}$. + +Now fix $\tau_0$ satisfying \entryref{8aac9c08c058}. + Let $\gamma_1$ be the +restriction of $\gamma$ to the subinterval $[0,\tau_0]$, and let +$y=\gamma_1(\tau_0)$. Again using the fact that $R(\gamma(\tau))\ge +-6$ for all $\tau$, we see that + +$$ +{\mathcal L}(\gamma_1)\le {\mathcal L}(\gamma)+4(T-T_{i-1})^{3/2}\le +3(T_{i+1})^{1/2}+4(T_{i+1})^{3/2}. +$$ + +Set $t'=T-\tau_0$. Notice that from the definition we +have $t'\le T_i$. Consider $B=B(y,t',\frac{r_i}{2C})$, and +define $\Delta=\mathit{min}(r_i^2/16C,\epsilon^2)$. According to +\entryref{0ddfe578961e} every point $z$ on a backward flow line +starting in $B$ and defined for time at most $ \Delta$ has the +property that $R(z)\le 2r_i^{-2}$. For any surgery time $\bar t$ in +$[t'-\Delta,t')\subset [T_{i-1},T)$ the scale $\bar h$ of the +surgery at time $\bar t$ is $\le \delta(\bar t)^2 r_i$, and hence +every point of the surgery cap has scalar curvature at least +$D^{-1}\delta(\bar t)^{-4}r_i^{-2}$. Since $\overline\delta(\bar t)\le +\overline \delta\le \delta_0\le \mathit{min}(D^{-1},1/10)$, it follows +that every point of the surgery cap has curvature at least +$\delta_0^{-3}r_i^{-2}\ge 1000r_i^{-2}$. Thus, + no point +$z$ as above can lie in a surgery cap. This means that the entire +backward parabolic neighborhood $P(y,t',\frac{r_i}{2C},-\Delta)$ +exists in ${\mathcal M}$, and the scalar curvature is bounded by $2 +r_i^{-2}$ on this backward parabolic neighborhood. Because of the +curvature pinching toward positive assumption, there is $C'<\infty$ +depending only on $r_i$ and such that the Riemann curvature is +bounded by $C'$ on $P(y,t',\frac{r_i}{2C},-\Delta)$. + +Consider the one-parameter family of metrics $g(\tau),\ 0\le \tau\le +\Delta$, on $B(y,t',\frac{r_i}{2C})$ obtained by restricting the +horizontal metric $G$ to the backward parabolic neighborhood. +There is $0<\Delta_1\le \Delta /2$ +depending only on $C'$ such that for every $\tau\in [0,\Delta_1]$ and +every non-zero tangent vector $v$ at a point of +$B(y,t',\frac{r_i}{2C})$ we have + +$$ +\frac{1}{2}\le \frac{|v|^2_{g(\tau)}}{|v|^2_{g(0)}}\le 2. +$$ + +Set $\hat r=\mathit{min}(\frac{r_i}{32C},\Delta_1/2)$, so that $\hat r$ depends +only on $r_i$, $C$, and $\epsilon$. Set $t''=t'-\Delta_1$. Clearly, +$B(y, t'',\hat r)\subset B(y,t',\frac{r_i}{2C})$ so that $B(y,t'',\hat r)\subset +P(y,t',\frac{r_i}{2C},-\Delta)$. Of course, it then follows that the parabolic neighborhood +$P(y,t'',\hat r,-\Delta_1)$ exists in ${\mathcal M}$ and + +$$ +P(y,t'',\hat r,-\Delta_1)\subset P(y,t',\frac{r_i}{2 C},-\Delta), +$$ + + so that the +Riemann curvature is bounded above by $C'$ on the parabolic +neighborhood $P(y,t'',\hat r,-\Delta_1)$. We set $r'=\mathit{min}(\hat r,(C')^{-1/2},\sqrt{\Delta_1}/2)$, so that $r'$ depends only on +$r_i$, $C$, and $\epsilon$. Then the parabolic neighborhood +$P(y,t'',r',-(r')^2)$ exists in ${\mathcal M}$ and $|\mathit{Rm}|\le +(r')^{-2}$ on $P(y,t'',r',-(r')^2)$. Hence, by the inductive +non-collapsing assumption either $y$ is contained in a component of $M_{t''}$ of positive sectional curvature or + +$$ +\mathit{Vol} B(y,t'',r')\ge +\kappa_i(r')^3. +$$ + +If $y$ is contained in a component of $M_{t''}$ of positive sectional curvature, then by + Hamilton's result, \entryref{86e5a5766bda}, +under Ricci flow the component of $M_{t''}$ containing $y$ flows forward as +a family of components of positive sectional curvature until it disappears. Since there is path moving backwards in time +from $x$ to $y$, this means that the original point $x$ is contained +in a component of its time-slice with positive sectional curvature. + +Let us consider the other possibility when $\mathit{Vol} B(y,t'',r')\ge +\kappa_i(r')^3$. + For each $z\in B(y,t'',r')$ let + +$$ +\mu_z\colon [T-t',T-t'']\to B(y,t',\frac{r_i}{2C}) +$$ + + be + the $G(t')$-geodesic connecting $y$ to $z$. Of course + +$$ +|X_{\mu_z}(\tau)|_{G(t')}\le \frac{r'}{\Delta_1} +$$ + + for every $\tau\in [0,\Delta_1]$. Thus, + +$$ +|X_{\mu_z}(\tau)|_{G(T-\tau)}\le + \frac{\sqrt{2}r'}{\Delta_1} +$$ + + for all $\tau\in [T-t',T-t'']$. + Now we let $\tilde \mu_z$ be the resulting path parameterized by backward time + on the time-interval $[T-t',T-t'']$. + We estimate + +$$ +\begin{aligned} +{\mathcal + L}(\tilde \mu_z) & = & + \int_{T-t'}^{T-t''}\sqrt{\tau}\left(R(\tilde\mu_z(\tau))+|X_{\tilde\mu_z}(\tau)|^2\right)d\tau + \\ +& \le & \sqrt{T-t''}\int_{T-t'}^{T-t''}\left(2 r_i^{-2}+\frac{2(r')^2}{\Delta_1^2}\right)d\tau +\\ +& \le & \sqrt{T-t''}(2r_i^{-2}\Delta_1+\frac{1}{2}) \le +\sqrt{T}\left(\frac{1}{16C}+\frac{1}{2}\right). +\end{aligned} +$$ + +In passing to the last inequality we use the fact, from the +definitions that $r'\le \sqrt{\Delta_1}/2$ and $\Delta\le +r_i^2/16C$, whereas $\Delta_1\le \Delta/2$. + +Since $C>1$, we see that + +$$ +{\mathcal L}(\tilde \mu_z)\le \sqrt{T}. +$$ + +Putting this together with the estimate, Equation (16.7), +for ${\mathcal L}(\gamma_1)$ tells us that for each $z\in +B(y,t'',r')$ we have + +$$ +{\mathcal L}(\gamma_1*\tilde\mu_z)\le +4(T_{i+1})^{1/2}+4(T_{i+1})^{3/2}\le L/2. +$$ + + Hence, by +\entryref{89a60eb69c3f} and the choice of $L$, there is a +minimizing ${\mathcal L}$-geodesic from $x$ to each point of +$B(y,t'',r')$ of length $\le L/2$, and these geodesics are contained +in the smooth part of ${\mathcal M}$. In fact, by +\entryref{d98ad25d9bc3} there is a compact subset $Y$ of +the open subset of smooth points of ${\mathcal M}$ that contains all +the minimizing ${\mathcal L}$-geodesics from $x$ to points of +$B(y,t'',r')$. + +Then, by \entryref{7aad50e49af2} (see also, +\entryref{db4ced24b05f}), the intersection, $B'$, of ${\mathcal U}_x$ +with $B(y,t'',r')$ + is an open subset of full measure in $B(y,t'',r')$. +Of course, $\mathit{Vol} B'=\mathit{Vol} B(y,t'',r')\ge \kappa_i(r')^3$ +and the function $l_x$ is bounded by $L/2$ on $B'$. It now follows +from \entryref{3defa9b2209c} that there is $\kappa>0$ depending only on +$\kappa_i$, $r'$, $\epsilon$ and $L$ such that $x$ is $\kappa$ +non-collapsed on scales $\le \epsilon$. Recall that $L$ depends only +on $T_{i+1}$, and $r'$ depends only on $r_i,C,C'$ and $\epsilon$, +whereas $C'$ depends only on $r_i$. Thus, in the final analysis, +$\kappa$ depends only on $\kappa_i$ and $r_i$ (and $C$ and +$\epsilon$ which are fixed). This entire analysis assumed that for +all $t\in [T_{i-1},T_{i+1})$ we have the inequality +$\overline\delta(t)\le +\overline\delta_1(L+4(t_{i+1})^{3/2},r_{i+1})$ as in +\entryref{86043191a3c2}. Since $L$ depends only on $i$ and $t_0$, this shows +that the upper bound for $\delta$ depends only on $r_{i+1}$ (and on +$i$, $t_0$, $C$, and $\epsilon$). This completes the proof of +\entryref{4ce8b5da9c58}. diff --git a/projects/poincare-conjecture/.astrolabe/docs-src/18-completion-of-the-proof-of-theorem.mdx b/projects/poincare-conjecture/.astrolabe/docs-src/18-completion-of-the-proof-of-theorem.mdx new file mode 100644 index 00000000..7dff45a9 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/docs-src/18-completion-of-the-proof-of-theorem.mdx @@ -0,0 +1,991 @@ + + +# Chapter 17 -- Completion of the proof of Theorem \entryref{703e4cfeece7} + +We have established the requisite non-collapsing result assuming the +existence of strong canonical neighborhoods. In order to complete +the proof of \entryref{703e4cfeece7} it remains for us to show the +existence of strong canonical neighborhoods. This is the result of +the next section. + +## Proof of the strong canonical neighborhood assumption + + +**Proposition.** Suppose that for some $i\ge 0$ we have surgery parameter sequences +$\delta_0\ge \delta_1\ge\cdots\ge \delta_i>0$, $\epsilon=r_0\ge +r_1\ge \cdots\ge r_i>0$ and $\kappa_0\ge \kappa_1\ge \cdots\ge +\kappa_i>0$. For any $r_{i+1}\le r_i$, let $\delta(r_{i+1})>0$ be +the constant in \entryref{4ce8b5da9c58} associated to these +three sequences and to $r_{i+1}$. + Then there are positive +constants $r_{i+1}\le r_i$ and $\delta_{i+1}\le\delta(r_{i+1})$ such that the +following holds. Suppose that $({\mathcal M},G)$ is a Ricci flow with surgery +defined for $0\le t + +**Proof.** Suppose that the result does not hold. + Then we can take a sequence of $r_a\rightarrow 0$ as + $a\rightarrow \infty$, all less than $r_i$, and for each $a$ a sequence + $\delta_{a,b}\rightarrow 0$ as $b\rightarrow \infty$ with each + $\delta_{a,b}\le \delta(r_a)$, where $\delta(r_a)\le \delta_i$ is the constant in \entryref{4ce8b5da9c58} + associated to the three sequences given in the statement of this proposition and $r_a$, such that for + each $a,b$ there is a Ricci flow with surgery $({\mathcal + M}_{(a,b)},G_{(a,b)})$ defined for $0\le t +**Lemma.** For each $a$, and given $a$, for all $b$ sufficiently large there is $t_{(a,b)}\in +[T_i,T_{(a,b)})$ such that the restriction of $({\mathcal +M}_{(a,b)},G_{(a,b)})$ to $\mathbf{t}^{-1}\left([0,t_{(a,b)})\right)$ +satisfies the strong $(C,\epsilon)$-canonical neighborhood +assumption with parameter $r_a$ and furthermore, there is +$x\in{\mathcal M}_{(a,b)}$ with $\mathbf{t}(x_{(a,b)})=t_{(a,b)}$ at +which the strong $(C,\epsilon)$-canonical neighborhood assumption +with parameter $r_a$ fails. + + +**Proof.** Fix $a$. + By supposition, for each $b$ there is a point $x\in {\mathcal M}_{(a,b)}$ at +which the strong $(C,\epsilon)$-canonical neighborhood assumption +fails for the parameter $r_a$. We call points at which this +condition fails *counterexample points*. Of course, since +$r_a\le r_i$ and since the restriction of $({\mathcal +M}_{(a,b)},g_{(a,b)})$ to $\mathbf{t}^{-1}([0,T_i))$ satisfies the +hypothesis of the proposition, we see that any counterexample point +$x$ has $\mathbf{t}(x)\ge T_i$. Take a sequence $x_n=x_{n,(a,b)}$ of +counterexample points with $\mathbf{t}(x_{n+1})\le \mathbf{t}(x_{n})$ for +all $n$ that minimizes $\mathbf{t}$ among all counterexample points in +the sense that for any $\xi>0$ and for any counterexample point +$x\in {\mathcal M}_{(a,b)}$ eventually $\mathbf{t}(x_{n})< \mathbf{t}(x)+\xi$. Let $t'=t_{(a,b)}'=\mathit{lim}_{n\rightarrow\infty}\mathbf{t}(x_{n})$. Clearly, $t'\in [T_i,T_{(a,b)})$, and by construction +the restriction of $({\mathcal M}_{(a,b)},G_{(a,b)})$ to $\mathbf{t}^{-1}([0,t'))$ satisfies the $(C,\epsilon)$-canonical neighborhood assumption +with parameter $r_a$. Since the surgery times are discrete, there is +$t''=t_{(a,b)}''$ with $t'0$ for all $b$ sufficiently large, we +have $\delta_{(a,b)}\le \delta''(A,\theta_1,\overline\delta)$, where +$\delta''(A,\theta_1,\overline\delta)$ +is the constant given in \entryref{0a4d6f17810d}. + + +**Claim.** Suppose that $b$ is sufficiently large so that +$\delta_{(a,b)}\le \delta''(A,\theta_1,\overline\delta_0)$, where $\overline\delta_0$ is the constant +given in \entryref{b92e1a990021}. Then +$\tilde t_{(a,b)}'\le \theta_1$. + + +**Proof.** In this proof we shall fix $(a,b)$, so we drop these indices from +the notation. + Consider +$s\le \theta_1$ maximal so that there is an embedding + +$$ +\psi=\psi_{(a,b)}\colon +B(p_0,0,A)\times [0,s)\to \widetilde{\mathcal +M}_{(a,b)} +$$ + + compatible with time and the vector field. First +suppose that $s<\theta_1$. Then according to +\entryref{ad4f67ea75bc} either the entire ball $B(p,0,A)$ +disappears at time $s$ or $s$ is the final time of the +time interval of definition for the flow $(\widetilde{\mathcal +M}_{(a,b)},\widetilde G_{(a,b)})$. Since we have the flow line from $z\in +B(p_0,0,A)$ extending to time $\tilde t'=\tilde t'_{(a,b)}$, in either case this +implies that $\tilde t'0$, for +all $b$ sufficiently large the pullback under this map of $\widetilde G_{(a,b)}$ is within $\delta$ +in the $C^{[1/\delta]}$-topology of the restriction of the standard solution. +Let $w_{(a,b)}$ be the result of flowing $x_{(a,b)}$ backward to time $0$. + + +**Claim.** For all $b$ sufficiently large, $w_{(a,b)}\in \psi_{(a,b)}(B(p_0,0,A)\times\{0\})$. + + +**Proof.** First notice that, by our choice of $\epsilon$, + every point in the $0$ time-slice of the closure of the strong $\epsilon$-neck +centered at $x_{(a,b)}$ is within distance $(1.1)\tilde Q_{(a,b)}^{-1}\epsilon^{-1}$ +of $w_{(a,b)}$. In particular, + +$$ +d_{\widetilde G_{(a,b)}}(w_{(a,b)},y_{(a,b)})<(1.1)\tilde Q_{(a,b)}^{-1/2}\epsilon^{-1}. +$$ + +Since $y_{(a,b)}$ is contained in the surgery cap and the scale of the surgery +at this time is $1$, $y_{(a,b)}$ is within distance $A_0+5$ of $p_{(a,b)}$. +Hence, by the triangle inequality and Inequality (17.1), we have + +$$ +\begin{aligned} +d_{\widetilde G_{(a,b)}}(w_{(a,b)},p_{(a,b)}) & < & (1.1)\tilde Q_{(a,b)}^{-1/2}\epsilon^{-1}+(A_0+5) \\ +& < & (1.1)\sqrt{5D}\epsilon^{-1}+(A_0+5). +\end{aligned} +$$ + +For $b$ sufficiently large, the image $\psi_{(a,b)}(B(p_0,0,A))$ +contains the ball of radius $(0.95)A$ +centered at $p_{(a,b)}$. Since by our choice of $A$ we have $(0.95)A>(1.1)\sqrt{5D}\epsilon^{-1}+(A_0+5)$, +the claim follows. + +We define $q_{(a,b)}\in B(p_0,0,A)$ so that +$\psi_{(a,b)}(q_{(a,b)},0)=w_{(a,b)}$. Of course, + +$$ +\psi_{(a,b)}(q_{(a,b)},\tilde t'_{(a,b)})=x_{(a,b)}. +$$ + + If follows from the +above computation that for all $b$ sufficiently large we have + +$$ +d_0(q_{(a,b)},p_0)<(1.15)\tilde Q_{(a,b)}^{-1/2}\epsilon^{-1}+(1.05)(A_0+5). +$$ + +Since the standard flow has non-negative curvature, it is a distance non-increasing flow. Therefore, + +$$ +d_{\tilde t'_{(a,b)}}(q_{(a,b)},p_0)<(1.15)\tilde Q_{(a,b)}^{-1/2}\epsilon^{-1}+(1.05)(A_0+5). +$$ + +Suppose that a point $(q,\tilde t'_{(a,b)})$ in the standard solution were the center of a +$\beta\epsilon/3$-neck, where $\beta$ is the constant from \entryref{a992d6a89479}. + Of course, for all $b$ sufficiently large, $R(q,\tilde +t'_{(a,b)})>(0.99)\tilde Q_{(a,b)}$. Since $\beta<1/2$ and +$\epsilon<\sqrt{5D}(A_0+5)/2$ and $\tilde Q_{(a,b)}\le 5D$, it follows from the +above distance estimate that this neck would contain $(p_0,\tilde t'_{(a,b)})$. +But this is impossible: since $(p_0,\tilde t'_{(a,b)})$ is an isolated fixed +point of an isometric $SO(3)$-action on the standard flow, all the sectional +curvatures at $(p_0,\tilde t'_{(a,b)})$ are equal, and this is in contradiction +with estimates on the sectional curvatures at any point of an $\epsilon$-neck +given in \entryref{dc5709757703}. We can then conclude from \entryref{6ff28dba4e56} +that for all $b$ sufficiently large, the point $(p_0,\tilde t'_{(a,b)})$ is +contained in the core of a $(C(\beta\epsilon/3),\beta\epsilon/3)$-cap +$Y_{(a,b)}$ in the $\tilde t'_{(a,b)}$ time-slice of the standard solution. +Now note that for +all $b$ sufficiently large, the scalar curvature of $(q_{(a,b)},\tilde +t'_{(a,b)})$ is at least $(0.99)\tilde Q_{(a,b)}$, since the scalar curvature of $x_{(a,b)}$ +is equal to $Q_{(a,b)}$. This implies that the +diameter of $Y_{(a,b)}$ is at most + +$$ +(1.01)\tilde Q^{-1/2}_{(a,b)}C(\beta\epsilon/3)<(1.1)\sqrt{5D}C(\beta\epsilon/3). +$$ + +Since $B(p_0,0,A)$ contains $B(p_0,\tilde t'_{(a,b)},A/2)$, and since +$C>C(\beta\epsilon/3)$, it follows from the definition of $A$, the above +distance estimate, and the triangle inequality that for all $b$ sufficiently +large $B(p_0,0,A)\times\{\tilde t'_{(a,b)}\}$ contains $Y_{(a,b)}$. + + Since $C> +C(\beta\epsilon/3)+1$ and since for $b$ sufficiently large +$\psi_{(a,b)}^*\widetilde G_{(a,b)}$ is arbitrarily close to the +restriction of the standard solution metric, it follows from +\entryref{7c2758f0e2b3} that for all $b$ sufficiently large, the image $\psi_{(a,b)}(Y_{(a,b)})$ + is a $(C,\epsilon)$-cap whose core contains $x_{(a,b)}$. +As we have already remarked, this contradicts the assumption that no $x_n$ +has a strong $(C,\epsilon)$-canonical neighborhood. + + This completes the proof in the last case and establishes +\entryref{778d690c4e7c}. + + +**Remark.** Notice that even though $x_{(a,b)}$ is the center of a strong +$\epsilon$-neck, the canonical neighborhoods of the $x_n$ constructed +in the second case are not a strong $\epsilon$-necks but rather are +$(C,\epsilon)$-caps coming from applying the flow to a +neighborhood of the surgery cap ${\mathcal C}$. + + +Now we return to the proof of \entryref{254e8dae97e6}. For each $a$, we pass +to a subsequence (in $b$) so that \entryref{778d690c4e7c} holds for all $(a,b)$. For +each $(a,b)$, let $t_{(a,b)}$ be as in that lemma. We fix a point +$x_{(a,b)}\in \mathbf{t}^{-1}(t_{(a,b)})\subset {\mathcal M}_{(a,b)}$ at which the +canonical neighborhood assumption with parameter $r_a$ fails. For each $a$ +choose $b(a)$ such that $\delta_{b(a)}\rightarrow 0$ as $a\rightarrow \infty$. For each +$a$ we set $({\mathcal M}_a,G_a)=({\mathcal M}_{(a,b(a))},G_{(a,b(a))})$, we +set $t_a=t_{(a,b(a))}$, and we let $x_a=x_{(a,b(a))}\in {\mathcal M}_a$. Let +$(\widetilde {\mathcal M}_a,\widetilde G_a)$ be the Ricci flow with surgery +obtained from $({\mathcal M}_a,G_a)$ by shifting $t_a$ to $0$ and rescaling the +metric and time by $R(x_a)$. We have the points $\tilde x_a$ in the $0$ +time-slice of $\widetilde {\mathcal M}_a$ corresponding to $x_a\in {\mathcal +M}_a$. Of course, by construction $R_{\widetilde G_a}(\tilde x_a)=1$ for all +$a$. + +We shall take limits of a subsequence of this sequence of based +Ricci flows with surgery. Since $r_a\rightarrow 0$ and $R(x_a)\ge +r_a^{-2}$, it follows that $R(x_a)\rightarrow \infty$. By +\entryref{4ce8b5da9c58}, since $\delta_{b(a)}\le \delta(r_a)$ +it follows that the restriction of $(\widetilde {\mathcal +M}_a,\widetilde G_a)$ to $\mathbf{t}^{-1}(-\infty,0)$ is +$\kappa$-non-collapsed on scales $\le \epsilon R_{G_a}^{1/2}(x_a)$. +By passing to a subsequence we arrange that one of the following two +possibilities holds: + +- **(i)** There is $A<\infty$ and $t'<\infty$ such that, +for each $a$ there is a flow line through a point $y_a$ of $B_{\widetilde +G_a}(\tilde x_a,0,A)$ that is not defined on all of $[-t',0]$. (See +Fig. 17.2.) +- **(ii)** For every $A<\infty$ and every $t'<\infty$, for all $a$ +sufficiently large all flow lines through points of $B_{\widetilde +G_a}(\tilde x_a,0,A)$ are defined on the interval $[-t', 0]$. + +Let us consider the second case. By \entryref{4ce8b5da9c58} +these rescaled solutions are $\kappa$-non collapsed on scales $\le +\epsilon R_{G_a}(x_a)^{1/2}$ for all $t<0$. Since this condition is +a closed constraint, the same is true if $t=0$. Since $R(x_a)\ge +r_a^{-2}$, by construction every point $\tilde +x\in(\widetilde{\mathcal M}_a,\widetilde G_a)$ with $R(\tilde x)\ge +1$ and $\mathbf{t}(\tilde x)<0$ has a strong $(C,\epsilon)$-canonical +neighborhood. + + +**Claim.** For all $a$ sufficiently large, +every point $\tilde x\in(\widetilde{\mathcal M}_a,\widetilde G_a)$ with $R(\tilde x)>1$ and +$\mathbf{t}(\tilde x)=0$ has a $(2C,2\epsilon)$-canonical neighborhood. + + +**Proof.** Assume that $\tilde x\in \widetilde{\mathcal M}$ has $R(\tilde +x)>1$. Suppose that $\tilde x$ is an exposed point. If $a$ is +sufficiently large, then $\delta_{b(a)}$ is arbitrarily close to +zero and hence by the last item in \entryref{cf61b731c6f2} and +the structure of the standard initial condition, we see that +$\tilde x$ is contained in the core of a $(2C,2\epsilon)$-cap. + +Suppose now that $\tilde x$ is not an exposed point. + Then we can take a sequence of points +$\tilde y_n\in \widetilde{\mathcal M}_a$ all lying on the flow line for the vector field through $\tilde x$ +converging to $\tilde x$ with $\mathbf{t}(\tilde y_n)<0$. Of course, for all $n$ sufficiently large +$R(\tilde y_n)>1$, +which implies that for all $n$ sufficiently large $\tilde y_n$ has a strong $(C,\epsilon)$-canonical +neighborhood. Passing to a subsequence, we can arrange that all of these canonical neighborhoods +are of the same type. If they are all $\epsilon$-round components, all $C$-components, or all +$(C,\epsilon)$-caps whose cores contain $y_n$, then by taking limits and arguing as in the proof of +\entryref{0760630d3990} we see that $\tilde x$ has a strong +$(2C,2\epsilon)$-canonical neighborhood of the same type. On the other hand, if $\tilde y_n$ is the center +of a strong $\epsilon$-neck for all $n$, then according to \entryref{d74749898b25}, +the limit point $\tilde x$ is the center +of a strong $2\epsilon$-neck. + +Since we have chosen $\epsilon>0$ sufficiently small so that \entryref{a9e023d082dc} +applies with $\epsilon$ replaced by $2\epsilon$, applying this theorem shows +that we can pass to a subsequence and take a smooth limiting +flow of a subsequence of the rescaled flows $(\widetilde {\mathcal +M}_a,\widetilde G_a)$ based at $\tilde x_a$ and defined for all +$t\in (-\infty,0]$. Because the $({\mathcal M}_a,G_a)$ all have +curvature pinched toward positive and since $R(x_a)\rightarrow \infty$ +as $a$ tends to infinity, this result says that the limiting flow +has non-negative, bounded curvature and is $\kappa$-non-collapsed on +all scales. That is to say, the limiting flow is a +$\kappa$-solution. By \entryref{e724298e5bc3} this contradicts +the fact that the strong $(C,\epsilon)$-canonical neighborhood +assumption fails at $x_a$ for every $a$. This contradiction shows +that in the second case there is a subsequence of the $a$ such that +$x_a$ has a strong canonical neighborhood and completes the proof of the second case. + +Let us consider the first case. In this case we will arrive at a +contradiction by showing that for all $a$ sufficiently large, the +point $x_a$ lies in a strong $(C,\epsilon)$-canonical neighborhood +coming from a surgery cap. Here is the basic result we use to find +that canonical neighborhood. + + +**Lemma.** Suppose that there are $A',D',t'<\infty$ such that the following +holds for all $a$ sufficiently large. There is a point $y_a\in +B_{\widetilde G_a}(\tilde x_a,0,A')$ and a flow line of $\chi$ + beginning at $y_a$, defined for backward time and ending at a + point $z_a$ in a +surgery cap ${\mathcal C}_a$ at time $-t_a$ for some $t_a\le t'$. We +denote this flow line by $y_a(t), -t_a\le t\le 0$. Furthermore, +suppose that the scalar curvature on the flow line from $y_a$ to +$z_a$ is bounded by $D'$. Then for all $a$ sufficiently large, $x_a$ +has a strong $(C,\epsilon)$-canonical neighborhood. + + +**Proof.** The proof is by contradiction. Suppose +the result does not hold. Then there are $A',D',t'<\infty $ and we +can pass to a subsequence (in $a$) such that the hypotheses of the +lemma hold for every $a$ but no $x_a$ has a strong +$(C,\epsilon)$-canonical neighborhood. The essential point of the +argument is to show that **in the units of the surgery scale** the +elapsed time between the surgery time and $0$ is less than $1$ and +the distance from the point $z_a$ to the tip of the surgery cap is +bounded independent of $a$. + + By \entryref{0b07beec84c0}, the fact +that the scalar curvature at $z_a$ is bounded by $D'$ implies that +for all $a$ sufficiently large the scale $\bar h_a$ of the surgery +at time $-t_a$ satisfies + +$$ +\bar h_a^2\ge (2D'D)^{-1}. +$$ + +(Recall that we are working in the rescaled flow $(\widetilde {\mathcal M}_a,\widetilde G_a)$.) + +Now we are ready to show that the elapsed time is bounded less than +one in the surgery scale. + + +**Claim.** There is $\theta_1<1$, depending on $D'$ and $t'$, such that for all +$a$ sufficiently large we have $t_a< \theta_1\bar h_a^2$. + + +**Proof.** We consider two cases: either $t_a\le \bar h_a^2/2$ or $\bar +h_a^2/2D'$ as + $t_a\ge \bar h_a^2/2$. Hence, +$R(y_a,(-t_a+(2\theta_1-1)\bar h_a^2))\le 2t_aD'\bar +h_a^{-2}\le 2t'D'\bar h_a^{-2}$. This contradiction +shows that if $a$ is sufficiently large then $t_a<\theta_1\bar +h_a^2$. + +We pass to a subsequence so that $t_a\bar h_a^{-2}$ converges to +some $\theta\le \theta_1$. We define $\widetilde C$ to be the +maximum of $C$ and $3\epsilon^{-1}\beta^{-1}$. Now, using Part 5 of +\entryref{f44d182ef27e} we set $A''\ge (9\widetilde +C+3A')\sqrt{2DD'}+6(A_0+5)$ sufficiently large so that in the + standard flow $B(p_0,0,A'')$ contains +$B(p_0,t,A''/2)$ for any $t\le (\theta_1+1)/2$. This constant is chosen only to +depend on $\theta_1$, $A'$, and $C$. As $a$ tends to infinity, $\delta_a$ tends +to zero which means, by \entryref{ad4f67ea75bc}, that for all $a$ sufficiently +large there is an embedding $\rho_a\colon B(p_0,-t_a,A''\bar h_a)\times +[-t_a,0]\to \widetilde {\mathcal M}_a$ compatible with time and the vector +field such that (after translating by $t_a$ to make the flow start at time $0$ +and scaling by $\bar h_a^{-2}$) the restriction of $\widetilde G_a$ to this +image is close in the $C^\infty$-topology to the restriction of the standard +flow to $B(p_0,0,A'')\times [0,\bar h_a^{-2}t_a]$. The image $\rho_a(p_0,-t_a)$ +is the tip $p_a$ of the surgery cap ${\mathcal C}_a$ in $\widetilde {\mathcal +M}_a$. In particular, for all $a$ sufficiently large the image +$\rho_a\left(B(p_0,-t_a,A''\bar h_a)\times\{0\}\right)$ contains the $A''\bar +h_a/3$-neighborhood of the image $\rho_a(p_0,0)$ of the tip of the surgery cap +under the flow forward to time $0$. By our choice of $A''$, and +Equation (17.2), this means that for all $a$ sufficiently large +$\rho_a\left(B(p_0,-t_a,A''\bar h_a)\times\{-t_a\}\right)$ contains the +$(3\widetilde C+A') +2(A_0+5)\bar h_a$ neighborhood of $p_a=\rho_a(p_0,-t_a)$. +Notice also that, since the standard solution has positive curvature and hence +the distance between points is non-increasing in time by +\entryref{51a2afee4465}, the distance at time $0$ between $\rho_a(p_0,0)$ and +$y_a$ is less than $2(A_0+5)\bar h_a$. By the triangle inequality, we conclude +that for all $a$ sufficiently large, $\rho_a\left(B(p_0,-t_a,A''\bar h_a)\times +\{0\}\right)$ contains the $3\widetilde C$-neighborhood of $x_a$. Since the +family of metrics on $\rho_a\left(B(p_0,-t_a,A''\bar h_a)\times +[-t_a,0]\}\right)$ (after time-shifting by $t_a$ and rescaling by $\bar +h_a^{-2}$) are converging smoothly to the ball $B(p_0,0,A'')\times [0,\theta]$ +in the standard flow, + for all $a$ sufficiently large then the flow from time $-t_a$ to $0$ on the +$3\widetilde C$-neighborhood of $x_a$ is, after rescaling by $\bar +h_a^{-2}$, + very nearly isometric to the restriction of the standard flow from time $0$ to $\bar h^{-2}_at_a$ on + the $3\widetilde C\bar h_a^{-1}$-neighborhood of some +point $q_a$ in the standard flow. Of course, since the scalar +curvature of $x_a$ is one, $R(q_a,\bar h_a^{-2}t_a)$ in the standard flow is +close to $\bar h_a^{-2}$. Hence, by \entryref{6ff28dba4e56} +there is a neighborhood $X$ of $(q_a,\bar h_a^{-2}t_a)$ in the standard solution +that either is a $(C,\epsilon)$-cap, or is an evolving +$\beta\epsilon/3$-neck centered at $(q_a,\bar h_a^{-2}t_a)$. In the latter case +either the evolving neck is defined for backward time +$(1+\beta\epsilon/3)$ or its initial time-slice is the zero +time-slice and this initial time-slice lies at distance at least $1$ +from the surgery cap. Of course, $X$ is contained in the +$CR(q_a,\bar h_a^{-2}t_a)^{-1/2}$ neighborhood of $(q_a,\bar h_a^{-2}t_a)$ in the standard +solution. Since $\widetilde C\ge C$ and $R(q_a,\bar h_a^{-2}t_a)$ is close to +$\bar h_a^{-2}$, the neighborhood $X$ is contained in the +$2\widetilde C\bar h_a^{-1}$-neighborhood of $(q_a,\bar h_a^{-2}t_a)$ in the +standard solution. Hence, after rescaling, the corresponding +neighborhood of $x_a$ is contained in the $3\widetilde +C$-neighborhood of $x_a$. If either of the first two cases in \entryref{6ff28dba4e56} occurs +for a subsequence of $a$ tending to infinity, then by +\entryref{7c2758f0e2b3} and the fact that $\widetilde C>\mathit{max}(C,\epsilon^{-1})$, we see that there is a subsequence of $a$ +for which $x_a$ either is contained in the core of a +$(C,\epsilon)$-cap or is the center of a strong $\epsilon$-neck. + +We must examine further the last case. We suppose that for every $a$ +this last case holds. Then for all $a$ sufficiently large we have an +$\beta\epsilon/3$-neck $N_a$ in the zero time-slice of $\widetilde{\mathcal +M}_a$ centered at $x_a$. It is an evolving neck and there is an +embedding $\psi\colon N_a\times [-t_a,0]\to \widetilde{\mathcal +M}_a$ compatible with time and the vector field so that the initial +time-slice $\psi(N_a\times \{-t_a\})$ is in the surgery time-slice +$M_{-t_a}$ and is disjoint from the surgery cap, so in fact it is +contained in the continuing region at time $-t_a$. As we saw above, +the image of the central $2$-sphere $\psi(S^2_a\times\{-t_a\})$ lies +at distance at most $ A''\bar h_a$ from the tip of the surgery cap +$p_a$ (where, recall, $A''$ is a constant independent of $a$). The +$2$-sphere, $\Sigma_a$, along which we do surgery, creating the surgery +cap with $p_a$ as its tip, is the central $2$-sphere of a strong +$\delta_{b(a)}$-neck. As $a$ tends to infinity the surgery control +parameter $\delta_{b(a)}$ tends to zero. Thus, for $a$ sufficiently +large this strong $\delta_{b(a)}$-neck will contain a strong +$\beta\epsilon/2$- neck $N'$ centered at $\psi(x_a,-t_a)$. Since we +know that the continuing region at time $-t_a$ contains a +$\beta\epsilon/3$-neck centered at $(x_a,-t_a)$, it follows that +$N'$ is also contained in $C_{-t_a}$. That is to say, $N'$ is +contained in the negative half of the $\delta_{b(a)}$-neck centered +at $\Sigma_a$. Now we are in the situation of +\entryref{a992d6a89479}. Applying this result tells us that $x_a$ +is the center of a strong $\epsilon$-neck. + +This completes the proof that for all $a$ sufficiently large, $x_a$ +has a $(C,\epsilon)$-canonical neighborhood in contradition to our +assumption. This contradiction completes the proof of +\entryref{111551d7831b}. + +There are several steps required to complete the proof of +\entryref{254e8dae97e6}. The first step helps us apply the previous claim to +find strong $(C,\epsilon)$-canonical neighborhoods. + + +**Claim.** Given any $A<\infty$ there is $D(A)<\infty$ and $\delta(A)>0$ such that for all $a$ +sufficiently large, $|\mathit{Rm}|$ is bounded by $D(A)$ along all backward flow lines +beginning at a point of $B_{\widetilde +G_a}(\tilde x_a,0,A)$ and defined for backward time at most $\delta(A)$. + + +**Proof.** Since all points $y\in ({\mathcal M}_a,G_a)$ with $R_{G_a}(y)\ge +r^{-2}_a$ and $\mathbf{t}(y)<\mathbf{t}(x_a)$ have strong +$(C,\epsilon)$-canonical neighborhoods, and since $R(x_a)=r^{-2}_a$, +we see that all points $y\in (\widetilde{\mathcal M}_a,\widetilde +G_a)$ with $\mathbf{t}(y_a)<0$ and with $R_{\tilde G_a}(y_a)\ge 1$ have +strong $(C,\epsilon)$-canonical neighborhoods. It follows that all +points in $(\widetilde {\mathcal M}_a,\widetilde G_a)$ with $\mathbf{t}(y)\le 0$ and $R(y)>1$ have strong $(2C,2\epsilon)$-canonical +neighborhoods. Also, since $\delta_a\le \delta(r_a)$, where +$\delta(r_a)$ is the constant given in \entryref{4ce8b5da9c58}, +and since the condition of being $\kappa$-non-collapsed is a closed +constraint, it follows from \entryref{4ce8b5da9c58} that these +Ricci flows with surgery are $\kappa$-non-collapsed for a fixed +$\kappa>0$. It is now immediate from \entryref{ecd62fc2ea2b} that there +is a constant $D_0(A)$ such that $R$ is bounded above on +$B_{\widetilde G_a}(\tilde x_a,0,A)$ by $D_0(A)$. Since every point +$y\in ({\mathcal M}_a,G_a)$ with $R(y)>1$ of the sequence of with +scalar curvature at least $1$ has a $(C,\epsilon)$ canonical +neighborhood, it follows from the definition that for every such +point $y$ we have $\left|\partial R(y)/\partial t\right|0$ and a bound $D'(A)$, both depending only in +$D_0(A)$, + for the scalar curvature at all points +of backward flow lines beginning in $B_{\widetilde +G_a}(\tilde x_a,0,A)$ and defined for backward time at most $\delta(A)$. Since the curvature is pinched toward positive, +it follows that there is a bound $D(A)$ depending only on $D'(A)$ to $|\mathit{Rm}|$ on the same flow lines. + + +**Claim.** After passing to a subsequence (in $a$), either: + +- **(1)** for each $A<\infty$ there are $D(A)<\infty$ and $t(A)>0$ such that for all $a$ +sufficiently large $P_{\widetilde G_a}(\tilde x_a,0,A,-t(A))$ exists in +$\widetilde {\mathcal M}_a$ and $|\mathit{Rm}|$ is bounded by $D(A)$ on this +backward parabolic neighborhood, or +- **(2)** each $x_a$ has a strong +$(C,\epsilon)$-canonical neighborhood. + + +**Proof.** First notice that if there is $t(A)>0$ for which the backwards parabolic neighborhood +$P=P_{\widetilde G_a}(\tilde x_a,0,A,-t(A))$ exists, +then, by \entryref{7549bace9302}, +there are constants $D(A)<\infty$ and $\delta(A)>0$ such that, replacing $t(A)$ by $\mathit{min}(t(A),\delta(A))$, + $|\mathit{Rm}|$ is bounded by $D(A)$ on $P$. +Thus, either Item (1) holds or +passing to a subsequence, we can suppose that there is some $A<\infty$ for which no $t(A)>0$ as +required by Item (1) exists. +Then, for each $a$ we find a point +$y_a\in B_{\widetilde G_a}(\tilde x_a,0,A)$ such that the backwards +flow line from $y_a$ meets a surgery cap at a time $-t_a$ where +$\mathit{lim}_{a\rightarrow \infty}(t_a)=0$. Then, by the previous claim, for all $a$ sufficiently large, the +sectional curvature along any backward flow line beginning in $B_{\widetilde G_a}(\widetilde x_a,0,A)$ +and defined for backward time $t_a$ is bounded by +a constant $D(A)$ independent of $a$. Under our assumption this +means that for all $a$ sufficiently large, there is a point $y_a\in +B_{\widetilde G_a}(\tilde x_a,0,A)$ and a backwards flow line +starting at $y_a$ ending at a point $z_a$ of a surgery cap, and the +sectional curvature along this entire flow line is bounded by $D(A)<\infty$. Thus, +applying \entryref{111551d7831b} produces the strong +$(C,\epsilon)$-canonical neighborhood around $x_a$, proving the claim. + +But we are assuming that no $x_a$ has a strong $(C,\epsilon)$-canonical +neighborhood. Thus, the consequence of the previous claim is that for each +$A<\infty$ there is a $t(A)>0$ such that for all $a$ sufficiently large +$P_{\widetilde G_a}(\tilde x_a,0,A,-t(A))$ exists in $\widetilde {\mathcal +M}_a$ and there is a bound, depending only on $A$ for $|\mathit{Rm}|$ on +this backward parabolic neighborhood. Applying \entryref{d9f3522dec12} we see that, after passing to a +subsequence, there is a smooth limit $(M_\infty,g_\infty,x_\infty)$ to the zero +time-slices $(\widetilde{\mathcal M}_a,\widetilde G_a,\tilde x_a)$. +Clearly, since the curvatures of the sequence are pinched toward positive, this +limit has non-negative curvature. + +Lastly, we show that $(M_\infty,g_\infty)$ has bounded curvature. By Part 3 of \entryref{7c2758f0e2b3} +each point of $(M_\infty,g_\infty)$ with scalar curvature +greater than one has a $(2C,2\epsilon)$-canonical neighborhood. If a point lies in an $2\epsilon$-component +or in a $2C$-component, then $M_\infty$ is compact, and hence clearly has bounded curvature. +Thus, we can assume that each $y\in M_\infty$ with $R(y)>1$ is either the center of a $2\epsilon$-neck +or is contained in the core of a $(2C,2\epsilon)$-cap. According to \entryref{cb9ca6d6db9f} + $(M_\infty,g_\infty)$ does not contain $2\epsilon$-necks of arbitrarily high curvature. + It now follows then that $(M_\infty,g_\infty)$ there is a bound to the scalar curvature of any + $2\epsilon$-neck and of any $(2C,2\epsilon)$-cap, and hence it follows that $(M_\infty,g_\infty)$ + has bounded curvature. + + +**Claim.** If the constant $t(A)>0$ cannot be chosen independent of $A$, then +after passing to a subsequence, the $x_a$ have strong +$(C,\epsilon)$-canonical neighborhoods. + + +**Proof.** Let $Q$ be the bound of the scalar curvature of +$(M_\infty,g_\infty,x_\infty)$. Then by \entryref{0ddfe578961e} + there is a constant +$\Delta t>0$ such that if $R_{\widetilde G_a}(y,0)\le 2Q$, then the +scalar curvature is bounded by $16Q$ on the backward flow line from +$y$ defined for any time $\le \Delta t$. Suppose that there is +$A<\infty$ and a subsequence of $a$ for which there is a flow line +beginning at a point $y_a \in B_{\widetilde G_a}(\tilde x_a,0,A)$ +defined for backward time at most $\Delta t$ and ending at a point +$z_a$ of a surgery cap. Of course, the fact that the scalar +curvature of $(M_\infty,g_\infty)$ is at most $Q$ implies that for +all $a$ sufficiently large, the scalar curvature of $B_{\widetilde +G_a}(\tilde x_a,0,A)$ is less than $2Q$. This implies that for all +$a$ sufficiently large the scalar curvature along the flow line +from $y_a$ to $z_a$ in a surgery cap is $\le 16Q$. Now invoking +\entryref{111551d7831b} we see that for all $a$ sufficiently large the +point $\widetilde x_a$ has a strong $(C,\epsilon)$-canonical +neighborhood. This is a contradiction, and this contradiction proves +that we can choose $t(A)>0$ independent of $A$. + +Since we are assuming that no $x_a$ has a strong +$(C,\epsilon)$-canonical neighborhood, this means that it is +possible to find a constant $t'>0$ such that $t(A)\ge t'$ for all +$A<\infty$. Now let $0T'$, after passing to a subsequence, for all +$a$ sufficiently large there is $A<\infty$ and a backwards flow +line, defined for a time less than $t$, starting at a point $y_a$ of +$B_{\widetilde G_a}(\tilde x_a,0,A)$ and ending at a point $z_a$ of +a surgery cap. Invoking \entryref{0ddfe578961e} again, we see that +for all $a$ sufficiently large, the scalar curvature is bounded on +the flow line from $y_a$ to $z_a$ by a constant independent of $a$. +Hence, as before, we see that for all $a$ sufficiently large $x_a$ +has a strong $(C,\epsilon)$-canonical neighborhood; again this is a contradiction. + +Hence, we have now shown that our assumption that the strong +$(C,\epsilon)$-canonical neighborhood assumption fails for all $r_a$ +and all $\delta_{a,b}$ leads to a contradiction and hence is false. + +This completes the proof of \entryref{254e8dae97e6}. + +## Surgery times don't accumulate + +Now we turn to the proof of \entryref{703e4cfeece7}. + Given surgery parameter +sequences + +$$ +\Delta_i=\{\delta_0,\ldots,\delta_i\} +$$ + +$$ +\mathbf{r_i}=\{r_0,\ldots,r_i\} +$$ + +$$ +\mathbf{K_i}=\{\kappa_0,\ldots,\kappa_i\}, +$$ + + we let +$r_{i+1}$ and $\delta_{i+1}$ be as in \entryref{254e8dae97e6} and +then set $\kappa_{i+1}=\kappa(r_{i+1})$ as in +\entryref{4ce8b5da9c58}. + Set + +$$ +{\mathbf{r}_{i+1}}=\{\mathbf{r_i},r_{i+1}\} +$$ + +$$ +{\mathbf{K}_{i+1}}=\{\mathbf{K_i},\kappa_{i+1}\} +$$ + +$$ +\Delta_{i+1}=\{\delta_0,\ldots,\delta_{i-1},\delta_{i+1},\delta_{i+1}\}. +$$ + +Of course, these are also surgery parameter sequences. + +Let $\overline \delta\colon [0,T]\to \Ar^+$ be any non-increasing positive +function and let $({\mathcal M},G)$ be a Ricci flow with surgery defined on +$[0,T)$ for some $T\in [T_i,T_{i+1})$ with surgery control parameter +$\overline\delta$. Suppose $\overline \delta\le \Delta_{i+1}$ and that this +Ricci flow with surgery satisfies the conclusion of \entryref{703e4cfeece7} with +respect to these sequences on its entire interval of definition. We wish to +extend this Ricci flow with surgery to one defined on $[0,T')$ for some $T'$ +with $T +**Lemma.** Given a Ricci flow with surgery $({\mathcal M},G)$ defined on $[0,T)$ with +$T\le T_{i+1}$ with surgery control parameter $\overline\delta$ a +non-increasing positive function defined on $[0,T_{i+1}]$ satisfying the +hypotheses of \entryref{703e4cfeece7} on its entire time-domain of definition, there +is a constant $N$ depending only on the volume of $(M_0,g(0))$, on $T_{i+1}$, +on $r_{i+1}$, and on $\overline\delta(T_{i+1})$ such that this Ricci flow with +surgery defined on the interval $[0,T)$ has at most $N$ surgery times. + + +**Proof.** Let $(M_t,g(t))$ be the $t$ time-slice of $({\mathcal M},G)$. If $t_0$ is not +a surgery time, then $\mathit{Vol}(t)=\mathit{Vol}(M_t,g(t))$ is a smooth function +of $t$ near $t_0$ and + +$$ +\frac{d\mathit{Vol}}{dt}(t_0)=-\int_{M_{t_0}}Rd\mathit{vol}, +$$ + + so that, because of the curvature +pinching toward positive hypothesis, we have $\frac{d\mathit{Vol}}{dt}(t_0)\le +6\mathit{Vol}(t_0)$. If $t_0$ is a surgery time, then either $M_{t_0}$ has fewer +connected components than $M_{t_0^-}$ or we do a surgery in an $\epsilon$-horn +of $M_{t_0^-}$. In the latter case we remove the end of the $\epsilon$-horn, +which contains the positive half of a $\overline\delta(t_0)$-neck of scale +$h(t_0)$. We then sew in a ball with volume at most $(1+\epsilon)Kh^3(t_0)$, +where $K<\infty$ is the universal constant given in \entryref{0b07beec84c0}. Since +$h(t_0)\le \overline\delta^2(t_0)r(t_0)\le \delta_0^2r(t_0)$ and since we have +chosen + $\overline\delta(t_0)\le\delta_00$ and the canonical +neighborhood parameter $r$ at time $t_0$ is at least $r_{i+1}>0$, it follows +that $h(t_0)\ge h(T_{i+1})>0$. Thus, each surgery at time $ t_0\le T_{i+1}$ +along a $2$-sphere removes at least a fixed amount of volume depending on +$\overline \delta(T_{i+1})$ and $r_{i+1}$. Since under Ricci flow the volume +grows at most exponentially, we see that there is a bound depending only on +$\overline\delta(T_{i+1})$, $T_{i+1}$, $r_{i+1}$ and $\mathit{Vol}(M_0,g(0))$ to +the number of $2$-sphere surgeries that we can do in this time interval. On the +other hand, the number of components at any time $t$ is at most $N_0+S(t)-D(t)$ +where $N_0$ is the number of connected components of $M_0$, $S(t)$ is the +number of $2$-sphere surgeries performed in the time interval $[0,t)$ and +$D(t)$ is the number of connected components removed by surgeries at times in +the interval $[0,t)$. Hence, there is a bound on the number of components in +terms of $N_0$ and $S(T)$ that can be removed by surgery in the interval +$[0,T)$. Since the initial conditions are normalized, $N_0$ is bounded by the +volume of $(M_0,g(0))$. This completes the proof of the result. + +This lemma completes the proof of the fact that for any $T\le T_{i+1}$, we +encounter only a fixed bounded number surgeries in the Ricci flow with +surgery from $0$ to $T$. The bound depends on the volume of the initial +manifold as well as the surgery constants up to time $T_{i+1}$. In particular, +for a given initial metric $(M_0,g(0))$ there is a uniform bound, depending +only on the surgery constants up to time $T_{i+1}$, on the number of surgeries +in any Ricci flow with surgery defined on a subinterval of $[0,T_{i+1})$. It +follows that the surgery times cannot accumulate in any finite interval. This +completes the proof of \entryref{703e4cfeece7}. + +To sum up, we have sequences $\Delta$, $\mathbf{K}$ and $\mathbf{r}$ as given in +\entryref{703e4cfeece7}. Let $\overline \delta\colon [0,\infty)\to \Ar$ be a +positive, non-increasing function with $\overline \delta\le \Delta$. Let $M$ be +a compact $3$-manifold that contains no embedded $\Ar P^2$ with trivial normal +bundle. We have proved that for any normalized initial Riemannian metric +$(M_0,g_0)$ there is a Ricci flow with surgery with time-interval of definition +$[0,\infty)$ and with $(M_0,g_0)$ as initial conditions. This Ricci flow with +surgery is $\mathbf{K}$-non-collapsed and satisfies the strong +$(C,\epsilon)$-canonical neighborhood theorem with respect to the parameter +$\mathbf{r}$. It also has curvature pinched toward positive. Lastly, for any $T\in +[0,\infty)$ if there is a surgery at time $T$ then this surgery is performed +using the surgery parameters $\overline\delta(T)$ and $r(T)$, where if $T\in +[T_i,T_{i+1})$ then $r(T)=r_{i+1}$. In this Ricci flow with surgery, there are +only finitely many surgeries on each finite time interval. As far as we know +there may be infinitely many surgeries in all. diff --git a/projects/poincare-conjecture/.astrolabe/docs/14-energy1.mdx b/projects/poincare-conjecture/.astrolabe/docs-src/19-finite-time-extinction.mdx similarity index 72% rename from projects/poincare-conjecture/.astrolabe/docs/14-energy1.mdx rename to projects/poincare-conjecture/.astrolabe/docs-src/19-finite-time-extinction.mdx index 74d93932..8ba06060 100644 --- a/projects/poincare-conjecture/.astrolabe/docs/14-energy1.mdx +++ b/projects/poincare-conjecture/.astrolabe/docs-src/19-finite-time-extinction.mdx @@ -1,47 +1,50 @@ -# Finite-time extinction + + +# Chapter 18 -- Finite-time extinction Our purpose in this chapter is to prove the following finite-time extinction theorem for certain Ricci flows with surgery which, as we shall show below, when combined with the theorem on the existence of Ricci flows with surgery defined for all $t\in [0,\infty)$ -(Theorem~), immediately yields Theorem~, -thus completing the proof of the Poincar\'e Conjecture and the +(\entryref{703e4cfeece7}), immediately yields \entryref{9999efa2be2c}, +thus completing the proof of the Poincaré Conjecture and the $3$-dimensional space-form conjecture. ## The result -**Theorem.** -Let $(M,g(0))$ be a compact, connected normalized Riemannian $3$-manifold. + +**Theorem.** Let $(M,g(0))$ be a compact, connected normalized Riemannian $3$-manifold. Suppose that the fundamental group of $M$ is a free product of finite groups and infinite cyclic groups. Then $M$ contains no $\Ar P^2$ with trivial normal -bundle. Let $({\mathcal M},G)$ be the Ricci flow with surgery defined for all +bundle. Let $({\mathcal M},G)$ be the Ricci flow with surgery defined for all $t\in [0,\infty)$ with $(M,g(0))$ as initial conditions given by -Theorem~. This Ricci flow with surgery becomes extinct after a finite +\entryref{703e4cfeece7}. This Ricci flow with surgery becomes extinct after a finite time in the sense that the time-slices $M_T$ of ${\mathcal M}$ are empty for all $T$ sufficiently large. + Let us quickly show how this theorem implies our main result -Theorem~. +\entryref{9999efa2be2c}. -**Proof.** (of Theorem~ assuming -Theorem~). Fix a normalized metric $g(0)$ on $M$, and let +**Proof.** (of \entryref{9999efa2be2c} assuming +\entryref{4b8a44aa316f}). Fix a normalized metric $g(0)$ on $M$, and let $({\mathcal M},G)$ be the Ricci flow with surgery defined for all $t\in -[0,\infty)$ produced by Theorem~ with initial conditions $(M,g(0))$. -According to Theorem~ there is $T>0$ for which the time-slice -$M_T$ is empty. By Corollary~, if there is $T$ for which $M_T$ +[0,\infty)$ produced by \entryref{703e4cfeece7} with initial conditions $(M,g(0))$. +According to \entryref{4b8a44aa316f} there is $T>0$ for which the time-slice +$M_T$ is empty. By \entryref{7822d06083e0}, if there is $T$ for which $M_T$ is empty, then for any $T'T_0$ the time-slice $M_{T'}$ is empty. The argument here is similar in spirit. There are two approaches. The first -approach is due to Perelman . Here, one represents a -non-trivial element in $\pi_3(M_{T_0},x_0)$ by a non-trivial +approach is due to Perelman [Perelman3]. Here, one represents a +non-trivial element in $\pi_3(M_{T_0},x_0)$ by a non-trivial element in $\pi_2(\Lambda M,*)$, where $\Lambda M$ is the free loop space on $M$ and $*$ is the trivial loop at $x_0$. For any compact family $\Gamma$ of homotopically trivial loops in $M$ we consider @@ -85,33 +88,33 @@ family and set $W(\Gamma)$ equal to the maximal area of these minimal spanning disks. For a given element in $\gamma\in \pi_2(\Lambda M)$ we set $W(\gamma)$ equal to the infimum over all representative $2$-sphere families $\Gamma$ -for $\gamma$ of $W(\Gamma)$. Under Ricci flow, the forward +for $\gamma$ of $W(\Gamma)$. Under Ricci flow, the forward difference quotient of this invariant satisfies an inequality and the distance-decreasing property of surgery -(Proposition~) says that the inequality remains +(\entryref{9021a4eca03f}) says that the inequality remains valid for Ricci flow with surgery. The inequality implies that the value $W(\gamma)$ goes negative in finite time, which is impossible. -The other approach, by Colding-Minicozzi , is +The other approach, by Colding-Minicozzi [ColdingMinicozzi], is to represent a non-trivial element in $\pi_3(M_T)$ as a non-trivial -element in $\pi_1(Maps(S^2,M))$, and associate to such an +element in $\pi_1(\mathit{Maps}(S^2,M))$, and associate to such an element the infimum over all representative families of the maximal energy of the $2$-spheres in the family. Again, one shows that under Ricci flow the forward difference quotient of this minimax satisfies an inequality that implies that it goes negative in finite time. As before, the distance-decreasing property of surgery -(Proposition~) implies that this inequality is +(\entryref{9021a4eca03f}) implies that this inequality is valid for Ricci flows with surgery. This tells us that the manifold must completely disappear in finite time. Our first reaction was that, of the two approaches, the one considered by Colding-Minicozzi was preferable since it seemed more natural and it had the advantage of avoiding the boundary issues -that occupy most of Perelman's analysis in . +that occupy most of Perelman's analysis in [Perelman3]. In the Colding-Minicozzi approach one must construct paths of $2$-spheres with the property that when the energy of the $2$-sphere is close to the maximum value along the path, then the $2$-sphere in question - represents a point in the space $Maps(S^2,M)$ that is + represents a point in the space $\mathit{Maps}(S^2,M)$ that is close to a (usually) non-minimal critical point for the energy functional on this space. Such paths are needed in order to establish the forward difference quotient result alluded to above. @@ -120,25 +123,25 @@ that one avoids having to deal with non-minimal critical points at the expense of dealing with the technical issues related to the boundary. Since the latter are one-dimensional in nature, they are much easier to handle. In the end we decided to follow Perelman's -approach, and that is the one we present here. In +approach, and that is the one we present here. In [Perelman3] there were two points that we felt required quite a bit of argument -beyond what Perelman presented. In \S 2.2 on page 4 of -, Perelman asserts that there is a local, pointwise +beyond what Perelman presented. In Section 2.2 on page 4 of +[Perelman3], Perelman asserts that there is a local, pointwise curvature estimate that can be obtained by adapting arguments in the -literature; see Lemmas~ and~ for the precise +literature; see Lemmas \entryref{38712e8cea3d} and \entryref{ff5d51a483d9} for the precise statement. To implement this adaption required further non-trivial -arguments. We present these arguments in Section~. In \S -2.5 on page 5 of Perelman asserts that an +arguments. We present these arguments in Section 18.7. In Section +2.5 on page 5 of [Perelman3] Perelman asserts that an elementary argument establishes a lower bound on the length of a boundary curve of a minimal annulus; see -Proposition~ for a precise statement. While the +\entryref{6e9739b0c59b} for a precise statement. While the statement seems intuitively clear, we found the argument, while elementary, was quite intricate. We present this argument in -Section~. +Section 18.6. The first use of these types of ideas to show that geometric objects must disappear in finite time under Ricci flow is due to Hamilton -. He was considering a situation where a +[HamiltonNSRF3M]. He was considering a situation where a time-slice $(M,g(t_0))$ of a $3$-dimensional Ricci flow had submanifolds on which the metric was close to (a truncated version) of a hyperbolic metric of finite volume. He wished to show that @@ -148,7 +151,7 @@ disk in $M$ whose boundary was a non-trivial loop on the torus. He represented this relative homotopy class by a minimal energy disk in $(M,g(t_0))$ and proved the same sort of forward difference quotient estimate for the area of the minimal disk in the relative homotopy -class. The same contradiction -- the forward difference quotient +class. The same contradiction -- the forward difference quotient implies that after a finite time the area would go negative if the disk continued to exist --- implies that after a finite amount of time this compressing disk must disappear. Using this he showed that @@ -156,14 +159,14 @@ for sufficiently large time all the boundary tori of almost hyperbolic submanifolds in $(M,g(t))$ were incompressible. In the next section we deal with $\pi_2$ and, using $W_2$, we show -that given a Ricci flow with surgery as in Theorem~ there +that given a Ricci flow with surgery as in \entryref{703e4cfeece7} there is $T_1<\infty$ such that for all $T\ge T_1$ every connected component of $M_T$ has trivial $\pi_2$. Then in the section after that, by analyzing $W_3$, we show that, under the group-theoretic -hypothesis of Theorem~, there is a $T_2<\infty$ such +hypothesis of \entryref{4b8a44aa316f}, there is a $T_2<\infty$ such that $M_T=\emptyset$ for all $T\ge T_2$. In both these arguments we need the same type of results -- a forward difference inequality for -the energy function; the statement that away from surgery times +the energy function; the statement that away from surgery times this function is continuous; and lastly, the statement that the value of the energy function at a surgery time is at most the liminf of its values at a sequence of times approaching the surgery time @@ -171,7 +174,7 @@ from below. ### Existence of the Ricci flow with surgery -Let $(M,g(0))$ be as in the statement of Theorem~, so that $M$ is +Let $(M,g(0))$ be as in the statement of \entryref{4b8a44aa316f}, so that $M$ is a compact, connected $3$-manifold whose fundamental group is a free product of finite groups and infinite cyclic groups. By scaling $g(0)$ by a sufficiently large constant, we can assume that $g(0)$ is normalized. Let us show that such @@ -183,12 +186,12 @@ Also, any non-trivial loop in $\Ar P^2$ has non-trivial normal bundle in $M$ so that inclusion of $\RR P^2$ into $M$ induces an injection on fundamental groups. Under the fundamental group hypotheses, $M$ decomposes as a connected sum of $3$-manifolds with finite fundamental groups and $2$-sphere bundles over -$S^1$, see . +$S^1$, see [Hempel]. Given an $\RR P^2$ with trivial normal bundle embedded in a connected sum, it can be replaced by one contained in one of the connected factors. [Proof: Let $\Sigma=\Sigma_1\cup\cdots\cup\Sigma_n$ be the spheres -giving the connected sum decomposition of $M$. Deform the $\Ar P^2$ +giving the connected sum decomposition of $M$. Deform the $\Ar P^2$ until it is transverse to $\Sigma$ and let $\gamma$ be a circle of intersection of $\Ar P^2$ with one of the $\Sigma_i$ that is innermost on $\Sigma_i$ in the sense that $\gamma$ bounds a disk $D$ @@ -205,7 +208,7 @@ embedded disjointly from $\Sigma$, and hence embedded in one of the prime factors of $M$. Since it does not separate this factor, by the Mayer-Vietoris sequence (see p. 149 of -) the first homology of the factor in question maps onto $\Zee$ +[Hatcher]) the first homology of the factor in question maps onto $\Zee$ and hence the factor in question has infinite fundamental group. But this group also contains the cyclic subgroup of order two, namely the image of $\pi_1(\RR P^2)$ under the map induced by the inclusion. Thus, the fundamental group of @@ -216,26 +219,28 @@ more elementary in the sense that it uses no $3$-manifold topology. Since it is a more complicated, and to us, a less illuminating argument, we decided to present the topological argument.) -Thus, by Theorem~, for any compact $3$-manifold $M$ whose +Thus, by \entryref{703e4cfeece7}, for any compact $3$-manifold $M$ whose fundamental group is a free product of finite groups and infinite cyclic groups and for any normalized metric $g(0)$ on $M$ there is a Ricci flow with surgery $({\mathcal M},G)$ defined for all time $t\in [0,\infty)$ satisfying the -conclusion of Theorem~ with $(M,g(0))$ as the initial conditions. +conclusion of \entryref{703e4cfeece7} with $(M,g(0))$ as the initial conditions. + **Definition.** Let $I$ be an interval (which is allowed to be open or closed at each end and finite or infinite at each end). By a *path of components* of a Ricci flow with surgery $({\mathcal M},G)$ defined for all $t\in I$ we mean a connected, open subset ${\mathcal -X}\subset **t**^{-1}(I)$ with the property that for every $t\in I$ +X}\subset \mathbf{t}^{-1}(I)$ with the property that for every $t\in I$ the intersection ${\mathcal X}(t)$ of ${\mathcal X}$ with each time-slice $M_t$ is a connected component of $M_t$. + Let ${\mathcal X}$ be a path of components in a Ricci flow with surgery $({\mathcal M},G)$, a path defined for all $t\in I$. Let $I'$ be a subinterval of $I$ with the property that no point of $I'$ except possibly its initial -point is a surgery time. Then the intersection of ${\mathcal X}$ with $**t**^{-1}(I')$ is the Ricci flow on the time interval $I'$ applied to ${\mathcal -X}(t)$ for any $t\in I'$. Thus, for such intervals $I'$ the intersection, -${\mathcal X}(I')$, of ${\mathcal X}$ with $**t**^{-1}(I')$ is determined by +point is a surgery time. Then the intersection of ${\mathcal X}$ with $\mathbf{t}^{-1}(I')$ is the Ricci flow on the time interval $I'$ applied to ${\mathcal +X}(t)$ for any $t\in I'$. Thus, for such intervals $I'$ the intersection, +${\mathcal X}(I')$, of ${\mathcal X}$ with $\mathbf{t}^{-1}(I')$ is determined by the time-slice ${\mathcal X}(t)$ for any $t\in I'$. That is no longer necessarily the case if some point of $I'$ besides its initial point is a surgery time. Let $t\in I$ be a surgery time, distinct from the initial point @@ -259,7 +264,7 @@ ${\mathcal X}(t)$ can be any one of these. ## Disappearance of components with non-trivial $\pi_2$ Let $({\mathcal M},G)$ be a Ricci flow with surgery satisfying the -conclusions of Theorem~. We make no assumptions about the +conclusions of \entryref{703e4cfeece7}. We make no assumptions about the fundamental group of the initial manifold $M_0$. In this section we shall show that at some finite time $T_1$ every connected component of $M_{T_1}$ has trivial $\pi_2$ and that this condition persists @@ -279,7 +284,7 @@ following. Consider a path of components ${\mathcal X}$ defined for $t\in [T_0,t']$ with the property that each time-slice ${\mathcal X}(t)$ has non-trivial $\pi_2$. Using the fact that after time $T_0$ all the $2$-sphere surgeries are along homotopically trivial -$2$-spheres, one shows easily that ${\mathcal X}$ is determined by +$2$-spheres, one shows easily that ${\mathcal X}$ is determined by its initial time-slice ${\mathcal X}(T_0)$. Also, it is easy to see that if there is a component of $M_{t'}$ with non-trivial $\pi_2$, then it is the final time-slice of some path of components defined @@ -294,12 +299,16 @@ and to consider the function $W_2=W_2^{\mathcal X}$ that assigns to each $t\in [T_0,t']$ the minimal area of a homotopically non-trivial $2$-sphere mapping into ${\mathcal X}(t)$. We show that this function is continuous except at the surgery times. Furthermore, we -show that if $t$ is a surgery time, then $W_2(t)\le liminf_{t'\rightarrow t^-}W_2(t)$. Lastly, we show that at any +show that if $t$ is a surgery time, then $W_2(t)\le \mathit{liminf}_{t'\rightarrow t^-}W_2(t)$. Lastly, we show that at any point $t\ge T_0$ we have -$$\frac{dW_2}{dt}(t)\le -4\pi-\frac{1}{2}R_min(t)W_2(t),$$ + +$$ +\frac{dW_2}{dt}(t)\le -4\pi-\frac{1}{2}R_\mathit{min}(t)W_2(t), +$$ + in the sense of forward difference quotients. It follows easily from -the bound $R_min(t)\ge -6/(4t+1)$ that there is $T_1({\mathcal -X})$ such that $W_2$ with these three properties cannot be +the bound $R_\mathit{min}(t)\ge -6/(4t+1)$ that there is $T_1({\mathcal +X})$ such that $W_2$ with these three properties cannot be non-negative for all $t\in [T_0,T_1({\mathcal X})]$ and hence $t' +**Lemma.** Suppose that $G$ is a finitely generated group, say generated by $k$ elements. Let $G=G_1*\cdots*G_\ell$ be a free product decomposition of $G$ with non-trivial free factors, i.e., with $G_i\not=\{1\}$ for each $i=1,\ldots,\ell$. Then $\ell\le k$. + -**Proof.** This is a consequence of Grushko's theorem , which +**Proof.** This is a consequence of Grushko's theorem [Stallings], which says that given a map of a finitely generated free group $F$ onto -the free product $G$, one can decompose the free group as a free +the free product $G$, one can decompose the free group as a free product of free groups $F=F_1*\cdots*F_\ell$ with $F_i$ mapping onto $G_i$. ### Homotopically non-trivial families of $2$-spheres + **Definition.** Let $X$ be a compact $3$-manifold (possibly disconnected). An embedded $2$-sphere in $X$ is said to be *homotopically essential* if the inclusion of the $2$-sphere into $X$ is not homotopic to a point map of the $2$-sphere to $X$. More generally, let $F=\{\Sigma_1,\ldots,\Sigma_n\}$ be a family of disjointly embedded $2$-spheres in $X$. We say that the family is *homotopically essential* if -\begin{enumerate} -\item[(i)] each $2$-sphere in the family is homotopically essential, and -\item[(ii)] for any $1\le i Notice that if $F=\{\Sigma_1,\ldots,\Sigma_n\}$ is a homotopically -essential family of disjointly embedded $2$-spheres in $X$, then +essential family of disjointly embedded $2$-spheres in $X$, then any subset $F$ is also homotopically essential. + **Lemma.** Let $X$ be a compact $3$-manifold (possibly disconnected). Then there is a finite upper bound to the number of spheres in any homotopically essential family of disjointly embedded $2$-spheres. + **Proof.** Clearly, without loss of generality we can assume that $X$ is connected. If $F$ is a homotopically essential family of $2$-spheres -in $X$, then by van Kampen's theorem, see p. 40 of , +in $X$, then by van Kampen's theorem, see p. 40 of [Hatcher], there is an induced graph of groups decomposition of $\pi_1(X)$ with all the edge groups being trivial. Since the family is homotopically essential, it follows that the group associated with each vertex of @@ -360,47 +374,54 @@ the number of vertices of order $i$ and by $E$ the number of edges of the graph. The number $E$ is the number of $2$-spheres in the family $F$. An elementary combinatorial argument shows that -$$2V_1+V_2\ge E+3(1-k).$$ + +$$ +2V_1+V_2\ge E+3(1-k). +$$ + Thus, we have a free product decomposition of $\pi_1(X)$ with at least $E+3(1-k)$ non-trivial free factors. Since $k$ is bounded by the rank of $H_1(X)$, applying -Lemma~ and using the fact that the fundamental group of a +\entryref{858af0fef413} and using the fact that the fundamental group of a compact manifold is finitely presented establishes the result. -### Two-sphere surgeries are trivial after finite -time +### Two-sphere surgeries are trivial after finite time + **Definition.** Let $({\mathcal M},G)$ be a Ricci flow with surgery. We say that a surgery along a $2$-sphere $S_0(t)$ at time $t$ in $({\mathcal M},G)$ is a *homotopically essential surgery* if, for every -$t' -**Proposition.** -Let $({\mathcal M},G)$ be a Ricci flow with surgery satisfying -Assumptions (1) -- (7) in Chapter~. Then there can + +**Proposition.** Let $({\mathcal M},G)$ be a Ricci flow with surgery satisfying +Assumptions (1) -- (7) in Chapter 14. Then there can be only finitely many homotopically essential surgeries along $2$-spheres in $({\mathcal M},G)$. + **Proof.** Associate to each compact $3$-manifold $X$ the invariant $s(X)$ which is the maximal number of spheres in any homotopically essential family of embedded $2$-spheres in $X$. The main step in establishing the corollary is the following: + **Claim.** Let $({\mathcal M},G)$ be a Ricci flow with surgery and for each $t$ set -$s(t)=s(M_t)$. If $t' -\begin{proof} -Clearly, for any $t_0$ we have $s(t)=s(t_0)$ for $t\ge t_0$ +**Proof.** Clearly, for any $t_0$ we have $s(t)=s(t_0)$ for $t\ge t_0$ sufficiently close to $t_0$. Also, if $t$ is not a surgery time, then $s(t)=s(t')$ for all $t' +**Proposition.** Let $({\mathcal M},G)$ be a Ricci flow with surgery satisfying the conclusion +of \entryref{703e4cfeece7}. Then there is some time $T_1<\infty$ such that every component of $M_T$ for any $T\ge T_1$ has trivial $\pi_2$. For every $T\ge T_1$, each component of $M_T$ either has finite fundamental group, and hence -has a homotopy $3$-sphere as universal covering, or has contractible universal +has a homotopy $3$-sphere as universal covering, or has contractible universal covering. + If $M$ is a connected $3$-manifold with $\pi_2(M)=0$, then the universal covering, $\widetilde M$, of $M$ is a $2$-connected @@ -510,7 +531,7 @@ $\pi_1(M)$ is finite. In this case $\widetilde M$ is a homotopy $3$-sphere. If $\widetilde M$ is non-compact then $H_3(\widetilde M)=0$, so that all its homology groups and hence, by the Hurewicz theorem, all its homotopy groups vanish. It follows from the -Whitehead theorem that $\widetilde M$ is contractible in this case. +Whitehead theorem that $\widetilde M$ is contractible in this case. This proves the last assertion in the proposition modulo the first assertion. @@ -521,27 +542,34 @@ X}$ defined for all $t\in [T_0,\infty)$ with the property that every time-slice has non-trivial $\pi_2$. We must rule out this possibility. To achieve this we introduce the area functional. + **Lemma.** Let $X$ be a compact Riemannian manifold with $\pi_2(X)\not=0$. Then there is a positive number $e_0=e_0(X)$ with the following two properties: -\begin{enumerate} \item Any map $f\colon S^2\to X$ with area less -than $e_0$ is homotopic to a point map. \item There is a minimal + +1. Any map $f\colon S^2\to X$ with area less +than $e_0$ is homotopic to a point map. +1. There is a minimal $2$-sphere $f\colon S^2\to X$, which is a branched immersion, with the property that the area of $f(S^2)=e_0$ and with the property that $f$ is not homotopic to a point map. -\end{enumerate} + -**Proof.** The first statement is Theorem 3.3 in . As for +**Proof.** The first statement is Theorem 3.3 in [SacksUhlenbeck]. As for the second, following Sacks-Uhlenbeck, for any $\alpha>1$ we consider the perturbed energy $E_\alpha$ given by -$$E_\alpha(s)=\int_{S^2}\left(1+|ds|^2\right)^\alpha da.$$ + +$$ +E_\alpha(s)=\int_{S^2}\left(1+|ds|^2\right)^\alpha da. +$$ + According to - this energy functional is Palais-Smale on the +[SacksUhlenbeck] this energy functional is Palais-Smale on the space of $H^{1,2\alpha}$ maps and has an absolute minimum among homotopically non-trivial maps, realized by a map $s_\alpha\colon S^2\to X$. We consider a decreasing sequence of $\alpha$ tending to $1$ and the minimizers $s_\alpha$ among homotopically non-trivial -maps. According to , after passing to a +maps. According to [SacksUhlenbeck], after passing to a subsequence, there is a weak limit which is a strong limit on the complement of a finite set of points in $S^2$. This limit extends to a harmonic map of $S^2\to M$, and its energy is less than or equal @@ -557,8 +585,8 @@ complete. We must examine the case when the limit is truly a weak limit. Let $s_n$ be a sequence as above with a weak limit $s$. If the limit is truly a weak limit, then there is bubbling. Let $x\in S^2$ be a -point where the limit $s$ is not a strong limit. Then according to - pre-composing with a sequence of conformal +point where the limit $s$ is not a strong limit. Then according to +[SacksUhlenbeck] pre-composing with a sequence of conformal dilations $\rho_n$ centered at this point leads to a sequence of maps $s_n'$ converging uniformly on compact subsets of $\Ar ^2$ to a non-constant harmonic map $s'$ that extends over the one-point @@ -571,7 +599,7 @@ possibility that $s'$ is homotopically trivial. Let $\alpha$ be the area, or equivalently the energy, of $s'$. Let $D\subset \Ar^2$ be a disk centered at the origin which contains three-quarters of the energy of $s'$ (or equivalently three-quarters of the area of $s'$), -and let $D'$ be the complementary disk to $D$ in $S^2$. For all $n$ +and let $D'$ be the complementary disk to $D$ in $S^2$. For all $n$ sufficiently large the area of $s_n'|D$ minus the area of $s_n'|D'$ is at least $\alpha/3$. The restrictions of $s_n'$ on $\partial D$ are converging smoothly to $s'|\partial D'$. Let $D_n\subset S^2$ be @@ -582,7 +610,7 @@ image $s_n(D_n)$ converges smoothly, after reparameterization, to $s'(\partial D)$. Thus, for all $n$ large, we can connect $s_n(\partial D_n)$ to $s'(\partial D')$ by an annulus $A_n$ contained in a small neighborhood of $s'(\partial D')$ and whose -area tends to $0$ as $n$ goes to infinity. For all $n$ sufficiently +area tends to $0$ as $n$ goes to infinity. For all $n$ sufficiently large, the resulting $2$-sphere $\Sigma_n$ made out of $s_n|(S^2\setminus D_n)\cup A_n\cup S'(D')$ is homotopic to $s(S^2)$ since $s'$ is homotopically trivial. Also, for all $n$ sufficiently @@ -605,115 +633,139 @@ all $t\in [T_0,\infty)$ with $\pi_2({\mathcal X}(t))\not= 0$ for all $t\in [T_0,\infty)$. For each $t\ge T_0$ we define $W_2(t)$ to be $e_0({\mathcal X}(t))$, where $e_0$ is -the invariant given in the previous lemma. Our assumption on ${\mathcal X}$ +the invariant given in the previous lemma. Our assumption on ${\mathcal X}$ means that $W_2(t)$ is defined and positive for all $t\in [T_0,\infty)$. + **Lemma.** -$$\frac{d}{dt}W_2(t)\le -4\pi-\frac{1}{2}R_min(t)W_2(t)$$ + +$$ +\frac{d}{dt}W_2(t)\le -4\pi-\frac{1}{2}R_\mathit{min}(t)W_2(t) +$$ + in the sense of forward difference quotients. If $t$ is not a surgery time, then $W_2(t)$ is continuous at $t$, and if $t$ is a -surgery time, then $$W_2(t)\le liminf_{t'\rightarrow -t^-}W_2(t').$$ +surgery time, then + +$$ +W_2(t)\le \mathit{liminf}_{t'\rightarrow +t^-}W_2(t'). +$$ + -Let us show how this lemma implies Proposition~. Because +Let us show how this lemma implies \entryref{4415092c2079}. Because the curvature is pinched toward positive, we have -$$R_min(t)\ge (-6)/(1+4t).$$ + +$$ +R_\mathit{min}(t)\ge (-6)/(1+4t). +$$ + Let $w_2(t)$ be the function satisfying the differential equation -$$\frac{dw_2}{dt}=-4\pi+\frac{3w_2}{1+4t}$$ -and $w_2(T_0)=W_2(T_0)$. Then by Lemma~ and -Lemma~ we have $W_2(t)\le w_2(t)$ for all $t\ge T_0$. On + +$$ +\frac{dw_2}{dt}=-4\pi+\frac{3w_2}{1+4t} +$$ + +and $w_2(T_0)=W_2(T_0)$. Then by \entryref{67c378f87457} and +\entryref{7bb0fa0c9412} we have $W_2(t)\le w_2(t)$ for all $t\ge T_0$. On the other hand, we can integrate to find -$$w_2(t)=w_2(T_0)\frac{(4t+1)^{3/4}}{(4T_0+1)^{3/4}}+4\pi -(4T_0+1)^{1/4}(4t+1)^{3/4}-4\pi (4t+1).$$ Thus, for $t$ sufficiently + +$$ +w_2(t)=w_2(T_0)\frac{(4t+1)^{3/4}}{(4T_0+1)^{3/4}}+4\pi +(4T_0+1)^{1/4}(4t+1)^{3/4}-4\pi (4t+1). +$$ + + Thus, for $t$ sufficiently large, $w_2(t)<0$. This is a contradiction since $W_2(t)$ is always positive, and $W_2(t)\le w_2(t)$. -This shows that to complete the proof of Proposition~ we -need only establish Lemma~. +This shows that to complete the proof of \entryref{4415092c2079} we +need only establish \entryref{7bb0fa0c9412}. -**Proof.** (of Lemma~) +**Proof.** (of \entryref{7bb0fa0c9412}) Let $f\colon S^2\to (X(t_0),g(t_0))$ be a minimal $2$-sphere. + **Claim.** -$$\frac{dArea_{g(t)}(f(S^2))}{dt}(t_0)\le -4\pi -\frac{1}{2} -R_min(g(t_0))Area_{g(t_0)}f(S^2).$$ - -\begin{proof} - Recall that, for any immersed surface $f\colon S^2\to (M,g(t_0))$, we -have ([Ha]) $$ +\frac{d\mathit{Area}_{g(t)}(f(S^2))}{dt}(t_0)\le -4\pi -\frac{1}{2} +R_\mathit{min}(g(t_0))\mathit{Area}_{g(t_0)}f(S^2). +$$ + + +**Proof.** Recall that, for any immersed surface $f\colon S^2\to (M,g(t_0))$, we +have ([Ha]) $$ \begin{aligned} - - \f1{d}{dt}Area_{g(t)}(f(S^2))\bigl|_{t=t_0}\bigr. & = & - \int_{S^2}\frac{1}{2}Tr|_{S^2}\Bigl(\frac{\partial g}{\partial t}\Bigr)\Bigl|_{t=t_0}\Bigr.da \\ +\frac{d}{dt}\mathit{Area}_{g(t)}(f(S^2))\bigl|_{t=t_0}\bigr. & = & + \int_{S^2}\frac{1}{2}\mathit{Tr}|_{S^2}\Bigl(\frac{\partial g}{\partial t}\Bigr)\Bigl|_{t=t_0}\Bigr.da \\ & = & - -\int_{S^2}(R-Ric(**n**,**n**))da \nonumber + -\int_{S^2}(R-\mathit{Ric}(\mathbf{n},\mathbf{n}))da \nonumber \end{aligned} $$ -$$ - - where $R$ denotes the scalar curvature of $M$, $Ric$ is the -Ricci curvature of $M$, and $**n**$ is the unit normal vector field of $\g1$ + where $R$ denotes the scalar curvature of $M$, $\mathit{Ric}$ is the +Ricci curvature of $M$, and $\mathbf{n}$ is the unit normal vector field of $\Sigma$ in $M$. Now suppose that $f(S^2)$ is minimal. We can rewrite this as -$$ - $$ \begin{aligned} -\f1{d}{dt}Area_{g(t)}(f(S^2))\big |_{t=0} +\frac{d}{dt}\mathit{Area}_{g(t)}(f(S^2))\big |_{t=0} &= & - -\int_{S^2}K_{S^2}da-\f1{1}{2}\int_{S^2}(|A|^2+R)da, + -\int_{S^2}K_{S^2}da-\frac{1}{2}\int_{S^2}(|A|^2+R)da, \end{aligned} $$ -$$ - where $K_{S^2}$ is the Gaussian curvature of $S^2$ and $A$ is the second fundamental form of $f(S^2)$ in $M$. (Of course, since $f(S^2)$ is minimal, the determinant of its second fundamental form is $-|A|^2/2$.) Even if $f$ is only a branched minimal surface, -() still holds when the integral on the right is replaced +(18.2) still holds when the integral on the right is replaced by the integral over the immersed part of $f(S^2)$. Then by the Gauss-Bonnet theorem we have -$$ - $$ \begin{aligned} -\f1{d}{dt}Area_{g(t)}(f(S^2))\bigl|_{t=t_0}\bigr. & \leq & --4\pi-\f1{1}{2}Area_{g(t_0)}(S^2)\min_{x\in M}\{R_{g}(x,t_0)\}. +\frac{d}{dt}\mathit{Area}_{g(t)}(f(S^2))\bigl|_{t=t_0}\bigr. & \leq & +-4\pi-\frac{1}{2}\mathit{Area}_{g(t_0)}(S^2)\min_{x\in M}\{R_{g}(x,t_0)\}. \end{aligned} $$ -$$ - Since $f(S^2)$ is a homotopically non-trivial sphere in ${\mathcal X}(t)$ for -all $t$ sufficiently close to $t_0$ we see that $W_2(t)\le Area_{g(t)}f(S^2)$. Since $Area_{g(t)}f(S^2))$ is a smooth function of -$t$, the forward difference quotient statement in Lemma~ follows -immediately from Claim~. +all $t$ sufficiently close to $t_0$ we see that $W_2(t)\le \mathit{Area}_{g(t)}f(S^2)$. Since $\mathit{Area}_{g(t)}f(S^2))$ is a smooth function of +$t$, the forward difference quotient statement in \entryref{7bb0fa0c9412} follows +immediately from \entryref{db9ef2c8d5c1}. We turn now to continuity at non-surgery times. Fix $t'\ge T_0$ -distinct from all surgery times. We show that the function +distinct from all surgery times. We show that the function $e_0(t')$ is continuous at $t'$. If $f\colon S^2\to {\mathcal X}(t')$ is the minimal area, homotopically non-trivial sphere, then the area of $f(S^2)$ with respect to a nearby metric $g(t)$ is close to the area of $f(S^2)$ in the metric $g(t')$. Of course, the area of $f(S^2)$ in the metric $g(t)$ is greater than or equal to -$W_2(t)$. This proves that $W_2(t)$ is upper semi-continuous at -$t'$. Let us show that it is lower semi-continuous at $t'$. +$W_2(t)$. This proves that $W_2(t)$ is upper semi-continuous at +$t'$. Let us show that it is lower semi-continuous at $t'$. + **Claim.** Let $(M,g(t)),\ t_0\le t\le t_1$, be a Ricci flow on a compact -manifold. Suppose that $|Ric_{g(t)}|\le D$ for all $t\in +manifold. Suppose that $|\mathit{Ric}_{g(t)}|\le D$ for all $t\in [t_0,t_1]$ Let $f\colon S^2\to (M,g(t_0))$ be a $C^1$-map. Then -$$Area_{g(t_1)}f(S^2)\le Area_{g(t_0)}f(S^2)e^{4D(t_1-t_0)}.$$ + +$$ +\mathit{Area}_{g(t_1)}f(S^2)\le \mathit{Area}_{g(t_0)}f(S^2)e^{4D(t_1-t_0)}. +$$ + **Proof.** The rate of change of the area of $f(S^2)$ at time $t$ is -$$\int_{f(S^2)}\frac{\partial g}{\partial t}(t)da=-2\int_{f(S^2)}Tr|_{TS^2}(Ric_{g(t)})da\le -4DArea_{g(t)}f(S^2) .$$ Integrating from $t_0$ to $t_1$ gives + +$$ +\int_{f(S^2)}\frac{\partial g}{\partial t}(t)da=-2\int_{f(S^2)}\mathit{Tr}|_{TS^2}(\mathit{Ric}_{g(t)})da\le +4D\mathit{Area}_{g(t)}f(S^2) . +$$ + + Integrating from $t_0$ to $t_1$ gives the result. Now suppose that we have a family of times $t_n$ converging to a @@ -727,12 +779,12 @@ the above claim, for any $\delta>0$ for all $n$ sufficiently large, the area of $f_n(S^2)$ with respect to the metric $g(t')$ is at most the area of $f_n(S^2)$ plus $\delta$. This shows that for any $\delta>0$ we have $W_2(t')\le W_2(t_n)+\delta$ for all $n$ -sufficiently large, and hence $W_2(t')\le liminf_{n\rightarrow\infty}W_2(t_n)$. This is the lower +sufficiently large, and hence $W_2(t')\le \mathit{liminf}_{n\rightarrow\infty}W_2(t_n)$. This is the lower semi-continuity. The last thing to check is the behavior of $W_2$ near a surgery time $t$. According to the description of the surgery process given in -Section~, we write ${\mathcal X}(t)$ as the union +Section 15.4, we write ${\mathcal X}(t)$ as the union of a compact subset $C(t)$ and a finite number of surgery caps. For every $t'0$ for all $t'0$ for all $t'0$ for all $t'0$, it follows that - $$W_2(t)\le liminf_{t'\rightarrow t^-}W_2(t').$$ -This establishes all three statements in Proposition~ and +$$ +W_2(t)\le \mathit{liminf}_{t'\rightarrow t^-}W_2(t'). +$$ + +This establishes all three statements in \entryref{4415092c2079} and completes the proof of the proposition. -As an immediate corollary of Proposition~, we obtain the +As an immediate corollary of \entryref{4415092c2079}, we obtain the sphere theorem for closed $3$-manifolds. -**Corollary.** -Suppose that $M$ is a closed, connected $3$-manifold containing no + +**Corollary.** Suppose that $M$ is a closed, connected $3$-manifold containing no embedded $\Ar P^2$ with trivial normal bundle, and suppose that $\pi_2(M)\not= 0$. Then either $M$ can be written as a connected sum $M_1\# M_2$ where neither of the $M_i$ is homotopy equivalent to $S^3$ or $M_1$ has a prime factor that is a $2$-sphere bundle over $S^1$. In either case, $M$ contains an embedded $2$-sphere which is homotopically non-trivial. + **Proof.** Let $M$ be as in the statement of the corollary. Let $g$ be a normalized metric on $M$, and let $({\mathcal M},G)$ be the Ricci flow with surgery defined for all time with $(M,g)$ as initial -conditions. According to Proposition~ there is $T<\infty$ +conditions. According to \entryref{4415092c2079} there is $T<\infty$ such that every component of $M_T$ has trivial $\pi_2$. Thus, by the analysis above, we see that there must be surgeries that kill elements in $\pi_2$: either the removal of a component with @@ -792,13 +848,14 @@ first surgery killing an element in $\pi_2$ is removal of such a component, then, because all the earlier $2$-sphere surgeries are along homotopically trivial $2$-spheres, the homotopically non-trivial embedded $2$-sphere in this component deforms back to an -embedded, homotopically non-trivial $2$-sphere in $M$. The other +embedded, homotopically non-trivial $2$-sphere in $M$. The other possibility is that the first time an element in $\pi_2(M)$ is killed it is by surgery along a homotopically non-trivial $2$-sphere. Once again, using the fact that all previous surgeries are along homotopically trivial $2$-spheres, deform this $2$-sphere back to $M$ producing a homotopically non-trivial $2$-sphere in $M$. + **Remark.** Notice that it follows from the list of disappearing components that the only ones with non-trivial $\pi_2$ are those based on the geometry $S^2\times \Ar$; that is to say, $2$-sphere bundles over @@ -812,13 +869,14 @@ done a connected sum decomposition into components each of which has trivial $\pi_2$. Each of these components is either covered by a contractible $3$-manifold or by a homotopy $3$-sphere, depending on whether its fundamental group has infinite or finite order. + ## Extinction Now we assume that the Ricci flow with surgery $({\mathcal M},G)$ satisfies the -conclusion of Theorem~ and also has initial condition $M$ that is a +conclusion of \entryref{703e4cfeece7} and also has initial condition $M$ that is a connected $3$-manifold whose fundamental group satisfies the hypothesis of -Theorem~. The argument showing that components with non-trivial +\entryref{4b8a44aa316f}. The argument showing that components with non-trivial $\pi_3$ disappear after a finite time is, in spirit, very similar to the arguments above, though the technical details are more intricate in this case. @@ -832,30 +890,32 @@ classes of elements in $\pi_1(M,x_0)$. The connected component of the identity of $\Lambda M$ consists of all homotopically trivial loops in $M$. Let $*$ be the trivial loop at $x_0$. + **Claim.** Suppose that $\pi_2(M,x_0)=0$. Then $\pi_2(\Lambda M,*)\cong \pi_3(M,x_0)$ and $\pi_2(\Lambda M,*)$ is identified with the free homotopy classes of maps of $S^2$ to the component of $\Lambda M$ consisting of homotopically trivial loops. + **Proof.** An element in $\pi_2(\Lambda M,*)$ is represented by a map -$S^2\times S^1\to M$ that sends $\{pt\}\times S^1$ to $x_0$. +$S^2\times S^1\to M$ that sends $\{\mathit{pt}\}\times S^1$ to $x_0$. Hence, this map factors through the quotient of $S^2\times S^1$ -obtained by collapsing $\{pt\}\times S^1$ to a point. The +obtained by collapsing $\{\mathit{pt}\}\times S^1$ to a point. The resulting quotient space is homotopy equivalent to $S^2\vee S^3$, and a map of this space into $M$ sending the wedge point to $x_0$ is, up to homotopy, the same as an element of $\pi_2(M,x_0)\oplus \pi_3(M,x_0)$. But we are assuming that $\pi_2(M,x_0)=0$. The first statement follows. For the second, notice that since $\pi_2(M,x_0)$ is trivial, $\pi_3(M,x_0)$ is identified with $H_3$ of the universal -covering $\widetilde M$ of $M$. Hence, for any map of $S^2$ into +covering $\widetilde M$ of $M$. Hence, for any map of $S^2$ into the component of $\Lambda M$ containing the trivial loops, the -resulting map $S^2\times S^1\to M$ lifts to $\widetilde M$. The +resulting map $S^2\times S^1\to M$ lifts to $\widetilde M$. The corresponding element in $\pi_3(M,x_0)$ is the image of the fundamental class of $S^2\times S^1$ in $H_3(\widetilde M)=\pi_3(M)$. -**Definition.** -Fix a homotopically trivial loop $\gamma\in \Lambda M$. We set + +**Definition.** Fix a homotopically trivial loop $\gamma\in \Lambda M$. We set $A(\gamma)$ equal to the infimum of the areas of any spanning disks for $\gamma$, where by definition a spanning disk is a Lipschitz map $D^2\to M$ whose boundary is, up to reparameterization, $\gamma$. @@ -869,12 +929,13 @@ $A(\Gamma(c))$. More generally, given a homotopy class $\xi\in \pi_2(\Lambda M,*)$ we define $W(\xi)$ to be equal to the infimum over all (not necessarily based) maps $\Gamma\colon S^2\to\Lambda M$ into the component of $\Lambda M$ consisting of homotopically trivial loops representing $\xi$ of $W(\Gamma)$. + -Now let us formulate the analogue of Proposition~ for +Now let us formulate the analogue of \entryref{4415092c2079} for $\pi_3$. Suppose that ${\mathcal X}$ is a path of components of the Ricci flow with surgery $({\mathcal M},G)$ defined for $t\in [t_0,t_1]$. Suppose that $\pi_2({\mathcal X}(t_0),x_0)=0$ and that -$\pi_3({\mathcal X}(t_0),x_0)\not=0$. Then, the same two conditions +$\pi_3({\mathcal X}(t_0),x_0)\not=0$. Then, the same two conditions hold for ${\mathcal X}(t)$ for each $t\in [t_0,t_1]$. The reason is that at a surgery time $t$, since all the $2$-spheres in ${\mathcal X}(t')$ ($t' +**Proposition.** Suppose that $({\mathcal M},G)$ is a Ricci flow with surgery as in +\entryref{703e4cfeece7}. Let ${\mathcal X}$ be a path of components of ${\mathcal M}$ defined for all $t\in[t_0,t_1]$ with $\pi_2({\mathcal X}(t_0))=0$. Suppose that $\xi\in \pi_3( X(t_0),*)$ is a non-trivial element. Then the function $W_\xi(t)$ satisfies the following inequality in the sense of forward difference quotients: -$$\frac{d W_\xi(t)}{dt}\le -2\pi-\frac{1}{2}R_min(t)W_\xi(t).$$ Also, for every $t\in [t_0,t_1]$ that is not a + +$$ +\frac{d W_\xi(t)}{dt}\le -2\pi-\frac{1}{2}R_\mathit{min}(t)W_\xi(t). +$$ + + Also, for every $t\in [t_0,t_1]$ that is not a surgery time the function $W_\xi(t)$ is continuous at $t$. Lastly, if $t$ is a surgery time then -$$W_\xi(t)\le liminf_{t'\rightarrow t^-}W_\xi(t').$$ + +$$ +W_\xi(t)\le \mathit{liminf}_{t'\rightarrow t^-}W_\xi(t'). +$$ + In the next subsection we assume this result and use it to complete the proof. -### Proof of Theorem~\protect{ assuming -Proposition~\protect{}} +### Proof of Theorem \entryref{4b8a44aa316f} assuming Proposition \entryref{b187c06d1432} - According to Proposition~ there is $T_1$ such that + According to \entryref{4415092c2079} there is $T_1$ such that every component of $M_T$ has trivial $\pi_2$ for every $T\ge T_1$. -Suppose that Theorem~ does not hold for this Ricci flow +Suppose that \entryref{4b8a44aa316f} does not hold for this Ricci flow with surgery. We consider a path of components ${\mathcal X}(t)$ of -${\mathcal M}$ defined for $[T_1,T_2]$. We shall show that there is +${\mathcal M}$ defined for $[T_1,T_2]$. We shall show that there is a uniform upper bound to $T_2$. + **Claim.** ${\mathcal X}(T_1)$ has non-trivial $\pi_3$. + **Proof.** By hypothesis the fundamental group of $M_0$ is a free product of infinite cyclic groups and finite groups. This means that the same @@ -935,36 +1006,45 @@ is true for the fundamental group of each component of $M_t$ for every $t\ge 0$, and in particular it is true for ${\mathcal X}(T_0)$. But we know that $\pi_2({\mathcal X}(T_0))=0$. + **Claim.** Let $X$ be a compact $3$-manifold. If $\pi_1(X)$ is a non-trivial free product or if $\pi_1(X)$ is isomorphic to $\Zee$, then $\pi_2(X)\not=0$. + -\begin{proof} -See , Theorem 5.2 on page 56 (for the case of a copy of $\Zee$) -and Theorem 7.1 on page 66 (for the case of a free product +**Proof.** See [Hempel], Theorem 5.2 on page 56 (for the case of a copy of $\Zee$) +and [Hempel] Theorem 7.1 on page 66 (for the case of a free product decomposition). Thus, it follows that $\pi_1({\mathcal X}(T_1))$ is a finite group (possibly trivial). But a $3$-manifold with finite fundamental group has a universal covering that is a compact $3$-manifold with trivial -fundamental group. Of course, by Poincar\'e duality any simply -connected $3$-manifold is a homotopy $3$-sphere. It follows +fundamental group. Of course, by Poincaré duality any simply +connected $3$-manifold is a homotopy $3$-sphere. It follows immediately that $\pi_3({\mathcal X}(T_1))\cong \Zee$. This completes the proof of the claim. -\end{proof} -Now we can apply Proposition~ to our path of components +Now we can apply \entryref{b187c06d1432} to our path of components ${\mathcal X}$ defined for all $t\in [T_1,T_2]$. First recall by -Theorem~ that the curvature of $({\mathcal M},G)$ is pinched toward -positive which implies that $R_min(t)\ge (-6)/(1+4t)$. Let $w(t)$ be the +\entryref{703e4cfeece7} that the curvature of $({\mathcal M},G)$ is pinched toward +positive which implies that $R_\mathit{min}(t)\ge (-6)/(1+4t)$. Let $w(t)$ be the function satisfying the differential equation -$$w'(t)=-2\pi+\frac{3}{1+4t}w(t)$$ + +$$ +w'(t)=-2\pi+\frac{3}{1+4t}w(t) +$$ + with initial condition $w(T_1)=W_\xi(T_1)$. According to -Proposition~ and Proposition~ we +\entryref{b187c06d1432} and \entryref{67c378f87457} we see that $W_\xi(t)\le w(t)$ for all $t\in [T_1,T_2]$. But direct integration shows that -$$w(t)=W_\xi(T_1)\frac{(4t+1)^{3/4}}{(4T_1+1)^{3/4}}+2\pi -(4T_0+1)^{1/4}(4t+1)^{3/4}-2\pi (4t+1).$$ This clearly shows that + +$$ +w(t)=W_\xi(T_1)\frac{(4t+1)^{3/4}}{(4T_1+1)^{3/4}}+2\pi +(4T_0+1)^{1/4}(4t+1)^{3/4}-2\pi (4t+1). +$$ + + This clearly shows that $w(t)$ becomes negative for $t$ sufficiently large, how large depending only on $W_\xi(T_1)$ and $T_1$. On the other hand, since $W_\xi(t)$ is the infimum of areas of disks, $W_\xi(t)\ge 0$ for all @@ -973,23 +1053,24 @@ that depends only on $T_1$ and on the component ${\mathcal X}(T_1)$. Since there are only finitely many connected components of $M_{T_1}$, this shows that $T_2$ depends only on $T_1$ and the Riemannian manifold $M_{T_1}$. This completes the proof of -Theorem~ modulo Proposition~. -\end{proof} +\entryref{4b8a44aa316f} modulo \entryref{b187c06d1432}. -Thus, to complete the argument for Theorem~ it remains -only to prove Proposition~. +Thus, to complete the argument for \entryref{4b8a44aa316f} it remains +only to prove \entryref{b187c06d1432}. ### Continuity of $W_\xi(t)$ In this subsection we establish the two continuity conditions for -$W_\xi(t)$ stated in Proposition~. +$W_\xi(t)$ stated in \entryref{b187c06d1432}. + **Claim.** If $t$ is not a surgery time, then $W_\xi(t)$ is continuous at $t$. + **Proof.** Since $t$ is not a surgery time, a family $\Gamma(t)\colon S^2\to \Lambda {\mathcal X}(t)$ is also a family $\Gamma(t')\colon S^2\to\Lambda{\mathcal X}(t')$ for all nearby $t'$. The minimal -spanning disks for the elements of $\Gamma(t)(x)$ are also spanning +spanning disks for the elements of $\Gamma(t)(x)$ are also spanning disks in the nearby ${\mathcal X}(t')$ and their areas vary continuously with $t$. But the maximum of the areas of these disks is an upper bound for $W_\xi(t)$. This immediately implies that @@ -1006,46 +1087,51 @@ ${\mathcal X}(t')$, where $\eta(|t-t'|)$ is a function going to zero as $|t-t'|$ goes to zero. This immediately implies the lower semi-continuity at the non-surgery time $t$. + **Claim.** Suppose that $t$ is a surgery time. Then -$$W_\xi(t)\le liminf_{t'\rightarrow t^-}W_\xi(t').$$ -**Proof.** This is immediate from the fact from Proposition~ that for any +$$ +W_\xi(t)\le \mathit{liminf}_{t'\rightarrow t^-}W_\xi(t'). +$$ + + +**Proof.** This is immediate from the fact from \entryref{9021a4eca03f} that for any $\eta>0$ for every $t' **Definition.** Let $(M,g(t)),\ t_0\le t\le t_1$, be a Ricci flow on a compact $3$-manifold. For any $a$ and any $t'\in [t_0,t_1]$ let $w_{a,t'}(t)$ be the solution to the differential equation $$ - -\frac{dw_{a,t'}}{dt}=-2\pi-\frac{1}{2}R_min(t)w_{a,t'}(t) +\frac{dw_{a,t'}}{dt}=-2\pi-\frac{1}{2}R_\mathit{min}(t)w_{a,t'}(t) $$ with initial condition $w_{a,t'}(t')=a$. We also denote $w_{a,t_0}$ by $w_a$. + -**Proposition.** -Let $(M,g(t)),\ t_0\le t\le t_1$, be a Ricci flow on a compact -$3$-manifold. Fix a map $\Gamma$ of $S^2$ to $\Lambda M$ + +**Proposition.** Let $(M,g(t)),\ t_0\le t\le t_1$, be a Ricci flow on a compact +$3$-manifold. Fix a map $\Gamma$ of $S^2$ to $\Lambda M$ whose image consists of homotopically trivial loops and $\zeta>0$. Then there is a continuous family $\widetilde\Gamma(t),\ t_0\le t\le t_1$, of maps $S^2\to \Lambda M$ whose image consists of homotopically @@ -1053,13 +1139,13 @@ trivial loops with $[\widetilde\Gamma(t_0)]=[\Gamma]$ in $\pi_3(M,*)$ such that for each $c\in S^2$ we have $|A(\widetilde\Gamma(t_0)(c))-A(\Gamma(c))|<\zeta$ and furthermore, one of the following two alternatives holds: -\begin{enumerate} -\item[(i)] The length of $\widetilde\Gamma(t_1)(c)$ is less than $\zeta$. -\item[(ii)] $A(\widetilde\Gamma(t_1)(c))\le w_{A(\widetilde\Gamma(t_0)(c))}(t_1)+\zeta$. -\end{enumerate} + +- **(i)** The length of $\widetilde\Gamma(t_1)(c)$ is less than $\zeta$. +- **(ii)** $A(\widetilde\Gamma(t_1)(c))\le w_{A(\widetilde\Gamma(t_0)(c))}(t_1)+\zeta$. + Before proving this result we shall show it implies the forward difference -quotient result in Proposition~. Let ${\mathcal X}$ be a path +quotient result in \entryref{b187c06d1432}. Let ${\mathcal X}$ be a path of components. Suppose that $\pi_2({\mathcal X}(t),x_0)=0$ for all $t$. Fix $t_0$ and fix a non-trivial element $\xi\in \pi_3({\mathcal X}(t_0),x_0)$, which we identify with a non-trivial element in $\xi\in \pi_2(\Lambda{\mathcal @@ -1068,31 +1154,44 @@ surgery times in the interval $(t_0,t_1]$. Restricting to this interval the family ${\mathcal X}(t)$ is a Ricci flow on ${\mathcal X}(t_0)$. In particular, all the ${\mathcal X}(t)$ are identified under the Ricci flow. Let -$w(t)$ be the solution to Equation~() with value +$w(t)$ be the solution to Equation (18.4) with value $w(t_0)=W_\xi({\mathcal X},t_0)$. We shall show that $W_\xi(t_1)\le w(t_1)$. Clearly, once we have this estimate, taking limits as $t_1$ approaches $t_0$ establishes the forward difference quotient result at $t_0$. -**Definition.** Let $A(t)=\int_{t'}^t\frac{1}{2}R_min(s)ds$. + +**Definition.** Let $A(t)=\int_{t'}^t\frac{1}{2}R_\mathit{min}(s)ds$. + Direct integration shows the following: -**Claim.** We have -$$w_{a,t'}(t'')=exp(-A(t''))\left(a-2\pi\int_{t'}^{t''}exp(A(t))dt\right).$$ If $a'>a$, then for $t_0\le t' +**Claim.** We have + +$$ +w_{a,t'}(t'')=\mathit{exp}(-A(t''))\left(a-2\pi\int_{t'}^{t''}\mathit{exp}(A(t))dt\right). +$$ + + If $a'>a$, then for $t_0\le t' + +The next thing to establish is the following. -**Lemma.** -Given a compact Riemannian manifold $(X,g)$ with $\pi_2(X)=0$. Then + +**Lemma.** Given a compact Riemannian manifold $(X,g)$ with $\pi_2(X)=0$. Then there is $\zeta>0$ such that if $\xi\in \pi_3({\mathcal X})$ is represented by a family $\Gamma\colon S^2\to\Lambda X$ with the property that for every $c\in S^2$ the length of the loop $\Gamma(c)$ is less than $\zeta$, then $\xi$ is the trivial homotopy element. + **Proof.** We choose $\zeta$ smaller than the injectivity radius of $(X,g)$. Then any pair of points at distance less than $\zeta$ apart are @@ -1113,83 +1212,85 @@ $\hat\Gamma(c)$ to $\Gamma(c)(x_0)$ shows that the family $\Gamma$ is homotopic to a $2$-sphere family of constant loops at different points of $X$. Since we are assuming that $\pi_2(X)$ is trivial, this means the family of loops is in fact trivial as an element of -$\pi_2(\Lambda X,*)$, which means that the original element $\xi\in +$\pi_2(\Lambda X,*)$, which means that the original element $\xi\in \pi_3(X)$ is trivial. Notice that this argument also shows the following: -**Corollary.** -Let $(X,g)$ be a compact Riemannian manifold. Given $\eta>0$ there is a + +**Corollary.** Let $(X,g)$ be a compact Riemannian manifold. Given $\eta>0$ there is a $0<\zeta<\eta/2$ such that any $C^1$-loop $c\colon S^1\to X$ of length less than $\eta$ bounds a disk in $X$ of area less than $\eta$. + -Now we return to the proof that Proposition~ implies -Proposition~. We consider the restriction of the +Now we return to the proof that \entryref{fd85732525c1} implies +\entryref{b187c06d1432}. We consider the restriction of the path ${\mathcal X}$ to the time interval $[t_0,t_1]$. As we have already remarked, since there are no surgery times in $(t_0,t_1]$, this restriction is a Ricci flow and all the ${\mathcal X}(t)$ are identified with each other under the flow. - Let $w(t)$ be the solution to Equation~() + Let $w(t)$ be the solution to Equation (18.4) with initial condition $w(t_0)=W_\xi(t_0)$. There are two cases to consider: (i) $w(t_1)\ge 0$ and $w(t_1)< 0$. -Suppose that $w(t_1)\ge 0$. Let $\eta>0$ be given. Then by Claim~ -and Corollary~, there is $0<\zeta<\eta/2$ such that the +Suppose that $w(t_1)\ge 0$. Let $\eta>0$ be given. Then by \entryref{eaf31795ca89} +and \entryref{37b374f0934d}, there is $0<\zeta<\eta/2$ such that the following two conditions hold: -\begin{enumerate} -\item[(a)] Any loop in ${\mathcal X}(t_1)$ of length less than $\zeta$ + +- **(a)** Any loop in ${\mathcal X}(t_1)$ of length less than $\zeta$ bounds a disk of area less than $\eta$. -\item[(b)] For every $a\in [0,W_\xi(t_0)+2\zeta]$ +- **(b)** For every $a\in [0,W_\xi(t_0)+2\zeta]$ the solution $w_{a}$ satisfies $w_{a}(t_1)0$ we can find a family $\widetilde\Gamma(t)$ with $\widetilde\Gamma(t_0)$ representing $\xi$ and with $W(\widetilde\Gamma(t_1))< w(t_1)+\eta$. This completes the proof of -Proposition~ when $w(t_1)\ge 0$. +\entryref{fd85732525c1} when $w(t_1)\ge 0$. Now suppose that $w(t_1)<0$. In this case, we must derive a contradiction since clearly it must be the case that for any one-parameter family $\widetilde\Gamma(t)$ we have $W(\widetilde\Gamma(t_1))\ge 0$. We fix $\eta>0$ -such that $w(t_1)+\eta<0$. Then using Lemma~ and -Claim~, we fix $\zeta$ with $0<\zeta<\eta/2$ such that: -\begin{enumerate} -\item[(i)] If $\Gamma\colon S^2\to \Lambda {\mathcal X}(t_1)$ is a family +such that $w(t_1)+\eta<0$. Then using \entryref{ed0e92ce8f57} and +\entryref{eaf31795ca89}, we fix $\zeta$ with $0<\zeta<\eta/2$ such that: + +- **(i)** If $\Gamma\colon S^2\to \Lambda {\mathcal X}(t_1)$ is a family of loops and each loop in the family is of length less than $\zeta$, then the family is homotopically trivial. -\item[(ii)] For any $a\in [0,W_\xi(t_0)+2\zeta]$ we have $w_a(t_1) +**Claim.** For any immersed $C^2$-curve $c\colon S^1\to (M,g(t_0))$ there is a curve-shrinking flow $c(x,t)$ defined for $t\in [t_0,t_1')$ for some $t_1'>t_0$ with the property that each $c(\cdot,t)$ is an immersion. Either the curve-shrinking flow extends to a curve-shrinking flow that is a family of immersions defined at $t_1'$ and beyond, or -$max_{x\in S^1}k(x,t)$ blows up as $t$ approaches $t_1'$ from +$\mathit{max}_{x\in S^1}k(x,t)$ blows up as $t$ approaches $t_1'$ from below. + -For a proof of this result, see Theorem 1.13 in . +For a proof of this result, see Theorem 1.13 in [AG]. -### The proof of Proposition~\protect{ in a simple case} +### The proof of Proposition \entryref{fd85732525c1} in a simple case The main technical hurdle to overcome is that in general the curve shrinking flow may not exist if the original curve is not immersed and even if the original curve is immersed the curve-shrinking flow can develop singularities, where the curvature of the curve goes to -infinity. Thus, we may not be able to define the curve-shrinking +infinity. Thus, we may not be able to define the curve-shrinking flow as a flow defined on the entire interval $[t_0,t_1]$, even though the Ricci flow is defined on this entire interval. But to -show the idea of the proof, let us suppose for a moment that the +show the idea of the proof, let us suppose for a moment that the starting curve is embedded and that no singularities develop in the curve-shrinking flow and show how to prove the result. -**Lemma.** -Suppose that $c\in \Lambda M$ is a homotopically trivial, embedded + +**Lemma.** Suppose that $c\in \Lambda M$ is a homotopically trivial, embedded $C^2$-loop. and suppose that there is a curve-shrinking flow $c(x,t)$ defined for all $t\in [t_0,t_1]$ with each $c(\cdot,t)$ being an embedded smooth curve. Consider the function $A(t)$ which assigns to $t$ the minimal area of a spanning disk for $c(\cdot,t)$. Then $A(t)$ is a continuous function of $t$ and -$$\frac{dA}{dt}(t)\le -2\pi-\frac{1}{2}R_min(t)A(t)$$ + +$$ +\frac{dA}{dt}(t)\le -2\pi-\frac{1}{2}R_\mathit{min}(t)A(t) +$$ + in the sense of forward difference quotients. + -**Proof.** According to results of Hildebrandt and Morrey, -and , for each $t\in [t_0,t_1]$, there is a smooth +**Proof.** According to results of Hildebrandt and Morrey, [Hildebrandt] +and [Morrey], for each $t\in [t_0,t_1]$, there is a smooth minimal disk spanning $c(\cdot,t)$. Fix $t'\in [t_0,t_1)$ and consider a smooth minimal disk $D\to (M,g(t'))$ spanning -$c(\cdot,t)$. It is immersed, see or . +$c(\cdot,t)$. It is immersed, see [HardtSimon] or [GL]. The family $c(\cdot,t)$ for $t$ near $t'$ is an isotopy of $c(\cdot,t')$. We can extend this to an ambient isotopy -$\varphi_t\colon M\to M$ with $\varphi_{t'}=Id$. We impose +$\varphi_t\colon M\to M$ with $\varphi_{t'}=\mathit{Id}$. We impose coordinates $\{x_\alpha\}$ on $D$; we let $h_{\alpha\beta}(t')$ be -the metric induced on $\varphi_{t'}(D)$ by $g(t')$, and we let $da$ +the metric induced on $\varphi_{t'}(D)$ by $g(t')$, and we let $da$ be the area form induced by the Euclidean coordinates on $D$. We compute -$$\frac{d}{dt}\bigl|_{t=t'}\bigr.Area(\varphi_t(D)) -=\frac{d}{dt}\bigl|_{t=t'}\bigr.\int_{\varphi_t(D)}\sqrt{det(h_{\alpha\beta})(t)}da.$$ Of course, +$$ +\frac{d}{dt}\bigl|_{t=t'}\bigr.\mathit{Area}(\varphi_t(D)) +=\frac{d}{dt}\bigl|_{t=t'}\bigr.\int_{\varphi_t(D)}\sqrt{\mathit{det}(h_{\alpha\beta})(t)}da. $$ + Of course, + $$ \begin{aligned} -\frac{d}{dt}\bigl|_{t=t'}\bigr.\int_{\varphi_t(D)}\sqrt{det(h_{\alpha\beta}(t))}da & = & -\int_{\varphi_{t'}(D)}\Bigl(Tr\,Ric^T\Bigr)\sqrt{det(h_{\alpha\beta}(t))}da \\ -& & +\int_{\varphi_{t'}(D)}div\left(\frac{\partial -\varphi_{t'}}{dt}\right)^T\sqrt{det(h_{\alpha\beta}(t))}da. +\frac{d}{dt}\bigl|_{t=t'}\bigr.\int_{\varphi_t(D)}\sqrt{\mathit{det}(h_{\alpha\beta}(t))}da & = & -\int_{\varphi_{t'}(D)}\Bigl(\mathit{Tr} \mathit{Ric}^T\Bigr)\sqrt{\mathit{det}(h_{\alpha\beta}(t))}da \\ +& & +\int_{\varphi_{t'}(D)}\mathit{div}\left(\frac{\partial +\varphi_{t'}}{dt}\right)^T\sqrt{\mathit{det}(h_{\alpha\beta}(t))}da. \end{aligned} $$ -$$ - - Here, $Ric^T$ + Here, $\mathit{Ric}^T$ denotes the restriction of the Ricci curvature of $g(t')$ to the tangent planes of $\varphi_{t'}(D)$ and $\frac{\partial(\varphi_{t'})^T}{\partial t}$ is the component of $\varphi_{t'}$ tangent to $\varphi_{t'}(D)$. Setting $\hat A$ equal to the second fundamental form of $\varphi_{t'}(D)$, using the fact that $\varphi_{t'}(D)$ is minimal and arguing as in the proof of -Claim~, we have - -$$ +\entryref{db9ef2c8d5c1}, we have $$ \begin{aligned} -\lefteqn{-\int_{\varphi_{t'}(D)}\Bigl(Tr\,Ric^T\Bigr)\sqrt{det(h_{\alpha\beta}(t'))}da } \\ +-\int_{\varphi_{t'}(D)}\Bigl(\mathit{Tr} \mathit{Ric}^T\Bigr)\sqrt{\mathit{det}(h_{\alpha\beta}(t'))}da \\ & = & -\int_{\varphi_{t'}(D)}K_{\varphi_{t'}(D)}da --\f1{1}{2}\int_{\varphi_{t'}(D)}(|\hat A|^2+R)da \\ -& \le & -\int_{\varphi_{t'}(D)}K_{\varphi_{t'}(D)}\sqrt{det(h_{\alpha\beta}(t'))}da-\f1{1}{2}Area{\varphi_{t'}(D)}\min_{x\in M}\{R(x,t')\}. +-\frac{1}{2}\int_{\varphi_{t'}(D)}(|\hat A|^2+R)da \\ +& \le & -\int_{\varphi_{t'}(D)}K_{\varphi_{t'}(D)}\sqrt{\mathit{det}(h_{\alpha\beta}(t'))}da-\frac{1}{2}\mathit{Area}{\varphi_{t'}(D)}\min_{x\in M}\{R(x,t')\}. \end{aligned} $$ -$$ - Integration by parts shows that -$$\int_{\varphi_{t'}(D)}div\left(\frac{\partial -\varphi_{t'}}{dt}\right)^T\sqrt{det(h_{\alpha\beta}(t'))}da=-\int_{ \varphi_{t'}(\partial + +$$ +\int_{\varphi_{t'}(D)}\mathit{div}\left(\frac{\partial +\varphi_{t'}}{dt}\right)^T\sqrt{\mathit{det}(h_{\alpha\beta}(t'))}da=-\int_{ \varphi_{t'}(\partial D)}\left(\frac{d\varphi_t}{dt}\bigl|_{t=t'}\bigr.\right)\cdot n -ds,$$ where $n$ is the inward pointing normal vector to +ds, +$$ + + where $n$ is the inward pointing normal vector to $\varphi_{t'}(D)$ along $\varphi_{t'}(\partial D)$. Of course, by definition, if the variation along the boundary is given by the curve-shrinking flow, then along $\varphi_{t'}(\partial D)$ we have -$$\left(\frac{d\varphi_t}{dt}\bigl|_{t=t'}\bigr.\right)\cdot n=k_geod.$$ - -Thus, we have $$ +\left(\frac{d\varphi_t}{dt}\bigl|_{t=t'}\bigr.\right)\cdot n=k_\mathit{geod}. +$$ + +Thus, we have $$ \begin{aligned} -\lefteqn{\frac{d}{dt}\bigl|_{t=t'}\bigr.\int_{\varphi_t(D)}\sqrt{det(h_{\alpha\beta}(t))}da } \\\ & \le & +\frac{d}{dt}\bigl|_{t=t'}\bigr.\int_{\varphi_t(D)}\sqrt{\mathit{det}(h_{\alpha\beta}(t))}da \\\ & \le & -\int_{\varphi_{t'}(D)}K_{\varphi_{t'}(D)}da-\int_{\varphi_{t'}(\partial -D)}k_geodds-\frac{1}{2}R_min(t')Area(\varphi_{t'}(D)). +D)}k_\mathit{geod}ds-\frac{1}{2}R_\mathit{min}(t')\mathit{Area}(\varphi_{t'}(D)). \end{aligned} $$ -$$ Of course, the Gauss-Bonnet theorem allows us to rewrite this as -$$\frac{d}{dt}\bigl|_{t=t'}\bigr.\int_{\varphi_t(D)}\sqrt{det(h_{\alpha\beta}(t))}da \le -2\pi-\frac{1}{2}R_min(t')Area(\varphi_{t'}(D)).$$ + +$$ +\frac{d}{dt}\bigl|_{t=t'}\bigr.\int_{\varphi_t(D)}\sqrt{\mathit{det}(h_{\alpha\beta}(t))}da \le -2\pi-\frac{1}{2}R_\mathit{min}(t')\mathit{Area}(\varphi_{t'}(D)). +$$ Let $\psi(t)$ be the solution to the ODE -$$\psi'(t)=-2\pi-\frac{1}{2}R_min(t)\psi(t)$$ + +$$ +\psi'(t)=-2\pi-\frac{1}{2}R_\mathit{min}(t)\psi(t) +$$ + with $\psi(t^-)=A(t^-)$. The following is immediate from the -previous lemma and Lemma~. +previous lemma and \entryref{67c378f87457}. + **Corollary.** With notation and assumptions as above, if the curve-shrinking flow is defined on the interval $[t^-,t^+]$ and if the curves $c(\cdot,t)$ are embedded for all $t\in [t^-,t^+]$ then -$$A(t^+)\le \psi(t^+).$$ + +$$ +A(t^+)\le \psi(t^+). +$$ + Actually, the fact that the loops in the curve-shrinking flow are embedded is not essential in dimensions $\ge 3$. -**Lemma.** -Suppose that the dimension of $M$ is at least $3$ and that + +**Lemma.** Suppose that the dimension of $M$ is at least $3$ and that $c(\cdot,t)$ is a $C^2$-family of homotopically trivial, immersed curves satisfying the curve-shrinking equation defined for $t^-\le t\le t^+$. For each $t$, let $A(t)$ be the infimum of the areas of -spanning disks for $c(\cdot,t)$. Then $A(t)$ is a continuous +spanning disks for $c(\cdot,t)$. Then $A(t)$ is a continuous function and, with $\psi$ as above, we have -$$A(t^+)\le \psi(t^+).$$ + +$$ +A(t^+)\le \psi(t^+). +$$ + **Proof.** We first remark that continuity has already been established. -To show the inequality, we begin with a claim. +To show the inequality, we begin with a claim. + **Claim.** It suffices to prove the following for every $\delta>0$. There is a $C^2$-family $\hat c(x,t)$ of immersions within $\delta$ in the $C^2$-topology to $c(x,t)$ defined on the interval $[t^-,t^+]$ such that -$$A(t^+)\le \psi_{\delta,\hat c}(t^+)$$ + +$$ +A(t^+)\le \psi_{\delta,\hat c}(t^+) +$$ + where $\psi_{\delta,\hat c}$ is the solution of the ODE -$$\psi_{\delta,\hat c}'(t)=-2\pi+2\delta L_{\hat c}(t)-\frac{1}{2}R_min(t)\psi_{\delta,\hat c}(t)$$ with value $A(\hat c(t^-))$ at + +$$ +\psi_{\delta,\hat c}'(t)=-2\pi+2\delta L_{\hat c}(t)-\frac{1}{2}R_\mathit{min}(t)\psi_{\delta,\hat c}(t) +$$ + + with value $A(\hat c(t^-))$ at $t^-$, and where $L_{\hat c}(t)$ denotes the length of the loop $\hat c(\cdot,t)$. + -\begin{proof} (of the claim) Suppose that for each $\delta$ there is such +**Proof.** (of the claim) Suppose that for each $\delta$ there is such a $C^2$-family as in the statement of the claim. Take a sequence $\delta_n$ tending to zero, and let $\hat c_n(\cdot,t)$ be a family as in the claim for $\delta_n$. Then by the continuity of the infimum of areas of the spanning disk in the $C^1$-topology, we see that -$$lim_{n\rightarrow \infty}A(\hat c_n(\cdot, t^\pm))=A(c(\cdot,t^\pm)).$$ + +$$ +\mathit{lim}_{n\rightarrow \infty}A(\hat c_n(\cdot, t^\pm))=A(c(\cdot,t^\pm)). +$$ + Since the $\hat c_n(x,t)$ converge in the $C^2$-topology to $c(x,t)$, the lengths $L(\hat c_n(t))$ are uniformly bounded and the $A(\hat c_n(t^-))$ converge to $A(c(t))$. Thus, the @@ -1397,47 +1546,65 @@ $\psi_{\delta_n,\hat c_n}$ converge uniformly to $\psi$ on $[t^-,t^+]$, and taking limits shows the required inequality for $A(c(t))$, thus proving the claim. -Now we return to the proof of the lemma. Let $\hat c(x,t)$ be a -generic $C^2$-immersion sufficiently close to $c(x,t)$ in the +Now we return to the proof of the lemma. Let $\hat c(x,t)$ be a +generic $C^2$-immersion sufficiently close to $c(x,t)$ in the $C^2$-topology so that the following hold: -\begin{enumerate} -\item[(1)] the difference of the curvature of $\hat c$ and of $c$ at + +- **(1)** the difference of the curvature of $\hat c$ and of $c$ at every $(x,t)$ is a vector of length less than $\delta$, -\item[(2)] the difference of $\partial \hat c/\partial t$ and $\partial +- **(2)** the difference of $\partial \hat c/\partial t$ and $\partial c/\partial t$ is a vector of length less than $\delta$, -\item[(3)] the ratio of the arc lengths of $\hat c$ and $c$ at every +- **(3)** the ratio of the arc lengths of $\hat c$ and $c$ at every $(x,t)$ is between $(1-\delta)$ and $(1+\delta)$. -\end{enumerate} + The generic family $\hat c(x,t)$ consists of embedded curves for all but a finite number of $t\in [t^-,t^+]$ and at the exceptional $t$ values the curve is immersed. Let $t_1 +**Lemma.** Assume that $(M,g(t)),\ t_0\le t\le t_1$, is a Ricci flow and that $c=c(x,t)$ is a solution to the curve-shrinking flow. We have vector fields $X=\partial /\partial x$ and $H=\partial /\partial t$ defined on the domain surface. We denote by $|X|_{c^*g}^2$ the function on the domain surface whose value at $(x,t)$ is $|(X(x,t))|_{g(t)}^2$. We define $S=|X|_{c^*g}^{-1}X$, the unit vector in the $x$-direction measured in the evolving metric. Then, -$$\frac{\partial}{\partial t}(|X|_{c^*g}^2)(x,t)=-2Ric_{g(t)}(X(x,t),X(x,t))-2k^2|X(x,t)|_{g(t)}^2,$$ + +$$ +\frac{\partial}{\partial t}(|X|_{c^*g}^2)(x,t)=-2\mathit{Ric}_{g(t)}(X(x,t),X(x,t))-2k^2|X(x,t)|_{g(t)}^2, +$$ + and -$$[H,S](x,t)=\left(k^2+Ric_{g(t)}(S(x,t),S(x,t))\right)S(x,t).$$ + +$$ +[H,S](x,t)=\left(k^2+\mathit{Ric}_{g(t)}(S(x,t),S(x,t))\right)S(x,t). +$$ + **Proof.** Notice that as $t$ varies $|X|_{c^*g}^2$ is not the norm of the vector field $X$ with respect to the pullback of a fixed metric @@ -1467,8 +1642,13 @@ In fact, there are two contributions to $H(|X|_{c^*g}^2)$: one, the usual Leibniz rule differentiating in a frozen metric $g(t)$ and the other coming from the effect on $|X|_{c^*g}^2$ of varying the metric with $t$. Thus, we have -$$H(|X|_{c^*g}^2)(x,t)=-2Ric_{c^*g(t)}(X(x,t),X(x,t)) -+2\langle\nabla_HX(x,t),X(x,t)\rangle_{c^*g(t)}.$$ Since $t$ and $x$ + +$$ +H(|X|_{c^*g}^2)(x,t)=-2\mathit{Ric}_{c^*g(t)}(X(x,t),X(x,t)) ++2\langle\nabla_HX(x,t),X(x,t)\rangle_{c^*g(t)}. +$$ + + Since $t$ and $x$ are coordinates on the surface swept out by the family of curves, $\nabla_HX=\nabla_XH$, and hence the second term on the right-hand side of the previous equation can be rewritten as @@ -1476,8 +1656,6 @@ $2\langle\nabla_XH(x,t),X(x,t)\rangle_{c^*g(t)}$. Since $X(x,t)$ and $H(x,t)$ are orthogonal in $c^*g(t)$ and since $X=|X|_{c^*g}S$, computing covariant derivatives in the metric $c^*g(t)$, we have -$$ - $$ \begin{aligned} 2\langle\nabla_XH,X\rangle_{c^*g(t)} & = & -2\langle H,\nabla_XX\rangle_{c^*g(t)}\\ @@ -1488,48 +1666,50 @@ $$ \end{aligned} $$ -$$ - This proves the first inequality. As for the second, since $X$ and $H$ commute we have -$$[H,S]=[H,|X|_{c^*g}^{-1}X]=H\left((|X|_{c^*g}^2)^{-1/2}\right)X= -\frac{-1}{2\left(|X|^2_g\right)^{3/2}}H(|X|_{c^*g})^2)X.$$ According to the + +$$ +[H,S]=[H,|X|_{c^*g}^{-1}X]=H\left((|X|_{c^*g}^2)^{-1/2}\right)X= +\frac{-1}{2\left(|X|^2_g\right)^{3/2}}H(|X|_{c^*g})^2)X. +$$ + + According to the first equation, we can rewrite this as -$$[H,S](x,t)=\left(k^2+Ric_{c^*g(t)}(S(x,t),S(x,t))\right)S(x,t).$$ + +$$ +[H,S](x,t)=\left(k^2+\mathit{Ric}_{c^*g(t)}(S(x,t),S(x,t))\right)S(x,t). +$$ Now let us compute the time derivative of $k^2$. In what follows we drop the dependence on the metric $c^*g(t)$ from all the curvature terms, but it is implicitly there. + **Lemma.** -$$ - $$ \begin{aligned} \frac{\partial }{\partial t}k^2 & = & \frac{\partial ^2}{\partial s^2}(k^2)-2\langle(\nabla_XH)^\perp,(\nabla_SH)^\perp\rangle_{c^*g}+2k^4\\ -& & -2Ric(H,H) +4k^2Ric(S,S)+2Rm(H,S,H,S), +& & -2\mathit{Ric}(H,H) +4k^2\mathit{Ric}(S,S)+2\mathit{Rm}(H,S,H,S), \end{aligned} $$ -$$ where the superscript $\perp$ means the image under projection to the orthogonal complement of $X$. + **Proof.** Using the same conventions as above for the function $|H|_{c^*g}$ and but leaving the metric implicit, we have $$ - \frac{\partial }{\partial t}k^2=\frac{\partial }{\partial -t}(|H|^2_{c^*g})=-2Ric(H,H)+2\langle\nabla_HH,H\rangle_{c^*g}. +t}(|H|^2_{c^*g})=-2\mathit{Ric}(H,H)+2\langle\nabla_HH,H\rangle_{c^*g}. $$ Now we compute -(using the second equation from Lemma~) - -$$ +(using the second equation from \entryref{4fc24fa1d31e}) $$ \begin{aligned} @@ -1537,61 +1717,66 @@ $$ & = & \nabla_S\nabla_HS+\nabla_{[H,S]}S+ {\mathcal R}(H,S)S \\ & = & \nabla_S\nabla_SH+\nabla_S([H,S])+\nabla_{[H,S]}S+{\mathcal R}(H,S)S \\ -& = & \nabla_S\nabla_SH+\nabla_S\left((k^2+Ric(S,S))S\right)+(k^2+Ric(S,S))\nabla_SS+ +& = & \nabla_S\nabla_SH+\nabla_S\left((k^2+\mathit{Ric}(S,S))S\right)+(k^2+\mathit{Ric}(S,S))\nabla_SS+ {\mathcal R}(H,S)S \\ -& = & \nabla_S\nabla_SH+2(k^2+Ric(S,S))H+S(k^2+Ric(S,S))S+ {\mathcal R}(H,S)S. +& = & \nabla_S\nabla_SH+2(k^2+\mathit{Ric}(S,S))H+S(k^2+\mathit{Ric}(S,S))S+ {\mathcal R}(H,S)S. \end{aligned} $$ -$$ - Using this, and the fact that $\langle H,S\rangle_{c^*g}=0$, we have $$ - -2\langle\nabla_HH,H\rangle_{c^*g}=2g(\nabla_S\nabla_SH,H)+4k^4+4k^2Ric(S,S))+2Rm(H,S,H,S). +2\langle\nabla_HH,H\rangle_{c^*g}=2g(\nabla_S\nabla_SH,H)+4k^4+4k^2\mathit{Ric}(S,S))+2\mathit{Rm}(H,S,H,S). $$ On the other hand, $$ - S(S(\langle H,H\rangle_{c^*g}))=2\langle\nabla_S\nabla_SH,H\rangle_{c^*g}+2\langle\nabla_SH,\nabla_SH\rangle_{c^*g}. $$ We write -$$\nabla_SH=(\nabla_SH)^\perp+\langle\nabla_SH,S\rangle_{c^*g}S.$$ + +$$ +\nabla_SH=(\nabla_SH)^\perp+\langle\nabla_SH,S\rangle_{c^*g}S. +$$ + Since $H$ and $S$ are orthogonal, we have $\langle\nabla_SH,S\rangle_{c^*g}=-\langle H,\nabla_SS\rangle_{c^*g}=-\langle H,H\rangle$. Thus, we have -$$\nabla_SH=(\nabla_SH)^\perp-\langle H,H\rangle_{c^*g}S. $$ -It follows that -$$-2\langle \nabla_SH,\nabla_SH\rangle_{c^*g}= --2\langle(\nabla_SH)^\perp,(\nabla_SH)^\perp\rangle_{c^*g}-2k^4.$$ Substituting -this into Equation~() gives +$$ +\nabla_SH=(\nabla_SH)^\perp-\langle H,H\rangle_{c^*g}S. $$ -2\langle\nabla_S\nabla_SH,H\rangle_{c^*g}=S(S(|H|_{c^*g}^2)) +It follows that + +$$ +-2\langle \nabla_SH,\nabla_SH\rangle_{c^*g}= -2\langle(\nabla_SH)^\perp,(\nabla_SH)^\perp\rangle_{c^*g}-2k^4. $$ -Plugging this into Equation~() and using -Equation~() yields + Substituting +this into Equation (18.7) gives +$$ +2\langle\nabla_S\nabla_SH,H\rangle_{c^*g}=S(S(|H|_{c^*g}^2)) +-2\langle(\nabla_SH)^\perp,(\nabla_SH)^\perp\rangle_{c^*g}-2k^4. $$ +Plugging this into Equation (18.6) and using +Equation (18.5) yields + $$ \begin{aligned} -\frac{\partial}{\partial t}k^2 & = & -2Ric(H,H)+S(S\langle +\frac{\partial}{\partial t}k^2 & = & -2\mathit{Ric}(H,H)+S(S\langle H,H\rangle_{c^*g})-2\langle(\nabla_SH)^\perp,(\nabla_SH)^\perp\rangle_{c^*g} \\ -& & +2k^4+4k^2Ric(S,S)+2Rm(H,S,H,S). +& & +2k^4+4k^2\mathit{Ric}(S,S)+2\mathit{Rm}(H,S,H,S). \end{aligned} $$ -$$ Of course, $S(S(\langle H,H\rangle_{c^*g}))=(k^2)''$ so that this gives the result. @@ -1603,98 +1788,153 @@ $$ (k^2)''-2\langle(\nabla_SH)^\perp,(\nabla_SH)^\perp\rangle_{c^*g}+2k^4+\widehat Ck^2, $$ + where the primes refer to the derivative with respect to arc length along the curve and $\widehat C$ is a constant depending only on an upper bound for the norm of the sectional -curvatures of the ambient manifolds in the Ricci flow. +curvatures of the ambient manifolds in the Ricci flow. -**Claim.** - There is a constant $C_1<\infty$ depending only on an + +**Claim.** There is a constant $C_1<\infty$ depending only on an upper bound for the norm of the sectional curvatures of the ambient manifolds in the Ricci flow $(M,g(t)),\ t_0\le t\le t_1$, such that -$$\frac{\partial}{\partial t} k\le k''+k^3+C_1k.$$ + +$$ +\frac{\partial}{\partial t} k\le k''+k^3+C_1k. +$$ + **Proof.** We set $C_1=\widehat C/2$, where $\widehat C$ is as in -Inequality~. It follows from Inequality~() that +Inequality 18.9. It follows from Inequality (18.9) that $$ - 2k\frac{\partial k}{\partial t}\le +2k\frac{\partial k}{\partial t}\le 2kk''+2(k')^2+2k^4-2\langle(\nabla_SH)^\perp,(\nabla_SH)^\perp\rangle_{c^*g}+\hat Ck^2. $$ + Since $k^2=\langle H,H\rangle_{c^*g}$, we see that $(k^2)'=2\langle\nabla_SH,H\rangle_{c^*g}$. Since $H$ is perpendicular to $S$, this can be rewritten as $(k^2)'=2\langle(\nabla_SH)^\perp,H\rangle_{c^*g}$. It follows that -$$k'=\frac{\langle(\nabla_SH)^\perp,H\rangle_{c^*g}}{|H|_{c^*g}}.$$ + +$$ +k'=\frac{\langle(\nabla_SH)^\perp,H\rangle_{c^*g}}{|H|_{c^*g}}. +$$ + Hence, -$$(k')^2\le \frac{\langle(\nabla_SH)^\perp,H\rangle_{c^*g}^2}{|H|_{c^*g}^2}\le -\langle(\nabla_SH)^\perp,(\nabla_SH)^\perp\rangle_{ c^*g}.$$ Plugging this into -Equation~() gives -$$\frac{\partial k}{\partial t}\le k''+k^3+C_1k.$$ + +$$ +(k')^2\le \frac{\langle(\nabla_SH)^\perp,H\rangle_{c^*g}^2}{|H|_{c^*g}^2}\le +\langle(\nabla_SH)^\perp,(\nabla_SH)^\perp\rangle_{ c^*g}. +$$ + + Plugging this into +Equation (18.10) gives + +$$ +\frac{\partial k}{\partial t}\le k''+k^3+C_1k. +$$ Now we define the *total length* of the curve $c(x,t)$, -$$L(t)=\int |X|_{c^*g}dx=\int ds.$$ + +$$ +L(t)=\int |X|_{c^*g}dx=\int ds. +$$ + We also define the *total curvature* of the curve $c(x,t)$, -$$\Theta(t)=\int k|X|_{c^*g}dx=\int kds.$$ -**Lemma.** There is a constant $C_2<\infty$ depending only on an +$$ +\Theta(t)=\int k|X|_{c^*g}dx=\int kds. +$$ + + +**Lemma.** There is a constant $C_2<\infty$ depending only on an upper bound for the norm of the sectional curvatures of the ambient manifolds in the Ricci flow such that $$ - \frac{d}{dt}L\le \int (C_2-k^2)ds $$ + and -$$\frac{d}{dt}\Theta\le C_2\Theta.$$ -**Proof.** $$\frac{d}{dt}L=\int\frac{\partial}{\partial t}\sqrt{|X|_{c^*g}^2}dx.$$ -By Lemma~ we have $$\frac{d}{dt}L=\int -\frac{1}{2|X|_{c^*g}}\left(-2Ric(X,X)-2k^2|X|_{c^*g}^2\right)dx.$$ Thus, +$$ +\frac{d}{dt}\Theta\le C_2\Theta. +$$ + + +**Proof.** $$ -\frac{d}{dt}L=\int (-Ric(S,S)-k^2)|X|_{c^*g}dx=\int(-Ric(S,S)-k^2)ds. +\frac{d}{dt}L=\int\frac{\partial}{\partial t}\sqrt{|X|_{c^*g}^2}dx. $$ - The first inequality in the lemma -then follows by taking $C_2$ to be an upper bound for the norm of -$Ric_{g(t)}$. -Now let us consider the second inequality in the statement. -$$\frac{d}{dt}\Theta=\int\frac{\partial}{\partial t}(k|X|_{c^*g})dx=\int\left(\frac{\partial k}{\partial -t}|X|_{c^*g} +k\frac{\partial |X|_{c^*g}}{\partial t}\right)dx.$$ Thus, using -Claim~ and the first equation in Lemma~ we have +By \entryref{4fc24fa1d31e} we have $$ +\frac{d}{dt}L=\int +\frac{1}{2|X|_{c^*g}}\left(-2\mathit{Ric}(X,X)-2k^2|X|_{c^*g}^2\right)dx. +$$ + + Thus, + +$$ +\frac{d}{dt}L=\int (-\mathit{Ric}(S,S)-k^2)|X|_{c^*g}dx=\int(-\mathit{Ric}(S,S)-k^2)ds. +$$ + + The first inequality in the lemma +then follows by taking $C_2$ to be an upper bound for the norm of +$\mathit{Ric}_{g(t)}$. + +Now let us consider the second inequality in the statement. + +$$ +\frac{d}{dt}\Theta=\int\frac{\partial}{\partial t}(k|X|_{c^*g})dx=\int\left(\frac{\partial k}{\partial +t}|X|_{c^*g} +k\frac{\partial |X|_{c^*g}}{\partial t}\right)dx. +$$ + + Thus, using +\entryref{67b855d0d6b5} and the first equation in \entryref{4fc24fa1d31e} we have $$ \begin{aligned} \frac{d}{dt}\Theta & \le & \int (k''+k^3+C_1k)ds+\int -\frac{k}{2|X|_{c^*g}}(-2Ric(X,X)-2k^2|X|_{c^*g}^2)dx \\ -& = & \int(k''+k^3+C_1k)ds-\int k(Ric(S,S)+k^2)ds \\& = & -\int(k''+C_1k-k\,Ric(S,S))ds. +\frac{k}{2|X|_{c^*g}}(-2\mathit{Ric}(X,X)-2k^2|X|_{c^*g}^2)dx \\ +& = & \int(k''+k^3+C_1k)ds-\int k(\mathit{Ric}(S,S)+k^2)ds \\& = & +\int(k''+C_1k-k \mathit{Ric}(S,S))ds. \end{aligned} $$ -$$ - Since $\int k''ds=0$ by the fundamental theorem of calculus, we get -$$\frac{d}{dt}\Theta\le C_2\Theta, + +$$ +\frac{d}{dt}\Theta\le C_2\Theta, $$ + for an appropriate constant $C_2$ depending only on an upper bound for the norm of the sectional curvatures of the ambient family $(M,g(t))$. -**Corollary.** The following holds for the constant $C_2$ as in the previous + +**Corollary.** The following holds for the constant $C_2$ as in the previous lemma. Let $c(x,t)$ be a curve-shrinking flow, let $L(t)$ be the total length of $c(t)$ and let $\Theta(t)$ be the total curvature of -$c(t)$. Then for any $t_0\le t' ### Ramp solutions in $ M\times S^1$ -As we pointed out in the beginning of Section~ the +As we pointed out in the beginning of Section 18.4 the main obstacle we must overcome is that the curve-shrinking flow does not always exist for the entire time interval $[t_0,t_1]$. The reason is the following: Even though, as we shall see, it is @@ -1702,7 +1942,7 @@ possible to bound the total curvature of the curve-shrinking flow in terms of the total curvature of the initial curve and the ambient Ricci flow, there is no pointwise estimate on the curvature for the curve-shrinking flow. The idea for dealing with this problem, which -goes back to , is to replace the original situation of +goes back to [AG], is to replace the original situation of curves in a manifold with graphs by taking the product of the manifold with a circle and using ramps. We shall see that in this context the curve-shrinking flow always exists. The problem then @@ -1710,104 +1950,119 @@ becomes to transfer the information back from the flows of ramps to the original manifold. Now suppose that the Ricci flow is of the form $(M, g(t))\times -(S^1_\lambda,ds^2)$ where $(S^1_\lambda,ds^2)$ denotes the circle +(S^1_\lambda,ds^2)$ where $(S^1_\lambda,ds^2)$ denotes the circle of length $\lambda$. Notice that the sectional curvatures of this product flow depend only on the sectional curvatures of $(M, g(t))$ and, in particular, are independent of $\lambda$. Let $U$ denote vector field made up of unit tangent vectors in the direction of the circle factors. Let $u(x,t)=\langle S,U\rangle_{g(t)}$. + **Claim.** -$$\frac{\partial u}{\partial t}=u''+(k^2+Ric(S,S))u\ge u''-C'u,$$ + +$$ +\frac{\partial u}{\partial t}=u''+(k^2+\mathit{Ric}(S,S))u\ge u''-C'u, +$$ + where $C'$ is an upper bound for the norm of the Ricci curvature of $(M,g(t))$. + **Proof.** Since $U$ is a constant vector field and hence parallel along all curves and -since $Ric(V,U)=0$ for all tangent vectors $V$, by Lemma~ +since $\mathit{Ric}(V,U)=0$ for all tangent vectors $V$, by \entryref{4fc24fa1d31e} we have -$$ - $$ \begin{aligned} -\frac{\partial}{\partial t}\langle S,U\rangle_{g(t)} & = & -2Ric(S,U) +\frac{\partial}{\partial t}\langle S,U\rangle_{g(t)} & = & -2\mathit{Ric}(S,U) +\langle dc(\nabla_HS),U\rangle_{g(t)} \\ & = & \langle dc(\nabla_HS),U\rangle_{g(t)} = \langle dc([H,S]+\nabla_SH),U\rangle_{g(t)} \\ -& = & (k^2+Ric(S,S))u+\langle dc(\nabla_SH),U\rangle_{g(t)} \\ -& = & (k^2+Ric(S,S))u+S(dc(\langle H),U\rangle_{g}) \\ - & = & (k^2+Ric(S,S))u+S(\langle dc(\nabla_SS),U\rangle_g) \\ -& = & (k^2+Ric(S,S))u+S(S(u)) = (k^2+Ric(S,S))u+u''. +& = & (k^2+\mathit{Ric}(S,S))u+\langle dc(\nabla_SH),U\rangle_{g(t)} \\ +& = & (k^2+\mathit{Ric}(S,S))u+S(dc(\langle H),U\rangle_{g}) \\ + & = & (k^2+\mathit{Ric}(S,S))u+S(\langle dc(\nabla_SS),U\rangle_g) \\ +& = & (k^2+\mathit{Ric}(S,S))u+S(S(u)) = (k^2+\mathit{Ric}(S,S))u+u''. \end{aligned} $$ -$$ - + **Definition.** A curve $c\colon S^1\to M\times S^1_\lambda$ is said to be a *ramp* if $u$ is strictly positive. + The main results of this section show that the curve-shrinking flow -is much better behaved for ramps than for the general smooth curve. +is much better behaved for ramps than for the general smooth curve. First of all, as the next corollary shows, the curve-shrinking flow applied to a ramp produces a one-parameter families of ramps. The main result of this section shows that for any ramp as initial curve, the curve-shrinking flow does not develop singularities as long as the ambient Ricci flow does not. -**Corollary.** -If $c(x,t),\ t_0\le t< t'_1<\infty$, is a solution of the curve + +**Corollary.** If $c(x,t),\ t_0\le t< t'_1<\infty$, is a solution of the curve shrinking flow in $( M, g(t))\times (S^1_\lambda,ds^2)$ and if $c(t_0)$ a ramp, then $c(t)$ is a ramp for all $t\in [t_0,t'_1)$. + -**Proof.** From the equation in Claim~, we see that for $C'$ an +**Proof.** From the equation in \entryref{2f115406780a}, we see that for $C'$ an upper bound for the norm of the Ricci curvature, we have -$$\frac{\partial}{\partial t}\left(e^{C't}u\right)\ge \left(e^{C't}u\right)''.$$ + +$$ +\frac{\partial}{\partial t}\left(e^{C't}u\right)\ge \left(e^{C't}u\right)''. +$$ + It now follows from a standard maximum principle argument that the minimum value of $e^{C't}u$ is a non-decreasing function of $t$. Hence, if $c(t_0)$ is a ramp then each $c(t)$ is a ramp and in fact $u(x,t)$ is uniformly bounded away from zero in terms of the minimum of $u(x,t_0)$ and the total elapsed time $t_1-t_0$. -**Lemma.** -Let $(M,g(t)),\ t_0\le t\le t_1$, be a Ricci flow. Suppose that + +**Lemma.** Let $(M,g(t)),\ t_0\le t\le t_1$, be a Ricci flow. Suppose that $c\colon S^1\to ( M\times S^1_\lambda, g(t)\times ds^2)$ is a ramp. Then there is a curve-shrinking flow $c(x,t)$ defined for all $t\in [t_0,t_1]$ with $c$ as the initial condition at time $t=t_0$. The curves $c(\cdot,t)$ are all ramps. + **Proof.** The real issue here is to show that the curve-shrinking flow exists for all $t\in [t_0,t_1]$. Given this, the second part of the statement follows from the previous corollary. If the curve shrinking flow does not exist on all of $[t_0,t_1]$ then by -Claim~ there is a $t'_1\le t_1$ such that the +\entryref{656c8fbce129} there is a $t'_1\le t_1$ such that the curve-shrinking flow exists on $[t_0,t_1')$ but $k$ is unbounded on $S^1\times [t_0,t'_1)$. Thus, to complete the proof we need to see that for any $t'_1$ for which the curve-shrinking flow is defined on $[t_0,t'_1)$ we have a uniform bound on $k$ on this region. -Using Claim~ and Claim~ we compute - -$$ +Using \entryref{67b855d0d6b5} and \entryref{2f115406780a} we compute $$ \begin{aligned} \frac{\partial}{\partial t}\left(\frac{k}{u}\right) & = &\frac{1}{u}\frac{\partial k}{\partial t}-\frac{k}{u^2}\frac{\partial u}{\partial t} \\ -& \le & \frac{k''+k^3+C_1k}{u}-\frac{k}{u^2}\left(u''+(k^2+Ric(S,S))u\right) +& \le & \frac{k''+k^3+C_1k}{u}-\frac{k}{u^2}\left(u''+(k^2+\mathit{Ric}(S,S))u\right) \\ -& = & \frac{k''}{u}-\frac{ku''}{u^2}+\frac{C_1k}{u}-\frac{k}{u}Ric(S,S). +& = & \frac{k''}{u}-\frac{ku''}{u^2}+\frac{C_1k}{u}-\frac{k}{u}\mathit{Ric}(S,S). \end{aligned} $$ +On the other hand, + +$$ +\left(\frac{k}{u}\right)''=\frac{k''u-u''k}{u^2}-2\left(\frac{u'}{u}\right) +\left(\frac{k'u-u'k}{u^2}\right). $$ -On the other hand, -$$\left(\frac{k}{u}\right)''=\frac{k''u-u''k}{u^2}-2\left(\frac{u'}{u}\right) -\left(\frac{k'u-u'k}{u^2}\right).$$ Plugging this in, and using the + Plugging this in, and using the curvature bound on the ambient manifolds we get -$$\frac{\partial}{\partial + +$$ +\frac{\partial}{\partial t}\left(\frac{k}{u}\right)\le -\left(\frac{k}{u}\right)''+\left(\frac{2u'}{u}\right)\left(\frac{k}{u}\right)'+C'\frac{k}{u},$$ +\left(\frac{k}{u}\right)''+\left(\frac{2u'}{u}\right)\left(\frac{k}{u}\right)'+C'\frac{k}{u}, +$$ + for a constant $C'$ depending only on a bound for the norm of the sectional curvature of the ambient Ricci flow. A standard maximum principle argument shows that the maximum of $k/u$ at time $t$ grows @@ -1818,21 +2073,27 @@ interval, the value of $k$ is bounded. Next let us turn to the growth rate of the area of a minimal annulus connecting two ramp solutions. -**Lemma.** Suppose that the dimension $n$ of $ M$ + +**Lemma.** Suppose that the dimension $n$ of $ M$ is at least three. Let $c_1(x,t)$ and $c_2(x,t)$ be ramp solutions in $( M, g)\times (S^1_\lambda,ds^2)$ with the image under the projection to $S^1_\lambda$ of each $c_i$ being of degree one. Let $\mu(t)$ be the infimum of the areas of annuli in $( M\times S^1_\lambda, g(t)\times ds^2)$ with boundary $c_1(x,t)\cup c_2(x,t)$. Then $\mu(t)$ is a continuous function of $t$ and -$$\frac{d}{dt}\mu(t)\le (2n-1)max_{x\in M}|Rm(x,t)|\mu(t),$$ + +$$ +\frac{d}{dt}\mu(t)\le (2n-1)\mathit{max}_{x\in M}|\mathit{Rm}(x,t)|\mu(t), +$$ + in the sense of forward difference quotients. + **Proof.** Fix a time $t'$. First assume that the loops $c_1(\cdot,t')$ and $c_2(\cdot,t')$ are disjoint. Under Ricci flow the metrics on the -manifold immediately become real analytic (see ) and +manifold immediately become real analytic (see [Bando]) and furthermore, under the curve-shrinking flow the curves $c_1$ and -$c_2$ immediately become analytic (see ). +$c_2$ immediately become analytic (see [GageHamilton]). [Neither of these results is essential for this argument because we could approximate both the metric and the curves by real analytic objects.] Establishing the results for these and taking limits would @@ -1842,19 +2103,24 @@ there is an annulus connecting them and there is a positive lower bound to the length of any simple closed curve in any such annulus homotopic to a boundary component. Hence, there is a minimal annulus spanning $c_1(\cdot,t')\coprod c_2(\cdot, t')$ According to results -of Hildebrandt () and Morrey () any +of Hildebrandt ([Hildebrandt]) and Morrey ([Morrey]) any minimal annulus $A$ with boundary the union of these two curves is real analytic up to and including the boundary and is immersed except for finitely many branch points. By shifting the boundary curves slightly within the annulus, we can assume that there are no boundary branch points. Again, if we can prove the result for these perturbed curves taking limits will give the result for the original -ones. Given the deformation vector $H$ on the boundary of the +ones. Given the deformation vector $H$ on the boundary of the annulus, extend it to a deformation vector $\hat H$ on the entire annulus. The first order variation of the area at time $t'$ of the resulting deformed family of annuli is given by -$$\frac{d Area\,A}{d t}(t')=\int_{A}(-Tr(Ric^T(g(t'))))da -+\int_{\partial A}-k_geodds,$$ where $Ric^T$ is the + +$$ +\frac{d \mathit{Area} A}{d t}(t')=\int_{A}(-\mathit{Tr}(\mathit{Ric}^T(g(t'))))da ++\int_{\partial A}-k_\mathit{geod}ds, +$$ + + where $\mathit{Ric}^T$ is the Ricci curvature in the tangent directions to the annulus. (The first term is the change in the area of the fixed annulus as the metric deforms. The second term is the change in the area of the family of @@ -1862,18 +2128,34 @@ annuli in the fixed metric. There is no contribution from moving the annulus in the normal direction since the original annulus is minimal.) If $A$ is embedded, then by the Gauss-Bonnet theorem, we have -$$\int_{\partial A}-k_geodds =\int_{A}Kda$$ + +$$ +\int_{\partial A}-k_\mathit{geod}ds =\int_{A}Kda +$$ + where $K$ is the Gaussian curvature of $A$. More generally, if $A$ has interior branch points of orders $n_1,\ldots,n_k$ then there is a correction term and the formula is -$$\int_{\partial A}-k_geodds=\int_{A}Kda-\sum_{i=1}^k2\pi(n_i-1).$$ Thus, we see -$$\frac{d Area\, A}{d t}(t')\le\int_{A}(-Tr(Ric^T(g(t')))+K)da.$$ -On the other hand, since $A$ is a minimal surface, $K$ is at most + +$$ +\int_{\partial A}-k_\mathit{geod}ds=\int_{A}Kda-\sum_{i=1}^k2\pi(n_i-1). +$$ + + Thus, we see + +$$ +\frac{d \mathit{Area} A}{d t}(t')\le\int_{A}(-\mathit{Tr}(\mathit{Ric}^T(g(t')))+K)da. +$$ + +On the other hand, since $A$ is a minimal surface, $K$ is at most the sectional curvature of $( M, g(t'))\times (S^1_\lambda,ds^2)$ along the two-plane tangent to the annulus. Of course, the trace of -the Ricci curvature along $A$ is at most $2(n-1)|max_{x\in -M}Rm(x,t')|$. Hence, -$$\frac{dArea\, A}{dt}(t')\le (2n-1)|max_{x\in M}Rm(x,t')|\mu(t').$$ +the Ricci curvature along $A$ is at most $2(n-1)|\mathit{max}_{x\in +M}\mathit{Rm}(x,t')|$. Hence, + +$$ +\frac{d\mathit{Area} A}{dt}(t')\le (2n-1)|\mathit{max}_{x\in M}\mathit{Rm}(x,t')|\mu(t'). +$$ This computation was done assuming that $c_2(\cdot,t')$ is disjoint from $c_1(\cdot,t')$. In general, since the dimension of $ M$ is at @@ -1894,17 +2176,19 @@ minimal annulus connecting $c_1(\cdot,t')$ and $c_2(\cdot,t')$. This establishes the continuity of $\mu(t)$ at $t'$ and also establishes the forward difference quotient estimate in the general case. -**Corollary.** Given curve-shrinking flows $c_1(\cdot,t)$ and $c_2(\cdot,t)$ for + +**Corollary.** Given curve-shrinking flows $c_1(\cdot,t)$ and $c_2(\cdot,t)$ for ramps of degree one in $(\bar M,\bar g(t))\times (S^1_\lambda,ds^2)$ the minimal area of an annulus connecting $c_1(\cdot,t)$ and $c_2(\cdot,t)$ grows at most exponentially with time with an exponent determined by an upper bound on the sectional curvature of the ambient flow, which in particular is independent of $\lambda$. + -## Proof of Proposition~\protect{} +## Proof of Proposition \entryref{fd85732525c1} Now we are ready to use the curve-shrinking flow for ramps in -$M\times S_\lambda^1$ to establish Proposition~ for $M$. As +$M\times S_\lambda^1$ to establish \entryref{fd85732525c1} for $M$. As we indicated above, the reason for replacing the flow $(M,g(t))$ that we are studying with its product with $S^1_\lambda$ and studying ramps in the product is that the curve-shrinking flow @@ -1916,62 +2200,64 @@ This requires careful analysis. ### Approximations to the original family -The first step in the proof of Proposition~ is to identify +The first step in the proof of \entryref{fd85732525c1} is to identify the approximation to the family $\Gamma$ that we shall use. Here is the lemma that gives the needed approximation together with all the properties we shall use. Given a loop $c$ in $M$ and $\lambda>0$ we define a loop $c^\lambda$ in $M\times S^1_\lambda$. The loop $c^\lambda$ is obtained by -setting $c^\lambda(x)=(c(x),x)$ where we use a standard +setting $c^\lambda(x)=(c(x),x)$ where we use a standard identification of the domain circle (the unit circle) for the free loop space with $S^1_\lambda$, an identification that defines a loop in $S^1_\lambda$ of constant speed $\lambda/2\pi$. -**Lemma.** -Given a continuous map $\Gamma\colon S^2\to \Lambda M$ representing + +**Lemma.** Given a continuous map $\Gamma\colon S^2\to \Lambda M$ representing an element of $\pi_3( M,*)$ and $0<\zeta<1$, there is a continuous map $\widetilde \Gamma\colon S^2\to \Lambda M$ with the following properties: -\begin{enumerate} -\item[(1)] $[\widetilde \Gamma]=[\Gamma]$ in $\pi_3(M,*)$. -\item[(2)] For each $c\in S^2$ the loop $\widetilde \Gamma(c)$ is a + +- **(1)** $[\widetilde \Gamma]=[\Gamma]$ in $\pi_3(M,*)$. +- **(2)** For each $c\in S^2$ the loop $\widetilde \Gamma(c)$ is a $C^2$-loop. -\item[(3)] For each $c\in S^2$ the length of $\widetilde \Gamma(c)$ is +- **(3)** For each $c\in S^2$ the length of $\widetilde \Gamma(c)$ is within $\zeta$ of the length of $\Gamma(c)$. -\item[(4)] For each $c\in S^2$, we have $|A(\widetilde +- **(4)** For each $c\in S^2$, we have $|A(\widetilde \Gamma(c))-A(\Gamma(c))|<\zeta$. -\item[(5)] There is a constant $C_0<\infty$ depending only on $\Gamma$, +- **(5)** There is a constant $C_0<\infty$ depending only on $\Gamma$, on the bounds for the norm of the Riemann curvature operator of the ambient Ricci flow, and on $\zeta$ such that for each $c\in S^2$ and each $\lambda\in (0,1)$ the total length and the total curvature of the ramp $\widetilde \Gamma(c)^\lambda$ are both bounded by $C_3$. -\end{enumerate} + Before proving this lemma we need some preliminary definitions and constructions. -**Definition.** Let $c\colon S^1\to M$ be a $C^1$-map. Fix a positive integer $n$. + +**Definition.** Let $c\colon S^1\to M$ be a $C^1$-map. Fix a positive integer $n$. By a *regular $n$-polygonal approximation* to $c$ we mean the -following. Let $\xi_n=exp(2\pi i/n)$, and consider the points +following. Let $\xi_n=\mathit{exp}(2\pi i/n)$, and consider the points $p_k=c(\xi^k_n)$ for $k=1,\ldots,n+1$. For each $1\le k\le n$, let $A_k$ be a minimal geodesic in $M$ from $p_k$ to $p_{k+1}$. We parameterize $A_k$ by the interval $[\xi_n^k,\xi_n^{k+1}]$ in the -circle at constant speed. This gives a piecewise geodesic map +circle at constant speed. This gives a piecewise geodesic map $c_n\colon S^1\to M$. + The following is immediate from the definition. -**Claim.** -Given $\zeta>0$ and a $C^1$-map $c\colon S^1\to M$ then for all $n$ + +**Claim.** Given $\zeta>0$ and a $C^1$-map $c\colon S^1\to M$ then for all $n$ sufficiently large the following hold for the $n$-polygonal approximation $c_n$ of $c$. -\begin{enumerate} -\item[(a)] the length of $c_n$ is within $\zeta$ of the length of $c$. -\item[(b)] there is a map of the annulus $S^1\times I$ to $M$ connecting + +- **(a)** the length of $c_n$ is within $\zeta$ of the length of $c$. +- **(b)** there is a map of the annulus $S^1\times I$ to $M$ connecting $c_n$ to $c$ with the property that the image is piecewise smooth and of area less than $\zeta$. -\end{enumerate} + **Proof.** The length of $c$ is the limit of the lengths of the $n$-polygonal approximations as $n$ goes to infinity. The first item is immediate @@ -1987,37 +2273,38 @@ As the next result shows, for $\zeta>0$, the integer $n(c)$ associated by the previous claim to a $C^1$-map $c$ can be made uniform as $c$ varies over a compact subset of $\Lambda M$. -**Claim.** -Let $X\subset \Lambda M$ be a compact subset and let $\zeta>0$ be + +**Claim.** Let $X\subset \Lambda M$ be a compact subset and let $\zeta>0$ be fixed. Then there is $N$ depending only on $X$ and $\zeta$ such the conclusion of the previous claim holds for every $c\in X$ and every $n\ge N$. + **Proof.** Suppose the result is false. Then for each $N$ there is $c_N\in X$ and $n\ge N$ so that the lemma does not hold for $c_N$ and $n$. Passing to a subsequence, we can suppose that the $c_N$ converge to -$c_\infty\in X$. Applying Claim~ we see that there is $N$ -such that the conclusion of Claim~ holds with $\zeta$ +$c_\infty\in X$. Applying \entryref{981d63c19e50} we see that there is $N$ +such that the conclusion of \entryref{981d63c19e50} holds with $\zeta$ replaced by $\zeta/2$ for $c_\infty$ and all $n\ge N$. Clearly, then -by continuity for all $n\ge N$ the conclusion of Claim~ +by continuity for all $n\ge N$ the conclusion of \entryref{981d63c19e50} holds for the $n$-polygonal approximation for every $c_l$ for all $l$ sufficiently large. This is a contradiction. -**Corollary.** -Let $\Gamma\colon S^2\to \Lambda M$ be a continuous map with the + +**Corollary.** Let $\Gamma\colon S^2\to \Lambda M$ be a continuous map with the property that $\Gamma(c)$ is homotopically trivial for all $c\in S^2$. Fix $\zeta>0$. For any $n$ sufficiently large denote by $\Gamma_n$ the family of loops defined by setting $\Gamma_n(c)$ -equal to the $n$-polygonal approximation to $\Gamma(c)$. There is +equal to the $n$-polygonal approximation to $\Gamma(c)$. There is $N$ such that for all $n\ge N$ we have -\begin{enumerate} -\item[(1)] $\Gamma_n$ is a continuous family of $n$-polygonal loops in + +- **(1)** $\Gamma_n$ is a continuous family of $n$-polygonal loops in $M$. -\item[(2)] For each $c\in S^2$, the loop +- **(2)** For each $c\in S^2$, the loop $\Gamma_n(c)$ is a homotopically trivial loop in $M$ and its length is within $\zeta$ of the length of $\Gamma(c)$. -\item[(3)] For each $c\in S^2$, we have $|A(\Gamma_n(c))-A(\Gamma(c))|<\zeta$. -\end{enumerate} +- **(3)** For each $c\in S^2$, we have $|A(\Gamma_n(c))-A(\Gamma(c))|<\zeta$. + **Proof.** Given $\Gamma$ there is a uniform bound over all $c\in S^2$ on the maximal speed of $\Gamma(c)$. Hence, for all $n$ sufficiently large, @@ -2032,24 +2319,28 @@ sufficiently large, for all $c\in S^2$ there is an annulus connecting $\Gamma(c)$ and $\Gamma_n(c)$. Hence, these loops are homotopic in $M$. The first statement in the second item follows immediately. The last statement in the second item and third item -follow immediately from Claim~. +follow immediately from \entryref{028d3b5182b5}. -The next step is to turn these $n$-polygonal approximations into +The next step is to turn these $n$-polygonal approximations into $C^2$-curves. We fix, once and for all, a $C^\infty$ function $\psi_n$ from the unit circle to $[0,\infty]$ with the following properties: -\begin{enumerate} -\item[(1)] $\psi_n$ is non-negative and vanishes to infinite order at the point $1$ on the unit circle. -\item[(2)] $\psi_n$ is periodic with period $2\pi i/n$. - \item[(3)] $\psi_n$ is positive on the interior of the interval $[1,\xi_n]$ on the unit circle, + +- **(1)** $\psi_n$ is non-negative and vanishes to infinite order at the point $1$ on the unit circle. +- **(2)** $\psi_n$ is periodic with period $2\pi i/n$. +- **(3)** $\psi_n$ is positive on the interior of the interval $[1,\xi_n]$ on the unit circle, and the restriction of $\psi_n$ to this interval is symmetric about -$exp(\pi i/n)$, and is increasing from $1$ to $exp(\pi +$\mathit{exp}(\pi i/n)$, and is increasing from $1$ to $\mathit{exp}(\pi i/n)$. -\item[(4)] $\int_1^{\xi_n}\psi_n(s)ds =2\pi/n$. -\end{enumerate} +- **(4)** $\int_1^{\xi_n}\psi_n(s)ds =2\pi/n$. Now we define a map $\tilde\psi_n\colon S^1\to S^1$ by -$$\tilde \psi_n(x)= \int_{1}^x\psi_n(y)dy.$$ It is easy to see that + +$$ +\tilde \psi_n(x)= \int_{1}^x\psi_n(y)dy. +$$ + + It is easy to see that the conditions on $\psi_n$ imply that this defines a $C^\infty$-map from $S^1$ to $S^1$ which is a homeomorphism and is a diffeomorphism on the complement of the $n^{th}$ roots of unity. @@ -2060,18 +2351,18 @@ smoothing $\tilde c_n$ is a $C^\infty$-loop in $M$ with the same length as the original polygonal loop $c_n$. Notice that the curvature of $\tilde c_n$ is not itself a continuous function: just like the polygonal map it replaces, it has a $\delta$-function at -the `corners' of $c_n$. +the 'corners' of $c_n$. -**Proof.** (of Lemma~) +**Proof.** (of \entryref{6bb9851ce528}) Given a continuous map $\Gamma\colon S^2\to \Lambda M$ and $\zeta>0$ -we fix $n$ sufficiently large so that Corollary~ +we fix $n$ sufficiently large so that \entryref{7a8f55be175e} holds for these choices of $\Gamma$ and $\zeta$. Let $\widetilde \Gamma=\widetilde \Gamma_n$ be the family of smoothings of the family $\Gamma_n$ of $n$-polygonal loops. Since this smoothing operation changes neither the length nor the area of a minimal -spanning disk, it follows immediately from the construction and -Corollary~ that $\widetilde \Gamma$ satisfies the -conclusions of Lemma~ except possibly the last one. +spanning disk, it follows immediately from the construction and +\entryref{7a8f55be175e} that $\widetilde \Gamma$ satisfies the +conclusions of \entryref{6bb9851ce528} except possibly the last one. To establish the last conclusion we must examine the lengths and total curvatures of the ramps $\widetilde \Gamma(c)^\lambda$ @@ -2080,13 +2371,14 @@ $0<\lambda<1$, and consider the product Ricci flow $(M,g(t))\times (S^1_\lambda,ds^2)$ where the metric on $S^1_\lambda$ has length $\lambda$. + **Claim.** For any $0<\lambda<1$, the length of the ramp $\widetilde \Gamma(c)^\lambda$ is at most $\lambda$ plus the length of $\Gamma(c)$. The total curvature of $\widetilde\Gamma(c)^\lambda$ is at most $n\pi$. + -\begin{proof} -The arc length element for $\widetilde \Gamma(c)^\lambda$ is +**Proof.** The arc length element for $\widetilde \Gamma(c)^\lambda$ is $\sqrt{a(x)^2+(\lambda/2\pi)^2}dx\le (a(x)+\lambda/2\pi)dx$ where $a(x)dx$ is the arc length element for $\widetilde\Gamma(c)$. Integrating gives the length estimate. @@ -2103,9 +2395,9 @@ These are parallel vector fields on the flat surface. vector $X(x)$ to the restriction of $\widetilde \Gamma(c)^\lambda$ to this interval is $L\psi_n(x) u+(\lambda/2\pi)v$, where $L$ is the length of the geodesic segment we are considering. Consider the -first-half subinterval $[\xi_n^k,\xi_n^k\cdotexp(\pi i/n)]$. +first-half subinterval $[\xi_n^k,\xi_n^k\cdot\mathit{exp}(\pi i/n)]$. The tangent vector $X(x)$ is $(\lambda/2\pi) v$ at the initial point -of this subinterval and is $L\psi_n(\xi_n^k\cdotexp(\pi +of this subinterval and is $L\psi_n(\xi_n^k\cdot\mathit{exp}(\pi i/2))u+(\lambda/2\pi) v$ at the final point. Throughout this interval the vector is of the form $a(x)u+(\lambda/2\pi) v$ where $a(x)$ is an increasing function of $x$. Hence, the tangent vector @@ -2116,7 +2408,7 @@ interval is the absolute value of the difference of the angles at the endpoints. This difference is less than $\pi/2$ and tends to $\pi/2$ as $\lambda$ tends to zero, unless $L=0$ in which case there is zero turning for any $\lambda>0$. By symmetry, the total turning -on the second-half subinterval $[\xi_n^k\cdotexp(\pi +on the second-half subinterval $[\xi_n^k\cdot\mathit{exp}(\pi i/n),\xi_n^{k+1}]$ is also bounded above by $\pi/2$. Thus, for any $\lambda>0$, the total turning on one of the segments is bounded above by $\pi$. Since there are $n$ segments this gives the upper @@ -2125,19 +2417,18 @@ bound of $n\pi$ on the total turning of $\widetilde This claim completes the proof of the last property required of $\widetilde\Gamma=\widetilde \Gamma_n$ and hence completes the proof -of Lemma~. -\end{proof} +of \entryref{6bb9851ce528}. Having fixed $\Gamma$ and $\zeta>0$, we fix $n$ and set $\widetilde\Gamma=\widetilde \Gamma_n$. We choose $n$ sufficiently -large so that $\widetilde \Gamma$ satisfies Lemma~. +large so that $\widetilde \Gamma$ satisfies \entryref{6bb9851ce528}. Fix $\lambda\in (0,1)$ and define $\widetilde \Gamma^\lambda\colon S^2\to (M\times S^1_\lambda)$, by setting $\widetilde \Gamma^\lambda(c)=\widetilde \Gamma(c)^\lambda$. -Fix $c\in S^2$, and let $\widetilde\Gamma_c^\lambda(t),\ t_0\le +Fix $c\in S^2$, and let $\widetilde\Gamma_c^\lambda(t),\ t_0\le t\le t_1$, be the curve-shrinking flow given in -Lemma~ with initial data the ramp +\entryref{11ec367eae93} with initial data the ramp $\widetilde\Gamma^\lambda(c)$ . As $c$ varies over $S^2$ these fit together to produce a one-parameter family $\widetilde\Gamma^\lambda(t)$ of maps $S^2\to\Lambda(M\times @@ -2147,72 +2438,80 @@ $\widetilde\Gamma^\lambda_c(t_0)=\widetilde\Gamma^\lambda(c)$, so that $p_1\widetilde \Gamma_c^\lambda(t_0)=\widetilde \Gamma(c)$. We shall show that for $\lambda>0$ sufficiently small, the family $p_1\widetilde\Gamma^\lambda(t)$ satisfies the conclusion of -Proposition~ for the fixed $\Gamma$ and $\zeta>0$. We do +\entryref{fd85732525c1} for the fixed $\Gamma$ and $\zeta>0$. We do this in steps. First, we show that fixing one $c\in S^2$, for -$\lambda$ sufficiently small (depending on $c$) an analogue of -Proposition~ holds for the one-parameter family of loops -$p_1\widetilde\Gamma_c^\lambda(t)$. By this we mean that either +$\lambda$ sufficiently small (depending on $c$) an analogue of +\entryref{fd85732525c1} holds for the one-parameter family of loops +$p_1\widetilde\Gamma_c^\lambda(t)$. By this we mean that either $p_1\widetilde\Gamma^\lambda_c(t_1)$ has length less than $\zeta$ or $A(p_1\widetilde\Gamma^\lambda_c(t_1))$ is at most the value $v(t_1)+\zeta$, where $v$ is the solution to the -Equation~() with initial condition +Equation (18.4) with initial condition $v(t_0)=A(\widetilde\Gamma(c))$. (Actually, we establish a slightly -stronger result, see Lemma~.) The next step in the +stronger result, see \entryref{9df99ae6f4cb}.) The next step in the argument is to take a finite subset ${\mathcal S}\subset S^2$ so that for every $c\in S^2$ there is $\hat c\in {\mathcal S}$ such that $\widetilde\Gamma(c)$ and $\widetilde\Gamma(\hat c)$ are sufficiently close. Then, using the result of a single $c$, we fix $\lambda>0$ sufficiently small so that the analogue of -Proposition~ for individual curves (or rather the slightly +\entryref{fd85732525c1} for individual curves (or rather the slightly stronger version of it) holds for every $\hat c\in {\mathcal S}$. -Then we complete the proof of Proposition~ using the fact +Then we complete the proof of \entryref{fd85732525c1} using the fact that for every $c$ the curve $\widetilde\Gamma(c)$ is sufficiently close to a curve $\widetilde\Gamma(\hat c)$ associated to an element $\hat c\in {\mathcal S}$. ### The case of a single $c\in S^2$ -According to Lemma~, for all $\lambda\in (0,1)$ the +According to \entryref{6bb9851ce528}, for all $\lambda\in (0,1)$ the lengths and total curvatures of the $\widetilde\Gamma^\lambda(c)$ are uniformly bounded for all $c\in S^2$. Hence, by -Corollary~ the same is true for +\entryref{156a3da68fbd} the same is true for $\widetilde\Gamma_c^\lambda(t)$ for all $c\in S^2$ and all $t\in [t_0,t_1]$. -**Claim.** -There is a constant $C_4$ depending on $t_1-t_0$, on the curvature + +**Claim.** There is a constant $C_4$ depending on $t_1-t_0$, on the curvature bound of the sectional curvature of the Ricci flow $(M,g(t)),\ t_0\le t\le t_1$, on the original family $\Gamma$ and on $\zeta$ such that for any $c\in S^2$ and any $t_0\le t' **Proof.** All the constants in this argument are allowed to depend on $t_1-t_0$, on the curvature bound of the sectional curvature of the Ricci flow $(M,g(t)),\ t_0\le t\le t_1$, on the original family $\Gamma$ and on $\zeta$ but are independent of $\lambda$, $c\in S^2$, and $t'1$) depending only on the curvature bound of the ambient manifolds and the family $\Gamma$ such that for any $c\in S^2$ we have @@ -2248,20 +2547,25 @@ $$ \int_{t_0}^{t_1}\int_{\widetilde\Gamma_c^\lambda(t)} k^2dsdt\le C_5. $$ + Thus, for any constant $1 +**Lemma.** Let $(M,g(t)),\ t_0\le t\le t_1$, be a Ricci flow. Then there exist constants $\delta>0$ and $\widetilde C_i<\infty$ for -$i=0,1,2,\ldots$, depending only on $t_1-t_0$ and a bound for the +$i=0,1,2,\ldots$, depending only on $t_1-t_0$ and a bound for the norm of the curvature of the Ricci flow, such that the following holds. Let $c(x,t)$ be a curve-shrinking flow that is an immersion for each $t$. Suppose that at a time $t'$ for some $0 The first statement follows from arguments very similar to those in -Section 4 of . Once $k^2$ is bounded by $\widetilde +Section 4 of [AG]. Once $k^2$ is bounded by $\widetilde C_0/(t-t')$ the higher derivative statements are standard, see -. For completeness we have included the proof of +[Altschuler]. For completeness we have included the proof of the first inequality in the last section of this chapter. We now fix $\delta>0$ (and also $\delta<1$) as described in the last -lemma for the Ricci flow $(M,g(t)),\ t_0\le t\le t_1$. By +lemma for the Ricci flow $(M,g(t)),\ t_0\le t\le t_1$. By Cauchy-Schwarz it follows that for every $t\in I_B(c,\lambda)$, - and for any arc $J$ in $\widetilde\Gamma_c^\lambda(\cdot,t)$ of + and for any arc $J$ in $\widetilde\Gamma_c^\lambda(\cdot,t)$ of length at most $\delta^2B^{-1}$ we have -$$\int_{J\times\{t\}}k\le \delta.$$ -Applying the previous lemma, for each $a\in I_B(c,\lambda)$ with + +$$ +\int_{J\times\{t\}}k\le \delta. +$$ + +Applying the previous lemma, for each $a\in I_B(c,\lambda)$ with $a\le t_1-B^{-1}-\delta^5B^{-2}$ we set $J(a)=[a+\delta^5B^{-2}/2,a+\delta^5B^{-2}]\subset [t_0,t_1-B^{-1}]$. Then for all $t\in \cup_{a\in @@ -2303,7 +2621,7 @@ $J(a_i)$ that cover all of $[t_0,t_1]$ except a subset of measure at most $3C_5B^{-1}$. Once we have a finite number of $J(a_i)$, we order them along the interval -$[t_0,t_1]$ so that their initial points form an increasing +$[t_0,t_1]$ so that their initial points form an increasing sequence. (Recall that they all have the same length.) Then if we have $J_i\cap J_{i+2}\not=\emptyset$, then $J_{i+1}$ is contained in the union of $J_i$ and $J_{i+2}$ and hence can be removed from the @@ -2313,20 +2631,19 @@ every point of $[t_0,t_1]$ is contained in at most $2$ of the intervals in the collection. Once we have arranged this we have a uniform bound, independent of $\lambda$ and $c\in S^2$, on the number of these intervals. We let $J_B(c,\lambda)$ be the union of -these intervals. According to the construction and Lemma~ +these intervals. According to the construction and \entryref{38712e8cea3d} these sets $J_B(c,\lambda)$ satisfy the following: -\begin{enumerate} -\item[(1)] $J_B(c,\lambda)\subset [t_0,t_1-B^{-1}]$ is a union of a bounded number of intervals + +- **(1)** $J_B(c,\lambda)\subset [t_0,t_1-B^{-1}]$ is a union of a bounded number of intervals (the bound being independent of $c\in S^2$ and of $\lambda$) of length $\delta^5B^{-2}/2$. -\item[(2)] The measure of $J_B(c,\lambda)$ is at least +- **(2)** The measure of $J_B(c,\lambda)$ is at least $t_1-t_0-3C_5B^{-1}$. -\item[(3)] For every $t\in J_B(c,\lambda)$ either the length of $\widetilde\Gamma_c^\lambda(t)$ is +- **(3)** For every $t\in J_B(c,\lambda)$ either the length of $\widetilde\Gamma_c^\lambda(t)$ is less than $\delta^2 B^{-1}$ or there are uniform bounds, depending only on the curvature bounds of the ambient Ricci flow and the initial family $\Gamma$, on the curvature and its higher spatial derivatives of $\widetilde\Gamma_c^\lambda(t)$. -\end{enumerate} Now we fix $c\in S^2$ and $10$, there is $1(t_1-t_0)^{-1}$, + +**Lemma.** Given $\zeta>0$, there is $1(t_1-t_0)^{-1}$, depending only on $\Gamma$ and the curvature bounds on the ambient Ricci flow $(M,g(t)),\ t_0\le t\le t_1$, such that the following holds. Let $t_2=t_1-B^{-1}$. Fix $ c\in S^2$. Let $v_c$ be the -solution to Equation~() with initial condition +solution to Equation (18.4) with initial condition $v_{c}(t_0)=A(p_1(\widetilde\Gamma(c)))$, so that in our previous notation $v_c=w_{A(p_1(\widetilde\Gamma(c)))}$. Then for all $\lambda>0$ sufficiently small, either $A(p_1\widetilde\Gamma_c^\lambda(t_1)) -**Proof.** In order to establish this lemma we need a couple of claims about +**Proof.** In order to establish this lemma we need a couple of claims about functions on $[t_0,t_1]$ that are approximately dominated by -solutions to Equation~(). In the first claim the +solutions to Equation (18.4). In the first claim the function in question is dominated on a finite collection of subintervals by solutions to these equations and the subintervals fill up most of the interval. In the second, we also allow the function to only be approximately dominated by the solutions to -Equation~() on these sub-intervals. In both claims the +Equation (18.4) on these sub-intervals. In both claims the result is that on the entire interval the function is almost dominated by the solution to the equation with the same initial value. -**Claim.** -Fix $C_4$ as in Claim~ and fix a constant + +**Claim.** Fix $C_4$ as in \entryref{52123e3c5a27} and fix a constant $\widetilde A>0$. Given $\zeta>0$ there is $\delta'>0$ depending on $C_4$, $t_1-t_0$, and $\widetilde A$ as well as the curvature bound of the ambient Ricci flow such that the following holds. Suppose that $f\colon [t_0,t_1]\to \Ar$ is a function and suppose that $J\subset [t_0,t_1]$ is a finite union of intervals. Suppose that on each interval $[a,b]$ of $J$ the function $f$ satisfies -$$f(b)\le w_{f(a),a}(b).$$ Suppose further that for any $t' + +**Proof.** We write $J$ as a union of disjoint intervals $J_1,\ldots,J_k$ so +that $J_i0$, $A$ and $C_6, C_4$ as in the last claim, and let + +**Claim.** Fix $\zeta>0$, $A$ and $C_6, C_4$ as in the last claim, and let $\delta'>0$ be as in the last claim. Suppose that we have $J\subset [t_0,t_1]$ which is a finite disjoint union of intervals with $t_1-t_0-|J|\le \delta'$. @@ -2479,51 +2847,56 @@ $t_1-t_0-|J|\le \delta'$. is $\delta''>0$ ($\delta''$ is allowed to depend on $J$) such that the following holds. Suppose that we have a function $f\colon [t_0,t_1]\to \Ar$ such that: -\begin{enumerate} -\item[(1)] For all $t' **Proof.** We define $C_7$ as in the previous proof. We use the notation $J=J_1\coprod\cdots\coprod J_k$ with $J_1C_2/(log4-log3)$. -\end{enumerate} -The first step in the proof of Lemma~ is the + +- **(1)** $B\ge 3C_5(\delta')^{-1}$, +- **(2)** $B\ge 3e^{C_2(t_1-t_0)}\zeta^{-1}$, and +- **(3)** $B>C_2/(\mathit{log}4-\mathit{log}3)$. + +The first step in the proof of \entryref{9df99ae6f4cb} is the following: -**Claim.** -After passing to a subsequence of $\{\lambda_n\}$, either: -\begin{enumerate} -\item[(1)] for each $n$ sufficiently large there is $t_n\in J_B(c)$ with + +**Claim.** After passing to a subsequence of $\{\lambda_n\}$, either: + +- **(1)** for each $n$ sufficiently large there is $t_n\in J_B(c)$ with the length of $\widetilde\Gamma_c^{\lambda_n}(t_n)<\delta^2B^{-1}$, or -\item[(2)] for each component $J_i=[t_i^-,t_i^+]$ of $ J_B(c)$, after +- **(2)** for each component $J_i=[t_i^-,t_i^+]$ of $ J_B(c)$, after composing $\widetilde\Gamma_c^{\lambda_n}(x,t)$ by a reparameterization of the domain circle (fixed in $t$ but a different reparameterization for each $n$) so that the $\widetilde @@ -2531,7 +2904,7 @@ different reparameterization for each $n$) so that the $\widetilde limiting curve-shrinking flow denoted $\widetilde\Gamma_c(t)$, for $t\in J_i$ for the sequence $p_1\widetilde\Gamma_c^{\lambda_n}(t),\ t_i^-\le t\le t_i^+$. The limiting flow consists of immersions. -\end{enumerate} + **Proof.** Suppose that the first case does not hold for any subsequence. Fix a component $J_i$ of $J_B(c)$. Then, by passing to a subsequence, by @@ -2562,7 +2935,7 @@ $p_1\widetilde\Gamma_c^{\lambda_n}(t_i^-)$. The result is an immersed curve in $(M,g(t_i^-))$ parameterized at unit speed. Since all the spatial and time derivatives of the $p_1\widetilde \Gamma_c^{\lambda_n}(t)$ are uniformly bounded, by passing to a -further subsequence, there is a smooth map $f\colon S^1\times +further subsequence, there is a smooth map $f\colon S^1\times [t_i^-,t_i^+]\to M$ which is a smooth limit of the sequence $\widetilde\Gamma_c^{\lambda_n}(t),\ t_i^-\le t\le t_i^+$. If for some $t\in [t_i^-,t_i^+]$ the curve $f|_{S^1\times\{t\}}$ is @@ -2572,7 +2945,7 @@ restriction of $f$ to the interval $[t_i^-,t]$ is a curve-shrinking flow. We claim that $f$ is a curve-shrinking flow on the entire interval $[t_i^-,t_i^+]$. Suppose not. Then there is a first $t'\le t_i^+$ for which $f|_{S^1\times \{t'\}}$ is not an immersion. -According to Lemma~ the maximum of the norms of +According to \entryref{656c8fbce129} the maximum of the norms of the curvature of the curves $f(t)$ must tend to infinity as $t$ approaches $t'$ from below. But the curvatures of $f(t)$ are the limits of the curvatures of the family $p_1\widetilde @@ -2580,75 +2953,94 @@ limits of the curvatures of the family $p_1\widetilde entire interval $[t_i^-,t_i^+]$. This contradiction shows that the entire limiting surface -$$f\colon S^1\times [t_i^-,t_i^+]\to (M,g(t))$$ + +$$ +f\colon S^1\times [t_i^-,t_i^+]\to (M,g(t)) +$$ + is a curve-shrinking flow of immersions. -**Remark.** -Notice that if the first case holds then by the choice of $B$ we + +**Remark.** Notice that if the first case holds then by the choice of $B$ we have a point $t_n\in J_B(c)$ for which the length of $\widetilde\Gamma_c^{\lambda_n}(t_n)$ is less than $e^{-C_2(t_1-t_0)}\zeta/3$. + -For each $n$, the family of curves -$p_1\widetilde\Gamma_c^{\lambda_n}(t)$ in $M$ all have +For each $n$, the family of curves +$p_1\widetilde\Gamma_c^{\lambda_n}(t)$ in $M$ all have $p_1\widetilde\Gamma_c^{\lambda_n}(t_0)=\widetilde\Gamma(c)$ as their initial member. Thus, these curves are all homotopically trivial. Hence, for each $t\in J_B(c)$ the limiting curve $\widetilde\Gamma(c)(t)$ of the $p_1\widetilde\Gamma_c^{\lambda_n}(t)$ is then also homotopically -trivial. It now follows from Lemma~, -Claim~ and Remark~ that one of the +trivial. It now follows from \entryref{27b452d6cb48}, +\entryref{07ac984fbe88} and \entryref{012eefbbc44a} that one of the following two conditions holds: -\begin{enumerate} -\item[(1)] for some $t\in J_B(c)$ the length of $\widetilde\Gamma(c)(t)$ is less than or equal to + +- **(1)** for some $t\in J_B(c)$ the length of $\widetilde\Gamma(c)(t)$ is less than or equal to $e^{-C_2(t_1-t_0)}\zeta/3$ or -\item[(2)] the function $A(t)$ that assigns to each $t\in +- **(2)** the function $A(t)$ that assigns to each $t\in J_B(c)$ the area of the minimal spanning disk for $p_1\widetilde\Gamma(c)(t)$ satisfies -$$\frac{dA(t)}{dt}\le -2\pi-\frac{1}{2}R_min(t)A(t)$$ + +$$ +\frac{dA(t)}{dt}\le -2\pi-\frac{1}{2}R_\mathit{min}(t)A(t) +$$ + in the sense of forward difference quotients. -\end{enumerate} -By continuity, for any $\delta''>0$ then for all $n$ + +By continuity, for any $\delta''>0$ then for all $n$ sufficiently large one of the following two conditions holds: -\begin{enumerate} -\item[(1)] there is $t_n\in J_B(c)$ such that the length of $\widetilde\Gamma_c^{\lambda_n}(t_n)$ + +- **(1)** there is $t_n\in J_B(c)$ such that the length of $\widetilde\Gamma_c^{\lambda_n}(t_n)$ is less than $e^{-C_2(t_1-t_0)}\zeta/2$, or -\item[(2)] for every $t\in J_B(c)$, the areas of the minimal +- **(2)** for every $t\in J_B(c)$, the areas of the minimal spanning disks for $p_1(\widetilde\Gamma_c^{\lambda_n}(t))$ satisfy -$$\frac{dA(p_1\widetilde\Gamma_c^{\lambda_n}(t))}{dt}\le -2\pi-\frac{1}{2}R_min(t)A(p_1\widetilde\Gamma_c^{\lambda_n}(t))+\delta''$$ in the + +$$ +\frac{dA(p_1\widetilde\Gamma_c^{\lambda_n}(t))}{dt}\le -2\pi-\frac{1}{2}R_\mathit{min}(t)A(p_1\widetilde\Gamma_c^{\lambda_n}(t))+\delta'' +$$ + + in the sense of forward difference quotients. -\end{enumerate} -Suppose that for every $n$ sufficiently large, for every $t\in +Suppose that for every $n$ sufficiently large, for every $t\in J_B(c)$ the length of $\widetilde\Gamma_c^{\lambda_n}(t)$ is at least $e^{-C_2(t_1-t_0)}\zeta/2$. We have already seen in -Claim~ that for every $t'0$ such that the following holds. Let $c,\hat c\in + +**Lemma.** There is a $\mu>0$ such that the following holds. Let $c,\hat c\in S^2$. Suppose that there is an arc $\omega$ in $S^2$ connecting $c$ to $\hat c$ with the area of the annulus $\widetilde\Gamma_\omega^\lambda(t_0)$ in $M\times S^1_\lambda$ less than $\mu$. Let $v_{\hat c}$, resp., $v_c$, be the solution to -Equation~() with initial condition $v_{\hat +Equation (18.4) with initial condition $v_{\hat c}(t_0)=A(\widetilde\Gamma(\hat c))$, resp., $v_c(t_0)=A(\widetilde\Gamma(c))$. If - $$A(p_1\widetilde\Gamma^\lambda_{\hat c}(t_1))\le v_{\hat c}+\zeta/2,$$ + +$$ +A(p_1\widetilde\Gamma^\lambda_{\hat c}(t_1))\le v_{\hat c}+\zeta/2, +$$ + then -$$A(p_1(\widetilde\Gamma^\lambda_c(t_1))\le v_c+\zeta.$$ + +$$ +A(p_1(\widetilde\Gamma^\lambda_c(t_1))\le v_c+\zeta. +$$ + **Proof.** First of all we require that $\mu0$ such that for any $r>0$ there is $\bar\mu>0$, + +**Lemma.** There is $\delta>0$ such that for any $r>0$ there is $\bar\mu>0$, depending on $r$ and on the curvature bound for the ambient Ricci flow such that the following holds. Suppose that $\gamma$ and $\hat \gamma$ are ramps in $(M,g(t))\times S^1_\lambda$. Suppose that the length of $\gamma$ is at least $r$ and suppose that on any sub-interval $I$ of $\gamma$ of length $r$ we have -$$\int_Ikds<\delta.$$ + +$$ +\int_Ikds<\delta. +$$ + Suppose also that there is an annulus connecting $\gamma$ and $\hat\gamma$ of area less than $\bar\mu$. Then the length of $\hat\gamma$ is at least $3/4$ the length of $\gamma$. + We give a proof of this lemma in the next section. Here we finish -the proof of Proposition~ assuming it. +the proof of \entryref{fd85732525c1} assuming it. -**Claim.** -There is $\mu>0$ such that the following holds. Suppose that $c,\hat + +**Claim.** There is $\mu>0$ such that the following holds. Suppose that $c,\hat c\in S^2$ are such that there is an arc $\omega$ in $S^2$ connecting $c$ and $\hat c$ such that the area of the annulus -$\widetilde\Gamma^\lambda(\omega)$ is at most $\mu$. Set +$\widetilde\Gamma^\lambda(\omega)$ is at most $\mu$. Set $t_2=t_1-B^{-1}$. If the length of $\widetilde\Gamma_{\hat c}^\lambda(t)$ is less than $\zeta/2$ for all $t\in [t_2,t_1]$, then the length of $p_1\widetilde\Gamma_c^\lambda(t_1)$ is less than $\zeta$. + **Proof.** The proof is by contradiction: Suppose that the length of $p_1\widetilde\Gamma_c^\lambda(t_1)$ is at least $\zeta$ and the @@ -2740,45 +3154,73 @@ length of $\widetilde\Gamma_{\hat c}^\lambda(t)$ is less than $\zeta/2$ for all $t\in [t_2,t_1]$. Of course, it follows that the length of $\widetilde\Gamma_c^\lambda(t_1)$ is also at least $\zeta$. The third condition on $B$ is equivalent to -$$e^{C_2B^{-1}}<4/3.$$ + +$$ +e^{C_2B^{-1}}<4/3. +$$ + It then follows from - Corollary~ that for every $t\in + \entryref{156a3da68fbd} that for every $t\in [t_2,t_1]$ the length of $\widetilde\Gamma^c_\lambda(t)$ is at least $3\zeta/4$. On the other hand, by hypothesis for every such $t$, the length of $\widetilde\Gamma_{\hat c}^\lambda(t)$ is less than $\zeta/2$. It -follows from Equation~() that -$$\int_{t_2}^{t_1}\left(\int k^2ds\right)dt\le +follows from Equation (18.11) that + +$$ +\int_{t_2}^{t_1}\left(\int k^2ds\right)dt\le C_2\left(\int_{t_2}^{t_1}L(\widetilde\Gamma_c^\lambda(t))dt\right)- -L(\widetilde\Gamma_c^\lambda(t_1))+L(\widetilde\Gamma_c^\lambda(t_2)).$$ +L(\widetilde\Gamma_c^\lambda(t_1))+L(\widetilde\Gamma_c^\lambda(t_2)). +$$ + (Here $L$ is the length of the curve.) From this and -Corollary~ we see that there is a constant $C_8$ +\entryref{156a3da68fbd} we see that there is a constant $C_8$ depending on the original family $\Gamma$ and on the curvature of the ambient Ricci flow such that -$$\int_{t_2}^{t_1}\left(\int_{\widetilde\Gamma_c^\lambda(t)} k^2ds\right)dt\le C_8.$$ + +$$ +\int_{t_2}^{t_1}\left(\int_{\widetilde\Gamma_c^\lambda(t)} k^2ds\right)dt\le C_8. +$$ + Since $t_1-t_2=B^{-1}$, this implies that there is $t'\in [t_2,t_1]$ with -$$\int_{\widetilde\Gamma_c^\lambda(t')}k^2ds\le C_8B.$$ + +$$ +\int_{\widetilde\Gamma_c^\lambda(t')}k^2ds\le C_8B. +$$ + By Cauchy-Schwarz, for any subinterval $I$ of length $\le r$ in $\widetilde\Gamma_c^\lambda(t')$ we have -$$\int_Ikds\le \sqrt{C_8Br}.$$ + +$$ +\int_Ikds\le \sqrt{C_8Br}. +$$ + We choose $00$ sufficiently small so that Lemma~ and -Claim~ hold. Then we choose a $\mu/2$-net $X$ for +**Proof.** (of \entryref{fd85732525c1}.) +Fix $\mu>0$ sufficiently small so that \entryref{10dace7fb165} and +\entryref{52f36e456355} hold. Then we choose a $\mu/2$-net $X$ for $\widetilde\Gamma$. We take $\lambda$ sufficiently small so that -Lemma~ holds for every $\hat c\in {\mathcal S}$. We +\entryref{9df99ae6f4cb} holds for every $\hat c\in {\mathcal S}$. We also choose $\lambda$ sufficiently small so that $X$ is a $\mu$-net for $\widetilde\Gamma^\lambda$. Let $c\in S^2$. Then there is $\hat c\in {\mathcal S}$ and an arc $\omega$ connecting $c$ and $\hat c$ such that the area of $\widetilde\Gamma^\lambda(\omega)<\mu$. Let $v_{\hat c}$, resp., $v_c$ be the solution to -Equation~ with initial condition $v_{\hat +Equation 18.4 with initial condition $v_{\hat c}(t_0)=A(\widetilde\Gamma({\hat c}))$, resp., $v_c(t_0)=A(\widetilde\Gamma(c))$. According to -Lemma~ either $A(p_1\widetilde\Gamma_{\hat +\entryref{9df99ae6f4cb} either $A(p_1\widetilde\Gamma_{\hat c}^\lambda(t_1)) **Claim.** Let $\psi\colon A\subset M\times S^1_\lambda$ be an area-minimizing annulus of area at most $\mu$ with smoothly embedded boundary as constructed above. Let $h$ be the induced (possibly singular) metric on $A$ induced by pulling back $g\times ds^2$ by $\psi$, and let $C''>0$ be an upper bound on the Gaussian curvature of $h$ (away -from the branch points). Then there is a deformation $\tilde h$ of +from the branch points). Then there is a deformation $\tilde h$ of $h$, supported near the interior branch points, to a smooth metric with the property that the area of the deformed smooth metric is at most $2\mu$ and where the upper bound for the curvature of $\tilde h$ is $2C''$. + **Proof.** Fix an interior branch point $p$. - Since $\psi$ is smooth and + Since $\psi$ is smooth and conformal onto its image, there is a disk in $A$ centered at $p$ in which $h=f(z,\bar z)|dz|^2$ for a smooth function $f$ on the disk. The function $f$ vanishes at the origin and is positive on the complement of the origin. Direct computation shows that the Gaussian curvature $K(h)$ of $h$ in this disk is given by -$$K(h)=\frac{-\triangle f}{2f^2}+\frac{|\nabla f|^2}{2f^3}\le C,$$ + +$$ +K(h)=\frac{-\triangle f}{2f^2}+\frac{|\nabla f|^2}{2f^3}\le C, +$$ + where $\triangle$ is the usual Euclidean Laplacian on the disk and $|\nabla -f|^2=(\partial f/\partial x)^2+(\partial f/\partial y)^2$. Now consider the +f|^2=(\partial f/\partial x)^2+(\partial f/\partial y)^2$. Now consider the metric $(f+\epsilon)|dz|^2$ on the disk. Its Gaussian curvature is -$$\frac{-\triangle f}{2(f+\epsilon)^2}+\frac{|\nabla f|^2}{2(f+\epsilon)^3}.$$ -**Claim.** For all $\epsilon>0$ the Gaussian curvature of $(f+\epsilon)|dz|^2$ -is at most $2C''$. +$$ +\frac{-\triangle f}{2(f+\epsilon)^2}+\frac{|\nabla f|^2}{2(f+\epsilon)^3}. +$$ -\begin{proof} -We see that $-\triangle f\le C''f^2$, so that + +**Claim.** For all $\epsilon>0$ the Gaussian curvature of $(f+\epsilon)|dz|^2$ +is at most $2C''$. + -$$ +**Proof.** We see that $-\triangle f\le C''f^2$, so that $$ \begin{aligned} @@ -2917,12 +3372,15 @@ C''+\frac{\epsilon f^2C''}{(f+\epsilon)^3}\le 2C''. \end{aligned} $$ -$$ - Now we fix a smooth function $\rho(r)$ which is identically one on a subdisk $D'$ of $D$ and vanishes near $\partial D$ and we replace the metric $h$ on the disk by -$$h_\epsilon=(f+\epsilon \rho(r))|dz|^2.$$ The above computation shows that the + +$$ +h_\epsilon=(f+\epsilon \rho(r))|dz|^2. +$$ + + The above computation shows that the Gaussian curvature of $h_\epsilon$ on $D'$ is bounded above by $2C''$. As $\epsilon$ tends to zero the restriction of the metric $h_\epsilon$ to $D\setminus D'$ converges uniformly in the @@ -2935,13 +3393,12 @@ $D$. Performing this construction near each of the finite number of interior branch points and taking $\epsilon$ sufficiently small gives the perturbation $\tilde h$ as required. -\end{proof} Thus, if $\gamma$ and $\hat \gamma$ are ramps as in -Lemma~, then replacing $\hat \gamma$ by a close $C^2$ +\entryref{8e814eab5ddf}, then replacing $\hat \gamma$ by a close $C^2$ approximation we have an abstract smooth annulus with a Riemannian metric connecting $\gamma$ and $\hat \gamma$. Taking limits shows -that establishing the conclusion of Lemma~ for a +that establishing the conclusion of \entryref{8e814eab5ddf} for a sequence of better and better approximations to $\hat \gamma$ will also establish it for $\hat \gamma$. This allows us to assume that $\gamma$ and $\hat \gamma$ are disjoint. The area of this annulus is @@ -2952,20 +3409,24 @@ Finally, the integral of the absolute value of the geodesic curvature over any interval of length $r$ of $\gamma$ is at most $\delta$. -With all these preliminary remarks, we see that Lemma~ +With all these preliminary remarks, we see that \entryref{8e814eab5ddf} follows from: -**Proposition.** -Fix $0<\delta<1/100$. For each $0 +**Proposition.** Fix $0<\delta<1/100$. For each $00$ such that the following holds. Suppose that $A$ is an annulus with boundary components $c_0$ and $c_1$. Denote by $l(c_0)$ and $l(c_1)$ the lengths of $c_0$ and $c_1$, respectively. Suppose that the Gaussian curvature of $A$ is bounded above by $C''$. -Suppose that $l(c_0)>r$ and that for each sub-interval $I$ of $c_0$ +Suppose that $l(c_0)>r$ and that for each sub-interval $I$ of $c_0$ of length $r$, the integral of the absolute value of the geodesic curvature along $I$ is less than $\delta$. Suppose that the area of $A$ is less than $\mu$. Then -$$l(c_1)\ge \frac{3}{4}l(c_0).$$ + +$$ +l(c_1)\ge \frac{3}{4}l(c_0). +$$ + To us, this statement was intuitively extremely reasonable but we could not find a result along these lines stated in the literature. @@ -2977,8 +3438,8 @@ $c_0$ using the family of geodesics perpendicular to the boundary. The bounds on the Gaussian curvature and local bounds on the geodesic curvature of $c_0$ imply that the exponential mapping will be an immersion out to some fixed distance $\delta$ or until the -geodesics meet the other boundary, whichever comes first. -Furthermore, the metric induced by this immersion will be close to +geodesics meet the other boundary, whichever comes first. +Furthermore, the metric induced by this immersion will be close to the product metric. Thus, if there is not much area, it must be the case that, in the measure sense, most of the geodesics in this family must meet the other boundary before distance $\delta$. One @@ -3019,53 +3480,63 @@ finishes as indicated above. ### First reductions Of course, if the hypothesis of the proposition holds for $r>0$ then -it holds for any $00$ such that the proposition holds provided that the area of the annulus is less than $\mu$. -The function $k_geod\colon c_0\to \Ar$ is smooth. We choose a -regular value $\alpha$ for $k_geod$ with $1<\alpha<1.1$. In +The function $k_\mathit{geod}\colon c_0\to \Ar$ is smooth. We choose a +regular value $\alpha$ for $k_\mathit{geod}$ with $1<\alpha<1.1$. In this way we divide $c_0$ into two disjoint subsets, $Y$ where -$k_geod>\alpha$, and $X$ where $k_geod\le \alpha$. The +$k_\mathit{geod}>\alpha$, and $X$ where $k_\mathit{geod}\le \alpha$. The subset $Y$ is a union of finitely many disjoint open intervals and $X$ is a disjoint union of finitely many closed intervals. -**Remark.** -The condition on $k_geod$ implies that for any arc $J$ in + +**Remark.** The condition on $k_\mathit{geod}$ implies that for any arc $J$ in $c_0$ of length $1$ the total length of $J\cap Y$ is less than $\delta$. + Fix $\delta'>0$. For each $x\in X$ there is a geodesic $D_x$ in $A$ whose initial point is $x$ and whose initial direction is orthogonal to $c_0$. Let $f(x)$ be the minimum of $\delta$ and the distance along $D_x$ to the first point (excluding $x$) of its intersection with $\partial A$. We set -$$S_X(\delta') =\{(x,t)\in X\times [0,\delta']\bigl|\bigr.\ t\le f(x)\}.$$ + +$$ +S_X(\delta') =\{(x,t)\in X\times [0,\delta']\bigl|\bigr.\ t\le f(x)\}. +$$ + The subset $S_X(\delta')$ inherits a Riemannian metric from the -product of the metric on $X$ induced by the embedding $X\subset +product of the metric on $X$ induced by the embedding $X\subset c_0$ and the standard metric on the interval $[0,\delta']$. -**Claim.** -There is $\delta'>0$ such that the following holds. + +**Claim.** There is $\delta'>0$ such that the following holds. The -exponential mapping defines a map $exp\colon S_X(\delta')\to +exponential mapping defines a map $\mathit{exp}\colon S_X(\delta')\to A$ which is a local diffeomorphism and the pullback of the metric on $A$ defines a metric on $S_X(\delta')$ which is at least $(1-\delta)^2$ times the given product metric. + **Proof.** This is a standard computation using the Gaussian curvature upper bound and the geodesic curvature bound. -Now we fix $0<\delta'<1/10$ so that Claim~ holds, and +Now we fix $0<\delta'<1/10$ so that \entryref{47e16059fb12} holds, and we set $S_X=S_X(\delta')$. We define -$$\partial_+S_X=\{(x,t)\in S_X \bigl|\bigr.\, t=f(x)\}.$$ + +$$ +\partial_+S_X=\{(x,t)\in S_X \bigl|\bigr. t=f(x)\}. +$$ + Then the boundary of $S_X$ is made up of $X$, the arcs $\{x\}\times [0,f(x)]$ for $x\in \partial X$ and $\partial_+(S_X)$. For any subset $Z\subset X$ we denote by $S_Z$ the intersection $(Z\times @@ -3091,48 +3562,72 @@ $\xi*D'_y*(D'_x)^{-1}$ bounds a disk $B$ in $A$. The arc $\xi$ is called the *base* of the focusing triangle and with, respect to an orientation of $c_0$, if $x$ is the initial point of $\xi$ then $D'_x$ is called the *left-hand side* of the focusing triangle -and $D'_y$ is called its *right-hand side*. See Fig.~. +and $D'_y$ is called its *right-hand side*. See Fig. 18.1. Our main goal here is the following lemma which gives an upper bound for the length of the base, $\xi$, of a focusing triangle in terms of the turning along the base and the area of the region $B$ enclosed by the triangle. -**Lemma.** - Suppose that we have a focusing triangle ${\mathcal T}$ with base $\xi$ bounding a disk $B$ + +**Lemma.** Suppose that we have a focusing triangle ${\mathcal T}$ with base $\xi$ bounding a disk $B$ in $A$. Suppose that the length of $\xi$ is at most one. Then -$$l(\xi)\le \left(\int_\xi k_geodds+Area(B)\right).$$ + +$$ +l(\xi)\le \left(\int_\xi k_\mathit{geod}ds+\mathit{Area}(B)\right). +$$ + **Proof.** We begin with a preliminary computation. We denote by $a(B)$ the area of $B$. We define -$$t_{\xi}=\int_{\xi}k_geodds \ \ \ and \ \ \ T_\xi=\int_{\xi}|k_geod|ds.$$ + +$$ +t_{\xi}=\int_{\xi}k_\mathit{geod}ds \ \ \ \mathit{and} \ \ \ T_\xi=\int_{\xi}|k_\mathit{geod}|ds. +$$ Recall that given a piecewise smooth curve, its *total turning* is the integral of the geodesic curvature over the smooth part of the boundary plus the sum over the break points of $\pi$ minus the -interior angle at the break point. The Gauss-Bonnet theorem tells +interior angle at the break point. The Gauss-Bonnet theorem tells us that for any compact surface with piecewise smooth boundary the integral of the Gaussian curvature over the interior of the surface plus the total turning around the boundary equals $2\pi$ times the Euler characteristic of the surface. -**Claim.** -The angle $\theta_B$ between $D'_x$ and $D'_y$ at the vertex $v$ -satisfies $$\theta_B\le t_\xi+a(B)$$ and for any measurable subset + +**Claim.** The angle $\theta_B$ between $D'_x$ and $D'_y$ at the vertex $v$ +satisfies + +$$ +\theta_B\le t_\xi+a(B) +$$ + + and for any measurable subset $B'\subset B$ we have -$$\theta_B-t_\xi-a(B)\le \int_{B'}Kda + +**Proof.** Since $D'_x$ and $D'_y$ meet $\partial A$ in right angles, the total turning around -the boundary of $B$ is -$$t_\xi+2\pi-\theta_B.$$ +the boundary of $B$ is + +$$ +t_\xi+2\pi-\theta_B. +$$ + Thus, by Gauss-Bonnet, we have -$$\theta_B=\int_BKda+ t_\xi.$$ + +$$ +\theta_B=\int_BKda+ t_\xi. +$$ + But $K\le 1$, giving the first stated inequality. On the other hand -$\int_BKda=\int_BK^+da+\int_BK^-da$, where $K^+=max(K,0)$ and +$\int_BKda=\int_BK^+da+\int_BK^-da$, where $K^+=\mathit{max}(K,0)$ and $K^-=K-K^+$. Since $0\le \int_BK^+da\le a(B)$ and $\int_BK^-\le 0$, the second string of inequalities follows. @@ -3140,10 +3635,11 @@ In order to make the computation we need to know that this triangle is the image under the exponential mapping of a spray of geodesics out of the vertex $v$. Establishing that requires some work. -**Claim.** -Let $a\in int\,\xi$. There is a shortest path in $B$ from $a$ + +**Claim.** Let $a\in \mathit{int} \xi$. There is a shortest path in $B$ from $a$ to $v$. This shortest path is a geodesic meeting $\partial B$ only in its end points. It has length $\le(1/2)+\delta'$. + **Proof.** The length estimate is obvious: Since $\xi$ has length at most $1$, a path along $\partial A\cap B $ from $a$ to the closest of $x$ and @@ -3152,8 +3648,8 @@ most $\delta'$. Thus, there is a path from $a$ to $v$ in $B$ of length at most $(1/2)+\delta'$. Standard convergence arguments show that there is a shortest path in -$B$ from $a$ to $v$. Fix $a\in int(\partial A\cap B)$. It is -clear that the shortest path cannot meet either of the `sides' +$B$ from $a$ to $v$. Fix $a\in \mathit{int}(\partial A\cap B)$. It is +clear that the shortest path cannot meet either of the 'sides' $D'_x$ and $D'_y$ at any point other than $v$. If it did, then there would be an angle at this point and a local shortcut, cutting off a small piece of the angle, would provide a shorter path. We must rule @@ -3175,8 +3671,10 @@ $B'$ is at most the area of $B$, which is less than $\mu<1/20<(\pi/4)-\delta$. This contradicts the Gauss-Bonnet theorem. + **Claim.** For any $a\in (\partial A\cap B)$ there is a unique minimal geodesic in $B$ from $a$ to $v$. + **Proof.** Suppose not; suppose there are two $\gamma$ and $\gamma'$ from $v$ to $a$. Since they are both minimal in $B$, each is embedded, and @@ -3207,14 +3705,14 @@ $1/2+\delta$, it follows that the exponential map is a local diffeomorphism near all such geodesics. Thus, $\beta'$ is an open subset of $\beta$. On the other, hand if the direction of $\gamma''\not=\gamma'$ is a point $b''\in \beta$ which is an -endpoint of an open interval $ \beta'$, and if this interval +endpoint of an open interval $ \beta'$, and if this interval separates $b''$ from the direction of $\gamma$ then the length of $\gamma''$ is less than the length of each point in the interval. Hence, the length of $\gamma''$ is less than $(1/2)+\delta'$. Invoking Gauss-Bonnet again we see that the angle between $\gamma''$ and $\gamma'$ is $<\mu$. -This proves that if $U$ is an open interval in $\beta'$ then the +This proves that if $U$ is an open interval in $\beta'$ then the endpoint of $U$ closest to the direction of $\gamma'$ is also contained in $\beta'$ (unless that endpoint is the direction of $\gamma'$). It is now elementary to see that $\beta'$ is all of @@ -3225,13 +3723,15 @@ any geodesic from $v$ whose initial direction is sufficiently close to that of $\gamma'$ and whose length is at most $(1/2)+\delta'$ will not cross $\gamma'$. -See Fig.~ +See Fig. 18.2 + **Remark.** The same argument shows that from any $a\in (\partial A\cap B)$ there is a unique embedded geodesic in $B$ from $v$ to $a$ with length at most $(1/2)+\delta'$. (Such geodesics may cross more than once, but the argument given in the lemma applies to sub-geodesics from $v$ to the first point of intersection along $\gamma$.) + Let $E$ be the sub-interval of the circle of tangent directions at $v$ consisting of all tangent directions of geodesics pointing into @@ -3264,40 +3764,62 @@ not need it.) 1$. We set $l(t)$ equal to the length of $\lambda(t)$. -**Claim.** $$\frac{dl}{dt}(t)\le max_{\psi\in E}h(\psi)\int_{\lambda(t)}k_geodds.$$ + +**Claim.** + +$$ +\frac{dl}{dt}(t)\le \mathit{max}_{\psi\in E}h(\psi)\int_{\lambda(t)}k_\mathit{geod}ds. +$$ + -**Proof.** First of all notice that, by construction, the curve $\xi$, which +**Proof.** First of all notice that, by construction, the curve $\xi$, which is defined by $\{s=h(\psi)\}$, is orthogonal to the radial geodesics to the endpoints. As a consequence, $h'(\psi)=0$ at the endpoints. Thus, each of the curves $\lambda(t)$ is orthogonal to the radial geodesics through its end points. Therefore, as we vary the family $\lambda(t)$ the formula for the derivative of the length is -$$l'(t)=\int_{\lambda(t)}k_geod(\psi)h(\psi)|cos(\theta(\psi,t))|ds$$ where $\theta(\psi,t)$ is the angle at + +$$ +l'(t)=\int_{\lambda(t)}k_\mathit{geod}(\psi)h(\psi)|\mathit{cos}(\theta(\psi,t))|ds +$$ + + where $\theta(\psi,t)$ is the angle at $(th(\psi),\psi)$ between the curve $s=th(\psi)$ and the radial geodesic. The result follows immediately. Next, we must bound the turning of $\lambda(t)$. For this we invoke the Gauss-Bonnet theorem once again. Applying this to the wedge-shaped disk $W(t)$ cut out by $\lambda(t)$ gives -$$\int_{W(t)}Kda+\int_{\lambda(t)}k_geodds=\theta_B.$$ -From Claim~ we conclude that -$$\int_{\lambda(t)}k_geodds\le t_\xi+a(B).$$ -Of course, by Claim~ we have $max_{\psi\in +$$ +\int_{W(t)}Kda+\int_{\lambda(t)}k_\mathit{geod}ds=\theta_B. +$$ + +From \entryref{512c74b9ad1f} we conclude that + +$$ +\int_{\lambda(t)}k_\mathit{geod}ds\le t_\xi+a(B). +$$ + +Of course, by \entryref{ea7c58d04fc7} we have $\mathit{max}_{\psi\in E}h(\psi)\le (1/2)+\delta'$. Since $l(0)=0$, this implies that -$$l(\xi)=l(1)\le (a(B)+t_B)((1/2)+\delta') +**Corollary.** Suppose that ${\mathcal T}$ is a focusing triangle with base $\xi$ of length at most one. Then the length of $\xi$ is at most $\delta+\mu$. More generally, suppose we have a collection of focusing triangles ${\mathcal T}_1,\ldots,{\mathcal T}_n$ whose bases all lie in a fixed interval of length one in $c_0$. Suppose also that the interiors of disks bounded by these focusing triangles -are disjoint. Then the sum of the lengths of the bases is at most +are disjoint. Then the sum of the lengths of the bases is at most $\delta+\mu$. + **Proof.** The first statement is immediate from the previous lemma. The second comes from the fact that the sum of the areas of the disks bounded @@ -3310,21 +3832,22 @@ mapping from being an embedding out to distance $\delta'$. ### No $D_x$ is an embedded arc with both ends in $c_0$ -One thing that we must show is that the geodesics $D_x$ are +One thing that we must show is that the geodesics $D_x$ are embedded. Here is a special case that will serve some of our purposes. -**Lemma.** -For each $x\in X$, there is no non-trivial sub-geodesic of $D_x$ + +**Lemma.** For each $x\in X$, there is no non-trivial sub-geodesic of $D_x$ which is a homotopically trivial embedded loop in $A$. + -**Proof.** Were there such a loop, its total turning would be $\pi$ minus the +**Proof.** Were there such a loop, its total turning would be $\pi$ minus the angle it makes when the endpoints of the arc meet. Since $K\le 1$ and the area of the disk bounded by this loop is less than the area -of $A$ which in turn is less than $\mu<\pi$, one obtains a +of $A$ which in turn is less than $\mu<\pi$, one obtains a contradiction to the Gauss-Bonnet theorem. -Next, we rule out the possibility that one of the geodesics $D_x$ +Next, we rule out the possibility that one of the geodesics $D_x$ has both endpoints contained in $c_0$. This is the main result of this section. In a sense, what the argument we give here shows that if there is a $D_x$ with both ends on $c_0$, then under the @@ -3333,9 +3856,10 @@ annulus. But out near the end of this thin tentacle there must be a short arc with large total turning, violating our hypothesis on the integrals of the geodesic curvature over arcs of length at most one. -**Lemma.** -There is no $x$ for which $D_x$ is an embedded arc with both + +**Lemma.** There is no $x$ for which $D_x$ is an embedded arc with both endpoints on $c_0$ and otherwise disjoint from $\partial A$. + **Proof.** Suppose that there were such a $D_x$. Then $D_x$ separates $A$ into two components, one of which, $B$, is a topological disk. Let $c_0'$ @@ -3345,7 +3869,11 @@ be the intersection of $c_0$ with $B$. We consider two cases: Case Let us show that the first case is not possible. Since $D_x$ is a geodesic and $D_x$ is perpendicular to $c_0$ at one end, the total turning around the boundary of $B$ is at most -$$3\pi/2+\int_{c_0'}k_geodds<3\pi/2+\delta,$$ + +$$ +3\pi/2+\int_{c_0'}k_\mathit{geod}ds<3\pi/2+\delta, +$$ + where the last inequality uses the fact that the length of $c_0'$ is at most one. On the other hand, $\int_{B}Kda<\mu$, and $\mu<1/20<(\pi/2)-\delta$. This contradicts the Gauss-Bonnet @@ -3359,7 +3887,7 @@ each $y\in X_J$ we let $f_B(y)$ be the minimum of $\delta'$ and the distance along $D_y$ to the first point (excluding $y$) of $D_y$ contained in $\partial B$ and let $D_y(B)$ be the sub-geodesic of $D_y$ of this length starting at $y$. Then $S_{X_J}(B)\subset -S_{X_J}$ is the union over $y\in X_J$ of $[0,f_B(y)]$. Clearly, the +S_{X_J}$ is the union over $y\in X_J$ of $[0,f_B(y)]$. Clearly, the exponential mapping defines an immersion of $S_{X_J}(B)$ into $B$. We need to replace $X_ J$ by a slightly smaller subset in order to make the exponential mapping be an embedding. To do this we shall @@ -3367,24 +3895,24 @@ remove bases of a maximal focusing triangles in $B$. First notice that for each $y\in X_J$ the exponential mapping is an embedding on $D_y(B)$. The reason is that the image of $D_y(B)$ is a -geodesic contained in the ball $B$. Lemma~ then shows +geodesic contained in the ball $B$. \entryref{b6fb2b9c364b} then shows that this geodesic is embedded. This leads to: -**Claim.** -For any component $c$ of $X_J$, the restriction of the exponential + +**Claim.** For any component $c$ of $X_J$, the restriction of the exponential mapping to $S_c(B)=\left(c\times [0,\delta')\right)\cap S_{X_J}(B)$ is an embedding. + -\begin{proof} -Since the geodesics that make up $S_c(B)$ have length at most +**Proof.** Since the geodesics that make up $S_c(B)$ have length at most $\delta'<1/10$ and since the curvature of the annulus is bounded above by $1$, the restriction of the exponential mapping to $S_c(B)$ -is a local diffeomorphism. The restriction to each $\{y\}\times +is a local diffeomorphism. The restriction to each $\{y\}\times [0,f_B(y)]$ is an embedding onto $D_y(B)$. If the restriction of the exponential mapping to $S_c(B)$ is not an embedding, then there are $y\not= y'$ in $c$ such that the geodesics $D_y(B)$ and $D_{y'}(B)$ meet. When they meet, they meet at a positive angle and by the -Gauss-Bonnet theorem this angle is less than $\mu+\delta$. Thus, +Gauss-Bonnet theorem this angle is less than $\mu+\delta$. Thus, all the geodesics starting at points sufficiently close to $y'$ and between $y$ and $y'$ along $c$ must also meet $D_y(B)$. Of course, if a sequence of $D_{y_i}(B)$ meet $D_y(B)$, then the same is true @@ -3392,15 +3920,16 @@ for the limit. It now follows that $D_{y''}(B)$ meets $D_y(B)$ for all $y''$ between $y$ and $y'$. This contradicts the fact that $D_y(B)$ is embedded. -**Claim.** -Any focusing triangle for $J$ must contain a component of $J\setminus X_J $. If + +**Claim.** Any focusing triangle for $J$ must contain a component of $J\setminus X_J $. If $\{{\mathcal T}_n\}$ is an infinite sequence of focus triangles for $J$, then, after passing to a subsequence, there is a limiting focusing triangle for $J$. + -**Proof.** The first statement is immediate from Claim~. +**Proof.** The first statement is immediate from \entryref{14aeee986b49}. Since $X\cap J$ is compact, it is clear that after passing to a subsequence each of the sequence of left-hand sides and the - sequence of right-hand sides converge to a geodesic arc orthogonal to + sequence of right-hand sides converge to a geodesic arc orthogonal to $J$ at points of $X$. Furthermore, these limiting geodesics meet in a point at distance at most $\delta'$ from the end of each. The only thing remaining to show is that the limiting left- and right-hand @@ -3408,12 +3937,12 @@ after passing to a subsequence, there is a limiting focusing triangle for $J$. This is clear since each focusing triangle contains one of the finitely many components of $J\setminus X_J$. -Using Claim~ we see that if there is a focusing +Using \entryref{2ff35cef8589} we see that if there is a focusing triangle for $J$ there is a first point $x_1$ in $ X_J$ whose associated geodesic contains the left-hand side of a focusing triangle for $J$. Then since the base length of any focusing triangle is bounded by a fixed constant, invoking again -Claim~, that there is a focusing triangle ${\mathcal +\entryref{2ff35cef8589}, that there is a focusing triangle ${\mathcal T}_1$ for $J$ that has left-hand side contained in the geodesic $D_{x_1}$ and has a maximal base among all such focusing triangles, Maximal in the sense that the base of this focusing triangle @@ -3427,8 +3956,10 @@ the first left-hand side of such focusing triangles and then find the maximal focusing triangle ${\mathcal T}_2$ with this left-hand side. + **Claim.** The interior of ${\mathcal T}_2$ is disjoint from the interior of ${\mathcal T}_1$. + **Proof.** Since by construction the interiors of the bases of ${\mathcal T}_1$ and ${\mathcal T}_2$ are disjoint, if the interior of ${\mathcal @@ -3456,22 +3987,28 @@ of steps. Let ${\mathcal T}_1,\ldots,{\mathcal T}_k$ be the focusing triangles so constructed, and denote by $\xi_i$ the base of ${\mathcal T}_i$. Let $X_J'= X_J\setminus \cup_{i=1}^k\xi_i$. + **Definition.** We call the triangles ${\mathcal T}_1,\ldots,{\mathcal T}_k$ constructed above, the *maximal set of focusing triangles* for $J$ relative to $B$. + + **Claim.** The length of $X'_J$ is at least $1-2\delta-\mu$. + **Proof.** Since the interiors of the ${\mathcal T}_i$ are disjoint, according -to Corollary~, we have $\sum_il(\xi_i)<\delta+\mu$. We -also know by Remark~ that the length of $X_J$ is at +to \entryref{9c8d4163f700}, we have $\sum_il(\xi_i)<\delta+\mu$. We +also know by \entryref{bb88301ceeda} that the length of $X_J$ is at least $(1-\delta)$. Putting these together gives the result. We define $S_{X'_J}(B)$ to be the intersection of $S_{X_J}(B)$ with $S_{X'_J}$. + **Claim.** The restriction of the exponential mapping to $S_{X'_J}(B)$ is an embedding. + **Proof.** Suppose that we have distinct points $x',y'$ in $X'_J$ such that $D_{x'}(B)\cap D_{y'}(B)\not=\emptyset$. We assume that $x' **Lemma.** For any $x\in X$ there is no sub-geodesic of $D_x$ that is an embedded loop in $A$. + **Proof.** We have already treated the case when the loop bounds a disk. Now we need to treat the case when the loop is homotopically non-trivial in $A$. Let $D'_x\subset D_x$ be the minimal compact sub-geodesic -containing $x$ that is not an embedded arc. Let $int\,B$ be +containing $x$ that is not an embedded arc. Let $\mathit{int} B$ be the complementary component of $D'_x$ in $A$ that contains -$c_0\setminus\{x\}$. There is a natural compactification of $int\, B$ as a disk and an immersion of this disk into $A$, an +$c_0\setminus\{x\}$. There is a natural compactification of $\mathit{int} B$ as a disk and an immersion of this disk into $A$, an immersion that is two-to-one along the shortest sub-geodesic of $D'_x$ from $x$ to the point of intersection of $D_x$ with itself. We do exactly the same construction as before. Take a sub-arc $J$ of @@ -3570,7 +4121,7 @@ Furthermore, the length of the arcs that these final ends sweep out is at least $0.86$. Hence, since the total length of the part of the boundary of $B$ coming from $D_x'$ is at most $2\delta'<0.2$, there must be a $z\in Z$ for which $D_z(B)$ has both ends in $c_0$. This -puts us back in the case ruled out in Lemma~. +puts us back in the case ruled out in \entryref{0d6f74e37efb}. ### Far apart $D_x$'s don't meet @@ -3579,45 +4130,48 @@ complement of the focusing triangles from being an embedding is that geodesics $D_x$ and $D_y$ might meet even though $x$ and $y$ are far apart along $c_0$. Our next goal is to rule this out. + **Lemma.** Let $x,y$ be distinct points of $X$. Suppose that there are sub-geodesics $D'_x\subset D_x$ and $D'_y\subset D_y$ with a common endpoint. Then the arc $D'_x*(D'_y)^{-1}$ cuts $A$ into two complementary components, exactly one of which is a disk, denoted -$B$. Then it is not possible for $B\cap c_0$ to contain an arc of +$B$. Then it is not possible for $B\cap c_0$ to contain an arc of length $1$. + -**Proof.** The proof is exactly the same as in Lemma~ except that +**Proof.** The proof is exactly the same as in \entryref{0d6f74e37efb} except that the part of the boundary of $B$ that one wants to avoid has length at most $2\delta'<0.2$ instead of $\delta'$. Still, since (in the -notation of the proof of Lemma~) the total length of +notation of the proof of \entryref{0d6f74e37efb}) the total length of $Z$ is at least $0.87$ so that the lengths of the other ends of the $D_z$ as $z$ ranges over $Z$ is at least $0.86$, there is $z\in Z$ for which both ends of $D_z$ lie in $c_0$. Again this puts us back -in the case ruled out by Lemma~. +in the case ruled out by \entryref{0d6f74e37efb}. As a special case of this result we have the following. -**Corollary.** -Suppose that we have an arc $\xi$ of length at most $1$ on $c_0$. -Denote the endpoints of $\xi$ by $x$ and $y$ and suppose that + +**Corollary.** Suppose that we have an arc $\xi$ of length at most $1$ on $c_0$. +Denote the endpoints of $\xi$ by $x$ and $y$ and suppose that $D_x\cap D_y\not=\emptyset$. Let $D'_x$ and $D'_y$ be sub-geodesics containing $x$ and $y$ respectively ending at the same point, $v$, and otherwise disjoint. Then the loop $\xi*D'_y*(D'_x)^{-1}$ bounds a disk in $A$. + **Proof.** If not, then it is homotopically non-trivial in $A$ and replacing -$\xi$ by its complement, $c_0\setminusint\,\xi$, gives us +$\xi$ by its complement, $c_0\setminus\mathit{int} \xi$, gives us exactly the situation of the previous lemma. (The length of -$c_0\setminusint\,\xi$ is at least one since the length of +$c_0\setminus\mathit{int} \xi$ is at least one since the length of $c_0$ is at least $2$.) Let us now summarize what we have established so far about the intersections of the geodesics $\{D_x\}_{x\in X}$. -**Corollary.** -For each $x\in X$, the geodesic $D_x$ is an embedded arc in $A$. + +**Corollary.** For each $x\in X$, the geodesic $D_x$ is an embedded arc in $A$. Either it has length $\delta'$ or its final point lies on $c_1$. -Suppose there are $x\not=x'$ in $X$ with $D_x\cap +Suppose there are $x\not=x'$ in $X$ with $D_x\cap D_{x'}\not=\emptyset$. Then there is an arc $\xi$ on $c_0$ connecting $x$ to $x'$ with the length of $\xi$ at most $\delta+\mu$. Furthermore, for sub-geodesics $D'_x\subset D_x$, @@ -3625,33 +4179,34 @@ containing $x$, and $D'_{x'}\subset D_{x'}$, containing $x'$, that intersect exactly in an endpoint of each, the loop $\xi*D'_{x'}*(D'_x)^{-1}$ bounds a disk $B$ in $A$, and the length of $\xi$ is at most the turning of $\xi$ plus the area of $B$. + ### Completion of the proof We have now completed all the technical work on focusing and we have -also shown that the restriction of the exponential mapping to the +also shown that the restriction of the exponential mapping to the complement of the bases of the focusing regions is an embedding. We are now ready to complete the proof of -Proposition~. +\entryref{6e9739b0c59b}. Let $J$ be an interval of length one in $c_0$. Because of -Corollary~ we can construct the maximal focusing +\entryref{8abfaea5aa9a} we can construct the maximal focusing triangles for $J$ as follows. Orient $J$, and begin at the initial point of $J$. At each step we consider the first $x$ (in the subinterval of $J$ under consideration) which intersects a $D_y$ for some later $y\in J$. If we have such $y$, then we can construct the sides of the putative triangle for sub-geodesics of $D_x$ and $D_y$. But we need to know that we have a focusing triangle. This is the -content of Corollary~. The same reasoning works when we +content of \entryref{ad36aef968b6}. The same reasoning works when we construct the maximal such focusing triangle with a given left-hand side, and then when we show that in the complement of the focusing triangles the map is an embedding. Thus, as before, for an interval -$J$ of length $1$, we construct a subset $X_J'\subset X\cap J$ of +$J$ of length $1$, we construct a subset $X_J'\subset X\cap J$ of length at least $0.97$ such that the restriction of the exponential mapping to $S_{X'_J}$ is an embedding. Again the area estimate shows that there is a subset $Z\subset X'_J$ whose length is at least $0.87$ with the property that for every $z\in Z$ the geodesic $D_z$ -has both endpoints in $\partial A$. By Lemma~, the +has both endpoints in $\partial A$. By \entryref{0d6f74e37efb}, the only possibility for the final endpoints of all these $D_z$'s is that they lie in $c_1$. @@ -3669,7 +4224,7 @@ focusing triangles for a subinterval $[x'_0,x]$, we consider the first point $y$ in the complementary interval $[x,x''_0]$ with the property that there is $y'$ in this same interval, further along with $D_y\cap D_{y'}\not=0$. Then, using -Corollary~ we construct the maximal focusing +\entryref{8abfaea5aa9a} we construct the maximal focusing triangle on $[x,x''_0]$ with left-hand side being a sub-geodesic of $D_y$. We then continue the construction inductively until we reach $x''_0$. Denote by $\xi_1,\ldots,\xi_k$ the bases of these focusing @@ -3678,25 +4233,30 @@ triangles and let $X'$ be $X\setminus\cup_i\xi_i$. The arguments above show that the exponential mapping is an embedding of $S_{X'}$ to the annulus. + **Claim.** For every subinterval $J$ of length one in $c_0$ the total length of the bases $\xi_i$ that meet $J$ is at most $2\delta+\mu<0.03$. + -**Proof.** Since, by Corollary~, every base of a focusing triangle +**Proof.** Since, by \entryref{9c8d4163f700}, every base of a focusing triangle has length at most $\delta+\mu$, we see that the union of the bases of focusing triangles meeting $J$ is contained in an interval of length $1+2(\delta+\mu)<2$. Hence, the total turning of the bases of these focusing triangles is at most $2\delta$ whereas the sum of their areas is at most $\mu$. The result now follows from -Corollary~. +\entryref{9c8d4163f700}. By hypothesis there is an integer $n\ge 1$ such that the length $l(c_0)$ of $c_0$ is greater than $n$ but less than or equal to $n+1$. Then it follows from the above that the total length of the bases of all the focusing triangles in our family is at most -$$(n+1)(2\delta+\mu)< 0.03(n+1)\le 0.06n\le 0.06 l(c_0).$$ + +$$ +(n+1)(2\delta+\mu)< 0.03(n+1)\le 0.06n\le 0.06 l(c_0). +$$ Since the restriction of the exponential mapping to $S_{X'}$ is an -embedding, it follows from Claim~ and the choice of +embedding, it follows from \entryref{47e16059fb12} and the choice of $\delta'$ that, for any open subset $Z$ of $X'$, the area of the image under the exponential mapping of $S_Z$ is at least $(1-\delta)^2\int_Zf(x)ds$, where $ds$ is the arc length along $Z$. @@ -3711,22 +4271,26 @@ $f(x)<\delta'$ for all $x\in X''$. This means that for every $x\in X''$ the geodesic $D_x$ spans from $c_0$ to $c_1$, and hence the exponential mapping embeds $\partial_+S_{X''}$ into $c_1$. But we have just seen that the length of the image under the exponential -mapping of $\partial_+S_{X''}$ is at least -$$(1-\delta)l(X'')>(0.99)l(X'')>(0.83)l(c_0).$$ -It follows that the length of $c_1$ is at least +mapping of $\partial_+S_{X''}$ is at least + +$$ +(1-\delta)l(X'')>(0.99)l(X'')>(0.83)l(c_0). +$$ + +It follows that the length of $c_1$ is at least $(0.83)l(c_0)>3(l(c_0))/4$. This completes the proof. -## Proof of the first inequality in Lemma~\protect{} +## Proof of the first inequality in Lemma \entryref{38712e8cea3d} Here is the statement that we wish to establish when the manifold $(W,h(t))$ is the product of $(M,g(t))\times (S^1_\lambda,ds^2)$. -**Lemma.** -Let $(W,h(t)),\ t_0\le t\le t_1$, be a Ricci flow and fix + +**Lemma.** Let $(W,h(t)),\ t_0\le t\le t_1$, be a Ricci flow and fix $\Theta<\infty$. Then there exist constants $\delta>0$ and $0 The rest of this section is devoted to the proof of this lemma. In - such a local estimate was established when the ambient +[AG] such a local estimate was established when the ambient manifold was Euclidean space and the curve in question is a graph. A related result for hypersurfaces that are graphs appears in -. The passage from Euclidean space to a general +[EckerHuisken]. The passage from Euclidean space to a general Ricci flow is straightforward, but it is more delicate to use the bound on total curvature on initial sub-arcs of length $r$ to show that in appropriate coordinates the evolving curve can be written as an evolving graph, so that the analysis in [2] can be applied. We fix $\delta>0$ sufficiently small. We fix $t'\in [t_0,t_1-\delta -r^2]$ for which the hypotheses of the lemma hold. The strategy of +r^2]$ for which the hypotheses of the lemma hold. The strategy of the proof is to first restrict to the maximum subinterval of $[t',t_2]$ of $[t',t'+\delta r^2]$ on which $k$ is bounded by $\sqrt{2/(t-t')}$. If $t_2 +**Claim.** There is a constant $D_0<\infty$, depending only on $\Theta$ and the curvature bound of the ambient Ricci flow such that for every $t\in [t',t'+\delta r^2]$ and every sub-arc $\gamma_{t'}$ whose length is at most $r$, we have $\int_{\gamma_t} kds -**Proof.** This is immediate from Corollary~ applied to all +**Proof.** This is immediate from \entryref{156a3da68fbd} applied to all of $c(\cdot,t)$. Now we fix $t_2\le t'+\delta r^2$ maximal subject to the condition @@ -3803,8 +4372,13 @@ $k(x,t_2)=\sqrt{\frac{2}{(t_2-t')}}$. Now consider a curve $\gamma_{t'}$ of length $r$. From the integral estimate in the previous claim and the assumed pointwise estimate on $k$, we see that -$$\int_{\gamma_t}k^2 ds\le max_{x\in\gamma_t}k(x,t)\int_{\gamma_{t}} kds -<\sqrt{\frac{2}{t-t'}}\cdot D_0.$$ Using Equation~(), + +$$ +\int_{\gamma_t}k^2 ds\le \mathit{max}_{x\in\gamma_t}k(x,t)\int_{\gamma_{t}} kds +<\sqrt{\frac{2}{t-t'}}\cdot D_0. +$$ + + Using Equation (18.12), it follows easily that, provided that $\delta>0$ is sufficiently small, the length of $\gamma_t$ is at least $(0.9)r$ for all $t\in [t',t_2]$, and more generally for any subinterval $\gamma'_{t'}$ of @@ -3817,37 +4391,73 @@ on $[-0.50,-0.45]$ and on $[0.45,0.50]$, and is identically $1$ on $[-3/8,3/8]$. There is a constant $D'$ such that $|\psi'|\le D'$ and $|\psi''|\le D'$. Now we fix the midpoint $x_0\in \gamma_{t'}$ and define the signed distance from $(x_0,t)$, denoted -$$s\colon \gamma_{t'}\times [t',t_2]\to \Ar,$$ + +$$ +s\colon \gamma_{t'}\times [t',t_2]\to \Ar, +$$ + as follows: -$$s(x,t)=\int _{x_0}^x|X(y,t)|dy.$$ + +$$ +s(x,t)=\int _{x_0}^x|X(y,t)|dy. +$$ We define the cut-off function -$$\varphi(x,t)=\psi\left(\frac{s(x,t)}{r}\right).$$ -**Claim.** -There is a constant $D_1$ depending only on the curvature bound for +$$ +\varphi(x,t)=\psi\left(\frac{s(x,t)}{r}\right). +$$ + + +**Claim.** There is a constant $D_1$ depending only on the curvature bound for the ambient Ricci flow such that for any sub-arc $\gamma_{t'}$ of length $r$, defining $\varphi(x,t)$ as above, for all $x\in \gamma_t$ and all $t\in [t',t_2]$ we have -$$\left|\frac{\partial \varphi(x,t)}{\partial t}\right|\le -\frac{D_1}{r\sqrt{t-t'}}+D_1.$$ + +$$ +\left|\frac{\partial \varphi(x,t)}{\partial t}\right|\le +\frac{D_1}{r\sqrt{t-t'}}+D_1. +$$ + **Proof.** Clearly, -$$\frac{\partial \varphi(x,t)}{\partial + +$$ +\frac{\partial \varphi(x,t)}{\partial t}=\psi'\left(\frac{s(x,t)}{r}\right)\cdot \frac{1}{r}\frac{\partial -s(x,t)}{\partial t}.$$ We know that $|\psi'|\le D'$ so that -$$\left|\frac{\partial \varphi(x,t)}{\partial +s(x,t)}{\partial t}. +$$ + + We know that $|\psi'|\le D'$ so that + +$$ +\left|\frac{\partial \varphi(x,t)}{\partial t}\right|\le \frac{D'}{r}\left|\frac{\partial s(x,t)}{\partial -t}\right|.$$ On the other hand, -$$s(x,t)=\int_{x_0}^x|X(y,t)|dy,$$ -so that $$ \left|\frac{\partial s(x,t)}{\partial t}\right| = -\left|\int_{x_0}^x\frac{\partial |X(y,t)|}{\partial t}dy\right|,$$ -By Lemma~ we have $$\frac{\partial -|X(y,t)|}{\partial y}dy = \left(-Ric(S(y,t),S(y,t))-k^2(y,t)\right)ds,$$ - so that there is a constant $D$ depending only on the bound of the -sectional curvatures of the ambient Ricci flow with +t}\right|. +$$ + + On the other hand, $$ +s(x,t)=\int_{x_0}^x|X(y,t)|dy, +$$ + +so that + +$$ +\left|\frac{\partial s(x,t)}{\partial t}\right| = +\left|\int_{x_0}^x\frac{\partial |X(y,t)|}{\partial t}dy\right|, +$$ + +By \entryref{4fc24fa1d31e} we have + +$$ +\frac{\partial +|X(y,t)|}{\partial y}dy = \left(-\mathit{Ric}(S(y,t),S(y,t))-k^2(y,t)\right)ds, +$$ + + so that there is a constant $D$ depending only on the bound of the +sectional curvatures of the ambient Ricci flow with $$ \begin{aligned} @@ -3857,99 +4467,168 @@ Dl(\gamma_t)+\int_{x_0}^xk^2(y,t)ds(y,t), \end{aligned} $$ -$$ and hence -by Claim~ -$$\left|\frac{\partial s(x,t)}{\partial t}\right|\le +by \entryref{1b8d34cae3a8} + +$$ +\left|\frac{\partial s(x,t)}{\partial t}\right|\le DD_0r+ -\int_{x_0}^xk^2(y,t)ds(y,t).$$ - Using -the fact that $k^2\le 2/(t-t')$, we have -$$\int_{x_0}^xk^2(y,t)ds(y,t)\le\sqrt{\frac{2}{t-t'}}\int_{x_0}^xkds\le -\frac{\sqrt{2}D_0}{\sqrt{t-t'}}.$$ Putting all this together, we see +\int_{x_0}^xk^2(y,t)ds(y,t). +$$ + + Using +the fact that $k^2\le 2/(t-t')$, we have + +$$ +\int_{x_0}^xk^2(y,t)ds(y,t)\le\sqrt{\frac{2}{t-t'}}\int_{x_0}^xkds\le +\frac{\sqrt{2}D_0}{\sqrt{t-t'}}. +$$ + + Putting all this together, we see that there is a constant $D_1$ such that -$$\left|\frac{\partial \varphi(x,t)}{\partial t}\right|\le -D_1(\frac{1}{r\sqrt{t-t'}}+1).$$ -**Claim.** -There is a constant $D_2$ depending only on the curvature bound of +$$ +\left|\frac{\partial \varphi(x,t)}{\partial t}\right|\le +D_1(\frac{1}{r\sqrt{t-t'}}+1). +$$ + + +**Claim.** There is a constant $D_2$ depending only on the curvature bound of the ambient Ricci flow and $\Theta$ and a constant $D_3$ depending only on the curvature bound of the ambient Ricci flow, such that for any $t\in [t',t_2]$ and any sub-arc $\gamma_{t'}$ of length $r$, we have -$$\left|\frac{d}{dt}\int_{\gamma_t}\varphi + +$$ +\left|\frac{d}{dt}\int_{\gamma_t}\varphi kds\right|\le D_2\left(1+\frac{1}{r\sqrt{t-t'}}\right)+\frac{D_2}{r^2}+D_3\int_{\gamma_t}\varphi -kds.$$ +kds. +$$ + **Proof.** We have -$$\left|\frac{d}{dt}\int_{\gamma_t}\varphi + +$$ +\left|\frac{d}{dt}\int_{\gamma_t}\varphi kds\right|\le\left|\int_{\gamma_t}\frac{\partial \varphi(x,t)}{\partial t}kds\right|+\left|\int_{\gamma_t}\varphi\frac{\partial -(kds)}{\partial t}\right|.$$ Using Claim~ for the first -term and Claim~ and arguing as in the proof of -Lemma~ for the second term, we have -$$\left|\frac{d}{dt}\int_{\gamma_t}\varphi +(kds)}{\partial t}\right|. +$$ + + Using \entryref{80e5fc6759b2} for the first +term and \entryref{67b855d0d6b5} and arguing as in the proof of +\entryref{88b8b3366b19} for the second term, we have + +$$ +\left|\frac{d}{dt}\int_{\gamma_t}\varphi kds\right|\le D_1\left(1+\frac{1}{r\sqrt{t-t'}}\right)\int_{\gamma_t}kds+\left|\int_{\gamma_t} -\varphi k''ds\right|+\int_{\gamma_t}C'_1\varphi kds,$$ where $C_1'$ +\varphi k''ds\right|+\int_{\gamma_t}C'_1\varphi kds, +$$ + + where $C_1'$ depends only on the ambient curvature bound. We bound the first term by -$$D_1D_0\left(\frac{1}{r\sqrt{t-t'}}+1\right),$$ + +$$ +D_1D_0\left(\frac{1}{r\sqrt{t-t'}}+1\right), +$$ + where $D_0$ is the constant depending on $\Theta$ and the ambient -curvature bound from Claim~. Since the ends of +curvature bound from \entryref{1b8d34cae3a8}. Since the ends of $\gamma_t$ are at distance at least $(0.45)r$ from $x_0$ all $t\in [t',t_2]$, we see that for all $t\in [t',t_2]$ -$$\int_{\gamma_t}\varphi k''=\int_{c(\cdot,t)}\varphi k''.$$ + +$$ +\int_{\gamma_t}\varphi k''=\int_{c(\cdot,t)}\varphi k''. +$$ + Integrating by parts we have -$$\int_{c(\cdot,t)} \varphi k'' ds=\int_{c(\cdot,t)}\varphi'' k ds,$$ + +$$ +\int_{c(\cdot,t)} \varphi k'' ds=\int_{c(\cdot,t)}\varphi'' k ds, +$$ + where the prime here refers to the derivative along $c(\cdot,t)$ with respect to arc length. Of course $|\varphi''|\le \frac{D'}{r^2}$. Thus, we see that -$$\left|\int_{\gamma_t}\varphi k''ds\right|\le -\frac{D'}{r^2}\int_{c(\cdot,t)}kds\le \frac{D'D_0}{r^2}.$$ Putting + +$$ +\left|\int_{\gamma_t}\varphi k''ds\right|\le +\frac{D'}{r^2}\int_{c(\cdot,t)}kds\le \frac{D'D_0}{r^2}. +$$ + + Putting all this together, we have -$$\left|\frac{d}{dt}\int_{\gamma_t}\varphi + +$$ +\left|\frac{d}{dt}\int_{\gamma_t}\varphi kds\right|\le D_2\left(1+\frac{1}{r\sqrt{t-t'}}\right)+\frac{D_2}{r^2}+D_3\int_{\gamma_t}\varphi -kds$$ for $D_2=D_0\,max(D',D_1)$ and $D_3=C_1'$. This gives +kds +$$ + + for $D_2=D_0 \mathit{max}(D',D_1)$ and $D_3=C_1'$. This gives the required estimate. -**Corollary.** For any $t\in [t',t_2]$ and any sub-arc $\gamma_{t'}$ of length $r$ + +**Corollary.** For any $t\in [t',t_2]$ and any sub-arc $\gamma_{t'}$ of length $r$ we have -$$\int_{\gamma_t}\varphi kds -\le D_4\sqrt{\delta}$$ for a constant $D_4$ that depends only on + +$$ +\int_{\gamma_t}\varphi kds +\le D_4\sqrt{\delta} +$$ + + for a constant $D_4$ that depends only on the sectional curvature bound of the ambient Ricci flow and $\Theta$. + **Proof.** This is immediate from the previous result by integrating from $t'$ to $t_2\le t'+\delta r^2$, and using the fact that $\delta<1$ and -$r<1$, and using the fact that $$\int_{\gamma_{t'}}\varphi kds\le -\int_{\gamma_{t'}}kds<\delta$$ since $\gamma_{t'}$ has length at +$r<1$, and using the fact that + +$$ +\int_{\gamma_{t'}}\varphi kds\le +\int_{\gamma_{t'}}kds<\delta +$$ + + since $\gamma_{t'}$ has length at most $r$. This gives: -**Corollary.** -For $\gamma_{t'}\subset c(\cdot,t')$ a sub-arc of length at most $r$ + +**Corollary.** For $\gamma_{t'}\subset c(\cdot,t')$ a sub-arc of length at most $r$ and for any $t\in [t',t_2]$, we have -$$\int_{\gamma_t}kds\le -2D_4\sqrt{\delta}.$$ For any $t\in [t',t_2]$ and any sub-arc + +$$ +\int_{\gamma_t}kds\le +2D_4\sqrt{\delta}. +$$ + + For any $t\in [t',t_2]$ and any sub-arc $J\subset c(\cdot,t)$ of length at most $r/2$ with respect to the metric $h(t)$, we have -$$\int_Jk(x,t)ds(x,t)\le 2D_4\sqrt{\delta}.$$ -**Proof.** We divide an interval $\gamma_{t'}\subset c(\cdot,t')$ of length at +$$ +\int_Jk(x,t)ds(x,t)\le 2D_4\sqrt{\delta}. +$$ + + +**Proof.** We divide an interval $\gamma_{t'}\subset c(\cdot,t')$ of length at most $r$ into two subintervals $\gamma'_{t'}$ and $\gamma''_{t'}$ of lengths at most $r/2$. Let $\hat \gamma'_{t'}$ and $\hat \gamma''_{t'}$ be intervals of length $r$ containing $\gamma'_{t'}$ -and $\gamma''_{t'}$ respectively as middle subintervals. We then -apply the previous corollary to $\hat \gamma'_{t'}$ and $\hat +and $\gamma''_{t'}$ respectively as middle subintervals. We then +apply the previous corollary to $\hat \gamma'_{t'}$ and $\hat \gamma''_{t'}$ using the fact that $\varphi k\ge 0$ everywhere and $\varphi k=k$ on the middle subintervals of $\hat \gamma'_{t'}$ and $\hat \gamma ''_{t'}$. For an interval $J\subset \gamma_t$ of length -$r/2$, according to Lemma~ the length of +$r/2$, according to \entryref{156a3da68fbd} the length of $\gamma_{t'}|_J$ with respect to the metric $h(t')$ is at most $r$, and hence this case follows from the previous case. @@ -3961,21 +4640,21 @@ an arc of length $r$ in $c(\cdot,t')$ and let $x_0$ be the central point of $\gamma_{t'}$. Denote $\gamma_{t'}(x_0)=p\in W$. We take the $h(t')$-exponential mapping from $T_pW\to W$. This map will be a local diffeomorphism out to a distance determined by the curvature -of $h(t')$. For an appropriate choice of the ball (depending on +of $h(t')$. For an appropriate choice of the ball (depending on the ambient curvature bound) the metric on the ball induced by pulling back $h(t)$ for all $t\in [t',t_2]$ will be within $\delta$ in the $C^1$-topology to the Euclidean metric $h'=h(t')_p$. By this we mean that -\begin{enumerate} -\item[(1)] $\left|\langle X,Y\rangle_{h(t)}-\langle + +- **(1)** $\left|\langle X,Y\rangle_{h(t)}-\langle X,Y\rangle_{h'}\right|<\delta |X|_{h'}|Y|_{h'}$ for all tangent vectors in the -coordinate system, and \item[(2)] viewing the connection $\Gamma$ as a bilinear +coordinate system, and +- **(2)** viewing the connection $\Gamma$ as a bilinear map on the coordinate space with values in the coordinate space we have $ |\Gamma(X,Y)|_{h'}<\delta |X|_{h'}|Y|_{h'}$. -\end{enumerate} We choose $00$ is sufficiently small, for every $t\in [t',t_3]$ there is @@ -4048,37 +4735,56 @@ curve $\gamma_t|_{I'}$ has length between $(0.8)r$ and $r$. The curve-shrinking flow is not defined on this product because under the curve-shrinking flow the $z$-coordinate of any given point is not constant. But the graph flow defined above, and studied in - (in the case of Euclidean background metric), is defined +[AG] (in the case of Euclidean background metric), is defined on $I'\times [t',t_3]$ since this flow preserves the $z$-coordinate. The time partial derivative in the curve-shrinking flow is given by $$ - \nabla_SS=\frac{\nabla_ZZ}{|Z|^2}-\frac{1}{|Z|^4}\langle \nabla_ZZ,Z\rangle Z. $$ + The time partial derivative in the graph-flow is given by $Y=\partial\widetilde \gamma/\partial t$. The tangent vector $Y$ is characterized by being $h'$-orthogonal to the $z$-axis and differing from $\nabla_SS$ by a functional multiple of $Z$. + **Claim.** -$$Y=\frac{\nabla_ZZ-\langle\Gamma(Z,Z),\partial_z\rangle_{h'} Z}{|Z|^2} + +$$ +Y=\frac{\nabla_ZZ-\langle\Gamma(Z,Z),\partial_z\rangle_{h'} Z}{|Z|^2} =\nabla_SS+\left(\frac{\langle \nabla_ZZ,Z\rangle}{|Z|^4} --\frac{\langle\Gamma(Z,Z),\partial_z\rangle_{h'} }{|Z|^2}\right)Z.$$ +-\frac{\langle\Gamma(Z,Z),\partial_z\rangle_{h'} }{|Z|^2}\right)Z. +$$ + **Proof.** In our Euclidean coordinates, $Z=(1,f_z)$ so that $\nabla_ZZ=(0,f_{zz})+\Gamma(Z,Z)$. Thus, -$$\langle \nabla_ZZ,\partial_z\rangle_{h'}=\langle -\Gamma(Z,Z),\partial_z\rangle_{h'}.$$ Since $\langle + +$$ +\langle \nabla_ZZ,\partial_z\rangle_{h'}=\langle +\Gamma(Z,Z),\partial_z\rangle_{h'}. +$$ + + Since $\langle Z,\partial_z\rangle_{h'}=1$, it follows that -$$\frac{ \nabla_ZZ-\langle\Gamma(Z,Z),\partial_z\rangle_{h'} Z}{|Z|^2}$$ is + +$$ +\frac{ \nabla_ZZ-\langle\Gamma(Z,Z),\partial_z\rangle_{h'} Z}{|Z|^2} +$$ + + is $h'$-orthogonal to the $z$-axis and hence is a multiple of $Y$. Since it differs by a multiple of $Z$ from $\nabla_SS$, it follows that it is $Y$. This gives the first equation; the second follows from this and -Equation~(). +Equation (18.14). To simplify the notation we set -$$\psi(Z)=\frac{\langle\Gamma(Z,Z),\partial_z\rangle_{h'}}{|Z|^2}.$$ + +$$ +\psi(Z)=\frac{\langle\Gamma(Z,Z),\partial_z\rangle_{h'}}{|Z|^2}. +$$ + Notice that from the conditions on $\Gamma$ and $h'$ it follows immediately that $|\psi(Z)|<(1.5)\delta$. @@ -4087,14 +4793,19 @@ immediately that $|\psi(Z)|<(1.5)\delta$. At this point we have a product coordinate system on which the metric is almost the Euclidean metric in the $C^1$-sense, and we have the graph flow given by -$$Y=\frac{\partial\widetilde \gamma}{\partial -t}=\frac{\nabla_ZZ}{|Z|^2}-\psi(Z)Z$$ defined on $[t',t_3]$ with + +$$ +Y=\frac{\partial\widetilde \gamma}{\partial +t}=\frac{\nabla_ZZ}{|Z|^2}-\psi(Z)Z +$$ + + defined on $[t',t_3]$ with image always contained in the given coordinate patch and written as a graph over a fixed interval $I'$ in the $z$-axis. For every $t\in [t',t_3]$ the length of $\gamma_{t'}|_{I'}$ in the metric $h(t')$ is between $(0.8)r$ and $r$. The function $f(z,t)$ whose graphs give the flow satisfies $|f_z|_{h'}\le 1/10$. Our next goal is to -estimate $|f_z|_{h'}$ and show that it is always less than $1/10$ +estimate $|f_z|_{h'}$ and show that it is always less than $1/10$ as long as $k^2\le 2/(t-t')$ and $t-t'\le \delta r^2$ for a sufficiently small $\delta$, i.e., for all $t\in [t',t_2]$; that is to say, our next goal is to prove that $t_3=t_2$. In all the @@ -4105,15 +4816,35 @@ allowed to change from line to line. The first step in doing this is to consider the angle between $\nabla_ZZ$ and $Z$. -**Claim.** Provided that $\delta>0$ is sufficiently small, + +**Claim.** Provided that $\delta>0$ is sufficiently small, the angle (measured in $h(t)$) between $Y$ and $Z=(1,f_z)$ is greater than $\pi/4$. Also, -\begin{enumerate} -\item[(1)] $$k\le |Y|<\sqrt{2}k.$$ -\item[(2)] $$\left|\langle \nabla_ZZ,Z\rangle\right|<(k+ 2\delta) |Z|^3.$$ -\item[(3)] $$\left|\nabla_ZZ\right|<2(|Y|+\delta).$$ -\item[(4)] $$|\langle Y,Z\rangle |\le |Y||f_z|(1+3\delta).$$ -\end{enumerate} + +- **(1)** + +$$ +k\le |Y|<\sqrt{2}k. +$$ + +- **(2)** + +$$ +\left|\langle \nabla_ZZ,Z\rangle\right|<(k+ 2\delta) |Z|^3. +$$ + +- **(3)** + +$$ +\left|\nabla_ZZ\right|<2(|Y|+\delta). +$$ + +- **(4)** + +$$ +|\langle Y,Z\rangle |\le |Y||f_z|(1+3\delta). +$$ + **Proof.** Under the hypothesis that $|f_z|_{h'}\le 1/10$, it is easy to see that the Euclidean angle between $(0,f_{zz})$ and $(1,f_z)$ is at @@ -4121,7 +4852,7 @@ most $\pi/2-\pi/5$. From this, the first statement follows immediately provided that $\delta$ is sufficiently small. Since $Y$ is the sum of $\nabla_SS$ and a multiple of $Z$ and since $\nabla_SS$ is $h(t)$-orthogonal to $Z$, it follows that -$|Y|=|\nabla_SS|\left(cos(\theta)\right)^{-1}$, where $\theta$ +$|Y|=|\nabla_SS|\left(\mathit{cos}(\theta)\right)^{-1}$, where $\theta$ is the angle between $\nabla_SS$ and $Y$. Since $Y$ is a multiple of $(0,f_{zz})$, it follows from the first part of the claim that the $h(t)$-angle between $Y$ and $\nabla_SS$ is less than $\pi/4$. Item @@ -4129,24 +4860,43 @@ $h(t)$-angle between $Y$ and $\nabla_SS$ is less than $\pi/4$. Item $|\nabla_SS|=k$. Since -$$\frac{\nabla_ZZ}{|Z|^2}=Y+\psi(Z)Z,$$ and $|Z|^2\le (1.01)(1+\delta)$, the third item is + +$$ +\frac{\nabla_ZZ}{|Z|^2}=Y+\psi(Z)Z, +$$ + + and $|Z|^2\le (1.01)(1+\delta)$, the third item is immediate. For Item (4), since $Y$ is $h'$-orthogonal to the $z$-axis, we have -$$\left|\langle Y,Z\rangle_{h'}\right|=\left|\langle Y,(0,f_z)\rangle_{h'}\right|\le |Y|_{h'}|f_z|_{h'}.$$ + +$$ +\left|\langle Y,Z\rangle_{h'}\right|=\left|\langle Y,(0,f_z)\rangle_{h'}\right|\le |Y|_{h'}|f_z|_{h'}. +$$ + From this and the comparison of $h(t)$ and $h'$, the Item (4) is immediate. Lastly, let us consider Item (2). We have -$$\langle Y,Z\rangle =\frac{\langle \nabla_ZZ,Z\rangle}{|Z|^2}-\langle -\Gamma(Z,Z),\partial_z\rangle_{h'}.$$ Thus, from Item (4) we have -$$\frac{\langle \nabla_ZZ,Z\rangle}{|Z|^2}\le |Y||f_z|(1+3\delta)+(1.5)\delta.$$ + +$$ +\langle Y,Z\rangle =\frac{\langle \nabla_ZZ,Z\rangle}{|Z|^2}-\langle +\Gamma(Z,Z),\partial_z\rangle_{h'}. +$$ + + Thus, from Item (4) we have + +$$ +\frac{\langle \nabla_ZZ,Z\rangle}{|Z|^2}\le |Y||f_z|(1+3\delta)+(1.5)\delta. +$$ + Since $Y<\sqrt{2}k$ and $|f_z|<1/10$, the Item (2) follows. -**Claim.** -The following hold provided that $\delta>0$ is sufficiently small: -\begin{enumerate} -\item[(1)] $|Z(\psi(Z))| +**Claim.** The following hold provided that $\delta>0$ is sufficiently small: + +- **(1)** $|Z(\psi(Z))| **Proof.** For the first item, we write $Z(\psi(Z))$ as a sum of terms where the differentiation by $Z$ acts on the various. When the $Z$-derivative acts on @@ -4167,55 +4917,54 @@ these terms together establishes the second inequality above. Now we wish to compute $\int_{I'\times\{t\}}|Z|^2dz.$ To do this we first note that using the definition of $Y$, and arguing as in the -proof of the first equation in of Lemma~ we have we +proof of the first equation in of \entryref{4fc24fa1d31e} we have we have -$$ - $$ \begin{aligned} \frac{\partial}{\partial t}|Z|^2 -& = & -2Ric(Z,Z)+2\langle \nabla_YZ,Z\rangle \\ -& = & -2Ric(Z,Z)+2\langle \nabla_ZY,Z\rangle +& = & -2\mathit{Ric}(Z,Z)+2\langle \nabla_YZ,Z\rangle \\ +& = & -2\mathit{Ric}(Z,Z)+2\langle \nabla_ZY,Z\rangle \end{aligned} $$ -$$ - Direct computation shows that -$$2\langle -\nabla_Z\left(\frac{\nabla_ZZ}{|Z|^2}\right),Z\rangle -=Z\left(\frac{Z(|Z|^2)}{|Z|^2}\right)-2\frac{|\nabla_ZZ|^2}{|Z|^4}|Z|^2.$$ Thus -from the Claim~, we have +$$ +2\langle +\nabla_Z\left(\frac{\nabla_ZZ}{|Z|^2}\right),Z\rangle +=Z\left(\frac{Z(|Z|^2)}{|Z|^2}\right)-2\frac{|\nabla_ZZ|^2}{|Z|^4}|Z|^2. $$ + Thus +from the \entryref{188acafae968}, we have + $$ \begin{aligned} - -\frac{\partial}{\partial t}|Z|^2 & = & 2\langle\nabla_ZY,Z\rangle -2Ric(Z,Z)\nonumber \\ +\frac{\partial}{\partial t}|Z|^2 & = & 2\langle\nabla_ZY,Z\rangle -2\mathit{Ric}(Z,Z)\nonumber \\ & = & Z\left(\frac{Z(|Z|^2)}{|Z|^2}\right)-2\frac{|\nabla_ZZ|^2}{|Z|^4}|Z|^2-2\langle -\nabla_Z(\psi(Z)Z),Z\rangle-2Ric(Z,Z) \nonumber \\ +\nabla_Z(\psi(Z)Z),Z\rangle-2\mathit{Ric}(Z,Z) \nonumber \\ & = & Z\left(\frac{Z(|Z|^2)}{|Z|^2}\right)-2|Y|^2|Z|^2 +V, \end{aligned} $$ +where + +$$ +V=-4|Z|^2\langle Y,\psi(Z)Z\rangle-2\psi^2(Z)|Z|^4-2\langle +\nabla_Z(\psi(Z)Z),Z\rangle-2\mathit{Ric}(Z,Z). $$ -where $$V=-4|Z|^2\langle Y,\psi(Z)Z\rangle-2\psi^2(Z)|Z|^4-2\langle -\nabla_Z(\psi(Z)Z),Z\rangle-2Ric(Z,Z).$$ By Item (1) in -Claim~ and Item (4) in Claim~ we have + By Item (1) in +\entryref{d3f828726e2b} and Item (4) in \entryref{8973f9acd2ae} we have $$ - |V| +**Claim.** + +$$ +|\langle \nabla_ZY,Z\rangle|\le (|f_z|(|\nabla_ZY|+2\delta|Y|)+\delta|Z|^2|Y|)(1+\delta). +$$ + **Proof.** Since $Y=(0,\phi)$ for some function $\phi$, we have $\nabla_ZY=(0,\phi_z)+\Gamma(Z,Y)$ and hence -$$|\langle\nabla_ZY,Z\rangle_{h'}| = |\langle \nabla_ZY,(1,f_z)\rangle_{h'}|\le |\langle + +$$ +|\langle\nabla_ZY,Z\rangle_{h'}| = |\langle \nabla_ZY,(1,f_z)\rangle_{h'}|\le |\langle f_z ,\phi_z\rangle_{h'}|+|\langle \Gamma(Z,Y),Z\rangle_{h'}|\le -|f_z|_{h'}|\phi_z|_{h'}+\delta|Z|^2|Y|.$$ On the other hand +|f_z|_{h'}|\phi_z|_{h'}+\delta|Z|^2|Y|. +$$ + + On the other hand $\nabla_ZY=(0,\phi_z)+\Gamma(Z,Y)$ so that $|\phi_z|_{h'}\le|\nabla_ZY|+\delta |Z||Y|$. From this the claim follows. Now for $\delta>0$ sufficiently small, using the fact that $1-\delta<|Z|^2<(1+\delta)(1.01)$ we can rewrite this as -$$ - $$ \begin{aligned} Q_t & \le & \frac{Q_{zz}}{|Z|^2}+\frac{8(1+3\delta)|Y|^2|f_z| @@ -4483,15 +5252,11 @@ Q_t & \le & \frac{Q_{zz}}{|Z|^2}+\frac{8(1+3\delta)|Y|^2|f_z| \end{aligned} $$ -$$ - where $\tilde A\le C'(|Y|^2+\delta|Y||\nabla_ZY|+\delta|Y|^3)$. Of course, $|Y||\nabla_ZY|+|Y|^3\le 2|Y|^2+|\nabla_ZY|^2+|Y|^4$. Using this, provided that $\delta$ is sufficiently small, we can rewrite this as -$$ - $$ \begin{aligned} Q_t & \le & \frac{Q_{zz}}{|Z|^2} +\frac{1}{(2-|Z|^2)}\cdot @@ -4502,13 +5267,9 @@ Q_t & \le & \frac{Q_{zz}}{|Z|^2} +\frac{1}{(2-|Z|^2)}\cdot \end{aligned} $$ -$$ - where $\tilde A''\le C'(|Y|^2)$. We denote the quantity within the brackets by $B$ and we estimate -$$ - $$ \begin{aligned} B & \le & @@ -4521,10 +5282,12 @@ B & \le & \end{aligned} $$ +Therefore, + +$$ +Q_t\le \frac{Q_{zz}}{|Z|^2}-Q^2+|\tilde A|\le \frac{Q_{zz}}{|Z|^2}-(Q-C_1')^2+(C_1')^2, $$ -Therefore, -$$Q_t\le \frac{Q_{zz}}{|Z|^2}-Q^2+|\tilde A|\le \frac{Q_{zz}}{|Z|^2}-(Q-C_1')^2+(C_1')^2,$$ for some constant $C'_1>1$ depending only on the curvature bound for the ambient Ricci flow. @@ -4535,23 +5298,45 @@ point $x_0$ then corresponds to $z=l/2$. Consider the function $g=l^2/(z^2(l-z)^2)$ on $I'\times [t',t_2]$. Direct computation shows that $g_{zz}\le 12g^2$. Now set -$$\widetilde Q=Q-C_1'$$ + +$$ +\widetilde Q=Q-C_1' +$$ + and -$$h=\frac{1}{t-t'}+\frac{4(1-\delta)^{-1}l^2}{z^2(l-z)^2}+C_1'.$$ + +$$ +h=\frac{1}{t-t'}+\frac{4(1-\delta)^{-1}l^2}{z^2(l-z)^2}+C_1'. +$$ + Then -$$ -h_t+(1-\delta)^{-1}h_{zz}+(C_1')^2\le h^2,$$ + +$$ +-h_t+(1-\delta)^{-1}h_{zz}+(C_1')^2\le h^2, +$$ + so that -$$(\widetilde Q-h)_t\le \frac{\widetilde Q_{zz}}{|Z|^2}-\frac{h_{zz}}{1-\delta}-\widetilde -Q^2+h^2.$$ Since both $h$ and $h_{zz}$ are positive, at any point + +$$ +(\widetilde Q-h)_t\le \frac{\widetilde Q_{zz}}{|Z|^2}-\frac{h_{zz}}{1-\delta}-\widetilde +Q^2+h^2. +$$ + + Since both $h$ and $h_{zz}$ are positive, at any point where $\widetilde Q -h\ge 0$ and $\widetilde Q_{zz}<0$, we have $(\widetilde Q-h)_t<0$. At any point where $Q_{zz}\ge 0$, using the fact that $|Z|^2\ge (1-\delta)$ we have -$$(\widetilde Q-h)_t\le (1-\delta)^{-1}(\widetilde Q-h)_{zz}-\widetilde -Q^2+h^2.$$ Thus, for any fixed $t$, at any local maximum for + +$$ +(\widetilde Q-h)_t\le (1-\delta)^{-1}(\widetilde Q-h)_{zz}-\widetilde +Q^2+h^2. +$$ + + Thus, for any fixed $t$, at any local maximum for $(\widetilde Q-h)(\cdot,t)$ at which $(\widetilde Q-h)$ is $\ge 0$ we have $(\widetilde Q-h)_t\le 0$. Since $\widetilde Q-h$ equals $-\infty$ at the end points of $I'$ for all times, there is a -continuous function $f(t)=max_{z\in I'}(\widetilde Q-h)(z,t)$, +continuous function $f(t)=\mathit{max}_{z\in I'}(\widetilde Q-h)(z,t)$, defined for all $t\in (t',t_2]$ approaching $-\infty$ uniformly as $t$ approaches $t'$ from above. By the previous discussion, at any point where $f(t)\ge 0$ we have $f'(t)\le 0$ in the sense of forward @@ -4559,16 +5344,30 @@ difference quotients. It now follows that $f(t)\le 0$ for all $t\in (t',t_2]$. This means that for all $t\in (t',t_2]$ at the $h'$-midpoint $x_0$ of $I'$ (the point where $z=l/2$) we have -$$Q(x_0,t)\le \frac{1}{t-t'}+\frac{16\cdot 4(1-\delta)^{-1}}{l^2}+C_1'.$$ + +$$ +Q(x_0,t)\le \frac{1}{t-t'}+\frac{16\cdot 4(1-\delta)^{-1}}{l^2}+C_1'. +$$ + Since $l\ge (0.8)r$ and since $t-t'\le \delta r^2$, we see that provided $\delta$ is sufficiently small (depending on the bound of the curvature of the ambient flow) we have -$$Q(x_0,t)< \frac{3}{2(t-t')}$$ for all $t\in [t',t_2].$ + +$$ +Q(x_0,t)< \frac{3}{2(t-t')} +$$ + + for all $t\in [t',t_2].$ Of course, since $|Z|^2\ge 1-\delta$ everywhere, this shows that -$$k^2(x_0,t)\le |Y(x_0,t)|^2= -(2-|Z(x_0,t)|^2)Q(x_0,t)< \frac{2}{(t-t')}$$ for all $t\in + +$$ +k^2(x_0,t)\le |Y(x_0,t)|^2= +(2-|Z(x_0,t)|^2)Q(x_0,t)< \frac{2}{(t-t')} +$$ + + for all $t\in [t',t_2]$. Since $x_0$ was an arbitrary point of $c(\cdot,t')$, this shows that $k(x,t)<\sqrt{\frac{2}{t-t'}}$ for all $x\in c(\cdot,t)$ and all $t\in [t',t_2]$. By the definition of $t_2$ this implies that $t_2=t'+\delta r^2$ and completes the - proof of Lemma~. + proof of \entryref{ff5d51a483d9}. diff --git a/projects/poincare-conjecture/.astrolabe/docs/15-canonnbhd.mdx b/projects/poincare-conjecture/.astrolabe/docs-src/20-appendix-canonical-neighborhoods.mdx similarity index 76% rename from projects/poincare-conjecture/.astrolabe/docs/15-canonnbhd.mdx rename to projects/poincare-conjecture/.astrolabe/docs-src/20-appendix-canonical-neighborhoods.mdx index c304532a..9562a6b2 100644 --- a/projects/poincare-conjecture/.astrolabe/docs/15-canonnbhd.mdx +++ b/projects/poincare-conjecture/.astrolabe/docs-src/20-appendix-canonical-neighborhoods.mdx @@ -1,4 +1,6 @@ -# Appendix: Canonical neighborhoods + + +# Chapter 19 -- Appendix: Canonical neighborhoods Recall that an $\epsilon$-neck structure on a Riemannian manifold $(N,g)$ centered at a point $x\in N$ is a diffeomorphism $\psi\colon @@ -14,8 +16,8 @@ followed by the projection to the second factor. ## Shortening curves -**Lemma.** -The following holds for all $\epsilon>0$ sufficiently small. Suppose + +**Lemma.** The following holds for all $\epsilon>0$ sufficiently small. Suppose that $(M,g)$ is a Riemannian manifold and that $N\subset M$ is an $\epsilon$-neck centered at $x$. Let $S(x)$ be the central two-sphere of this neck and suppose that $S(x)$ separates $M$. Let @@ -26,6 +28,7 @@ $s^{-1}(-\epsilon^{-1},-\epsilon^{-1}/2)$ then there is a rectifiable curve from $x$ to $y$ contained in the closure of the positive side of $S(x)$ whose length is at most the length of $\gamma$ minus $\frac{1}{2}\epsilon^{-1}R(x)^{-1/2}$. + **Proof.** Since $\gamma$ contains a point on the negative side of $S(x)$ and it ends on the positive side of $S(x)$, there is a $c\in (0,a)$ such @@ -41,28 +44,33 @@ followed by $\gamma|_{[c,a]}$ is the required shorter path. ## The geometry of an $\epsilon$-neck -**Lemma.** -For any $0<\alpha<1/8$ there is $\epsilon_1=\epsilon_1(\alpha)>0$ such that + +**Lemma.** For any $0<\alpha<1/8$ there is $\epsilon_1=\epsilon_1(\alpha)>0$ such that the following two conditions hold for all $0<\epsilon\le \epsilon_1$. -\begin{enumerate} \item[(1)] If $(N,g)$ is an $\epsilon$-neck centered at $x$ of + +- **(1)** If $(N,g)$ is an $\epsilon$-neck centered at $x$ of scale one (i.e., with $R(x)=1$) then the principal sectional curvatures at any point of $N$ are within $\alpha/6$ of $\{1/2,0,0\}$. In particular, for any $y\in N$ we have -$$(1-\alpha)\le R(y)\le (1+\alpha).$$ -\item[(2)] There is unique two-plane of maximal sectional curvature at + +$$ +(1-\alpha)\le R(y)\le (1+\alpha). +$$ + +- **(2)** There is unique two-plane of maximal sectional curvature at every point of an $\epsilon$-neck, and the angle between the distribution of two-planes of maximal sectional curvature and the two-plane field tangent to the family of two-spheres of the $\epsilon$-neck structure is everywhere less than $\alpha$. -\end{enumerate} + **Proof.** The principal curvatures and their directions are continuous functions of the metric $g$ in the space of metrics with the $C^2$-topology. The statements follow immediately. -**Corollary.** -The following holds for any $\epsilon>0$ sufficiently small. Suppose -that $(N,g)$ is an $\epsilon$-neck and we have and an embedding + +**Corollary.** The following holds for any $\epsilon>0$ sufficiently small. Suppose +that $(N,g)$ is an $\epsilon$-neck and we have and an embedding $f\colon S^2 \to N$ with the property that the restriction of $g$ to the image of this embedding is within $\epsilon$ in the $C^{[1/\epsilon]}$-topology to the round metric $h_0$ of scalar @@ -70,9 +78,10 @@ curvature one on $S^2$ and with the norm of the second fundamental form less than $\epsilon$. Then the two-sphere $f(S^2)$ is isotopic in $N$ to any member of the family of two-spheres coming from the $\epsilon$-neck structure on $N$. + **Proof.** By the previous lemma, if $\epsilon$ is sufficiently small for every -$n\in N$ there is a unique two-plane, $P_n$, at each point on which +$n\in N$ there is a unique two-plane, $P_n$, at each point on which the sectional curvature is maximal. The sectional curvature on this two-plane is close to $1/2$ and the other two eigenvalues of the curvature operator at $n$ are close to zero. Furthermore, $P_n$ @@ -94,20 +103,25 @@ crosses each line $\{x\}\times (-\epsilon^{-1},\epsilon^{-1})$ transversely and in exactly one point. Clearly then, it is isotopic in $N$ to any two-sphere of the form $S^2\times \{s\}$. -**Lemma.** - For any $\alpha>0$ there is $\epsilon_2=\epsilon_2(\alpha)>0$ + +**Lemma.** For any $\alpha>0$ there is $\epsilon_2=\epsilon_2(\alpha)>0$ such that the following hold for all $0<\epsilon\le \epsilon_2$. Suppose that $(N,g)$ is an $\epsilon$-neck centered at $x$ and -$R(x)=1$. Suppose that $\gamma$ is a minimal geodesic in $N$ from +$R(x)=1$. Suppose that $\gamma$ is a minimal geodesic in $N$ from $p$ to $q$. We suppose that $\gamma$ is parameterized by arc length, is of length $\ell>\epsilon^{-1}/100$, and that $s(p) **Proof.** Let us consider a geodesic $\mu$ in the product Riemannian manifold $S^2\times \Ar$ with the metric on $S^2$ being of constant Gaussian @@ -131,41 +145,52 @@ $k\ge 2$. The second statement is obvious since the diameter of any member of the family of two-spheres in the standard metric is $\sqrt{2}\pi$. -**Corollary.** -For any $\alpha>0$ there is $\epsilon_3=\epsilon_3(\alpha)>0$ such -that the following hold for any $0<\epsilon\le \epsilon_3$ and any + +**Corollary.** For any $\alpha>0$ there is $\epsilon_3=\epsilon_3(\alpha)>0$ such +that the following hold for any $0<\epsilon\le \epsilon_3$ and any $\epsilon$-neck $N$ of scale $1$ centered at $x$. -\begin{enumerate} -\item[(1)] Suppose that $p$ and $q$ are points of $N$ + +- **(1)** Suppose that $p$ and $q$ are points of $N$ with either $|s(q)-s(p)|\ge \epsilon^{-1}/100$ or $d(p,q)\ge -\epsilon^{-1}/100$. Then we have -$$(1-\alpha)|s(q)-s(p)|\le d(p,q)\le (1+\alpha)|s(q)-s(p)|.$$ -\item[(2)] -$$B(x,(1-\alpha)\epsilon^{-1})\subset -N\subset B(x,(1+\alpha)\epsilon^{-1}).$$ -\item[(3)] Any geodesic that exits from both ends of $N$ has length at least $2(1-\alpha) -\epsilon^{-1}$.\end{enumerate} - -**Corollary.** -The following holds for all $\epsilon>0$ sufficiently small. Let $N$ +\epsilon^{-1}/100$. Then we have + +$$ +(1-\alpha)|s(q)-s(p)|\le d(p,q)\le (1+\alpha)|s(q)-s(p)|. +$$ + +- **(2)** + +$$ +B(x,(1-\alpha)\epsilon^{-1})\subset +N\subset B(x,(1+\alpha)\epsilon^{-1}). +$$ + +- **(3)** Any geodesic that exits from both ends of $N$ has length at least $2(1-\alpha) +\epsilon^{-1}$. + + + +**Corollary.** The following holds for all $\epsilon>0$ sufficiently small. Let $N$ be an $\epsilon$-neck centered at $x$. If $\gamma$ is a shortest geodesic in $N$ between its endpoints and if $|\gamma|>R(x)^{-1/2}\epsilon^{-1}/100$, then $\gamma$ crosses each two-sphere in the neck structure on $N$ at most once. + There is a closely related lemma. -**Lemma.** -The following holds for every $\epsilon>0$ sufficiently small. + +**Lemma.** The following holds for every $\epsilon>0$ sufficiently small. Suppose that $(M,g)$ is a Riemannian manifold and that $N\subset M$ is an $\epsilon$-neck centered at $x$ and suppose that $\gamma$ is a shortest geodesic in $M$ between its endpoints and that the length of every component of $N\cap|\gamma|$ has length at least $ R(x)^{-1/2}\epsilon^{-1}/8$. Then $\gamma$ crosses each two-sphere -in the neck structure on $N$ at most once; see Fig.~. +in the neck structure on $N$ at most once; see Fig. 19.1. + **Proof.** We parameterize $\gamma$ as a map from $[a,b]\to M$. By -Corollary~, provided that $\epsilon>0$ is sufficiently +\entryref{82f43e137897}, provided that $\epsilon>0$ is sufficiently small, each component of $\gamma\cap N$ crosses each two-sphere of the neck structure at most once. Suppose that there is some two-sphere $S^2\times\{x\}$ that is crossed by two different @@ -175,22 +200,22 @@ $\gamma$ with $S^2\times\{s\}$. There are two cases to consider. Suppose that the two components of $\gamma\cap N$ cross $S^2\times \{x\}$ in opposite directions. In this case, since each component of $\gamma\cap N$ has length at -least $\epsilon^{-1}/8$, then applying Corollary~ we +least $\epsilon^{-1}/8$, then applying \entryref{b74cd360f72f} we can take the two-sphere that they both cross to be $S^2\times \{s\}$ for some $s\in (-(0.9)\epsilon^{-1},(0.9)\epsilon^{-1})$. Applying -Corollary~ again we see that the distance from this -sphere to the complement of $N$ is at least +\entryref{b74cd360f72f} again we see that the distance from this +sphere to the complement of $N$ is at least $R(x)^{-1/2}\epsilon^{-1}/20$. Let $c0$ as close to zero as we want by making $\epsilon$ smaller. Clearly, @@ -198,8 +223,8 @@ then in this case as well, replacing $\gamma([c,d])$ with a path of length less than $2\pi R(x)^{-1/2}$ on $S^2\times\{s\}$ will shorten the length of $\gamma$. -**Corollary.** -The following holds for all $\epsilon>0$ sufficiently small and any + +**Corollary.** The following holds for all $\epsilon>0$ sufficiently small and any $C<\infty$. Let $X$ be an $(C,\epsilon)$-cap in a complete Riemannian manifold $(M,g)$, and let $Y$ be its core and let $S$ be the central two-sphere of the $\epsilon$-neck $N=X-\overline Y$. We @@ -209,19 +234,20 @@ closed negative half of $N$ and let $S$ be the boundary of $\widehat Y$. Suppose that $\gamma$ is a minimal geodesic in $(M,g)$ that contains a point of the core $Y$. Then the intersection of $\gamma$ with $\widehat Y$ is an interval containing an endpoint of $\gamma$; -see Fig.~. +see Fig. 19.1. + **Proof.** If $\gamma$ is completely contained in $\widehat Y$ then the result is clear. Suppose that the path is $\gamma\colon [a,b]\to M$ and $\gamma(d)\in Y$ for some $d\in [a,b]$. Suppose that there are $a'0$ such + +**Lemma.** Given $0<\alpha<1$ there is $\epsilon_4=\epsilon_4(\alpha)>0$ such that the following holds for any $0<\epsilon\le \epsilon_4$. Suppose that $N$ is an $\epsilon$-neck centered at $x$ in a connected manifold $M$ (here we are not assuming that $R(x)=1$). We suppose that the central $2$-sphere of $N$ separates $M$. Let $z$ be a point -outside of the middle two-thirds of $N$ and lying on the negative +outside of the middle two-thirds of $N$ and lying on the negative side of the central $2$-sphere of $N$. (We allow both the case when $z\in N$ and when $z\not\in N$.) Let $p$ be a point in the middle half of $N$. Let $\mu\colon [0,a]\to N$ be a straight line segment (with respect to the standard product metric) in the positive $s$-direction in $N$ beginning at $p$ and ending at a point $q$ of $N$. Then -$$(1-\alpha)(s(q)-s(p))\le d(z,q)-d(z,p)\le (1+\alpha)(s(q)-s(p)).$$ + +$$ +(1-\alpha)(s(q)-s(p))\le d(z,q)-d(z,p)\le (1+\alpha)(s(q)-s(p)). +$$ + **Proof.** This statement is clearly true for the product metric on an infinite cylinder, and hence by continuity, for any given $\alpha$, the @@ -266,8 +296,8 @@ result holds for all $\epsilon>0$ sufficiently small. $N$ separates the ambient manifold $M$. Otherwise, there may be shorter geodesics from $z$ to $q$ entering the other end of $N$. -**Lemma.** -Given any $\alpha>0$ there is $\epsilon(\alpha)>0$ such that the + +**Lemma.** Given any $\alpha>0$ there is $\epsilon(\alpha)>0$ such that the following holds for any $0<\epsilon\le \epsilon(\alpha)$. Suppose that $N$ is an $\epsilon$-neck centered at $x$ in a connected manifold $M$ (here we are not assuming that $R(x)=1$) and that $z$ @@ -279,13 +309,14 @@ topological $2$-sphere contained in the middle quarter of $N$ that maps homeomorphically onto $S^2$ under the projection mapping $N\to S^2$ determined by the $\epsilon$-neck structure. Furthermore, if $p'\in \partial B(z,d)$ then $|s(p)-s(p')|<\alpha -R(x)^{-1/2}\epsilon^{-1}$; see Fig.~. +R(x)^{-1/2}\epsilon^{-1}$; see Fig. 19.2. + **Proof.** The statement is scale-invariant, so we can assume that $R(x)=1$. Denote by $S(z,d)$ the boundary of the metric ball $B(z,d)$. We orient $s$ so that $z$ lies to the negative side of the central two-sphere of $N$. It follows immediately from the previous result -that, provided that $\epsilon>0$ is sufficiently small, $S(z,d)$ +that, provided that $\epsilon>0$ is sufficiently small, $S(z,d)$ intersects any line ${y}\times (-\epsilon^{-1}/3,\epsilon^{-1}/3)$ in at most one point. To complete the proof we need only show that $S(z,d)$ is contained @@ -294,7 +325,7 @@ distance from $d$ to any point in the two-sphere factor of $N$ containing $p$ is contained in the interval $[d-2\pi,d+2\pi]$. Provided that $\epsilon$ is sufficiently small depending on $\alpha$, the result follows immediately from -Lemma~. +\entryref{6cf1cbc39ada}. ## Overlapping $\epsilon$-necks @@ -302,51 +333,74 @@ The subject of this section is the internal geometric properties of $\epsilon$-necks and of intersections of $\epsilon$-necks. We assume that $\epsilon\le 1/200$. -**Proposition.** -Given $0<\alpha\le 10^{-2}$, there is + +**Proposition.** Given $0<\alpha\le 10^{-2}$, there is $\epsilon_5=\epsilon_5(\alpha)>0$ such that the following hold for all $0<\epsilon\le \epsilon_5$. Let $N$ and $N'$ be $\epsilon$-necks centered at $x$ and $x'$, respectively, in a Riemannian manifold $X$: -\begin{enumerate} -\item[(1)] If $N\cap N'\not=\emptyset$ then + +- **(1)** If $N\cap N'\not=\emptyset$ then $1-\alpha + +**Proof.** Fix $0<\epsilon_5(\alpha)\le\mathit{min}(\epsilon_1(\alpha_1),\epsilon_2(\alpha/3),\epsilon_3(\alpha),\alpha/3)$ +sufficiently small so that \entryref{98c6a95f8556} holds. The +first two items are then immediate from \entryref{dc5709757703}. The +third statement is immediate from \entryref{88a5a2233418}, and the +fourth statement from \entryref{98c6a95f8556}. Let us consider the last statement. Let $y\in N\cap N'$ have $-(0.9)\epsilon^{-1}\le s_N(y)\le (0.9)\epsilon^{-1}$. By reversing the $s$-directions of $N$ and/or $N'$ if necessary, we can assume that $0\le s_N(y)\le @@ -360,67 +414,76 @@ a point $y'$ with $(0.96)\epsilon^{-1}=s_{N'}(y')$ According to Item (0.96)\epsilon^{-1}$. There is a similar construction when $s_{N'}(y)<-(0.96)\epsilon^{-1}$. In all cases this allows us to find $y'$ such that both the following hold: -$$-(0.96)\epsilon^{-1}\le s_N(y')\le (0.96)\epsilon^{-1}$$ -$$-(0.96)\epsilon^{-1}\le s_{N'}(y')\le (0.96)\epsilon^{-1}.$$ + +$$ +-(0.96)\epsilon^{-1}\le s_N(y')\le (0.96)\epsilon^{-1} +$$ + +$$ +-(0.96)\epsilon^{-1}\le s_{N'}(y')\le (0.96)\epsilon^{-1}. +$$ Let $y'$ be any point satisfying both these inequalities. According -to Lemma~ and Part (1) of this result, the diameter -of $S(y')$ is at most $2\pi h$, where $h$ is the scale of $N$ and +to \entryref{88a5a2233418} and Part (1) of this result, the diameter +of $S(y')$ is at most $2\pi h$, where $h$ is the scale of $N$ and $N'$. Since $\epsilon^{-1}\ge 200$, it follows from -Corollary~ that $S(y')$ is contained in $N'$. +\entryref{b74cd360f72f} that $S(y')$ is contained in $N'$. Symmetrically $S'(y')$ is contained in $N$. Now consider the intersection of any straight line in the neck structure on $N$ with $N'$. According to Part (3), this intersection is connected. Thus, $N\cap N'$ is a union of open arcs in the $s_N$-directions thought the points of $S(y')$. These arcs can be -used to define a diffeomorphism from $N\cap N'$ to $S^2\times +used to define a diffeomorphism from $N\cap N'$ to $S^2\times (0,1)$ sending $S(y')$ to $S^2\times \{1/2\}$. Also, we have the straight line isotopy from $S'(y')$ to $S(y')$ contained in $N\cap N'$. -## Regions covered by $\epsilon$-necks and -$(C,\epsilon)$-caps +## Regions covered by $\epsilon$-necks and $(C,\epsilon)$-caps Here we fix $0<\epsilon\le 1/200$ sufficiently small so that all the results in the previous two sections hold with $\alpha=10^{-2}$. ### Chains of $\epsilon$-necks + **Definition.** Let $(X,g)$ be a Riemannian manifold. By a *finite chain* of $\epsilon$-necks in $(X,g)$, we mean a sequence $N_a,\ldots,N_{b}$, -of $\epsilon$-necks in $(X,g)$ such that: \begin{enumerate} \item -for all $i,\ a\le i Notice that in an $\epsilon$-chain $N_i\cap N_j=\emptyset$ if $|i-j|\ge 5$. -**Lemma.** -The union $U$ of the $N_i$ in a finite or infinite chain of + +**Lemma.** The union $U$ of the $N_i$ in a finite or infinite chain of $\epsilon$-necks is diffeomorphic to $S^2\times (0,1)$. In particular, it is an $\epsilon$-tube. + **Proof.** Let us first prove the result for finite chains. The proof that $U$ is diffeomorphic to $S^2\times (0,1)$ is by induction on $b-a+1$. If $b=a$, then the result is clear. Suppose that we know the result for chains of smaller cardinality. Then $N_a\cup\cdots\cup N_{b-1}$ is diffeomorphic to $S^2\times (0,1)$. Hence by Part (5) of -Proposition~, $U$ is the union of two manifolds +\entryref{a46d72b87e07}, $U$ is the union of two manifolds each diffeomorphic to $S^2\times (0,1)$ meeting in an open subset diffeomorphic to $S^2\times (0,1)$. Furthermore, by the same result in the intersection there is a two-sphere isotopic to each of the -two-sphere factors from the two pieces. It now follows easily that +two-sphere factors from the two pieces. It now follows easily that the union is diffeomorphic to $S^2\times (0,1)$. Now consider an infinite chain. It is an increasing union of finite chains each diffeomorphic to $S^2\times (0,1)$ and with the two-spheres of one @@ -432,18 +495,20 @@ chain, $U=\cup_{a\le i\le b}N_i$, in $M$ is equal to the frontier of the positive end of $N_b$ union the frontier of the negative end of $N_a$. Thus, we have: -**Corollary.** -Let $\{N_a,\ldots,N_b\}$ be a chain of $\epsilon$-necks. If a + +**Corollary.** Let $\{N_a,\ldots,N_b\}$ be a chain of $\epsilon$-necks. If a connected set $Y$ meets both $U=\cup_{a\le i\le b}N_i$ and its -complement, then $Y$ either contains points of the frontier of the -negative end $N_a$ or of the positive end of $N_b$. +complement, then $Y$ either contains points of the frontier of the +negative end $N_a$ or of the positive end of $N_b$. + The next result shows there is no frontier at an infinite end. -**Lemma.** -Suppose that $\{N_0,\cdots\}$ is an infinite chain of + +**Lemma.** Suppose that $\{N_0,\cdots\}$ is an infinite chain of $\epsilon$-necks in $M$. Then the frontier of $U=\cup_{i=0}^\infty N_i$ is the frontier of the negative end of $N_0$. + **Proof.** Suppose that $x$ is a point of the frontier of $U$. Let $x_i\in U$ be a sequence converging to $x$. If the $x_i$ were contained in a @@ -465,14 +530,15 @@ $j\ge i+5$. The length of such a path is at least the minimum of the width of $N_0$ and the width of $N_{k(j)}$ for some $j$ sufficiently large. But we have just seen that there is a positive lower bound to the scales of the $N_{k(j)}$ independent of $j$, and hence by -Corollary~ there is a positive lower bound, independent +\entryref{b74cd360f72f} there is a positive lower bound, independent of $j$, to the widths of the $N_{k(j)}$. This shows that there is a positive lower bound, independent of $i$, to the distance from $x_i$ to $x$ .This is impossible since $x_i$ converges to $x$. - In fact, there is a geometric version of Lemma~. + In fact, there is a geometric version of \entryref{035b5e6ca144}. -**Lemma.** There is $\epsilon_0>0$ such that the following holds for all + +**Lemma.** There is $\epsilon_0>0$ such that the following holds for all $0<\epsilon\le \epsilon_0$. Suppose that $\{N_j\}_{j\in J}$ is a chain of $\epsilon$-necks in a Riemannian manifold $M$. Let $U=\cup_{j\in J}N_j$. Then there exist an interval $I$ and a smooth @@ -480,8 +546,9 @@ map $p\colon U\to I$ such that every fiber of $p$ is a two-sphere, and if $y$ is in the middle $7/8$'s of $N_j$ then the fiber $p^{-1}(p(y))$ makes a small angle at every point with the family of two-spheres in the $\epsilon$-neck $N_j$. + -**Proof.** Since according to Lemma~ the two-spheres for $N_j$ +**Proof.** Since according to \entryref{dc5709757703} the two-spheres for $N_j$ and $N_{j+1}$ almost line up, it is an easy matter to interpolate between the projection maps to the interval to construct a fibration of $U$ by two-spheres with the given property. The interval $I$ is @@ -492,7 +559,6 @@ A finite or infinite chain $\{N_j\}_{j\in J}$ of $\epsilon$-necks is element of $J$, we have $$ - (0.99)R(x_j)^{-1/2}\epsilon^{-1}\le d(x_j,x_{j+1})\le (1.01)R(x_j)^{-1/2}\epsilon^{-1}, $$ @@ -502,18 +568,19 @@ where, for each $j$, $x_j$ is the central point of $N_j$. Notice that in a balanced chain $N_j\cap N_{j'}=\emptyset$ if $|j-j'|\ge 3$. -**Lemma.** -There exists $\epsilon_0>0$ such that for all $0<\epsilon\le -\epsilon_0$ the following is true. Suppose that $N$ and $N'$ are + +**Lemma.** There exists $\epsilon_0>0$ such that for all $0<\epsilon\le +\epsilon_0$ the following is true. Suppose that $N$ and $N'$ are $\epsilon$-necks centered at $x$ and $x'$, respectively, in a Riemannian manifold $M$. Suppose that $x'$ is not contained in $N$ but is contained in the closure of $N$ in $M$. Suppose also that the two-spheres of the neck structure on $N$ and $N'$ separate $M$. Then, possibly after reversing the $\epsilon$-neck structures on $N$ and/or $N'$, the pair $\{N,N'\}$ forms a balanced chain. + -**Proof.** By Corollary~, -Inequality~() holds for $d(x,x')$. Once we have this +**Proof.** By \entryref{b74cd360f72f}, +Inequality (19.1) holds for $d(x,x')$. Once we have this inequality, it follows immediately from the same corollary that, possible after reversing, the $s$-directions $\{N,N'\}$ makes a balanced chain of $\epsilon$-necks. (It is not possible for the @@ -521,9 +588,9 @@ positive end of $N_b$ to meet $N_a$ for this would allow us to create a loop meeting the central two-sphere of $N_b$ transversely in a single point, so that this two-sphere would not separate $M$.) -**Lemma.** -There exists $\epsilon_0>0$ such that for all $0<\epsilon\le -\epsilon_0$ the following is true. Suppose that + +**Lemma.** There exists $\epsilon_0>0$ such that for all $0<\epsilon\le +\epsilon_0$ the following is true. Suppose that $\{N_a,\ldots,N_b\}$ is a balanced chain in a Riemannian manifold $M$ with $U=\cup_{i=a}^bN_i$. Suppose that the two-spheres of the neck structure of $N_a$ separate $M$. @@ -535,20 +602,21 @@ $\{N_a,\ldots,N_b,N\}$ is a balanced chain. Similarly, if $x$ is in the closure of the minus end of $N_a$, then (again after possibly reversing the direction of $N$) we have that $\{N,N_a,\ldots,N_b\}$ is a balanced $\epsilon$-chain. + **Proof.** The two cases are symmetric; we consider only the first. Since $x$ is contained in the closure of $N_b$, clearly $N_b\cap N\not=\emptyset$. Also, clearly, provided that $\epsilon>0$ is sufficiently small, $d(x_b,x)$ -satisfies Inequality~() so that Lemma~ +satisfies Inequality (19.1) so that \entryref{58804e22eff2} the pair $\{N_b,N\}$ forms an $\epsilon$-chain, and hence a balanced $\epsilon$-chain. It is not possible for $N$ to meet the negative end of $N_a$ since the central two-sphere of $N_a$ separates $M$. Hence $\{N_a,\ldots,N_b,N\}$ is a balanced chain of $\epsilon$-necks. -**Proposition.** -There exists $\epsilon_0>0$ such that for all $0<\epsilon\le + +**Proposition.** There exists $\epsilon_0>0$ such that for all $0<\epsilon\le \epsilon_0$ the following is true. Let $X$ be a connected subset of a Riemannian manifold $M$ with the property that every point $x\in X$ is the center of an $\epsilon$-neck $N(x)$ in $M$. Suppose that @@ -558,29 +626,29 @@ there is a subset $\{x_i\}$ of $X$ such that the necks $N(x_i)$ chain of $\epsilon$-necks $\{N(x_i)\}$ whose union $U$ contains $X$. The union $U$ is diffeomorphic to $S^2\times (0,1)$. It is an $\epsilon$-tube. + -**Proof.** According to Lemma~ for $\epsilon>0$ sufficiently +**Proof.** According to \entryref{6dc87ad116fc} for $\epsilon>0$ sufficiently small the following holds. Suppose that we have a balanced chain of $\epsilon$-necks $N_a\ldots,N_{b}$, with $N_i$ centered at $x_i\in -X$, whose union $U$ does not contain $X$. +X$, whose union $U$ does not contain $X$. Then one of the following holds: -\begin{enumerate} -\item[(1)] It is possible to find an $\epsilon$-neck $N_{b+1}$ centered + +- **(1)** It is possible to find an $\epsilon$-neck $N_{b+1}$ centered at a point of the intersection of $X$ with the closure of the positive end of $N_{b}$ so that $N_a,\ldots,N_{b+1}$ is a balanced $\epsilon$-chain. -\item[(2)] It is possible to find an $\epsilon$-neck $N_{a-1}$ centered at a +- **(2)** It is possible to find an $\epsilon$-neck $N_{a-1}$ centered at a point of the intersection of $X$ with the closure of the negative end of $N_a$ so that $N_{a-1},N_a,\ldots,N_b$ is a balanced $\epsilon$-chain. -\end{enumerate} Now assume that there is no finite balanced chain of $\epsilon$-necks $N(x_i)$ containing $X$. Then we can repeatedly lengthen a balanced chain of $\epsilon$-necks centered at points of $X$ by adding necks at one end or the other. Suppose that we have a half-infinite balanced chain $\{N_0,N_1,\ldots,\}$. By -Lemma~ the frontier of this union is the frontier +\entryref{f6496966d395} the frontier of this union is the frontier of the negative end of $N_0$. Thus, if we can construct a balanced chain which is infinite in both directions, then the union of the necks in this chain is a component of $M$ and hence contains the @@ -588,24 +656,25 @@ connected set $X$. If we can construct a balanced chain that is infinite at one end but not the other that cannot be further extended, then the connected set is disjoint from the frontier of the negative end of the first neck in the chain and, as we have see -above, the `infinite' end of the chain has no frontier. Thus, $X$ is +above, the 'infinite' end of the chain has no frontier. Thus, $X$ is disjoint from the frontier of $U$ in $M$ and hence is contained in $U$. Thus, in all cases we construct a balanced chain of -$\epsilon$-necks containing $X$. By Lemma~ the union +$\epsilon$-necks containing $X$. By \entryref{035b5e6ca144} the union of the necks in this chain is diffeomorphic to $S^2\times (0,1)$ and hence is an $\epsilon$-tube. -**Lemma.** -The following holds for every $\epsilon>0$ sufficiently small. Let + +**Lemma.** The following holds for every $\epsilon>0$ sufficiently small. Let $(M,g)$ be a connected Riemannian manifold. Suppose that every point of $M$ is the center of an $\epsilon$-neck. Then either $M$ is diffeomorphic to $S^2\times (0,1)$ and is an $\epsilon$-tube, or $M$ is diffeomorphic to an $S^2$-fibration over $S^1$. + **Proof.** If the two-spheres of the $\epsilon$-necks do not separate $M$, then it follows from the previous result that $M$ is an $\epsilon$-tube. If one of the two-spheres does separate, then take the universal -covering $\widetilde M$ of $M$. Every point of $\widetilde M$ is +covering $\widetilde M$ of $M$. Every point of $\widetilde M$ is the center of an $\epsilon$-neck (lifting an $\epsilon$-neck in $M$) and the two-spheres of these necks separate $\widetilde M$. Thus the first case applies, showing that $\widetilde M$ is diffeomorphic to @@ -615,31 +684,31 @@ fundamental group. This means that the quotient is fibered by $S^2$'s over $S^1$, and the fibers of this fibration are isotopic to the central two-spheres of the $\epsilon$-necks. -## Subsets of the union of cores of $(C,\epsilon)$-caps -and $\epsilon$-necks. +## Subsets of the union of cores of $(C,\epsilon)$-caps and $\epsilon$-necks. In this section we fix $0<\epsilon\le 1/200$ so that all the results of this section hold with $\alpha=0.01$. -**Proposition.** -For any $C<\infty$ the following holds. Suppose that $X$ is a + +**Proposition.** For any $C<\infty$ the following holds. Suppose that $X$ is a connected subset of a Riemannian three-manifold $(M,g)$. Suppose that every point of $X$ is either the center of an $\epsilon$-neck or is contained in the core of a $(C,\epsilon)$-cap. Then one of the following holds: -\begin{enumerate} -\item[(1)] $X$ is contained in a component of $M$ that is the union of + +- **(1)** $X$ is contained in a component of $M$ that is the union of two $(C,\epsilon)$-caps. This component is diffeomorphic to $S^3$, $\Ar P^3$ or $\Ar P^3\#\Ar P^3$. -\item[(2)] $X$ is contained in a component of $M$ that is a double $C$-capped $\epsilon$-tube. +- **(2)** $X$ is contained in a component of $M$ that is a double $C$-capped $\epsilon$-tube. This component is diffeomorphic to $S^3$, $\Ar P^3$ or $\Ar P^3\#\Ar P^3$. -\item[(3)] $X$ is contained in a single $(C,\epsilon)$-cap. -\item[(4)] $X$ is contained in a $C$-capped $\epsilon$-tube. -\item[(5)] $X$ is contained in an $\epsilon$-tube. -\item[(6)] $X$ is contained in a component of $M$ that is an +- **(3)** $X$ is contained in a single $(C,\epsilon)$-cap. +- **(4)** $X$ is contained in a $C$-capped $\epsilon$-tube. +- **(5)** $X$ is contained in an $\epsilon$-tube. +- **(6)** $X$ is contained in a component of $M$ that is an $\epsilon$-fibration, which itself is a union of $\epsilon$-necks. -\end{enumerate} -(See Fig.~.) + +(See Fig. 19.4.) + **Proof.** We divide the proof into two cases: Case I: There is a point of $X$ contained in the core of a $(C,\epsilon)$-cap. Case II: Every point @@ -647,13 +716,14 @@ of $X$ is the center of an $\epsilon$-neck. **Case I:** We begin the study of this case with a claim. + **Claim.** It is not possible to have an infinite chain of $(C,\epsilon)$-caps $C_0\subset C_1\subset \cdots$ in $M$ with the property that for each $i\ge 1$, the closure of the core of $C_i$ contains a point of the frontier of $C_{i-1}$ + -\begin{proof} -We argue by contradiction. Suppose there is such an infinite chain. +**Proof.** We argue by contradiction. Suppose there is such an infinite chain. Fix a point $x_0\in C_0$ and let $Q_0=R(x_0)$. For each $i\ge 1$ let $x_i$ be a point in the frontier of $C_{i-1}$ that is contained in the closure of the core of $C_i$. For each $i$ let $N_i$ be the @@ -664,7 +734,7 @@ boundary of the core of $C_i$. It is the central two-sphere of an $\epsilon$-neck $N'_i$ in $C_i$. We orient the $s$-direction of $N'_i$ so that the non-compact end of $C_i$ lies off the positive end of $N'_i$. We denote by $h_{i-1}$ the scale of $N_{i-1}$ and by -$h_i'$ the scale of $N'_i$. By Lemma~ the ratio +$h_i'$ the scale of $N'_i$. By \entryref{dc5709757703} the ratio $h_{i-1}/h'_i$ is between $0.99$ and $1.01$. Suppose that $S'_i$ is disjoint from $C_{i-1}$. Then one of the complementary components of $S'_i$ in $M$ contains $C_{i-i}$, and of course, one of the @@ -672,14 +742,14 @@ complementary components of $S'_i$ is the core of $C_i$. These complementary components must be the same, for otherwise $C_{i-1}$ would be disjoint from the core of $C_i$ and hence the intersection of $C_{i-1}$ and $C_i$ would be contained in $N_i$. This cannot -happen since $C_{i-1}$ is contained in $C_i$. Thus, if $S'_i$ is +happen since $C_{i-1}$ is contained in $C_i$. Thus, if $S'_i$ is disjoint from $C_{i-1}$, then the core of $C_i$ contains $C_{i-1}$. This means that the distance from $x_0$ to the complement of $C_i$ is greater than the distance of $x_0$ to the complement of $C_{i-1}$ by an amount equal to the width of $N_i$. Since the scale of $N_i$ is at least $C^{-1/2}R(x_0)^{-1/2}$ (see -(5) of Definition~), it follows from -Corollary~ that this width is at least +(5) of \entryref{757e83881e62}), it follows from +\entryref{b74cd360f72f} that this width is at least $2(0.99)C^{-1/2}R(x_0)^{-1/2}\epsilon^{-1}$. Next suppose that $S'_i$ is contained in $C_{i-1}$. Then one of the @@ -693,16 +763,16 @@ contain $A$, which is a contradiction since $C_i$ contains $C_{i-1}$ and $A\subset C_{i-1}$. Lastly, we suppose that $S'_i$ is neither contained in $C_{i-1}$ nor -in its complement. Then $S'_i$ must meet $N_{i-1}$. According to -Proposition~ the $s$-directions in $N_{i-1}$ and +in its complement. Then $S'_i$ must meet $N_{i-1}$. According to +\entryref{a46d72b87e07} the $s$-directions in $N_{i-1}$ and $N'_i$ either almost agree or are almost opposite. Let $x\in S'_i\cap \partial N_{i-1}$ so that $s_{N'_i}(x)=0$. Move from $x$ along the $s_{N'_i}$-direction that moves into $N_{i-1}$ to a point $x'$ with $|s_{N_i}(x')|=(0.05)\epsilon^{-1}$. According to -Proposition~ +\entryref{a46d72b87e07} $(0.94)\epsilon^{-1} +**Claim.** If $C'$ is a $(C,\epsilon)$-cap whose core contains a point of the frontier of $\widetilde C$, then $\widetilde C\cup C'$ is a component of $M$ containing $X$. + **Proof.** We suppose that $\widetilde C$ is the union of $C_0$ and a balanced chain $N_0,\ldots,N_k$ of $\epsilon$-necks. We orient this chain so @@ -806,18 +875,18 @@ that $C_0$ lies off the negative end of each of the $N_i$. Let $S'$ be the boundary of the core of $C'$ and let $N'$ be an $\epsilon$-neck contained in $C'$ whose central two-sphere is $S'$. We orient the direction $s_{N'}$ so that the positive direction -points away from the core of $C'$. The first step in proving this +points away from the core of $C'$. The first step in proving this claim is to establish the following. -**Claim.** -Suppose that there is a two-sphere $\Sigma\subset N'$ contained in + +**Claim.** Suppose that there is a two-sphere $\Sigma\subset N'$ contained in the closure of the positive half of $N'$ and also contained in $\widetilde C$. Suppose that $\Sigma$ is isotopic in $N'$ to the central two-sphere $S'$ of $N'$. Then $\widetilde C\cup C'$ is a component of $M$, a component containing $X$. + -\begin{proof} -$\Sigma$ separates $\widetilde C$ into two components: $A$, which +**Proof.** $\Sigma$ separates $\widetilde C$ into two components: $A$, which has compact closure in $\widetilde C$, and $B$, containing the end of $\widetilde C$. The two-sphere $\Sigma$ also divides $C'$ into two components. Since $\Sigma$ is isotopic in $N'$ to $S'$, the @@ -832,13 +901,13 @@ opposite sides of their common frontier. This means that $\overline A\cup \overline A'$ is a component of $M$. Clearly, this component is also equal to $\widetilde C\cup C'$. Since $X$ is connected and this component contains a point $x_0$ of $X$, it contains $X$. This -completes the proof of Claim~. +completes the proof of \entryref{1b6c3ac94c37}. -Now we return to the proof of Claim~. We consider +Now we return to the proof of \entryref{327067f8535b}. We consider three cases. **First Subcase: $S'\subset \widetilde C$.** In this case -we apply Claim~ to see that $\widetilde C\cup C'$ is a +we apply \entryref{1b6c3ac94c37} to see that $\widetilde C\cup C'$ is a component of $M$ containing $X$. **Second Subcase: $S'$ is disjoint from @@ -853,27 +922,27 @@ $\widetilde C$ both have $S'$ as their frontier and they lie on opposite sides of $S'$. Since the closure of the core of $C'$ contains a point of the frontier of $\widetilde C$, it must be the case that $S'$ also contains a point of this frontier. By -Proposition~, the neck $N'\subset C'$ meets +\entryref{a46d72b87e07}, the neck $N'\subset C'$ meets $N_{k}$ and there is a two-sphere $\Sigma\subset N'\cap N_{k}$ isotopic in $N'$ to $S'$ and isotopic in $N_{k}$ to the central two-sphere of $N_{k}$. Because $N_{k}\subset \widetilde C$ and $\widetilde C$ is disjoint from the core of $C'$, we see that $\Sigma$ is contained in the positive half of $N'$. Applying -Claim~ we see that $\widetilde C\cup C'$ is a component +\entryref{1b6c3ac94c37} we see that $\widetilde C\cup C'$ is a component of $M$ containing $X$. **Third Subcase: $S'\cap \widetilde C\not=\emptyset$ and $S'\not\subset \widetilde C$.** Clearly, in this case $S'$ contains a point of the frontier of $\widetilde C$ in $M$, i.e., a point of the frontier of the positive end of $N_k$ in $M$. Since $N_k\cap -N'\not=\emptyset$, by Lemma~ the scales of $N_k$ and +N'\not=\emptyset$, by \entryref{dc5709757703} the scales of $N_k$ and $N'$ are within $1\pm 0.01$ of each other, and hence the diameter of $S'$ is at most $2\pi$ times the scale of $N_k$. Since the central two-sphere $S'$ of $N'$ contains a point in the frontier of the -positive end of $N_k$, it follows from Lemma~ that $S'$ -is contained on the positive side of the central two-sphere of +positive end of $N_k$, it follows from \entryref{b74cd360f72f} that $S'$ +is contained on the positive side of the central two-sphere of $N_k$ and that the frontier of the positive end of $N_k$ is -contained in $N'$. By Proposition~ there is a +contained in $N'$. By \entryref{a46d72b87e07} there is a two-sphere $\Sigma$ in the neck structure for $N'$ that is contained in $N_k$ and is isotopic in $N_k$ to the central two-sphere from that neck structure. Let $A$ be the complementary component of @@ -897,13 +966,12 @@ core of $C'$. This then contradicts our choice of $C_0$. This completes the analysis of all the cases and hence completes the -proof of Claim~. -\end{proof} +proof of \entryref{327067f8535b}. The last thing to do in this case in order to prove the proposition in Case I is to show that $\widetilde C\cup C'$ is diffeomorphic to $S^3$, $\Ar P^3$, or $\Ar P^3\#\Ar P^3$. The reason for this is that -$\widetilde C$ is diffeomorphic to $C_0$; hence $\widetilde C$ +$\widetilde C$ is diffeomorphic to $C_0$; hence $\widetilde C$ either is diffeomorphic to an open three-ball or to a punctured $\Ar P^3$. Thus, the frontier of $C'$ in $\widetilde C$ is a two-sphere that bounds either a compact three-ball or the complement of an open @@ -912,7 +980,7 @@ to a three-ball or to a punctured $\Ar P^3$, the result follows. **Case II:** Suppose that every point of $X$ is the center of an $\epsilon$-neck. Then if the two-spheres of these necks -separate $M$, it follows from Proposition~ that $X$ +separate $M$, it follows from \entryref{ba681e15e1a6} that $X$ is contained in an $\epsilon$-tube in $M$. It remains to consider the case when the two-spheres of these necks @@ -923,7 +991,7 @@ terminates after a finite number of steps in a finite $\epsilon$-chain whose union contains $X$, or it can be continued infinitely often creating an infinite $\epsilon$ chain containing $X$ or at some finite stage (possibly after reversing the indexing -and the $s$-directions of the necks) we have a balanced +and the $s$-directions of the necks) we have a balanced $\epsilon$-chain $N_a\cup\cdots\cup N_{b-1}$ and a point of the intersection of $X$ with the frontier of the positive end of $N_{b-1}$ that is the center of an $\epsilon$-neck $N_b$ with the @@ -931,7 +999,7 @@ property that $N_b$ meets the negative end of $N_a$. Intuitively, the chain wraps around on itself like a snake eating its tail. If the intersection of $N_a\cap N_b$ contains a point $x$ with $s_{N_a}(x)\ge -(0.9)\epsilon^{-1}$, then according to -Proposition~ the intersection of $N_a\cap N_b$ +\entryref{a46d72b87e07} the intersection of $N_a\cap N_b$ is diffeomorphic to $S^2\times (0,1)$ and the two-sphere in this product structure is isotopic in $N_a$ to the central two-sphere of $N_a$ and is isotopic in $N_b$ to the central two-sphere of $N_b$. @@ -939,7 +1007,7 @@ In this case it is clear that $N_a\cup \cdots \cup N_b$ is a component of $M$ that is an $\epsilon$-fibration. We examine the possibility that the intersection $N_a\cap N_b$ -contains some points in the negative end of $N_a$ but is contained +contains some points in the negative end of $N_a$ but is contained in $s_{N_a}^{-1}((-\epsilon^{-1},-(0.9)\epsilon^{-1}))$. Set $A= s_{N_a}^{-1}((-\epsilon^{-1},-(0.8)\epsilon^{-1}))$. Notice that since $X$ is connected and $X$ contains a point in the frontier of @@ -947,10 +1015,10 @@ the positive end of $N_a$ (since we have added at least one neck at this end), it follows that $X$ contains points in $s_{N_a}^{-1}(s)$ for all $s\in [0,\epsilon^{-1})$. If there are -no points of $X$ in $A$, then we replace $N_a$ by an +no points of $X$ in $A$, then we replace $N_a$ by an $\epsilon$-neck $N_a'$ centered at a point of $s_{N_a}^{-1}\left((0.15)\epsilon^{-1}\right)\cap X$. Clearly, by -Lemma~ $N'_a$ contains +\entryref{b74cd360f72f} $N'_a$ contains $s_{N_a}^{-1}(-(0.8)\epsilon{-1},\epsilon^{-1})$ and is disjoint from $s_{N_a}^{-1}((-\epsilon^{-1},-(0.9)\epsilon^{-1})$, so that $N_a',N_{a+1}, \ldots, N_b$ is a chain of $\epsilon$-necks @@ -964,26 +1032,25 @@ point in the universal cover of $M_0$ is the center of an $\epsilon$-neck that is disjoint from all its non-trivial covering translates. Hence, the quotient $M_0$ is an $\epsilon$-fibration. -We have now completed the proof of Proposition~. -\end{proof} +We have now completed the proof of \entryref{0e9a1a167caa}. As an immediate corollary we have: -**Proposition.** -For all $\epsilon>0$ sufficiently small the following holds. Suppose + +**Proposition.** For all $\epsilon>0$ sufficiently small the following holds. Suppose that $(M,g)$ is a connected Riemannian manifold such that every point is either contained in the core of a $(C,\epsilon)$-cap in $M$ or is the center of an $\epsilon$-neck in $M$. Then one of the following holds: -\begin{enumerate} -\item[(1)] $M$ is diffeomorphic to $S^3$, $\Ar P^3$ or $\Ar P^3\#\Ar + +- **(1)** $M$ is diffeomorphic to $S^3$, $\Ar P^3$ or $\Ar P^3\#\Ar P^3$, and $M$ is either a double $C$-capped $\epsilon$-tube or is the union of two $(C,\epsilon)$-caps. -\item[(2)] $M$ is diffeomorphic to $\Ar^3$ or $\Ar P^3\setminus \{point\}$, +- **(2)** $M$ is diffeomorphic to $\Ar^3$ or $\Ar P^3\setminus \{\mathit{point}\}$, and $M$ is either a $(C,\epsilon$-cap or a $C$-capped $\epsilon$-tube. -\item[(3)] $M$ is diffeomorphic to $S^2\times \Ar$ and is an +- **(3)** $M$ is diffeomorphic to $S^2\times \Ar$ and is an $\epsilon$-tube. -\item[(4)] $M$ is diffeomorphic to an $S^2$-bundle over $S^1$ and is an $\epsilon$-fibration. -\end{enumerate} +- **(4)** $M$ is diffeomorphic to an $S^2$-bundle over $S^1$ and is an $\epsilon$-fibration. + diff --git a/projects/poincare-conjecture/.astrolabe/docs/01-introduction.mdx b/projects/poincare-conjecture/.astrolabe/docs/01-introduction.mdx new file mode 100644 index 00000000..9383bfec --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/docs/01-introduction.mdx @@ -0,0 +1,1639 @@ + + +# Chapter 0 -- Introduction + + In this book we present a complete +and detailed proof of the + +> **Poincaré Conjecture: +> a closed, smooth, simply connected $3$-manifold is +> diffeomorphic to $S^3$.** (Every topological $3$-manifold admits a +> differentiable structure and every homeomorphism between smooth +> $3$-manifolds can be approximated by a diffeomorphism. Thus, +> classification results about topological $3$-manifolds up to +> homeomorphism and about smooth $3$-manifolds up to diffeomorphism +> are equivalent. In this book 'manifold' means 'smooth manifold.') + +This conjecture was formulated by Henri Poincaré [Poincare] +in 1904 and has remained open until the recent work of Perelman. The +arguments we give here are a detailed version of those that appear +in Perelman's three preprints [Perelman1, Perelman2, Perelman3]. +Perelman's arguments rest on a foundation built by Richard Hamilton +with his study of the Ricci flow equation for Riemannian metrics. +Indeed, Hamilton believed that Ricci flow could be used to establish +the Poincaré Conjecture and more general topological +classification results in dimension $3$, and laid out a program to +accomplish this. The difficulty was to deal with singularities in +the Ricci flow. Perelman's breakthrough was to understand the +qualitative nature of the singularities sufficiently to allow him to +prove the Poincaré Conjecture (and \entryref{9999efa2be2c} below +which implies the Poincaré Conjecture). For +a detailed history of the Poincaré Conjecture, see Milnor's survey +article [Milnorsurvey]. + +i̱gskip +A class of examples closely related to the $3$-sphere are *the +$3$-dimensional spherical space-forms*, i.e., +the quotients of $S^3$ by free, linear actions of finite subgroups of the +orthogonal group $O(4)$. There is a generalization of the Poincaré +Conjecture, called the **$3$-dimensional spherical space-form +conjecture**, which conjectures +that any closed $3$-manifold with finite fundamental group is diffeomorphic to +a $3$-dimensional spherical space-form. Clearly, a special case of the +$3$-dimensional spherical space-form conjecture is the Poincaré Conjecture. + + As indicated in Remark 1.4 of [Perelman3], the arguments we present + here not only prove the Poincaré Conjecture, they prove the $3$-dimensional space-form conjecture. + In fact, the purpose of this book is to prove the following + more general theorem. + +\entryblock{9999efa2be2c} + +This immediately implies an affirmative resolution of the Poincaré +Conjecture and of the $3$-dimensional spherical space-form +conjecture. + +\entryblock{d08303132397} + +Before launching into a more detailed description of the contents of +this book, one remark on the style of the exposition is in order. +Because of the importance and visibility of the results discussed +here, and because of the number of incorrect claims of proofs of +these results in the past, we felt that it behooved us to work out +and present the arguments in great detail. Our goal was to make the +arguments clear and convincing and also to make them more easily +accessible to a wider audience. As a result, experts may find some +of the points are overly elaborated. + +## Overview of Perelman's argument + +In dimensions less than or equal to three, any Riemannian metric of constant +Ricci curvature has constant sectional curvature. Classical results in +Riemannian geometry show that the universal cover of a closed manifold of +constant positive curvature is diffeomorphic to the sphere and that the +fundamental group is identified with a finite subgroup of the orthogonal group +acting linearly and freely on the universal cover. Thus, one can approach the +Poincaré Conjecture and the more general $3$-dimensional spherical space-form +problem by asking the following question. Making the appropriate fundamental +group assumptions on $3$-manifold $M$, how does one establish the existence of +a metric of constant Ricci curvature on $M$? The essential ingredient in +producing such a metric is the Ricci flow equation +introduced by Richard Hamilton in [Hamilton3MPRC]: + +$$ +\frac{\partial g(t)}{\partial t}=-2\mathit{Ric}(g(t)), +$$ + +where $\mathit{Ric}(g(t))$ is the Ricci curvature of the metric $g(t)$. +The fixed points (up to rescaling) of this equation are the +Riemannian metrics of constant Ricci curvature. For a general +introduction to the subject of the Ricci flow see Hamilton's survey +paper [Hamiltonsurvey], the book by Chow-Knopf +[ChowKnopf], or the book by Chow, Lu, and Ni [ChowLuNi]. +The Ricci flow equation is a (weakly) parabolic partial differential +flow equation for Riemannian metrics on a smooth manifold. Following +Hamilton, one defines a Ricci flow to be a family +of Riemannian metrics $g(t)$ on a fixed smooth manifold, +parameterized by $t$ in some interval, satisfying this equation. One +considers $t$ as time and studies the equation as an initial value +problem: Beginning with any Riemannian manifold $(M,g_0)$ find a +Ricci flow with $(M,g_0)$ as initial metric; that is to say find a +one-parameter family $(M,g(t))$ of Riemannian manifolds with +$g(0)=g_0$ satisfying the Ricci flow equation. This equation is +valid in all dimensions but we concentrate here on dimension three. +In a sentence, the method of proof is to begin with any Riemannian +metric on the given smooth $3$-manifold and flow it using the Ricci +flow equation to obtain the constant curvature metric for which one +is searching. There are two examples where things work in exactly +this way, both due to Hamilton. (i) If the initial metric has +positive Ricci curvature, Hamilton proved over twenty years ago, +[Hamilton3MPRC], that under the Ricci flow the manifold shrinks +to a point in finite time, that is to say, there is a finite-time +singularity, and, as we approach the singular time, the diameter of +the manifold tends to zero and the curvature blows up at every +point. Hamilton went on to show that, in this case, rescaling by a +time-dependent function so that the diameter is constant produces a +one-parameter family of metrics converging smoothly to a metric of +constant positive curvature. (ii) At the other extreme, in +[HamiltonNSRF3M] Hamilton showed that if the Ricci flow exists +for all time and if there is an appropriate curvature bound together +with another geometric bound, then as $t\rightarrow\infty$, after +rescaling to have a fixed diameter, the metric converges to a metric +of constant negative curvature. + +The results in the general case are much more complicated to +formulate and much more difficult to establish. While Hamilton +established that the Ricci flow equation has short-term existence +properties, i.e., one can define $g(t)$ for $t$ in some interval +$[0,T)$ where $T$ depends on the initial metric, it turns out that +if the topology of the manifold is sufficiently complicated, say it +is a non-trivial connected sum, then no matter what the initial +metric is one must encounter finite-time singularities, forced by +the topology. More seriously, even if the manifold has simple +topology, beginning with an arbitrary metric one expects to (and +cannot rule out the possibility that one will) encounter finite-time +singularities in the Ricci flow. These singularities, unlike in the +case of positive Ricci curvature, occur along proper subsets of the +manifold, not the entire manifold. + Thus, to derive the topological +consequences stated above, it is not sufficient in general to stop +the analysis the first time a singularity arises in the Ricci flow. +One is led to study a more general evolution process + called *Ricci flow with surgery*, + first introduced by Hamilton in the context + of four-manifolds, [Hamilton4MPIC]. + This evolution process + is still parameterized by an interval in time, so that for each $t$ + in the interval of definition there is a compact Riemannian $3$-manifold + $M_t$. But there is a discrete set of +times at which the manifolds and metrics undergo topological and +metric discontinuities (surgeries). In each of the complementary +intervals to the singular times, the evolution is the usual Ricci +flow, though, because of the surgeries, the topological type of the +manifold $M_t$ changes as $t$ moves from one complementary interval +to the next. From an analytic point of view, the surgeries at the +discontinuity times are introduced in order to 'cut away' a +neighborhood of the singularities as they develop and insert by +hand, in place of the 'cut away' regions, geometrically nice +regions. This allows one to continue the Ricci flow (or more +precisely, restart the Ricci flow with the new metric constructed at +the discontinuity time). Of course, the surgery process also changes +the topology. To be able to say anything useful topologically about +such a process, one needs results about Ricci flow, and one also +needs to control both the topology and the geometry of the surgery +process at the singular times. For example, it is crucial for the +topological applications that we do surgery along $2$-spheres rather +than surfaces of higher genus. Surgery along $2$-spheres produces +the connected sum decomposition, which is well-understood +topologically, while, for example, Dehn surgeries along tori can +completely destroy the topology, changing any $3$-manifold into any +other. + +The change in topology turns out to be completely understandable and +amazingly, the surgery processes produce exactly the topological +operations needed to cut the manifold into pieces on which the Ricci +flow can produce the metrics sufficiently controlled so that the +topology can be recognized. + +The bulk of this book (Chapters 1-17 and the Appendix) concerns the +establishment of the following long-time existence result for Ricci +flow with surgery. + +\entryblock{847af3ecd4a8} + +While \entryref{847af3ecd4a8} is central for all applications of Ricci +flow to the topology of three-dimensional manifolds, the argument +for the $3$-manifolds described in \entryref{9999efa2be2c} is +simplified, and avoids all references to the nature of the flow as +time goes to infinity, because of the following finite-time +extinction result. + +\entryblock{127db1dc9315} + +This result is established in Chapter 18 following the argument +given by Perelman in [Perelman3], see also +[ColdingMinicozzi]. + +We immediately deduce \entryref{9999efa2be2c} from +Theorems \entryref{847af3ecd4a8} and \entryref{127db1dc9315} as follows: Let $M$ be a +$3$-manifold satisfying the hypothesis of \entryref{9999efa2be2c}. +Then there is a finite sequence $M=M_0,M_1,\ldots,M_k=\emptyset$ +such that for each $i,\ 1\le i\le k$, $M_i$ is obtained from +$M_{i-1}$ by a connected sum decomposition or $M_i$ is obtained +from $M_{i-1}$ by removing a component diffeomorphic to one of +$S^2\times S^1$, $\Ar P^3\#\Ar P^3$, a non-orientable $2$-sphere +bundle over $S^1$, or a $3$-dimensional spherical space-form. +Clearly, it follows by downward induction on $i$ that each +connected component of $M_i$ is diffeomorphic to a connected sum of +$3$-dimensional spherical space-forms, copies of $S^2\times S^1$, +and copies of the non-orientable $2$-sphere bundle over $S^1$. In +particular, $M=M_0$ has this form. Since $M$ is connected by +hypothesis, this proves the theorem. In fact, this argument proves +the following: + +\entryblock{7078952cb9e4} + +**Proof.** Repeated application of \entryref{847af3ecd4a8} shows that (1) implies +(2). The implication (2) implies (3) is immediate from van Kampen's +theorem. The fact that (3) implies (1) is \entryref{127db1dc9315}. +This shows that (1), (2) and (3) are all equivalent. Since +three-dimensional spherical space-forms and $S^2$-bundles over $S^1$ +are easily seen to be prime, (2) implies (4). Thus, it remains +only to see that (4) implies (3). We consider a manifold $M$ +satisfying (4), a prime factor $P$ of $M$, and universal covering +$\widetilde P$ of $P$. First suppose that $\pi_2(P)=\pi_2(\widetilde +P)$ is trivial. Then, by hypothesis $\pi_3(P)=\pi_3(\widetilde P)$ +is non-trivial. By the Hurewicz theorem this means that +$H_3(\widetilde P)$ is non-trivial, and hence that $\widetilde P$ is +a compact, simply connected three-manifold. It follows that +$\pi_1(P)$ is finite. Now suppose that $\pi_2(P)$ is non-trivial. +Then $P$ is not diffeomorphic to $\Ar P^3$. Since $P$ is prime and +contains no locally separating $\Ar P^2$, it follows that $P$ +contains no embedded $\Ar P^2$. Then by the sphere theorem there is +an embedded two-sphere in $P$ that is homotopically non-trivial. +Since $P$ is prime, this sphere cannot separate, so cutting $P$ open +along it produces a connected manifold $P_0$ with two boundary +two-spheres. Since $P_0$ is prime, it follows that $P_0$ is +diffeomorphic to $S^2\times I$ and hence $P$ is diffeomorphic to a +two-sphere bundle over the circle. + +\entryblock{48f3d2b5dea3} + +One can use Ricci flow in a more general study of three-manifolds +than the one we carry out here. There is a conjecture due to +Thurston, see [Thurstongeom], known as Thurston's +Geometrization Conjecture or simply as the Geometrization Conjecture for +three-manifolds. It conjectures that every $3$-manifold without +locally separating $\Ar P^2$'s (in particular every orientable +$3$-manifold) is a connected sum of prime $3$-manifolds each of +which admits a decomposition along incompressible (I.e., +embedded by a map that is injective on $\pi_1$.) tori into pieces +that admit locally homogeneous geometries of finite volume. Modulo +questions about cofinite-volume lattices in $SL_2(\Cee)$, proving +this conjecture leads to a complete classification + of $3$-manifolds without locally separating $\Ar P^2$'s, +and in particular to a complete classification of all orientable +$3$-manifolds. (See Peter Scott's survey article [Scott].) By +passing to the orientation double cover and working equivariantly, +these results can be extended to all $3$-manifolds. + + Perelman in +[Perelman2] has stated results which imply a positive +resolution of Thurston's Geometrization conjecture. Perelman's +proposed proof of Thurston's Geometrization Conjecture relies in an +essential way on \entryref{847af3ecd4a8}, namely the existence of +Ricci flow with surgery for all positive time. But it also involves +a further analysis of the limits of these Ricci flows as time goes +to infinity. This further analysis involves analytic arguments which +are exposed in Sections 6 and 7 of Perelman's second paper +([Perelman2]), following earlier work of Hamilton +([HamiltonNSRF3M]) in a simpler case of bounded curvature. They +also involve a result (Theorem 7.4 from [Perelman2]) from the +theory of manifolds with curvature locally bounded below that are +collapsed, related to results of Shioya-Yamaguchi [SY]. The +Shioya-Yamaguchi results in turn rely on an earlier, unpublished +work of Perelman proving the so-called 'Stability Theorem.' +Recently, Kapovich, [Kap] has put a preprint on the archive +giving a proof of the stability result. We have been examining +another approach, one suggested by Perelman in [Perelman2], +avoiding the stability theorem, cf, [KleinerLott2] and +[MorganTian2]. It is our view that the collapsing results +needed for the Geometrization Conjecture are in place, but that +before a definitive statement that the Geometrization Conjecture has +been resolved can be made these arguments must be subjected to the +same close scrutiny that the arguments proving the Poincaré +Conjecture have received. This process is underway. + +In this book we do not attempt to explicate any of the results +beyond \entryref{847af3ecd4a8} described in the previous paragraph +that are needed for the Geometrization Conjecture. Rather, we +content ourselves with presenting a proof of \entryref{9999efa2be2c} +above which, as we have indicated, concerns initial Riemannian +manifolds for which the Ricci flow with surgery becomes extinct +after finite time. We are currently preparing a detailed proof, +along the lines suggested by Perelman, of the further results that +will complete the proof of the Geometrization Conjecture. + +As should be clear from the above overview, Perelman's argument did +not arise in a vacuum. Firstly, it resides in a context provided by +the general theory of Riemannian manifolds. In particular, various +notions of convergence of sequences of manifolds play a crucial +role. The most important is geometric convergence (smooth +convergence on compact subsets). Even more importantly, Perelman's +argument resides in the context of the theory of the Ricci flow +equation, introduced by Richard Hamilton and extensively studied by +him and others. Perelman makes use of almost every previously +established result for $3$-dimensional Ricci flows. One exception is +Hamilton's proposed classification results for three-dimensional +singularities. These are replaced by Perelman's strong qualitative +description of singularity development for Ricci flows on compact +three-manifolds. + +The first five chapters of the book review the necessary background +material from these two subjects. Chapters 6 through 11 then explain +Perelman's advances. In Chapter 12 we introduce the standard +solution, which is the manifold +constructed by hand that one 'glues in' in doing surgery. Chapters +13 through 17 describe in great detail the surgery process and prove +the main analytic and topological estimates that are needed to show +that one can continue the process for all positive time. At the end +of Chapter 17 we have established \entryref{847af3ecd4a8}. Chapter 18 +discusses the finite-time extinction result. Chapter 19 is an +appendix on some topological results that were needed in the surgery +analysis in Chapters 13-17. + +## Background material from Riemannian geometry + +### Volume and injectivity radius + +One important general concept that is used throughout is the notion +of a manifold being non-collapsed at a point. Suppose that we have a +point $x$ in a complete Riemannian $n$-manifold. Then we say that +the manifold is *$\kappa$-non-collapsed* at $x$ +provided that the following holds: For any $r$ such that the norm of +the Riemannian curvature tensor, $|\mathit{Rm}|$, is $\le r^{-2}$ at +all points of the metric ball, $B(x,r)$, of radius $r$ centered at +$x$, we have $\mathit{Vol} B(x,r)\ge \kappa r^n$. There is a +relationship between this notion and the injectivity +radius of $M$ at $x$. Namely, if $|\mathit{Rm}|\le r^{-2}$ on $B(x,r)$ and if $B(x,r)$ is +$\kappa$-non-collapsed then the injectivity radius of $M$ at $x$ is +greater than or equal to a positive constant that depends only on +$r$ and $\kappa$. The advantage of working with the volume +non-collapsing condition is that, unlike for the injectivity radius, +there is a simple equation for the evolution of volume under Ricci +flow. + +Another important general result is the Bishop-Gromov volume +comparison result that says that if the +Ricci curvature of a complete Riemannian $n$-manifold $M$ is bounded +below by a constant $(n-1)K$ then for any $x\in M$ the ratio of the +volume of $B(x,r)$ to the volume of the ball of radius $r$ in the +space of constant curvature $K$ is a non-increasing function whose +limit as $r\rightarrow 0$ is $1$. + +All of these basic facts from Riemannian geometry are reviewed in +the first chapter. + +### Manifolds of non-negative curvature + +For reasons that should be clear from the above description and in +any event will become much clearer shortly, manifolds of +non-negative curvature play an extremely important role in the +analysis of Ricci flows with surgery. We need several general +results about them. The first is the soul theorem for manifolds of +non-negative sectional curvature. A *soul* is a compact, totally +geodesic submanifold. The entire manifold is diffeomorphic to the +total space of a vector bundle over any of its souls. If a +non-compact $n$-manifold has positive sectional curvature, then any +soul for it is a point, and in particular, the manifold is +diffeomorphic to Euclidean space. In addition, the distance function +$f$ from a soul has the property that for every $t>0$ the pre-image +$f^{-1}(t)$ is homeomorphic to an $(n-1)$-sphere and the pre-image +under this distance function of any non-degenerate interval +$I\subset \Ar ^+$ is homeomorphic to $S^{n-1}\times I$. + +Another important result is the splitting theorem, which says that, if a complete manifold of non-negative +sectional curvature has a geodesic line (an isometric copy of $\Ar$) +that is distance minimizing between every pair of its points, then +that manifold is a metric product of a manifold of one lower +dimension and $\Ar$. In particular, if a complete $n$-manifold of +non-negative sectional curvature has two ends then it is a metric +product $N^{n-1}\times \Ar$ where $N^{n-1}$ is a compact manifold. + +Also, we need some of the elementary comparison results from Toponogov +theory. These compare ordinary triangles in the +Euclidean plane with triangles in a manifold of non-negative sectional +curvature whose sides are minimizing geodesics in that manifold. + +### Canonical neighborhoods + +Much of the analysis of the geometry of Ricci flows revolves around +the notion of canonical neighborhoods. +Fix some $\epsilon>0$ sufficiently small. There are two types of +non-compact canonical neighborhoods: $\epsilon$-necks +and $\epsilon$-caps. An +$\epsilon$-neck in a Riemannian $3$-manifold $(M,g)$ centered at a +point $x\in M$ is a submanifold $N\subset M$ and a diffeomorphism +$\psi\colon S^2\times (-\epsilon^{-1},\epsilon^{-1})\to N$ such that +$x\in \psi(S^2\times \{0\})$ and such that the pullback of the +rescaled metric, $\psi^*(R(x)g)$, is within $\epsilon$ in the +$C^{[1/\epsilon]}$-topology of the product of the round metric of +scalar curvature $1$ on $S^2$ with the usual metric on the interval +$(-\epsilon^{-1},\epsilon^{-1})$. (Throughout, $R(x)$ denotes the +scalar curvature of $(M,g)$ at the point $x$.) An $\epsilon$-cap is +a non-compact submanifold ${\mathcal C}\subset M$ with the property +that a neighborhood $N$ of infinity in ${\mathcal C}$ is an +$\epsilon$-neck, such that every point of $N$ is the center of an +$\epsilon$-neck in $M$, and such that the *core*, ${\mathcal C}\setminus +\overline N$, of the $\epsilon$-cap is diffeomorphic to either a +$3$-ball or a punctured $\Ar P^3$. It will also be important to +consider $\epsilon$-caps that, after rescaling to make $R(x)=1$ for +some point $x$ in the cap, have bounded geometry (bounded diameter, +bounded ratio of the curvatures at any two points, and bounded +volume). If $C$ represents the bound for these quantities, then we +call the cap an $(C,\epsilon)$-cap. See +Fig. 0.1. + An +$\epsilon$-tube in $M$ is a submanifold of +$M$ diffeomorphic to $S^2\times (0,1)$ which is a union of +$\epsilon$-necks and with the property that each point of the +$\epsilon$-tube is the center of an $\epsilon$-neck in $M$. + +There are two other types of canonical neighborhoods in +$3$-manifolds -- (i) a $C$-component and (ii) +an $\epsilon$-round component. The +$C$-component is a compact, connected Riemannian manifold of +positive sectional curvature diffeomorphic to either $S^3$ or $\Ar +P^3$ with the property that rescaling the metric by $R(x)$ for any +$x$ in the component produces a Riemannian manifold whose diameter + is at most $C$, whose sectional curvature at any point and in any +$2$-plane direction is between $C^{-1}$ and $C$, and whose volume is +between $C^{-1}$ and $C$. An $\epsilon$-round component is a +component on which the metric rescaled by $R(x)$ for any $x$ in the +component is within $\epsilon$ in the $C^{[1/\epsilon]}$-topology of +a round metric of scalar curvature one. + +As we shall see, the singularities at time $T$ of a $3$-dimensional +Ricci flow are contained in subsets that are unions of canonical +neighborhoods with respect to the metrics at nearby, earlier times +$t'0$ sufficiently small, if a connected manifold is a union +of $\epsilon$-tubes and $\epsilon$-caps then it is diffeomorphic to +$\Ar^3$, $S^2\times \Ar$, $S^3$, $S^2\times S^1$, $\Ar P^3\#\Ar +P^3$, the total space of a line bundle over $\Ar P^2$, or the +non-orientable $2$-sphere bundle over $S^1$. This topological result +is proved in the appendix at the end of the book. **We shall fix +$\epsilon>0$ sufficiently small so that these results hold.** + +There is one result relating canonical neighborhoods and manifolds +of positive curvature of which we make repeated use: Any complete +$3$-manifold of positive curvature does not admit $\epsilon$-necks +of arbitrarily high curvature. In particular, if $M$ is a complete +Riemannian $3$-manifold with the property that every point of scalar +curvature greater than $r_0^{-2}$ has a canonical neighborhood, then +$M$ has bounded curvature. This turns out to be of central +importance and is used repeatedly. + +All of these basic facts about Riemannian manifolds of non-negative +curvature are recalled in the second chapter. + +## Background material from Ricci flow + +Hamilton [Hamilton3MPRC] introduced the Ricci flow +equation, + +$$ +\frac{\partial g(t)}{\partial t}=-2\mathit{Ric}(g(t)). +$$ + +This is an evolution equation for a one-parameter family of Riemannian metrics +$g(t)$ on a smooth manifold $M$. The Ricci flow equation is weakly parabolic +and is strictly parabolic modulo the 'gauge group', which is the group of +diffeomorphisms of the underlying smooth manifold. One should view this +equation as a non-linear, tensor version of the heat equation. From it, one can +derive the evolution equation for the Riemannian metric tensor, the Ricci +tensor, and the scalar curvature function. These are all parabolic equations. +For example, the evolution equation for scalar curvature $R(x,t)$ is + +$$ +\frac{\partial R}{\partial t}(x,t)=\triangle +R(x,t)+2|\mathit{Ric}(x,t)|^2, +$$ + + illustrating the +similarity with the heat equation. (Here $\triangle$ is the +Laplacian with non-positive spectrum.) + +### First results + +Of course, the first results we need are uniqueness and short-time existence +for solutions to the Ricci flow equation for compact manifolds. These results +were proved by Hamilton ([Hamilton3MPRC]) using the Nash-Moser inverse +function theorem, + ([HamiltonIFTNM]). These results are standard for +strictly parabolic equations. By now there is a fairly standard method for +working 'modulo' the gauge group (the group of diffeomorphisms) and hence +arriving at a strictly parabolic situation where the classical existence, +uniqueness and smoothness results apply. The method for the Ricci flow equation +goes under the name of 'DeTurck's trick.' + +There is also a result that allows us to patch together local solutions +$(U,g(t)),\ a\le t\le b$, and $(U,h(t)),\ b\le t\le c$, to form a smooth +solution defined on the interval $a\le t\le c$ provided that $g(b)=h(b)$. + +Given a Ricci flow $(M,g(t))$ we can always translate, replacing $t$ +by $t+t_0$ for some fixed $t_0$, to produce a new Ricci flow. We can +also rescale by any positive constant $Q$ by setting +$h(t)=Qg(Q^{-1}t)$ to produce a new Ricci flow. + +### Gradient shrinking solitons + +Suppose that $(M,g)$ is a complete Riemannian manifold, and suppose +that there is a constant $\lambda>0$ with the property that + +$$ +\mathit{Ric}(g)=\lambda g. +$$ + +In this case, it is easy to see that there is a Ricci flow given by + +$$ +g(t)=(1-2\lambda t)g. +$$ + + In particular, all the metrics in this flow +differ by a constant factor depending on time and the metric is a +decreasing function of time. These are called *shrinking +solitons*. Examples are compact +manifolds of constant positive Ricci curvature. + +There is a closely related, but more general, class of examples: the +*gradient shrinking solitons*. Suppose that $(M,g)$ is + a complete Riemannian manifold, and suppose that there is a constant +$\lambda>0$ and a function $f\colon M\to \Ar$ satisfying + +$$ +\mathit{Ric}(g)=\lambda g-\mathit{Hess}^{g}f. +$$ + + In this case, there is a Ricci flow which is a shrinking + family after we pull back by the one-parameter family of diffeomorphisms generated by + the time-dependent vector field $\frac{1}{1-2\lambda t}\nabla_{g} f$. + An example of a gradient shrinking soliton is the manifold + $S^2\times \Ar$ with the family of metrics being the product of the + shrinking family of round metrics on $S^2$ and the constant + family of standard metrics on $\Ar$. The function $f$ is $s^2/4$ where $s$ is the + Euclidean parameter on $\Ar$. + +### Controlling higher derivatives of curvature + +Now let us discuss the smoothness results for geometric limits. The +general result along these lines is Shi's theorem, see [Shi1, Shi2]. Again, this is a standard type of +result for parabolic equations. Of course, the situation here is +complicated somewhat by the existence of the gauge group. Roughly, +Shi's theorem says the following. Let us denote by $B(x,t_0,r)$ the +metric ball in $(M,g(t_0))$ centered at $x$ and of radius $r$. If we +can control the norm of the Riemannian curvature tensor on a +backward neighborhood of the form $B(x,t_0,r)\times [0,t_0]$, then +for each $k>0$ we can control the $k^{th}$ covariant derivative of +the curvature on $B(x,t_0,r/2^k)\times [0,t_0]$ by a constant over +$t^{k/2}$. This result has many important consequences in our study +because it tells us that geometric limits are smooth limits. Maybe +the first result to highlight is the fact (established earlier by +Hamilton) that if $(M,g(t))$ is a Ricci flow defined on $0\le +t0$ the Riemannian manifold $(\widetilde M,g(t))$ splits as a +Riemannian product of a surface of positive curvature and a +Euclidean line. Furthermore, the flow on the cover $\widetilde M$ is +the product of a $2$-dimensional flow and the trivial +one-dimensional Ricci flow on the line; see Sections 8 and 9 of +[Hamilton4MPCO]. +- **(5)** In particular, there is no Ricci flow of non-negative curvature tensor +$(U,g(t)),$ defined for $\ 0\le t\le T$ with $T>0$, such that +$(U,g(T))$ is isometric to an open subset in a non-flat, +$3$-dimensional metric cone. + +### Geometric limits + +In the fifth chapter we discuss geometric +limits of Riemannian manifolds and of Ricci flows. Let us review the +history of these ideas. The first results about geometric limits of +Riemannian manifolds go back to Cheeger in his thesis in 1967; see +[Cheeger]. Here Cheeger obtained topological results. In +[Gromov] Gromov proposed that geometric limits should exist in +the Lipschitz topology and suggested a result along these lines, +which also was known to Cheeger. + In +[GreeneWu], Greene-Wu gave a rigorous proof of the compactness +theorem suggested by Gromov and also enhanced the convergence to be +$C^{1,\alpha}$-convergence by using harmonic coordinates; see also +[SPeters]. Assuming that all the derivatives of curvature are +bounded, one can apply elliptic theory to the expression of +curvature in harmonic coordinates and deduce $C^\infty$-convergence. +These ideas lead to various types of compactness results that go +under the name Cheeger-Gromov compactness for Riemannian manifolds. +Hamilton in [Hamiltonlimits] extended these results to Ricci +flows. We shall use the compactness results for both Riemannian +manifolds and for Ricci flows. In a different direction, geometric +limits were extended to the non-smooth context by Gromov in +[Gromov] where he introduced a weaker topology, called the +Gromov-Hausdorff topology and proved a compactness theorem. + + Recall that a +sequence of based Riemannian manifolds $(M_n,g_n,x_n)$ is said to +*converge geometrically* to a +based, complete Riemannian manifold $(M_\infty,g_\infty,x_\infty)$ +if there is a sequence of open subsets $U_n\subset M_\infty$ with +compact closures, with $x_\infty\in U_1\subset \overline U_1\subset +U_2\subset \overline U_2\subset U_3\subset \cdots $ with +$\cup_nU_n=M_\infty$, and embeddings $\varphi_n\colon U_n\to M_n$ +sending $x_\infty$ to $x_n$ so that the pull back metrics, +$\varphi_n^*g_n$, converge uniformly on compact subsets of +$M_\infty$ in the $C^\infty$-topology to $g_\infty$. Notice that the +topological type of the limit can be different from the topological +type of the manifolds in the sequence. There is a similar notion of +geometric convergence for a sequence of based Ricci flows. + +Certainly, one of the most important consequences of Shi's results, +cited above, is that, in concert with Cheeger-Gromov compactness, +it allows us to form smooth geometric limits of sequences of based +Ricci flows. We have the following result of Hamilton's; see +[Hamiltonlimits]: + +\entryblock{56d435df5c6d} + +To emphasize, the two conditions that we must check in order to +extract a geometric limit of a subsequence based at points at time +zero are: (i) uniform non-collapsing at the base point in the time +zero metric, and (ii) for each $A<\infty$ uniformly bounded +curvature for the restriction of the flow to the metric balls of +radius $A$ centered at the base points. + +Most steps in Perelman's argument require invoking this result in order to form +limits of appropriate sequences of Ricci flows, often rescaled to make the +scalar curvatures at the base point equal to $1$. If, before rescaling, the +scalar curvature at the base points goes to infinity as we move through the +sequence, then the resulting limit of the rescaled flows has non-negative +sectional curvature. This is a consequence of the fact that the sectional +curvatures of the manifolds in the sequence are uniformly pinched toward +positive. It is for exactly this reason that non-negative curvature plays such +an important role in the study of singularity development in three-dimensional +Ricci flows. + +## Perelman's advances + +So far we have been discussing the results that were known before +Perelman's work. They concern almost exclusively Ricci flow (though +Hamilton in [Hamilton4MPIC] had introduced the notion of +surgery and proved that surgery can be performed preserving the +condition that the curvature is pinched toward positive, as in (2) +above). Perelman extended in two essential ways the analysis of +Ricci flow -- one involves the introduction of a new analytic +functional, *the reduced length*, which is the tool by which he +establishes the needed non-collapsing results, and the other is a +delicate combination of geometric limit ideas and consequences of +the maximum principle together with the non-collapsing results in +order to establish bounded curvature at bounded distance results. +These are used to prove in an inductive way the existence of +canonical neighborhoods, which is a crucial ingredient in proving +that is possible to do surgery iteratively, creating a flow defined +for all positive time. + +While it is easiest to formulate and consider these techniques in +the case of Ricci flow, in the end one needs them in the more +general context of Ricci flow with surgery since we inductively +repeat the surgery process, and in order to know at each step that +we can perform surgery we need to apply these results to the +previously constructed Ricci flow with surgery. We have chosen to +present these new ideas only once -- in the context of generalized +Ricci flows -- so that we can derive the needed consequences in all +the relevant contexts from this one source. + +### The reduced length function + +In Chapter 6 we come to +the first of Perelman's major contributions. Let us first describe +it in the context of an ordinary three-dimensional Ricci flow, but +viewing the Ricci flow as a horizontal metric on a space-time which +is the manifold $M\times I$, where $I$ is the interval of definition +of the flow. Suppose that $I=[0,T)$ and fix $(x,t)\in M\times +(0,T)$. We consider paths $\gamma(\tau),\ 0\le \tau\le +\overline\tau$, in space-time with the property that for every +$\tau\le \overline\tau$ we have $\gamma(\tau)\in M\times \{t-\tau\}$ +and $\gamma(0)=x$. These paths are said to be *parameterized by +backward time*. See Fig. 0.2. The ${\mathcal +L}$-*length* of such a path is given by + +$$ +{\mathcal L}(\gamma)=\int_0^{\overline +\tau}\sqrt{\tau}\left(R(\gamma(\tau))+|\gamma'(\tau)|^2\right)d\tau, +$$ + +where the derivative on $\gamma$ refers to the spatial derivative. +There is also the closely related *reduced length* + +$$ +\ell(\gamma)=\frac{{\mathcal L}(\gamma)}{2\sqrt{\overline\tau}}. +$$ + +There is a theory for the functional ${\mathcal L}$ analogous to the +theory for the usual energy function (Even though this +functional is called a length, the presence of the +$|\gamma'(\tau)|^2$ in the integrand means that it behaves more like +the usual energy functional for paths in a Riemannian manifold.). + In +particular, there is the notion of an ${\mathcal +L}$-geodesic, and the reduced length +as a function on space-time $\ell_{(x,t)}\colon M\times [0,t)\to +\Ar$. One establishes a crucial monotonicity for this reduced length along ${\mathcal +L}$-geodesics. Then one defines the reduced volume + +$$ +\widetilde +V_{(x,t)}(U\times\{\bar t\})=\int_{U\times\{\bar +t\}}\bar\tau^{-3/2}e^{-\ell_{(x,t)}(q,\bar\tau)}d\mathit{vol}_{g(\bar\tau}(q), +$$ + + where, as before $\bar\tau=t-\bar t$. +Because of the monotonicity of $\ell_{(x,t)}$ along ${\mathcal +L}$-geodesics, the reduced volume is also +non-increasing under the flow (forward in $\bar\tau$ and hence +backward in time) of open subsets along ${\mathcal L}$-geodesics. +This is the fundamental tool which is used to establish +non-collapsing results which in turn are essential in proving the +existence of geometric limits. + +The definitions and the analysis of the reduced length function and +the reduced volume as well as the monotonicity results are valid in +the context of the generalized Ricci flow. The only twist to be +aware of is that in the more general context one cannot always +extend ${\mathcal L}$-geodesics; they may run 'off the edge' of +space-time. Thus, the reduced length function and reduced volume +cannot be defined globally, but only on appropriate open subsets of +a time-slice (those reachable by minimizing ${\mathcal +L}$-geodesics). But as long as one can flow an open set $U$ of a +time-slice along minimizing ${\mathcal L}$-geodesics in the +direction of decreasing $\bar\tau$, the reduced volumes of the +resulting family of open sets form a monotone non-increasing +function of $\bar\tau$. This turns out to be sufficient to extend +the non-collapsing results to Ricci flow with surgery provided that +we are careful in how we choose the parameters that go into the +definition of the surgery process. + +### Application to non-collapsing results + +As we indicated in the previous paragraph, one of the main +applications of the reduced length function is to prove +non-collapsing results for three-dimensional Ricci flows with +surgery. In order to make this argument work, one takes a weaker +notion of $\kappa$-non-collapsed by +making a stronger curvature bound assumption: one considers + points $(x,t)$ and constants $r$ with the property that $|\mathit{Rm}|\le r^{-2}$ on +$P(x,t,r,-r^2)=B(x,t,r)\times (t-r^2,t]$. The +$\kappa$-non-collapsing condition applies to these balls and says +that $\mathit{Vol}(B(x,t,r))\ge \kappa r^3.$ The basic idea in proving +non-collapsing is to use the fact that as we flow forward in time +via minimizing ${\mathcal L}$-geodesics the reduced volume is a +non-decreasing function. Hence, a lower bound of the reduced volume +of an open set at an earlier time implies the same lower bound for +the corresponding open subset at a later time. This is contrasted +with direct computations (related to the heat kernel in $\Ar^3$) +that say if the manifold is highly collapsed near $(x,t)$ (i.e., +satisfies the curvature bound above but is not +$\kappa$-non-collapsed for some small $\kappa$) then the reduced +volume $\widetilde V_{(x,t)}$ is small at times close to $t$. Thus, +to show that the manifold is non-collapsed at $(x,t)$ we need only +find an open subset at an earlier time that is reachable by +minimizing ${\mathcal L}$-geodesics and that has a reduced volume +bounded away from zero. + +One case where it is easy to do this is when we have a Ricci flow of +compact manifolds or of complete manifolds of non-negative +curvature. Hence, these manifolds are non-collapsed at all points +with a non-collapsing constant that depends only on the geometry of +the initial metric of the Ricci flow. Non-collapsing results are +crucial and are used repeatedly in dealing with Ricci flows with +surgery in Chapters 10 -- 17, for these give one of the two +conditions required in order to take geometric limits. + +### Application to ancient $\kappa$-non-collapsed solutions + +There is another important application of the length function, which +is to the study of non-collapsed, ancient solutions in dimension three. In the case that the generalized Ricci +flow is an ordinary Ricci flow either on a compact manifold or on a +complete manifold (with bounded curvatures) one can say much more +about the reduced length function and the reduced volume. Fix a +point $(x_0,t_0)$ in space-time. First of all, one shows that every +point $(x,t)$ with $t0$ is sufficiently small so that all the results from the +appendix about manifolds covered by $\epsilon$-necks and $\epsilon$-caps hold, +the above results about gradient shrinking solitons lead to a rough qualitative +description of all $\kappa$-solutions. There are those which do not have +strictly positive curvature. These are gradient shrinking solitons, either an +evolving family of round $2$-spheres times $\Ar$ or the quotient of this family +by an involution. Non-compact $\kappa$-solutions of strictly positive curvature +are diffeomorphic to $\Ar^3$ and are the union of an $\epsilon$-tube and a core +of a $(C_1,\epsilon)$-cap. The compact ones of strictly positive curvature are +of two types. The first type are positive, constant curvature shrinking +solitons. Solutions of the second type are diffeomorphic to either $S^3$ or +$\Ar P^3$. Each time-slice of a $\kappa$-solution of the second type is either +of uniformly bounded geometry (curvature, diameter, and volume) when rescaled +so that the scalar curvature at a point is one, or admits an $\epsilon$-tube +whose complement is either a disjoint union of the cores of two +$(C_1,\epsilon)$-caps. + +This gives a rough qualitative understanding of $\kappa$-solutions. Either +they are round, or they are finitely covered by the product of a round surface +and a line, or they are a union of $\epsilon$-tubes and cores of +$(C_1,\epsilon)$-caps , or they are diffeomorphic to $S^3$ or $\Ar P^3$ and +have bounded geometry (again after rescaling so that there is a point of scalar +curvature $1$). This is the source of canonical neighborhoods for Ricci flows: + the point is that this qualitative result remains true for any point $x$ in a +Ricci flow that has an appropriate size neighborhood within +$\epsilon$ in the $C^{[1/\epsilon]}$-topology of a neighborhood in a +$\kappa$-solution. For example, if we have a sequence of based +generalized flows $({\mathcal M}_n,G_n,x_n)$ converging to a based +$\kappa$-solution, then for all $n$ sufficiently large $x$ will have +a canonical neighborhood, one that is either an $\epsilon$-neck +centered at that point, a $(C_1,\epsilon)$-cap whose core contains +the point, a $C_1$-component, or an $\epsilon$-round component. + +### Bounded curvature at bounded distance + +Perelman's other major breakthrough is his result establishing +bounded curvature at bounded distance for blow-up limits of +generalized Ricci flows. As we have alluded to several times, many +steps in the argument require taking (smooth) geometric limits of a +sequence of based generalized flows about points of curvature +tending to infinity. To study such a sequence we rescale each term +in the sequence so that its curvature at the base point becomes one. +Nevertheless, in taking such limits we face the problem that even +though the curvature at the point we are focusing on (the points we +take as base points) was originally large and has been rescaled to +be one, there may be other points in the same time-slice of much +larger curvature, which, even after the rescalings, can tend to +infinity. + If +these points are at uniformly bounded (rescaled) distance from the +base points, then they would preclude the existence of a smooth +geometric limit of the based, rescaled flows. In his arguments, +Hamilton avoided this problem by always focusing on points of +maximal curvature (or almost maximal curvature). That method will +not work in this case. The way to deal with this possible problem is +to show that a generalized Ricci flow satisfying appropriate +conditions satisfies the following. For each $A<\infty$ there are +constants $Q_0=Q_0(A)<\infty$ and $Q(A)<\infty$ such that any point +$x$ in such a generalized flow for which the scalar curvature +$R(x)\ge Q_0$ and for any $y$ in the same time-slice as $x$ with +$d(x,y)0$ sufficiently small and let $C=\mathit{max}(C_1,C_2)<\infty$, where $C_1$ is the constant associated to +$\epsilon$ for $\kappa$-solutions and $C_2$ is the constant +associated to $\epsilon$ for the standard solution. We say that a +point $x$ in a generalized Ricci flow has a $(C,\epsilon)$-canonical +neighborhood if one of the following holds: + +1. $x$ is contained in a connected component of a +time-slice that is a $C$-component. +1. $x$ is contained in a connected component of its time-slice +that is within $\epsilon$ of round in the +$C^{[1/\epsilon]}$-topology. +1. $x$ is contained in the core of a $(C,\epsilon)$-cap. +1. $x$ is the center of a strong $\epsilon$-neck. + +We shall study Ricci flows with surgery defined for $0\le +t0$ so that the associated generalized Ricci flow is +$\kappa$-non-collapsed on scales at most $\epsilon$, in the sense +that we require only that balls of radius $r\le \epsilon$ be +$\kappa$-non-collapsed. +1. There is $r_0>0$ such that any point of space-time at which +the scalar curvature is $\ge r_0^{-2}$ has an +$(C,\epsilon)$-canonical neighborhood. + +The main result is that, having a Ricci flow with surgery defined on +some time interval satisfying these conditions, it is possible to +extend it to a longer time interval in such a way that it still +satisfies the same conditions, possibly allowing the constants +$\kappa$ and $r_0$ defining these conditions to get closer to zero, +but keeping them bounded away from $0$ on each compact time +interval. We repeat this construction inductively, and, since it is +easy to see that on any compact time interval there can only be a +bounded number of surgeries. In the end, we create a Ricci flow with +surgery defined for all positive time. As far as we know, it may be +the case that in the entire flow defined all the way to infinity +there are infinitely many surgeries. + +### Surgery + +Let us describe how we extend a Ricci flow with surgery satisfying +all the conditions listed above and becoming singular at time +$T<\infty$. Fix $T^-0$ there +is $h=h(\delta,\rho,r_0)$ such that for any $\epsilon$-horn +${\mathcal H}$ whose boundary lies in $\Omega_\rho$ and for any +$x\in{\mathcal H}$ with $R(x)\ge h^{-2}$, the point $x$ is the +center of a strong $\delta$-neck. + +Now we are ready to describe the surgery procedure. It depends on +our choice of standard solution on $\Ar ^3$ and on a choice of +$\delta>0$ sufficiently small. For each $\epsilon$-horn in $\Omega$ +whose boundary is contained in $\Omega_\rho$ fix a point of +curvature $(h(\delta, \rho,r_0))^{-2}$ and fix a strong +$\delta$-neck centered at this point. Then we cut the +$\epsilon$-horn open along the central $2$-sphere $S$ of this neck +and remove the end of the $\epsilon$-horn that is cut off by $S$. +Then we glue in a ball of a fixed radius around the tip from the +standard solution, after scaling the metric on this ball by +$(h(\delta,\rho,r_0))^2$. To glue these two metrics together we must +use a partition of unity near the $2$-spheres that are matched. +There is also a delicate point that we first bend in the metrics +slightly so as to achieve positive curvature near where we are +gluing. This is an idea due to Hamilton, and it is needed in order +to show that the condition of curvature pinching toward positive is +preserved. In addition, we remove all components of $\Omega$ that do +not contain any points of $\Omega_\rho$. + +This operation produces a new compact $3$-manifold. One continues +the Ricci flow with surgery by letting this Riemannian manifold at +time $T$ evolve under the Ricci flow. See Fig. 0.6. + +### Topological effect of surgery + +Looking at the +situation just before the surgery time, we see a finite number of +disjoint submanifolds, each diffeomorphic to either $S^2\times I$ or +the $3$-ball, where the curvature is large. In addition there may be +entire components of where the scalar curvature is large. The effect +of $2$-sphere surgery is to do a finite number of ordinary +topological surgeries along $2$-spheres in the $S^2\times I$. This +simply effects a partial connected-sum decomposition and may +introduce new components diffeomorphic to $S^3$. We also remove +entire components, but these are covered by $\epsilon$-necks and +$\epsilon$-caps so that they have standard topology (each one is +diffeomorphic to $S^3$, $\Ar P^3$, $\Ar P^3\# \Ar P^3$, $S^2\times +S^1$, or the non-orientable $2$-sphere bundle over $S^1$). Also, we +remove $C$-components and $\epsilon$-round components (each of these +is either diffeomorphic to $S^3$ or $\Ar P^3$ or admits a metric of +constant positive curvature). Thus, the topological effect of +surgery is to do a finite number of ordinary $2$-sphere topological +surgeries and to remove a finite number of topologically standard +components. + +## Extending Ricci flows with surgery + +Now we come to the crux of the argument. We must show that if we +have a Ricci flow with surgery defined for some time $0\le +t0$ sufficiently small. +There is also the issue of whether the surgery times can accumulate. + +Of course, the initial metric does not change as we extend surgery +so that the condition that the normalized initial metric is clearly +preserved as we extend surgery. As we have already remarked, +Hamilton had proved earlier that one can do surgery in such a way as +to preserve the condition that the curvature is pinched toward +positive. The other two conditions require more work, and, as we +indicated above, the constants may decay to zero as we extend the +Ricci flow with surgery. + +If we have all the conditions for the Ricci flow with surgery up to time $T$, +then the analysis of the open subset on which the curvature remains bounded +holds, and given $\delta>0$ sufficiently small, we do surgery on the central +$S^2$ of a strong $\delta$-neck in each $\epsilon$-horn meeting $\Omega_\rho$. +In addition we remove entirely all components that do not contain points of +$\Omega_\rho$. We then glue in the cap from the standard solution. This gives +us a new compact $3$-manifold and we restart the flow from this manifold. + +The $\kappa$-non-collapsed result is extended to the new part of the +Ricci flow with surgery using the fact that it holds at times +previous to $T$. To establish this extension one uses ${\mathcal +L}$-geodesics in the associated generalized Ricci flow and reduced +volume as indicated before. In order to get this argument to work, +one must require $\delta>0$ to be sufficiently small; how small is +determined by $r_0$. + +The other thing that we must establish is the existence of canonical +neighborhoods for all points of sufficiently large scalar curvature. +Here the argument is by contradiction. We consider all Ricci flows +with surgery that satisfy all four conditions on $[0,T)$ and we +suppose that we can find a sequence of such containing points +(automatically at times $T'>T$) of arbitrarily large curvature where +there are not canonical neighborhoods. In fact, we take the points +at the first such time violating this condition. We base our flows +at these points. Now we consider rescaled versions of the +generalized flows so that the curvature at these base points is +rescaled to one. We are in a position to apply the bounded curvature +and bounded distance results to this sequence, and of course the +$\kappa$-non-collapsing results which have already been established. +There are two possibilities. The first is that the rescaled sequence +converges to an ancient solution. This ancient solution has +non-negative curvature by the pinching hypothesis. General results +about three-manifolds of non-negative curvature imply that it also +has bounded curvature. It is $\kappa$-non-collapsed. Thus, in this +case the limit is a $\kappa$-solution. This produces the required +canonical neighborhoods for the base points of the tail of the +sequence modeled on the canonical neighborhoods of points in a +$\kappa$-solution. This contradicts the assumption that none of +these points has a canonical neighborhood. + +The other possibility is that one can take a partial smooth limit +but that this limit does not extend all the way back to $-\infty$. +The only way this can happen is if there are surgery caps that +prevent extending the limit back to $-\infty$. This means that the +base points in our sequence are all within a fixed distance and time +(after the rescaling) of a surgery region. But in this case results +from the nature of the standard solution show that if we have taken +$\delta>0$ sufficiently small, then the base points have canonical +neighborhoods modeled on the canonical neighborhoods in the standard +solution, again contradicting our assumption that none of the base +points has a canonical neighborhood. In order to show that our base +points have neighborhoods near those of the standard solution, one +appeals to a geometric limit argument as $\delta\rightarrow 0$. This +argument uses the uniqueness of the Ricci flow for the standard +solution. (Actually, Bruce Kleiner pointed out to us that one only +needs a compactness result for the space of all Ricci flows with the +standard metric as initial metric, not uniqueness, and the +compactness result can be proved by the same arguments that prove +the compactness of the space of $\kappa$-solutions.) + +Interestingly enough, in order to establish the uniqueness of the +Ricci flow for the standard solution, as well as to prove that this +flow is defined for time $[0,1)$ and to prove that at infinity it is +asymptotic to an evolving cylinder requires the same results -- +non-collapsing and the bounded curvature at bounded distance that we +invoked above. For this reason, we order the material described here +as follows. First, we introduce generalized Ricci flows, and then +introduce the length function in this context and establish the +basic monotonicity results. Then we have a chapter on stronger +results for the length function in the case of complete manifolds +with bounded curvature. At this point we are in a position to prove +the needed results about the Ricci flow from the standard solution. +Then we are ready to define the surgery process and prove the +inductive non-collapsing results and the existence of canonical +neighborhoods. + +In this way, one establishes the existence of canonical neighborhoods. Hence, +one can continue to do surgery, producing a Ricci flow with surgery defined for +all positive time. Since these arguments are inductive, it turns out that the +constants in the non-collapsing and in the canonical neighborhood statements +decay in a predetermined rate as time goes to infinity. + +Lastly, there is the issue of ruling out the possibility that the surgery times +accumulate. The idea here is very simple: Under Ricci flow during an elapsed +time $T$, volume increases at most by a multiplicative factor which is a fixed +exponential of the time $T$. Under each surgery there is a removal of at least +a fixed positive amount of volume depending on the surgery scale $h$, which in +turns depends on $\delta$ and $r_0$. Since both $\delta$ and $r_0$ are bounded +away from zero on each finite interval, there can be at most finitely many +surgeries in each finite interval. Notice that this argument allows for the +possibility that in the entire flow all the way to infinity there are +infinitely many surgeries. It is still unknown whether that possibility ever +happens. + +This completes our outline of the proof of \entryref{847af3ecd4a8}. + +## Finite-time extinction + +The last topic we discuss is the proof of the finite-time +extinction for Ricci flows with +initial metrics satisfying the hypothesis of +\entryref{127db1dc9315}. + +As we present it, the finite extinction result has two steps. The +first step is to show that there is $T<\infty$ (depending on the +initial metric) such that for all $t\ge T$, all connected components +of the $t$-time-slice $M_t$ have trivial $\pi_2$. First, an easy +topological argument shows that only finitely many of the $2$-sphere +surgeries in a Ricci flow with surgery can be along homotopically +non-trivial $2$-spheres. Thus, after some time $T_0$ all $2$-sphere +surgeries are along homotopically trivial $2$-spheres. Such a +surgery does not affect $\pi_2$. Thus, after time $T_0$, the only +way that $\pi_2$ can change is by removal of components with +non-trivial $\pi_2$. (An examination of the topological types of +components that are removed shows that there are only two types of +such components with non-trivial $\pi_2$: $2$-sphere bundles over +$S^1$ and $\Ar P^3\#\Ar P^3$.) We suppose that at every $t\ge T_0$ +there is a component of $M_t$ with non-trivial $\pi_2$. Then we can +find a connected open subset ${\mathcal X}$ of $\mathbf{t}^{-1}([T_0,\infty))$ with the property that for each $t\ge T_0$ +the intersection ${\mathcal X}(t)= {\mathcal X}\cap M_t$ is a +component of $M_t$ with non-trivial $\pi_2$. We define a function +$W_2\colon [T_0,\infty)\to \Ar$ associated with such an ${\mathcal +X}$. The value $W_2(t)$ is the minimal area of all homotopically +non-trivial $2$-spheres mapping into ${\mathcal X}(t)$. This minimal +area $W_2(t)$ is realized by a harmonic map of $S^2$ into ${\mathcal +X}(t)$. The function $W_2$ varies continuously under Ricci flow and +at a surgery is lower semi-continuous. Furthermore, using an idea +that goes back to Hamilton (who applied it to minimal disks) one +shows that the forward difference quotient of the minimal area +satisfies + +$$ +\frac{dW_2(t)}{dt}\le -4\pi +\frac{3}{(4t+1)}W_2(t). +$$ + +(Here, the explicit form of the bound for the forward difference +quotient depends on the way we have chosen to normalize initial +metric and also on Hamilton's curvature pinching result.) + +But any function $W_2(t)$ with these properties and defined for all +$t>T_0$, becomes negative at some finite $T_1$ (depending on the +initial value). This is absurd since $W_2(t)$ is the minimum of +positive quantities. This contradiction shows that such a path of +components with non-trivial $\pi_2$ cannot exist for all $t\ge T_0$. +In fact, it even gives a computable upper bound on how long such a +component ${\mathcal X}$, with every time-slice having non-trivial +$\pi_2$, can exist in terms of the minimal area of a homotopically +non-trivial $2$-sphere mapping into ${\mathcal X}(T_0)$. It follows +that there is $T<\infty$ with the property that every component of +$M_T$ has trivial $\pi_2$. This condition then persists for all +$t\ge T$. + +Three remarks are in order. This argument showing that eventually +every component of the time-slice $t$ has trivial $\pi_2$ is not +necessary for the topological application (\entryref{127db1dc9315}), +or indeed, for any other topological application. The reason is the +sphere theorem (see [Hempel]), which says that if $\pi_2(M)$ is +non-trivial then either $M$ is diffeomorphic to an $S^2$ bundle over +$S^1$ or $M$ has a non-trivial connected sum decomposition. Thus, we +can establish results for all $3$-manifolds if we can establish them +for $3$-manifolds with $\pi_2=0$. Secondly, the reason for giving +this argument is that it is pleasing to see Ricci flow with surgery +implementing the connected sum decomposition required for +geometrization of $3$-manifolds. Also, this argument is a simpler +version of the one that we use to deal with components with +non-trivial $\pi_3$. Lastly, these results on Ricci flow do not use +the sphere theorem so that establishing the cutting into pieces with +trivial $\pi_2$ allows us to give a different proof of this result +(though admittedly one using much deeper ideas). + +Let us now fix $T<\infty$ such that for all $t\ge T$ all the +time-slices $M_t$ have trivial $\pi_2$. There is a simple +topological consequence of this and our assumption on the initial +manifold. If $M$ is a compact $3$-manifold whose fundamental group +is either a non-trivial free product or an infinite cyclic group, +then $M$ admits a homotopically non-trivial embedded $2$-sphere. +Since we began with a manifold $M_0$ whose fundamental group is a +free product of finite groups and infinite cyclic groups, it follows +that for $t\ge T$ every component of $M_t$ has finite fundamental +group. Fix $t\ge T$. Then each component of $M_t$ has a finite +cover that is simply connected, and thus, by an elementary argument +in algebraic topology, each component of $M_t$ has non-trivial +$\pi_3$. The second step in the finite-time extinction argument is +to use a non-trivial element in this group analogously to the way we +used homotopically non-trivial $2$-spheres to show that eventually +the manifolds have trivial $\pi_2$. + + There are two +approaches to this second step: the first is due to Perelman in +[Perelman3] and the other due to Colding-Minicozzi in +[ColdingMinicozzi]. In their approach Colding-Minicozzi +associate to a non-trivial element in $\pi_3(M)$ a non-trivial +element in $\pi_1(\mathit{Maps}(S^2,M))$. This element is represented +by a one-parameter family of $2$-spheres (starting and ending at +the constant map) representing a non-trivial element $\xi\in +\pi_3(M_0)$. They define the *width* of this homotopy class by +$W(\xi,t)$ by associating to each representative the maximal energy +of the $2$-spheres in the family and then minimizing over all +representatives of the homotopy class. Using results of Jost +[Jost], they show that this function satisfies the same forward +difference inequality that $W_2$ satisfies (and has the same +continuity property under Ricci flow and the same semi-continuity +under surgery). Since $W(\xi,t)$ is always $\ge 0$ if it is defined, +this forward difference quotient inequality implies that the +manifolds $M_t$ must eventually become empty. + +While this approach seemed completely natural to us, and while we +believe that it works, we found the technical details +daunting (Colding and Minicozzi tell us they plan to give an +expanded version of their argument with a more detailed proof.) +(because one is forced to consider index-one critical points of the +energy functional rather than minima). For this reason we chose to +follow Perelman's approach. He represents a non-trivial element in +$\pi_3(M)$ as a non-trivial element in $\xi\in \pi_2(\Lambda M,*)$ +where $\Lambda M$ is the free loop space of $M$. He then associates +to a family $\Gamma\colon S^2\to \Lambda M$ of homotopically trivial +loops an invariant $W(\Gamma)$ which is the maximum of the areas of +minimal spanning disks for the loops $\Gamma(c)$ as $c$ ranges over +$S^2$. The invariant of a non-trivial homotopy class $\xi$ is then +the infimum over all representatives $\Gamma$ for $\xi$ of +$W(\Gamma)$. As before, this function is continuous under Ricci flow +and is lower semi-continuous under surgery (unless the surgery +removes the component in question). It also satisfies a forward +difference quotient + +$$ +\frac{dW(\xi)}{dt}\le -2\pi+\frac{3}{4t+1}W(\xi). +$$ + +The reason for the term $-2\pi$ instead of $-4\pi$ which occurs in +the other cases is that we are working with minimal $2$-disks +instead of minimal $2$-spheres. Once this forward difference +quotient estimate (and the continuity) have been established the +argument finishes in the same way as the other cases: a function $W$ +with the properties we have just established cannot be non-negative +for all positive time. This means the component in question, and +indeed all components at later time derived from it, must disappear +in finite time. Hence, under the hypothesis on the fundamental group +in \entryref{127db1dc9315} the entire manifold must disappear at +finite time. + +Because this approach uses only minima for the energy or area +functional, one does not have to deal with higher index critical +points. But one is forced to face other difficulties though -- +namely boundary issues. Here, one must prescribe the deformation of +the family of boundary curves before computing the forward +difference quotient of the energy. The obvious choice is the +curve-shrinking flow (see [AG]). Unfortunately, this flow can +only be defined when the curve in question is immersed and even in +this case the curve-shrinking flow can develop singularities even if +the Ricci flow does not. Following Perelman, or indeed [AG], +one uses the device of taking the product with a small circle and +using loops, called *ramps*, that go around that +circle once. In this context the curve-shrinking flow remains +regular as long as the Ricci flow does. One then projects this flow +to a flow of families of $2$-spheres in the free loop space of the +time-slices of the original Ricci flow. Taking the length of the +circle sufficiently small yields the boundary deformation needed to +establish the forward difference quotient result. This requires a +compactness result which holds under local total curvature bounds. +This compactness result holds off of a set of time of small total +measure, which is sufficient for the argument. At the very end of +the argument we need an elementary but complicated result on annuli, +which we could not find in the literature. For more details on these +points see Chapter 18. + +## Acknowledgements + +In sorting out Perelman's arguments we have had the aid of many +people. First of all, during his tour of the United States in the +Spring of 2003, Perelman spent enormous amounts of time explaining +his ideas and techniques both in public lectures and in private +sessions to us and to many others. Since his return to Russia, he +has also freely responded to our questions by e-mail. Richard +Hamilton has also given unstintingly of his time, explaining to us +his results and his ideas about Perelman's results. We have +benefitted tremendously from the work of Bruce Kleiner and John +Lott. They produced a lengthy set of notes filling in the details in +Perelman's arguments [KleinerLott]. We have referred to those +notes countless times as we came to grips with Perelman's ideas. In +late August and early September of 2004, Kleiner, Lott and the two +of us participated in a workshop at Princeton University, supported +by the Clay Math Institute, going through Perelman's second paper +(the one on Ricci flow with surgery) in detail. This workshop played +a significant role in convincing us that Perelman's arguments were +complete and correct and also in convincing us to write this book. +We thank all the participants of this workshop and especially +Guo-Feng Wei, Peng Lu, Yu Ding, and X.-C. Rong who, together with +Kleiner and Lott, made significant contributions to the workshop. +Before, during, and after this workshop, we have benefitted from +private conversations too numerous to count with Bruce Kleiner and +John Lott. + +Several of the analytic points were worked out in detail by others, +and we have freely adapted their work. Rugang Ye wrote a set of +notes proving the Lipschitz properties of the length function and +proving the fact that the cut locus for the reduced length function +is of measure zero. Closely related to this, Ye gave detailed +arguments proving the requisite limiting results for the length +function required to establish the existence of a gradient shrinking +soliton. None of these points was directly addressed by Perelman, +and it seemed to us that they needed careful explanation. Also, the +proof of the uniqueness of the standard solution was established +jointly by the second author and Peng Lu, and the proof of the +refined version of Shi's theorem where one has control on a certain +number of derivatives at time zero was also shown to us by Peng Lu. +We also benefitted from Ben Chow's expertise and vast knowledge of +the theory of Ricci flow especially during the ClayMath/MSRI Summer +School on Ricci Flow in 2005. We had several very helpful +conversations with Tom Mrowka and also with Robert Bryant, +especially about annuli of small area. + +The second author gave courses at Princeton University and ran +seminars on this material at MIT and Princeton. Natasa Sesum and +Xiao-Dong Wang, see [STW], wrote notes for the seminars at MIT, +and Edward Fan and Alex Subotic took notes for the seminars and +courses at Princeton, and they produced preliminary manuscripts. We +have borrowed freely from these manuscripts, and it is a pleasure to +thank each of them for their efforts in the early stages of this +project. + +The authors thank all the referees for their time and effort which +helped us to improve the presentation. In particular, we wish to +thank Cliff Taubes who carefully read the entire manuscript and gave +us enumerable comments and suggestions for clarifications and +improvements. We also thank Terry Tao for much helpful feedback +which improved the exposition. We also thank Colin Rourke for his +comments on the finite-time extinction argument. + +We thank Lori Lejeune for drawing the figures and John Etnyre for technical +help in inserting these figures into the manuscript. + +Lastly, during this work we were both generously supported by the Clay +Mathematical Institute, and it is a pleasure to thank the Clay Mathematics +Institute for its support and to thank its staff, especially Vida Salahi, for +their help in preparing this manuscript. We also thank the National Science Foundation +for their continued support and the second author thanks the Jim Simons +for his support during the period that he was a faculty member at MIT. + +## List of related papers + +For the readers' convenience we gather here references to all the +closely related articles. + +First and foremost are Perelman's three preprints, [Perelman1], +[Perelman2], and [Perelman3]. The first of these +introduces the main techniques in the case of Ricci flow, the second +discusses the extension of these techniques to Ricci flow with +surgery, and the last gives the short-cut to the Poincaré +Conjecture and the $3$-dimensional spherical space-form conjecture, +avoiding the study of the limits as time goes to infinity and +collapsing space arguments. There are the detailed notes by Bruce +Kleiner and John Lott, [KleinerLott], which greatly expand and +clarify Perelman's arguments from the first two preprints. There is +also a note on Perelman's second paper by Yu Ding [yuding]. +There is the article by Colding-Minicozzi [ColdingMinicozzi], +which gives their alternate approach to the material in Perelman's +third preprint. Collapsing space arguments needed for the full +geometrization conjecture are discussed in Shioya-Yamaguchi +[SY]. Lastly, after we had submitted a preliminary version of +this manuscript for refereeing, H.-D. Cao and X.-P. Zhu published an +article on the Poincaré Conjecture and Thurston's Geometrization +Conjecture; see [CaoZhu]. diff --git a/projects/poincare-conjecture/.astrolabe/docs/02-prelim.mdx b/projects/poincare-conjecture/.astrolabe/docs/02-prelim.mdx deleted file mode 100644 index bd477ec5..00000000 --- a/projects/poincare-conjecture/.astrolabe/docs/02-prelim.mdx +++ /dev/null @@ -1,1742 +0,0 @@ -# Preliminaries from Riemannian -geometry - -In this chapter we will recall some basic facts in Riemannian -geometry. For more details we refer the reader to and -. Throughout, we always adopt Einstein's summation -convention on repeated indices and `manifold' means a paracompact, -Hausdorff, smooth manifold. - -## Riemannian metrics and the Levi-Civita connection - -Let $M$ be a manifold and let $p$ be a point of $M$. Then $TM$ -denotes the tangent bundle of $M$ and $T_pM$ is the tangent space at -$p$. Similarly, $T^*M$ denotes the cotangent bundle of $M$ and -$T^*_pM$ is the cotangent space at $p$. For any vector bundle -${\mathcal V}$ over $M$ we denote by $\Gamma({\mathcal V})$ the -vector space of smooth sections of ${\mathcal V}$. - -**Definition.** Let $M$ be an $n$-dimensional manifold. A \it Riemannian metric \rm -$g$ on $M$ is a smooth section of $T^*M\otimes T^*M$ defining a -positive definite symmetric bilinear form on $T_pM$ for each $p\in -M$. In local coordinates $(x^1,\cdots, x^n)$, one has a natural -local basis $\{\partial_1,\cdots,\partial_n\}$ for $TM$, where -$\partial_i=\frac{\partial}{\partial x^i}$. The metric tensor -$g=g_{ij} dx^i\otimes dx^j$ is represented by a smooth matrix-valued -function - -$$ -g_{ij}=g(\partial_i,\partial_j). -$$ - - The -pair $(M,g)$ is a *Riemannian manifold*. We denote by $(g^{ij})$ -the inverse of the matrix $(g_{ij})$. - -Using a partition of unity one can easily see that any manifold -admits a Riemannian metric. A Riemannian metric on $M$ allows us to -measure lengths of smooth paths in $M$ and hence to define a -distance function by setting $d(p,q)$ equal to the infimum of the -lengths of smooth paths from $p$ to $q$. This makes $M$ a metric -space. For a point $p$ in a Riemannian manifold $(M,g)$ and for $r> -0$ we denote the metric ball of radius $r$ centered at $p$ in $M$ by -$B(p,r)$ or by $B_g(p,r)$ if the metric needs specifying or -emphasizing. It is defined by -$$B(p,r)=\{q\in M\left|\right. d(p,q)0$ (resp., $K(P)<0$) for every -2-plane $P$. There are analogous notions of non-negative and -non-positive sectional curvature. - -In local coordinates, suppose that $X = X^{i}\partial_i$ and $Y = -Y^{i}\partial_i$. Then we have - -$$ -K(P) = R_{ijkl}X^{i}Y^{j}X^{k}Y^{l}. -$$ - -A Riemannian manifold is said to have *constant sectional -curvature* if $K(P)$ is the same for all $p\in M$ and all $2$-planes -$P\subset T_pM$. One can show that a manifold $(M,g)$ has constant -sectional curvature $\lambda$ if and only if -$$ -R_{ijkl} = -\lambda(g_{ik}g_{jl}-g_{il}g_{jk}). -$$ - Of course, the sphere of -radius $r$ in $\Ar^n$ has constant sectional curvature $1/r^2$, -$\Ar^n$ with the Euclidean metric has constant sectional curvature -$0$, and the hyperbolic space ${\mathbb H}^n$, which, in the -Poincar\'e model, is given by the unit disk with the metric -$$\frac{4(dx^{2}_{1}+\cdots+ -dx^{2}_{n})}{(1-\abs{x}^{2})^{2}},$$ or in the upper half-space -model with coordinates $(x^1,\ldots,x^n)$ is given by -$$\frac{ds^2}{(x^n)^2}$$ has constant sectional curvature $-1$. In all -three cases we denote the constant curvature metric by $g_st$. - -**Definition.** Using the metric, one can replace the Riemann curvature tensor -${\mathcal R}$ by a symmetric bilinear form $Rm$ on -$\wedge^2TM$. In local coordinates let -$\varphi=\varphi^{ij}\partial_i\wedge\partial _j$ and -$\psi=\psi^{kl}\partial _k\wedge\partial_l$ be local sections of -$\wedge^2TM$. The formula for $Rm$ is - -$$ -Rm(\varphi, \psi)=R_{ijkl}\varphi^{ij}\psi^{kl}. -$$ - -We call $Rm$ the *curvature -operator*. We say $(M,g)$ has -*positive curvature operator* if $Rm(\varphi,\varphi)>0$ for any nonzero 2-form $\varphi = -\varphi^{ij}\partial_{i}\wedge -\partial_{j}$ and has *nonnegative curvature operator* if -$Rm(\varphi,\varphi)\geq0$ for any $\varphi\in \wedge^2TM$. - -Clearly, if the curvature operator is a positive (resp., -non-negative) operator then the manifold is positively (resp., -non-negatively) curved. - -**Definition.** The *Ricci curvature tensor*, -denoted $Ric$ or $Ric_g$ when it is necessary to specify -the metric, is a symmetric contravariant two-tensor. In local -coordinates it is defined by - -$$ -Ric(X,Y)=g^{kl}R(X,\partial_k,Y,\partial_l). -$$ - -The value of this tensor at a point $p\in M$ is given by -$\sum_{i=1}^nR(X(p),e_i,Y(p),e_i)$ where $\{e_{1},\cdots,e_{n}\}$ is -an orthonormal basis of $T_pM$. Clearly $Ric$ is a symmetric -bilinear form on $TM$, given in local coordinates by -$$Ric=Ric_{ij}dx^i\otimes dx^j,$$ where -$Ric_{ij}=Ric(\partial_i,\partial _j)$. The -*scalar curvature* is defined by: -$$R=R_g=tr_g Ric=g^{ij}Ric_{ij}.$$ We will say that $Ric \geq -k$ (or $\leq k$) if all the eigenvalues of $Ric$ are $\geq k$ -(or $\leq k$). - -Clearly, the curvatures are natural in the sense that if $F\colon N -\rightarrow M$ is a diffeomorphism and if $g$ is a Riemannian metric -on $M$, then $F^*g$ is a Riemannian metric on $N$ and we have $Rm(F^*g)=F^*(Rm(g))$, $Ric(F^{*}g) = F^{*}(Ric(g))$, and $R(F^*g)=F^*(R(g))$. - -### Consequences of the Bianchi identities - -There is one consequence of the second Bianchi identity that will be -important later. For any contravariant two-tensor $\omega$ on $M$ -(such as $Ric$ or $Hess(f)$) we define the contravariant -one-tensor $div(\omega)$ as follows: For any vector field $X$ -we set -$$div(\omega)(X)=\nabla^*\omega(X)=g^{rs}\nabla_r(\omega)(X,\partial_s).$$ - -**Lemma.** -$$dR=2div(Ric)=2\nabla^*Ric.$$ - -For a proof see Proposition 6 of Chapter 2 on page 40 of . - -We shall also need a formula relating the connection Laplacian on -contravariant one-tensors with the Ricci curvature. Recall that for -a smooth function $f$, we defined the symmetric two-tensor -$\nabla^2f$ by -$$\nabla^2f(X,Y)=\nabla_X\nabla_Y(f)-\nabla_{\nabla_X(Y)}(f)=Hess(f)(X,Y),$$ -and then defined the Laplacian -$$\triangle f=tr\nabla^2f=g^{ij}(\nabla^2f)_{ij}.$$ These operators extend to tensors of any -rank. Suppose that $\omega$ is a contravariant tensor of rank $k$. Then we -define $\nabla^2\omega$ to be a contravariant tensor of rank $k+2$ given by -$$\nabla^2\omega(\cdot,X,Y)=(\nabla_{X}\nabla_{Y}\omega)(\cdot) --\nabla_{\nabla_X(Y)}\omega(\cdot).$$ This expression is not -symmetric in the vector fields $X,Y$ but the commutator is given by -evaluating the curvature operator ${\mathcal R}(X,Y)$ on $\omega$. -We define the connection Laplacian on the tensor $\omega$ to be -$$\triangle \omega=g^{ij}\nabla^2(\omega)(\partial_i,\partial_j).$$ -Direct computation gives the standard Bochner formula relating these -Laplacians with the Ricci curvature; see for example Proposition -4.36 on page 168 of . - -**Lemma.** - Let $f$ be a smooth function on a Riemannian manifold. -Then we have the following formula for contravariant one-tensors: -$$ \triangle df=d\triangle f+Ric((\nabla f)^*,\cdot).$$ - -### First examples - -The most homogeneous Riemannian manifolds are those of constant -sectional curvature. These are easy to classify; see Corollary 10 of -Chapter 5 on page 147 of . - -**Theorem.** **(Uniformization Theorem)** If -$(M^n,g)$ is a complete, simply-connected Riemannian manifold of -constant sectional curvature $\lambda$, then: -\begin{enumerate} -\item If $\lambda=0$, then $M^n$ is isometric to Euclidean $n$-space. -\item If $\lambda>0$ there is a diffeomorphism $\phi \colon M \rightarrow S^{n}$ - such that g = $\lambda^{-1}\phi^{*}(g_st)$ where $g_{st}$ is the usual metric on the unit sphere in $\Ar^{n+1}$. -\item If $\lambda<0$ there is a diffeomorphism $\phi \colon M \rightarrow {\mathbb H}^{n}$ - such that g = $\abs{\lambda}^{-1}\phi^{*}(g_st)$ where $g_{st}$ is the Poincar\'e metric of constant curvature -$-1$ on ${\mathbb H}^n$. -\end{enumerate} - -Of course, if $(M^n,g)$ is a complete manifold of constant sectional -curvature then its universal covering satisfies the hypothesis of -the theorem and hence is one of $S^n, \Ar^n$, or ${\mathbb H}^n$, up -to a constant scale factor. This implies that $(M,g)$ is isometric -to a quotient of one of these simply connected spaces of constant -curvature by the free action of a discrete group of isometries. Such -a Riemannian manifold is called a -*space-form*. - -**Definition.** The Riemannian manifold -$(M,g)$ is said to be an *Einstein manifold with Einstein -constant* $\lambda$ if $Ric(g)=\lambda g$. - -\begin{exam} Let $M$ be an $n$-dimensional manifold with $n$ being either $2$ or $3$. -If $(M,g)$ is Einstein with Einstein constant $\lambda$, one can -easily show that $M$ has constant sectional curvature -$\frac{\lambda}{n-1}$, so that in fact $M$ is a space-form. -\end{exam} - -### Cones - -Another class of examples that will play an important role in our -study of the Ricci flow is that of cones. - -**Definition.** -Let $(N,g)$ be a Riemannian manifold. We define the *open cone* -over $(N,g)$ to be the manifold $N\times (0,\infty)$ with the metric -$\tilde g$ defined as follows: For any $(x,s)\in N\times (0,\infty)$ -we have -$$\tilde g(x,s)=s^2g(x)+ds^2.$$ - -Fix local coordinates $(x^1,\ldots,x^n)$ on $N$. Let -$\Gamma_{ij}^k;\ 1\le i,j,k\le n$, be the Christoffel symbols for -the Levi-Civita connection on $N$. Set $x^0=s$. In the local -coordinates $(x^0,x^1,\ldots,x^n)$ for the cone we have the -Christoffel symbols $\widetilde \Gamma_{ij}^k;\ 0\le i,j,k\le n$, -for $\tilde g$. The relation between the metrics gives the following -relations between the two sets of Christoffel symbols: - -$$ - -$$ -\begin{aligned} -\widetilde -\Gamma_{ij}^k & = & \Gamma_{ij}^k; \ \ \ \ 1\le i,j,k\le n \\ -\widetilde \Gamma_{ij}^0 & = & -sg_{ij}; \ \ \ \ 1\le i,j\le n \\ -\widetilde\Gamma_{i0}^j & = & \widetilde\Gamma_{0i}^j = -s^{-1}\delta^j_i; \ \ \ \ 1\le i,j\le n \\ -\widetilde \Gamma_{i0}^0 & = & 0;\ \ \ \ 0\le i\le n \\ -\widetilde \Gamma_{00}^i & = & 0;\ \ \ \ 0\le i\le n. -\end{aligned} -$$ - -$$ - -Denote by ${\mathcal R}_g$ the curvature tensor for $g$ and by -${\mathcal R}_{\tilde g}$ the curvature tensor for $\tilde g$. Then -the above formulas lead directly to: - -$$ - -$$ -\begin{aligned} -{\mathcal R}_{\tilde g}(\partial_i,\partial_j)(\partial_0) & = & 0;\ \ \ \ 0\le i,j\le n \\ -{\mathcal R}_{\tilde g}(\partial_i,\partial_j)(\partial_i) & = & -{\mathcal -R}_g(\partial_i,\partial_j)(\partial_i)+g_{ii}\partial_j-g_{ji}\partial_i;\ -\ \ \ 1\le i,j\le n \\ -\end{aligned} -$$ - -$$ - -This allows us to compute the Riemann curvatures of the cone in -terms of those of $N$. - -**Proposition.** -Let $N$ be a Riemannian manifold of dimension $n-1$. Fix $(x,s)\in -c(N)=N\times (0,\infty)$. With respect to the coordinates -$(x^0,\ldots,x^n)$ the curvature operator ${Rm}_{\tilde -g}(p,s)$ of the cone decomposes as -$$\begin{pmatrix} 0 & 0 \\ s^2(Rm_g(p)-\wedge^2g(p)) & 0 -\end{pmatrix},$$ -where $\wedge^2g(p)$ is the symmetric form on $\wedge^2T_pN$ induced -by $g$. - -**Corollary.** For any $p\in N$ let $\lambda_1,\ldots,\lambda_{(n-1)(n-2)/2}$ be -the eigenvalues of $Rm_g(p)$. Then for any $s>0$ there are -$(n-1)$ zero eigenvalues of $Rm_{\tilde g}(p,s)$. The other -$(n-1)(n-2)/2$ eigenvalues of $Rm_{\tilde g}(p,s)$ are -$s^{-2}(\lambda_i-1)$. - -**Proof.** Clearly from Proposition~, we see that under the -orthogonal decomposition $\wedge^2T_{(p,s)}c(N)=\wedge^2T_pN\oplus -T_pN$ the second subspace is contained in the null space of $ Rm_{\tilde g}(p,s)$, and hence contributes $(n-1)$ zero -eigenvalues. Likewise, from this proposition we see that the -eigenvalues of the restriction of $Rm_{\tilde g}(p,s)$ to the -subspace $\wedge^2T_pN$ are given by -$s^{-4}(s^2(\lambda_i-1))=s^{-2}(\lambda_i-1)$. - -## Geodesics and the exponential map - -Here we review standard material about geodesics, Jacobi fields, and -the exponential map. - -### Geodesics and the energy functional - -**Definition.** Let $I$ be an open interval. -A smooth curve $\gamma\colon I \rightarrow M$ is called a -*geodesic* if $\nabla_{\dot \gamma}\dot -\gamma =0$. - -In local coordinates, we write $\gamma(t)=(x^1(t),\ldots,x^n(t))$ -and this equation becomes -$$0=\nabla_{\dot \gamma}\dot \gamma(t) = -\left(\sum\limits_k\left(\ddot x^k(t) +\dot x^i(t)\dot -x^j(t)\Gamma^{k}_{ij}(\gamma(t))\right)\partial_k\right).$$ This is -a system of $2^{nd}$ order ODE's. The local existence, uniqueness -and smoothness of a geodesic through any point $p\in M$ with initial -velocity vector $v\in T_pM$ follow from the classical ODE theory. -Given any two points in a complete manifold, a standard limiting -argument shows that there is a rectifiable curve of minimal length -between these points. Any such curve is a geodesic. We call -geodesics that minimize the length between their endpoints *minimizing geodesics*. - -We have the classical theorem showing that on a complete manifold -all geodesics are defined for all time (see Theorem 16 of Chapter 5 -on p. 137 of ). - -**Theorem.** (Hopf-Rinow) If $(M,g)$ is complete as a metric space, then every -geodesic extends to a geodesic defined for all time. - -Geodesics are critical points of the energy functional. Let $(M,g)$ -be a complete Riemannian manifold. Consider the space of $C^1$-paths -in $M$ parameterized by the unit interval. On this space we have the -energy functional -$$E(\gamma)=\frac{1}{2}\int_{0}^{1}\langle \gamma'(t),\gamma'(t)\rangle dt.$$ - Suppose that we have a one-parameter family of paths parameterized by - $[0,1]$, - all having the same initial point $p$ and the same final point $q$. - By this we mean that we have a surface $\tilde\gamma(t,u)$ with the property that for each - $u$ the path $\gamma_u=\tilde\gamma(\cdot,u)$ is a path from $p$ to $q$ parameterized by $[0,1]$. - Let $\widetilde X=\partial\tilde \gamma/\partial t$ and - $\widetilde Y=\partial\tilde \gamma/\partial u$ be the corresponding vector - fields along the surface swept out by $\tilde \gamma$, and denote by $X$ and $Y$ the restriction - of these vector fields along $\gamma_0$. - We compute - -$$ - -$$ -\begin{aligned} -\frac{dE(\gamma_u)}{du}\Bigl|_{u=0}\Bigr.& = & \left( \int_0^1\langle \nabla_{\widetilde Y}\widetilde X,\widetilde - X\rangle dt\right)\left|_{u=0}\right. \\ - & = & \left(\int_0^1\langle\nabla_{\widetilde X}\widetilde Y,\widetilde - X\rangle dt \right)\left|_{u=0}\right.\\ - & = & -\left( \int_0^1\langle\nabla_{\widetilde X}\widetilde X,\widetilde - Y\rangle dt\right)\left|_{u=0}\right.=-\int_0^1\langle\nabla_XX,Y\rangle, -\end{aligned} -$$ - -$$ - -where the first equality in the last line comes from integration by -parts and the fact that $\widetilde Y$ vanishes at the endpoints. -Given any vector field $Y$ along $\gamma_0$ there is a one-parameter -family $\tilde \gamma(t,u)$ of paths from $p$ to $q$ with $\tilde -\gamma(t,0)=\gamma_0$ and with $\widetilde Y(t,0)=Y$. Thus, from the -above expression we see that $\gamma_0$ is a critical point for the -energy functional on the space of paths from $p$ to $q$ -parameterized by the interval $[0,1]$ if and only if $\gamma_0$ is a -geodesic. - -Notice that it follows immediately from the geodesic equation that -the length of a tangent vector along a geodesic is constant. Thus, -if a geodesic is parameterized by $[0,1]$ we have -$$E(\gamma)=\frac{1}{2}L(\gamma)^2.$$ -It is immediate from the Cauchy-Schwarz inequality that for any -curve $\mu$ parameterized by $[0,1]$ we have -$$E(\mu)\ge \frac{1}{2}L(\mu)^2$$ -with equality if and only if $|\mu'|$ is constant. In particular, a -curve parameterized by $[0,1]$ minimizes distance between its -endpoints if it is a minimum for the energy functional on all paths -parameterized by $[0,1]$ with the given endpoints. - -### Families of geodesics and Jacobi fields - -Consider a family of geodesics $\tilde\gamma(u,t)=\gamma_u(t)$ -parameterized by the interval $[0,1]$ with $\gamma_u(0)=p$ for all -$u$. Here, unlike the discussion above, we allow $\gamma_u(1)$ to -vary with $u$. As before define vector fields along the surface -swept out by $\tilde \gamma$: $\widetilde X=\partial -\tilde\gamma/\partial t$ and let $\tilde -Y=\partial\tilde\gamma/\partial u$. We denote by $X$ and $Y$ the -restriction of these vector fields to the geodesic -$\gamma_0=\gamma$. - Since each -$\gamma_u$ is a geodesic, we have $\nabla_{\widetilde X}{\widetilde -X}=0$. Differentiating this equation in the $\widetilde Y$-direction -yields $\nabla_{\widetilde Y}\nabla_{\widetilde X}\widetilde X=0$. -Interchanging the order of differentiation, using -$\nabla_{\widetilde X}\widetilde Y=\nabla_{\widetilde Y}{\widetilde -X}$, and then restricting to $\gamma$, we get the *Jacobi -equation*: -$$\nabla_X\nabla_XY+{\mathcal R}(Y,X)X=0.$$ -Notice that the left-hand side of the equation depends only on the -value of $Y$ along $\gamma$, not on the entire family. We denote the -left-hand side of this equation by ${rm Jac}(Y)$, so that the Jacobi -equation now reads -$${rm Jac}(Y)=0.$$ - The fact that all the geodesics begin at the same point at time -$0$ means that $Y(0)=0$. A vector field $Y$ along a geodesic -$\gamma$ is said to be a *Jacobi field* if -it satisfies this equation and vanishes at the initial point $p$. A -Jacobi field is determined by its first derivative at $p$, i.e., by -$\nabla_XY(0)$. We have just seen that this is the equation -describing, to first order, variations of $\gamma$ by a family of -geodesics with the same starting point. - -Jacobi fields are also determined by the energy functional. Consider -the space of paths parameterized by $[0,1]$ starting at a given -point $p$ but free to end anywhere in the manifold. Let $\gamma$ be -a geodesic (parameterized by $[0,1]$) from $p$ to $q$. Associated to -any one-parameter family $\tilde \gamma(t,u)$ of paths parameterized -by $[0,1]$ starting at $p$ we associate the second derivative of the -energy at $u=0$. Straightforward computation gives -$$\frac{d^2E(\gamma_u)}{du^2}\Bigl|_{u=0}\Bigr.=\langle \nabla_XY(1),Y(1)\rangle+\langle -X(1),\nabla_Y\widetilde Y(1,0)\rangle-\int_0^1\langle {rm -Jac}(Y),Y\rangle dt.$$ - -Notice that the first term is a boundary term from the integration -by parts, and it depends not just on the value of $Y$ (i.e., on -$\widetilde Y$ restricted to $\gamma$) but also on the first-order -variation of $\widetilde Y$ in the $Y$ direction. There is the -associated bilinear form that comes from two-parameter families -$\tilde \gamma(t,u_1,u_2)$ whose value at $u_1=u_1=0$ is $\gamma$. -It is -$$\frac{d^2E}{du_1du_2}\Bigl|_{u_1=u_2=0}\Bigr.=\langle \nabla_XY_1(1),Y_2(1)\rangle+\langle -X(1),\nabla_{Y_1}\widetilde Y_2(1,0)\rangle-\int_0^1\langle {rm -Jac}(Y_1),Y_2\rangle dt.$$ Notice that restricting to the space of -vector fields that vanish at both endpoints, the second derivatives -depend only on $Y_1$ and $Y_2$ and the formula is -$$\frac{d^2E}{du_1du_2}\Bigl|_{u_1=u_2=0}\Bigr.=-\int_0^1\langle {rm Jac}(Y_1),Y_2\rangle dt,$$ -so that this expression is symmetric in $Y_1$ and $Y_2$. The -associated quadratic form on the space of vector fields along -$\gamma$ vanishing at both endpoints -$$-\int_0^1\langle {rm Jac}(Y),Y\rangle dt$$ -is the second derivative of the energy function at $\gamma$ for any -one-parameter family whose value at $0$ is $\gamma$ and whose first -variation is given by $Y$. - -### Minimal geodesics - -**Definition.** Let $\gamma$ be a geodesic beginning at $p\in M$. For any $t>0$ we -say that $q=\gamma(t)$ is a *conjugate point along -$\gamma$* if there is a non-zero Jacobi field -along $\gamma$ vanishing at $\gamma(t)$. - -**Proposition.** -Suppose that $\gamma\colon [0,1]\to M$ is a minimal geodesic. Then -for any $t<1$ the restriction of $\gamma$ to $[0,t]$ is the unique -minimal geodesic between its endpoints and there are no conjugate -points on $\gamma([0,1))$, i.e., there is no non-zero Jacobi field -along $\gamma$ vanishing at any $t\in [0,1)$. - -We shall sketch the proof. For more details see Proposition 19 and Lemma 14 of -Chapter 5 on pp. 139 and 140 of . - -**Proof.** (Sketch) -Fix $0 0$, such that the restriction of $exp_p$ to the ball -$B_{g|T_pM}(0,r_0)$ in $T_pM$ is a diffeomorphism onto -$B_g(p,r_0)$. Fix $g$-orthonormal linear coordinates on $T_pM$. -Transferring these coordinates via $exp_p$ to coordinates on -$B(p,r_0)$ gives us *Gaussian normal coordinates* on $B(p,r_0)\subset M$. - -Suppose now that $M$ is complete, and fix a point $p\in M$. For -every $q\in M$, there is a length-minimizing path from $p$ to $q$. -When parameterized at constant speed equal to its length, this path -is a geodesic with domain interval $[0,1]$. Consequently, $exp_p\colon T_pM\to M$ is onto. The differential of the exponential -mapping is given by Jacobi fields: - Let -$\gamma\colon [0,1]\to M$ be a geodesic from $p$ to $q$, and let - $X\in T_pM$ be $\gamma'(0)$. Then the exponential -mapping at $p$ is a smooth map from $T_p(M)\to M$ sending $X$ to -$q$. Fix $Z\in T_pM$. Then there is a unique Jacobi field $Y_Z$ -along $\gamma$ with $\nabla_XY_Z(0)=Z$. The association $Z\mapsto -Y_Z(1)\in T_qM$ is a linear map from $T_p(M)\to T_qM$. Under the -natural identification of $T_pM$ with the tangent plane to $T_pM$ at -the point $Z$, this linear mapping is the differential of $exp_p\colon T_pM\to M$ at the point $X\in T_pM$. - -**Corollary.** -Suppose that $\gamma$ is a minimal geodesic parameterized by $[0,1]$ -starting at $p$. Let $X(0)=\gamma'(0)\in T_pM$. Then for each -$t_0<1$ the restriction $\gamma|_{[0,t_0]}$ is a minimal geodesic -and $exp_p\colon T_pM\to M$ is a local diffeomorphism near -$t_0X(0)$. - -**Proof.** Of course, $exp_p(t_0X(0))=\gamma(t_0)$. According to the -previous discussion, the kernel of the differential of the -exponential mapping at $t_0X(0)$ is identified with the space of -Jacobi fields along $\gamma$ vanishing at $\gamma(t_0)$. According -to Proposition~ the only such Jacobi field is the -trivial one. Hence, the differential of $exp_p$ at $t_0X(0)$ -is an isomorphism, completing the proof. - -**Definition.** There is an open neighborhood $U_p\subset T_pM$ of $0$ consisting -of all $v\in T_pM$ for which: (i) $\gamma_v$ is the unique minimal -geodesic from $p$ to $\gamma_v(1)$, and (ii) $ exp_p$ is a -local diffeomorphism at $v$. We set ${\mathcal C}_p\subset M$ equal -to $M\setminus exp_p(U_p)$. Then ${\mathcal C}_p$ is called -the *cut locus from $p$*. It is a closed subset -of measure $0$. - -It follows from Corollary~ that $U\subset T_pM$ is a -star-shaped open neighborhood of $0\in T_pM$. - -**Proposition.** The map -$$exp_p\colon U_p\to M\setminus {\mathcal C}_p$$ -is a diffeomorphism. - - For a proof see p. 139 of . - -**Definition.** The *injectivity radius $inj_M(p)$ of $M$ at $p$* is the -supremum of the $r> 0$ for which the restriction of $exp_p\colon T_pM\to M$ to the ball $B(0,r)$ of radius $r$ in $T_pM$ -is a diffeomorphism into $M$. Clearly, $inj_M(p)$ is the -distance in $T_pM$ from $0$ to the frontier of $U_p$. It is also the -distance in $M$ from $p$ to the cut locus ${\mathcal C}_p$. - -Suppose that $inj_M(p)=r$. There are two possibilities: -Either there is a broken, closed geodesic through $p$, broken only -at $p$, of length $2r$, or there is a geodesic $\gamma$ of length -$r$ emanating from $p$ whose endpoint is a conjugate point along -$\gamma$. The first case happens when the exponential mapping is not -one-to-one of the closed ball of radius $r$ in $T_pM$, and the -second happens when there is a tangent vector in $T_pM$ of length -$r$ at which $exp_p$ is not a local diffeomorphism. - -## Computations in Gaussian normal coordinates - -In this section we compute the metric and the Laplacian (on -functions) in local Gaussian coordinates. A direct computation shows -that in Gaussian normal coordinates on a metric ball about $p\in M$ -the metric takes the form - -$$ - -$$ -\begin{aligned} - -g_{ij}(x)& = & \delta_{ij} + \frac{1}{3}R_{iklj}x^kx^l + -\frac{1}{6}R_{iklj,s}x^kx^lx^s \\ -& & +(\frac{1}{20}R_{iklj,st} +\frac{2}{45}\sum_m -R_{iklm}R_{jstm})x^kx^lx^sx^t + O(r^{5}), \nonumber -\end{aligned} -$$ - -$$ - -where $r$ is the distance from $p$. (See, for example Proposition -3.1 on page 41 of~, with the understanding that, with -the conventions there, the quantity $R_{ijkl}$ there differs by sign -from ours.) - -Let $\gamma$ be a geodesic in $M$ emanating from $p$ in the direction $v$. -Choose local coordinates $\theta^1,\ldots,\theta^{n-1}$ on the unit sphere in -$T_pM$ in a neighborhood of $v/|v|$. Then $(r, \theta^{1}, ... ,\theta^{n-1})$ -are local coordinates at any point of the ray emanating from the origin in the -$v$ direction (except at $p$). Transferring these via $exp_p$ produces -local coordinates $(r,\theta^{1}, ... ,\theta^{n-1})$ along $\gamma$. Using -Gauss's lemma (Lemma 12 of Chapter 5 on p. 133 of ), we can -write the metric locally as - -$$ -g=dr^{2}+ -r^{2}h_{ij}(r,\theta)d\theta^{i}\otimes d\theta^{j}. -$$ - Then the -volume form - -$$ - -$$ -\begin{aligned} -dV &= \sqrt{det(g_{ij})}dr\wedge d\theta^{1} -\wedge\cdots\wedge -d\theta^{n-1}\\ -& = r^{n-1}\sqrt{det(h_{ij})}dr\wedge d\theta^{1} -\wedge\cdots\wedge d\theta^{n-1}. -\end{aligned} -$$ - -$$ - -**Lemma.** The *Laplacian operator* acting on scalar -functions on $M$ is given in local coordinates by - -$$ -\triangle = \frac{1}{\sqrt{det(g)}}\partial_i -\left(g^{ij}\sqrt{det(g)}\partial_j\right). -$$ - -**Proof.** Let us compute the derivative at a point $p$. We have -$$\frac{1}{\sqrt{det(g)}}\partial_i -\left(g^{ij}\sqrt{det(g)}\partial_j\right)f=g^{ij}\partial_i\partial_jf+\partial_ig^{ij}\partial_jf+ -\frac{1}{2}g^{ij}\partial_iTr(\tilde g)\partial_jf,$$ where $\tilde -g=g(p)^{-1}g$. On the other hand from the definition of the -Laplacian, Equation~(), and -Equation~() we have -$$\triangle -f=g^{ij}Hess(f)(\partial_i,\partial_j)=g^{ij}\left(\partial_i\partial_j(f)- -\nabla_{\partial_i}\partial_jf\right) -=g^{ij}\partial_i\partial_jf-g^{ij}\Gamma_{ij}^k\partial_kf.$$ Thus, -to prove the claim it suffices to show that -$$g^{ij}\Gamma_{ij}^k=-(\partial_ig^{ik}+\frac{1}{2}g^{ik}Tr(\partial_i\tilde g)).$$ -From the definition of the Christoffel symbols we have -$$g^{ij}\Gamma_{ij}^k=\frac{1}{2}g^{ij}g^{kl}(\partial_ig_{jl}+\partial_jg_{il}-\partial_lg_{ij}).$$ -Of course, $g^{ij}\partial_ig_{jl}=-\partial_ig^{ij}g_{jl}$, so that - $g^{ij}g^{kl}\partial_ig_{jl}=-\partial_ig^{ik}$. It follows by symmetry that $g^{ij}g^{jl}\partial_jg_{il} - =-\partial_ig^{ik}$. The last term is clearly $-\frac{1}{2}g^{ik}Tr(\partial_i\tilde g).$ - -Using Gaussian local coordinates near $p$, we have - -$$ - -$$ -\begin{aligned} -\triangle r &= -\frac{1}{r^{n-1}\sqrt{det(h)}}\partial_r\left(r^{n-1}\sqrt{det(h)}\right)\\ -&= \frac{n-1}{r} - + \partial_r\log\left(\sqrt{det(h)}\right). -\end{aligned} -$$ - -$$ - -From this one computes directly that - -$$ -\triangle r - = \frac{n-1}{r} - \frac{r}{3}Ric(v,v)+ O(r^{2}), -$$ - - where $v = \dot r(0)$, cf, p.265-268 of . So - $$\triangle r \leq - \frac{n-1}{r}\ \ when\ \ r \ll 1\ \ and\ Ric > 0.$$ -This local computation has the following global analogue. - -\begin{ex}(E.Calabi, 1958) -Let $f(x)=d(p,x)$ be the distance function from $p$. If $(M, g)$ has -$Ric \geq 0$, then - -$$ -\triangle f \leq \frac{n-1}{f} -$$ - - in the sense of distributions. -\end{ex} - [Compare , p. 284 Lemma 42]. - -**Remark.** -The statement that $\triangle f\leq \frac{n-1}{f}$ in *the sense -of distributions* (or equivalently *in the weak sense*) means -that for any non-negative test function $\phi$, that is to say for -any compactly supported $C^\infty$-function $\phi$, we have -$$\int_Mf\triangle \phi dvol\le \int_M\left(\frac{n-1}{f}\right)\phi dvol.$$ -Since the triangle inequality implies that $|f(x)-f(y)|\le d(x,y)$, it follows -that $f$ is Lipschitz, and hence that the restriction of $\nabla f$ to any -compact subset of $M$ is an $L^2$ one-form. Integration by parts then shows -that -$$\int_Mf\triangle -\phi dvol=-\int_M\langle \nabla f,\nabla\phi\rangle dvol.$$ - -Since $\abs{\nabla f} = 1$ and $\triangle f$ is the mean curvature -of the geodesic sphere $\partial B(x,r)$, $Ric(v,v)$ measures -the difference of the mean curvature between the standard Euclidean -sphere and the geodesic sphere in the direction $v$. Another -important geometric object is the shape operator associated to $f$, -denoted $S$. By definition it is the Hessian of $f$; i.e., -$S=\nabla^2f=Hess(f)$. - -## Basic curvature comparison results - -In this section we will recall some of the basic curvature -comparison results in Riemannian geometry. The reader can refer to -, Section 1 of Chapter 9 for details. - -We fix a point $p\in M$. For any real number $k\ge 0$ let $H^n_k$ denote the -simply connected, complete Riemannian $n$-manifold of constant sectional -curvature $-k$. Fix a point $q_k\in H^n_k$, and consider the exponential map -$exp_{q_k}\colon T_{q_k}(H^n_k)\to H^n_k$. This map is a global -diffeomorphism. Let us consider the pullback, $\tilde h_k$, of the Riemannian -metric on $H^n_k$ to $T_{q_k}H^n_k$. A formula for this tensor is easily given -in polar coordinates on $T_{q_k}(H^n_k)$ in terms of the following function. - -**Definition.** We define a function $sn_k$ as follows:$$sn_k(r)=\begin{cases} -r \ \ & if\ \k=0 \\ -\frac{1}{\sqrt{k}}sinh(\sqrt{k}r) \ \ & if \ \ k>0. -\end{cases}$$ - -The function $sn_{k}(r)$ is the solution to the equation - -$$ - -$$ -\begin{aligned} -\varphi'' - k\varphi &= 0,\\ -\varphi(0) &= 0,\\ -\varphi'(0) &= 1. -\end{aligned} -$$ - -$$ - -We define - $ct_{k}(r) = \frac{sn_{k}^{'}(r)}{\sqrt{k}sn_{k}(r)}$. - -Now we can compare manifolds of varying sectional curvature with -those of constant curvature. - -**Theorem.** -(Sectional Curvature Comparison) Fix $k\ge 0$. Let $(M, g)$ be a -Riemannian manifold with the property that $-k \leq K(P)$ for every -$2$-plane $P$ in $TM$. Fix a minimizing geodesic $\gamma\colon -[0,r_0)\to M$ parameterized at unit speed with $\gamma(0)=p$. Impose -Gaussian polar coordinates $(r,\theta^1,\ldots,\theta^{n-1})$ on a -neighborhood of $\gamma$ so that $g=dr^2+g_{ij}\theta^i\otimes -\theta^j$. Then for all $00$. For every $\epsilon>0$ there is $\delta>0$ -depending on $n$ and $\epsilon$ such that the following holds. -Suppose that $(M^n,g)$ is a complete Riemannian manifold of -dimension $n$ and that $p\in M$. Suppose that $|Rm(x)|\le -r^{-2}$ for all $x\in B(p,r)$. If the injectivity radius of $M$ at -$p$ is at least $\epsilon r$, then $Vol(B(p,r))\ge \delta -r^n$. - -**Proof.** Suppose that $|Rm(x)|\le r^{-2}$ for all $x\in B(p,r)$. -Replacing $g$ by $r^2 g$ allows us to assume that $r=1$. Without -loss of generality we can assume that $\epsilon\le 1$. The map $exp_p$ is a diffeomorphism on the ball $B(0,\epsilon)$ in the -tangent space, and by Theorem~ the volume of -$B(p,\epsilon)$ is at least that of the ball of radius $\epsilon$ in -the $n$-sphere of radius $1$. This gives a lower bound to the volume -of $B(p,\epsilon)$, and a fortiori to $B(p,1)$, in terms of $n$ and -$\epsilon$. - - We shall normally work with volume, which behaves nicely under Ricci -flow, but in order to take limits we need to bound the injectivity -radius away from zero. Thus, the more important, indeed crucial, -result for our purposes is the converse to the previous proposition; -see Theorem 4.3, especially Inequality (4.22), on page 46 of -, or see Theorem 5.8 on page 96 of -. - -**Theorem.** -Fix an integer $n>0$. For every $\epsilon>0$ there is $\delta>0$ -depending on $n$ and $\epsilon$ such that the following holds. -Suppose that $(M^n,g)$ is a complete Riemannian manifold of -dimension $n$ and that $p\in M$. Suppose that $|Rm(x)|\le -r^{-2}$ for all $x\in B(p,r)$. If $Vol(B(p,r))\ge \epsilon r^n$ then -the injectivity radius of $M$ at $p$ is at least $\delta r$. - -# Manifolds of non-negative curvature - -In studying singularity development in $3$-dimensional Ricci flows -one forms blow-up limits. By this we mean the following. One -considers a sequence of points $x_k$ in the flow converging to the -singularity. It will be the case that $R(x_k)$ tends to $\infty$ as -$k$ tends to $\infty$. We form a sequence of based Riemannian -manifolds labeled by $k$, where the $k^{th}$ Riemannian manifold is -obtained by taking the time-slice of $x_k$, rescaling its metric by -$R(x_k)$, and then taking $x_k$ as the base point. This creates a -sequence with the property that for each member of the sequence the -scalar curvature at the base point is one. Because of a pinching -result of Hamilton's (see Chapter~), if there is a -geometric limit of this sequence, or of any subsequence of it, then -that limit is non-negatively curved. Hence, it is important to -understand the basic properties of Riemannian manifolds of -non-negative curvature in order to study singularity development. In -this chapter we review the properties that we shall need. We suppose -that $M$ is non-compact and of positive (resp., non-negative) -curvature. The key to understanding these manifolds is the Busemann -function associated to a minimizing geodesic ray. - -## Busemann functions - -A geodesic ray $\lambda\colon [0,\infty)\to M$ is said to be *minimizing* if the restriction of $\lambda$ to every compact -subinterval of $[0,\infty)$ is a length-minimizing geodesic arc, -i.e., a geodesic arc whose length is equal to the distance between -its endpoints. Likewise, a geodesic line $\lambda\colon -(-\infty,\infty)\to M$ is said to be *minimizing* if its -restriction to every compact sub-interval of $\Ar$ is a length -minimizing geodesic arc. - -Clearly, if a sequence of minimizing geodesic arcs $\lambda_k$ -converges to a geodesic arc, then the limiting geodesic arc is also -minimizing. More generally, if $\lambda_k$ is a sequence of length -minimizing geodesic arcs whose initial points converge and whose -lengths go to infinity, then, after passing to a subsequence, there -is a limit which is a minimizing geodesic ray. (The existence of a -limit of a subsequence is a consequence of the fact that a geodesic -ray is determined by its initial point and its initial tangent -direction.) Similarly, if $I_k$ is an sequence of compact intervals -with the property that every compact subset of $\Ar$ is contained in -$I_k$ for all sufficiently large $k$, if for each $k$ the map -$\lambda_k\colon I_k\to M$ is a minimizing geodesic arc, and if -$lim_{k\rightarrow \infty}\lambda_k(0)$ exists, then, after -passing to a subsequence there is a limit which is a minimizing -geodesic line. Using these facts one establishes the following -elementary lemma. - -**Lemma.** -Suppose that $M$ is a complete, connected, non-compact Riemannian -manifold and let $p$ be a point of $M$. Then $M$ has a minimizing -geodesic ray emanating from $p$. If $M$ has more than one end, then -it has a minimizing line. - -**Definition.** Suppose that $\lambda\colon [0,\infty)\to M$ is a minimizing -geodesic ray with initial point $p$. For each $t\ge 0$ we consider -$B_{\lambda,t}(x)=d(\lambda(t),x)-t$. This is a family of functions -satisfying $|B_{\lambda,t}(x)-B_{\lambda,t}(y)|\le d(x,y)$. Since -$\lambda$ is a minimizing geodesic, $B_{\lambda,t}(p)=0$ for all -$t$. It follows that $B_{\lambda,t}(x)\ge -d(x,p)$ for all $x\in M$. -Thus, the family of functions $B_{\lambda,t}$ is pointwise bounded -below. The triangle inequality shows that for each $x\in M$ the -function $B_{\lambda,t}(x)$ is a non-increasing function of $t$. - It follows that, for -each $x\in M$, $lim_{t\rightarrow \infty} B_{\lambda,t}(x)$ -exists. We denote this limit by $B_\lambda(x)$. This is the *Busemann function* for $\lambda$. - -Clearly, $B_\lambda(x)\ge -d(x,\lambda(0))$. By equicontinuity -$B_\lambda(x)$ is a continuous function of $x$ and in fact a -Lipschitz function satisfying $|B_\lambda(x)-B_\lambda(y)|\le -d(x,y)$ for all $x,y\in X$. Clearly $B_\lambda(\lambda(s))=-s$ for -all $s\ge 0$. Since $B_\lambda$ is Lipschitz, $\nabla B_\lambda$ is -well-defined as an $L^2$-vector field. - -**Proposition.** -Suppose that $M$ is complete and of non-negative Ricci curvature. -Then, for any minimizing geodesic ray $\lambda$, the Busemann -function $B_\lambda$ satisfies $\Delta B_\lambda\le 0$ in the weak -sense. - -**Proof.** First notice that since $B_\lambda$ is Lipschitz, $\nabla B_\lambda$ -is an $L^2$-vector field on $M$. That is to say, $B_\lambda\in -W_loc^{1,2}$, i.e., $B_\lambda$ locally has one derivative in -$L^2$. Hence, there is a sequence of $C^\infty$-functions $f_n$ -converging to $B_\lambda$ in $W^{1,2}_{loc}$. Let $\varphi$ be a -test function (i.e., a compactly supported $C^\infty$-function). -Integrating by parts yields -$$-\int_M\langle \nabla -f_n,\nabla\varphi\rangle dvol=\int_Mf_n\triangle \varphi dvol.$$ Using the fact that $f_n$ converges to $B_\lambda$ in -$W^{1,2}_{loc}$ and taking limits yields -$$-\int_M\langle \nabla B_\lambda,\nabla \varphi\rangle dvol= \int_M -B_\lambda\triangle \varphi dvol.$$ - -Thus, to prove the proposition we need only show that if $\varphi$ -is a non-negative test function, then -$$-\int_M \langle\nabla B_\lambda,\nabla\varphi\rangle dvol\le 0.$$ -For a proof of this see - Proposition 1.1 and its proof on pp. 7 and 8 in . - -## Comparison results in non-negative curvature - -Let us review some elementary comparison results for manifolds of -non-negative curvature. These form the basis for Toponogov -theory, . For any pair -of points $x,y$ in a complete Riemannian manifold $s_{xy}$ denotes a -minimizing geodesic from $x$ to $y$. We set $|s_{xy}|=d(x,y)$ and -call it the *length* of the side. A *triangle* in a -Riemannian manifold consists of three vertices $a,b,c$ and three -sides $s_{ab}$,$s_{ac}$,$s_{bc}$. We denote by $\angle_a$ the angle -of the triangle at $a$, i.e., the angle at $a$ between the geodesic -rays $s_{ab}$ and $s_{ac}$. - -**Theorem.** **(Length comparison)** -Let $(M,g)$ be a manifold of non-negative curvature. Suppose that -$\triangle(a,b,c)$ is a triangle in $M$ and let -$\triangle(a',b',c')$ be a Euclidean triangle. -\begin{enumerate} -\item Suppose that the corresponding -sides of $\triangle(a,b,c)$ and $\triangle(a',b',c')$ have the same -lengths. Then the angle at each vertex of the Euclidean triangle is -no larger than the corresponding angle of $\triangle(a,b,c)$. -Furthermore, for any $\alpha$ and $\beta$ less than $|s_{ab}|$ and -$|s_{ac}|$ respectively, let $x$, resp. $x'$, be the point on -$s_{ab}$, resp. $s_{a'b'}$, at distance $\alpha$ from $a$, resp. -$a'$, and let $y$, resp. $y'$, be the point on $s_{ac}$, resp. -$s_{a'c'}$, at distance $\beta$ from $a$, resp. $a'$. Then -$d(x,y)\ge d(x',y')$. -\item Suppose that $|s_{ab}|=|s_{a'b'}|$, that -$|s_{ac}|=|s_{a'c'}|$ and that $\angle_a=\angle_{a'}$. Then -$|s_{b'c'}|\ge |s_{bc}|$. -\end{enumerate} - -See Fig.~. For a proof of this result see -Theorem 4.2 on page 161 of , or Theorem 2.2 on page 42 -of . - -One corollary is a monotonicity result. Suppose that -$\triangle(a,b,c)$ is a triangle in a complete manifold of -non-negative curvature. Define a function $EA(u,v)$ defined for -$0\le u\le |s_{ab}|$ and $0\le v\le |s_{ac}|$ as follows. For $u$ -and $v$ in the indicated ranges, let $x(u)$ be the point on $s_{ab}$ -at distance $u$ from $a$ and let $y(v)$ be the point of $s_{ac}$ at -distance $v$ from $a$. Let $EA(u,v)$ be the angle at $a'$ of the -Euclidean triangle with side lengths $|s_{a'b'}|=u$, $|s_{a'c'}|=v$ -and $|s_{b'c'}|=d(x(u),y(v))$. - -**Corollary.** -Under the assumptions of the previous theorem, $EA(u,v)$ is a -monotone non-increasing function of each variable $u$ and $v$ when -the other variable is held fixed. - - Suppose that $\alpha, \beta,\gamma$ are three geodesics -emanating from a point $p$ in a Riemannian manifold. Let $\angle_p -(\alpha,\beta)$, $\angle_p(\beta,\gamma)$ and -$\angle_p(\alpha,\gamma)$ be the angles of these geodesics at $p$ as -measured by the Riemannian metric. Then of course -$$\angle_p(\alpha,\beta)+\angle_p(\beta,\gamma)+\angle_p(\alpha,\gamma)\le -2\pi$$ since this inequality holds for the angles between straight -lines in Euclidean $n$-space. There is a second corollary of -Theorem~ which gives an analogous result for the -associated Euclidean angles. - -**Corollary.** -Let $(M,g)$ be a complete Riemannian manifold of non-negative -curvature. Let $p,a,b,c$ be four points in $M$ and let -$\alpha,\beta,\gamma$ be minimizing geodesic arcs from the point -$p$ to $a,b,c$ respectively. Let $T(a,p,b)$, $T(b,p,c)$ and -$T(c,p,a)$ be the triangles in $M$ made out of these minimizing -geodesics and minimizing geodesics between $a,b,c$. Let -$T(a',p',b')$, $T(b',p',c')$ and $T(c',p',a')$ be planar triangles -with the same side lengths. Then -$$\angle_{p'}T(a',p',b')+\angle_{p'}T(b',p',c')+\angle_{p'}T(c',p',a')\le 2\pi.$$ - -**Proof.** Consider the sum of these angles as the geodesic arcs in $M$ are -shortened without changing their direction. By the first property of -Theorem~ the sum of the angles of these triangles -is a monotone decreasing function of the lengths. Of course, the -limit as the lengths all go to zero is the corresponding Euclidean -angle. The result is now clear. - -## The soul theorem - -A subset $X$ of a Riemannian manifold $(M,g)$ is said to be *totally convex* if every geodesic segment with endpoints in $X$ is -contained in $X$. Thus, a point $p$ in $M$ is totally convex if and -only if there is no broken geodesic arc in $M$ broken exactly at -$x$. - -**Theorem.** (Cheeger-Gromoll, see - and ) Suppose that -$(M,g)$ is a connected, complete, non-compact Riemannian manifold of -non-negative sectional curvature. Then $M$ contains a soul $S\subset -M$. By definition a *soul* is a compact, totally -geodesic, totally convex submanifold (automatically of positive -codimension). Furthermore, $M$ is diffeomorphic to the total space -of the normal bundle of the $S$ in $M$. If $(M,g)$ has positive -curvature, then any soul for it is a point, and consequently $M$ is -diffeomorphic to $\Ar^n$. - -**Remark.** We only use the soul theorem for manifolds with positive curvature -and the fact that any soul of such a manifold is a point. A proof of -this result first appears in . - -The rest of this section is devoted to a sketch of the proof of this -result. Our discussion follows closely that in -starting on p.~349. We shall need more information about complete, -non-compact manifolds of non-negative curvature, so we review a -little of their theory as we sketch the proof of the soul theorem. - -**Lemma.** Let $(M,g)$ be a complete, non-compact Riemannian manifold of -non-negative sectional curvature and let $p\in M$. For every -$\epsilon>0$ there is a compact subset $K=K(p,\epsilon)\subset M$ -such that for all points $q\notin K$, if $\gamma$ and $\mu$ are -minimizing geodesics from $p$ to $q$, then the angle that $\gamma$ -and $\mu$ make at $q$ is less than $\epsilon$. - -See Fig.~. - -**Proof.** The proof is by contradiction. Fix $0<\epsilon<1$ sufficiently small -so that $cos(\epsilon/2)<1-\epsilon^2/12$. Suppose that there -is a sequence of points $q_n$ tending to infinity such that for each -$n$ there are minimizing geodesics $\gamma_n$ and $\mu_n$ from $p$ -to $q_n$ making angle at least $\epsilon$ at $q_n$. For each $n$ let -$d_n=d(p,q_n)$. By passing to a subsequence we can suppose that for -all $n$ and $m$ the cosine of the angle at $p$ between $\gamma_n$ -and $\gamma_m$ at least $1-\epsilon^2/24$, and the cosine of the -angle at $p$ between $\mu_n$ and $\mu_m$ is at least -$1-\epsilon^2/24$. We can also assume that for all $n\ge 1$ we have -$d_{n+1}\ge (100/\epsilon^2) d_n$. Let $\delta_n=d(q_n,q_{n+1})$. -Applying the first Toponogov property at $p$, we see that -$\delta_n^2\le d_n^2+d_{n+1}^2-2d_nd_{n+1}(1-\epsilon^2/24)$. -Applying the same property at $q_n$ we have -$$d_{n+1}^2\le d_n^2+\delta_n^2-2d_n\delta_ncos(\theta),$$ where -$\theta\le \pi$ is the angle at $q_n$ between $\gamma_n$ and a -minimal geodesic joining $q_n$ to $q_{n+1}$. Thus, -$$cos(\theta)\le -\frac{d_n-d_{n+1}(1-\epsilon^2/24)}{\delta_n}.$$ By the triangle -inequality (and the fact that $\epsilon<1$) we have $\delta_n\ge -(99/\epsilon)d_n$ and $\delta_n\ge d_{n+1}(1-(\epsilon^2/100))$. -Thus, -$$cos(\theta)\le -\epsilon^2/99-(1-\epsilon^2/24)/(1-(\epsilon^2/100))<-(1-\epsilon^2/12).$$ -This implies that $cos(\pi-\theta)>(1-\epsilon^2/12)$, which -implies that $\pi-\theta<\epsilon/2$. That is to say, the angle at -$q_n$ between $\gamma_n$ and a shortest geodesic from $q_n$ to -$q_{n+1}$ is between $\pi-\epsilon/2$ and $\pi$. By symmetry, the -same is true for the angle between $\mu_n$ and the same shortest -geodesic from $q_n$ to $q_{n+1}$. Thus, the angle between $\gamma_n$ -and $\mu_n$ at $q_n$ is less than $\epsilon$, contradicting our -assumption. - -**Corollary.** -Let $(M,g)$ be a complete, non-compact manifold of non-negative -sectional curvature. Let $p\in M$ and define a function $f\colon -M\to \Ar$ by $f(q)=d(p,q)$. Then there is $R<\infty$ such that for -$R\le sR$, the pre-image $f^{-1}([s,s'])$ is homeomorphic to -$f^{-1}(s)\times[s,s']$ and that the end $f^{-1}\left([s,\infty)\right)$ is -homeomorphic to $f^{-1}(s)\times [s,\infty)$. - -In a complete, non-compact $n$-manifold of positive curvature any soul is a -point. While the proof of this result uses the same ideas as discussed above, -we shall not give a proof. Rather we refer the reader to Theorem 84 of - on p. 349. A soul has the property that if two minimal -geodesics emanate from $p$ and end at the same point $q\not= p$, then the angle -that they make at $q$ is less than $\pi/2$. Also, of course, the exponential -mapping is a diffeomorphism sufficiently close to the soul. Applying the above -lemma and a standard compactness argument, we see that in fact there is -$\epsilon>0$ such that all such pairs of minimal geodesics from $p$ ending at -the same point make angle less than $\pi/2-\epsilon$ at that point. Hence, in -this case there is a vector field $X$ on all of $M$ vanishing only at the soul, -and agreeing with the gradient of the distance function near the soul, so that -the distance function from $p$ is strictly increasing to infinity along each -flow line of $X$ (except the fixed point). Using $X$ one establishes that $M$ -is diffeomorphic to $\Ar^n$. It also follows that all the level surfaces -$f^{-1}(s)$ for $s>0$ are homeomorphic to $S^{n-1}$ and for $00$ -and for any $x\in M$ we have $d(x,\lambda(t))+d(x,\lambda(-s))\ge -s+t$, and hence $B_+(x)+B_-(x)\ge 0$. Clearly, $B_+(x)+B_-(x)=0$ for -any $x$ in the image of $\lambda$. Thus, the function $B_++B_-$ is -everywhere $\ge 0$, vanishes at at a least one point and satisfies -$\Delta (B_++B_-)\le 0$ in the weak sense. This is exactly the -set-up for the maximum principle, cf. , p. 279. - -**Theorem.** **(The Maximum Principle)** -Let $f$ be a real-valued continuous function on a connected -Riemannian manifold with $\Delta f\ge 0$ in the weak sense. Then $f$ -is locally constant near any local maximum. In particular, if $f$ -achieves its maximum then it is a constant. - -Applying this result to $-(B_++B_-)$, we see that $B_++B_-=0$, so that -$B_-=-B_+$. It now follows that $\Delta B_+=0$ in the weak sense. By standard -elliptic regularity results this implies that $B_+$ is a smooth harmonic -function. - -Next, we show that for all $x\in M$ we have $|\nabla B_+(x)|=1$. -Fix $x\in M$. Take a sequence $t_n$ tending to infinity and consider -minimizing geodesics $\mu_{+,n}$ from $x$ to $\lambda_+(t_n)$. By -passing to a subsequence we can assume that there is a limit as -$n\rightarrow \infty$. This limit is a minimizing geodesic ray -$\mu_+$ from $x$, which we think of as being `asymptotic at -infinity' to $\lambda_+$. Similarly, we construct a minimizing -geodesic ray $\mu_-$ from $x$ asymptotic at infinity to $\lambda_+$. - Since -$\mu_+$ is a minimal geodesic ray, it follows that for any $t$ the -restriction $\mu_+|_{[0,t]}$ is the unique length minimizing -geodesic from $x$ to $\mu_+(t)$ and that $\mu_+(t)$ is not a -conjugate point along $\mu_+$. It follows by symmetry that $x$ is -not a conjugate point along the reversed geodesic $-\mu_+|_{[0,t]}$ -and hence that $x\in U_{\mu_+(t)}$. This means that the function -$d(\mu_+(t),\cdot)$ is smooth at $x$ with gradient equal to the unit -tangent vector in the negative direction at $x$ to $\mu_+$, and -consequently that $B_{\mu_+,t}$ is smooth at $x$. Symmetrically, for -any $t>0$ the function $B_{\mu_-,t}$ is smooth at $x$ with the -opposite gradient. Notice that these gradients have norm one. We -have -$$B_{\mu_+,t}+B_+(x)\ge B_+=-B_-\ge -(B_{\mu_-,t}+B_-(x)).$$ -Of course, $B_{\mu_+,t}(x)=0$ and $B_{\mu_-,t}(x)=0$, so that -$$B_{\mu_+,t}(x)+B_+(x)=-(B_{\mu_-,t}(x)+B_-(x)).$$ -This squeezes $B_+$ between two smooth functions with the same value -and same gradient at $x$ and hence shows that $B_+$ is $C^1$ at $x$ -and $|\nabla B_+(x)|$ is of norm one. - -Thus, $B$ defines a smooth Riemannian submersion from $M\to \Ar$ -which implies that $M$ is isometric to a product of the fiber over -the origin with $\Ar$. - -This result together with Lemma~ shows that if $M$ -satisfies the hypothesis of the theorem, then it can be written as -a Riemannian product $M=N\times \Ar$. Since $M$ has at least two -ends, it follows immediately that $N$ is compact. This completes the -proof of the theorem. -\end{proof} - -## $\epsilon$-necks - -Certain types of (incomplete) Riemannian manifolds play an -especially important role in our analysis. The purpose of this -section is to introduce these manifolds and use them to prove one -essential result in Riemannian geometry. - -For all of the following definitions we fix $0<\epsilon<1/2$. Set -$k$ equal to the greatest integer less than or equal to -$\epsilon^{-1}$. In particular, $k\ge 2$. - -**Definition.** -Suppose that we have a fixed metric $g_0$ on a manifold $M$ and an -open submanifold $X\subset M$. We say that another metric $g$ on $X$ -is {\em within $\epsilon$ of $g_0|_X$ in the -$C^{[1/\epsilon]}$-topology} if, setting $k=[1/\epsilon]$ we have - -$$ -sup_{x\in X}\left(|g(x)-g_0(x)|_{g_0}^2+ -\sum_{\ell=1}^k|\nabla_{g_0}^\ell g(x)|_{g_0}^2\right)< -\epsilon^2, -$$ - where the covariant derivative -$\nabla^\ell_{g_0}$ is the Levi-Civita connection of $g_0$ and -norms are the pointwise $g_0$-norms on -$$Sym^2T^*M\otimes \underbrace{T^*M\otimes\cdots\otimes -T^*M}_{\ell-times}.$$ - -More generally, given two smooth families of metrics $g(t)$ and -$g_0(t)$ on $M$ defined for $t$ in some interval $I$ we say that -the family $g(t)|_X$ is within $\epsilon$ of the family $g_0(t)|_X$ -in the $C^{[1/\epsilon]}$-topology if we have -$$sup_{(x,t)\in X\times I}\left(\left| g(x,t) --g_0(x,t)\right|_{g_0(t)}^2 +\sum_{\ell=1}^k\left|\nabla_{g_0}^\ell -g(x,t)\right|_{g_0}^2\right)<\epsilon^2.$$ - -**Remark.** Notice that if we view a one-parameter family of metrics $g(t)$ as a -curve in the space of metrics on $X$ with the -$C^{[1/\epsilon]}$-topology then this is the statement that the two -paths are pointwise within $\epsilon$ of each other. It says nothing -about the derivatives of the paths, or equivalently about the time -derivatives of the metrics and of their covariant derivatives. We -will always be considering paths of metrics satisfying the Ricci -flow equation. In this context two one-parameter families of metrics -that are close in the $C^{2k}$-topology exactly when the $r^{th}$ -time derivatives of the $s^{th}$-covariant derivatives are close for -all $r,s$ with $s+2r\le 2k$. - -The first object of interest is one that, up to scale, is close to a -long, round cylinder. - -**Definition.** -Let $(N,g)$ be a Riemannian manifold and $x\in N$ a point. Then {\em -an $\epsilon$-neck structure on $(N,g)$ -centered at $x$} consists of a diffeomorphism -$$\varphi\colon S^2\times (-\epsilon^{-1},\epsilon^{-1})\to N,$$ -with $x\in \varphi(S^2\times\{0\})$, such that the metric -$R(x)\varphi^*g$ is within $\epsilon$ in the -$C^{[1/\epsilon]}$-topology of the product of the usual Euclidean -metric on the open interval with the metric of constant Gaussian -curvature $1/2$ on $S^2$. We also use the terminology *$N$ is an -$\epsilon$-neck centered at $x$*. The image under $\varphi$ of the -family of submanifolds $S^2\times \{t\}$ is called the *family -of $2$-spheres of the $\epsilon$-neck*. The submanifold -$\varphi(S^2\times \{0\})$ is called *the central $2$-sphere* of -the $\epsilon$-neck structure. We denote by $s_N\colon N\to \Ar$ -the composition $p_2\circ \varphi^{-1}$, where $p_2$ is the -projection of $S^2\times (-\epsilon^{-1},\epsilon^{-1})$ to the -second factor. There is also the vector field $\partial/\partial -s_N$ on $N$ which is $\varphi_*$ of the standard vector field in the -interval-direction of the product. We also use the terminology of -the *plus* and *minus* end of the $\epsilon$-neck in the -obvious sense. The opposite (or reversed) $\epsilon$-neck structure -is the one obtained by composing the structure map with $Id_{S^2}\times -1$. We define the - *positive half of the neck* to be the region -$s_N^{-1}(0,\epsilon^{-1})$ and the *negative half* to be the -region $s_N^{-1} (-\epsilon^{-1},0)$. For any other fraction, e.g., -the left-hand three-quarters, the right-hand one-quarter, there are -analogous notions, all measured with respect to $s_N\colon N\to -(-\epsilon^{-1},\epsilon^{-1})$. We also use the terminology the -middle one-half, or middle one-third of the $\epsilon$-neck; again -these regions have their obvious meaning when measured via $s_N$. - -*An $\epsilon$-neck* in a Riemannian manifold $X$ is a -codimension-zero submanifold $N$ and an $\epsilon$-structure on $N$ -centered at some point $x\in N$. - -The *scale* of an $\epsilon$-neck $N$ centered -at $x$ is $R(x)^{-1/2}$. The scale of $N$ is denoted $r_N$. Intuitively, this -is a measure of the radius of the cross-sectional $S^2$ in the neck. In fact, -the extrinsic diameter of any $S^2$ factor in the neck is close to $\sqrt{2}\pi -r_N$. See Fig.~ in the introduction. - -Here is the result that will be so important in our later arguments. - -**Proposition.** The following -holds for any $\epsilon>0$ sufficiently small. - Let $(M,g)$ be a complete, positively curved Riemannian -$3$-manifold. Then $(M,g)$ does not contain $\epsilon$-necks of -arbitrarily small scale. - -**Proof.** The result is obvious if $M$ is compact, so we assume that $M$ is -non-compact. Let $p\in M$ be a soul for $M$ (Theorem~), -and let $f$ be the distance function from $p$. Then $f^{-1}(s)$ is -connected for all $s>0$. - -**Lemma.** -Suppose that $\epsilon>0$ is sufficiently small that Lemma~ from -the appendix holds. Let $(M,g)$ be a non-compact $3$-manifold of positive -curvature and let $p\in M$ be a soul for it. Then for any $\epsilon$-neck $N$ -disjoint from $p$ - the central $2$-sphere of $N$ separates the soul from the end of the manifold. In particular, if - two $\epsilon$-necks $N_1$ and $N_2$ in $M$ are disjoint from each other and from $p$, then -the central $2$-spheres of $N_1$ and $N_2$ are the boundary -components of a region in $M$ diffeomorphic to $S^2\times I$. - -\begin{proof} -Let $N$ be an $\epsilon$-neck disjoint from $p$. By -Lemma~ for any point $z$ in the middle third of $N$, the -boundary of the metric ball $B(p,d(z,p))$ is a topological -$2$-sphere in $N$ isotopic in $N$ to the central $2$-sphere of $N$. -Hence, the central $2$-sphere separates the soul from the end of -$M$. The second statement follows immediately by applying this to -$N_1$ and $N_2$. - -Let $N_1$ and $N_2$ be disjoint $\epsilon$-necks, each disjoint from -the soul. By the previous lemma, the central $2$-spheres $S_1$ and -$S_2$ of these necks are smoothly isotopic to each other and they -are the boundary components of a region diffeomorphic to $S^2\times -I$. Reversing the indices if necessary we can assume that $N_2$ is -closer to $\infty$ than $N_1$, i.e., further from the soul. -Reversing the directions of the necks if necessary, we can arrange -that for $i=1,2$ the function $s_{N_i}$ is increasing as we go away -from the soul. We define $C^\infty$- functions $\psi_i$ on $N_i$, -functions depending only on $s_{N_i}$, as follows. The function -$\psi_1$ is zero on the negative side of the middle third of $N_1$ -and increases to be identically one on the positive side of the -middle third. The function $\psi_2$ is one on the negative side of -the middle third of $N_2$ and decreases to be zero on the positive -side. We extend $\psi_1,\psi_2$ to a function $\psi$ defined on all -of $M$ by requiring that it be identically one on the region $X$ -between $N_1$ and $N_2$ and to be identically zero on $M\setminus -(N_1\cup X\cup N_2)$. - -Let $\lambda$ be a geodesic ray from the soul of $M$ to infinity, -and $B_\lambda$ its Busemann function. Let $N$ be any -$\epsilon$-neck disjoint from the soul, with $ s_N$ direction chosen -so that it points away from the soul. At any point of the middle -third of $N$ where $B_\lambda$ is smooth, $\nabla B_\lambda$ is a -unit vector in the direction of the unique minimal geodesic ray from -the end of $\lambda$ to this point. Invoking Lemma~ -from the appendix we see that at such points $\nabla B_\lambda$ is -close to $-R(x)^{1/2}\partial/\partial s_N$, where $x\in N$ is the -center of the $\epsilon$-neck. Since $\nabla B_\lambda$ is $L^2$ its -non-smooth points have measure zero and hence, the restriction of -$\nabla B_\lambda$ to the middle third of $N$ is close in the -$L^2$-sense to $-R(x)^{1/2}\partial/\partial s_N$. - -Applying this to $N_1$ and $N_2$ we see that - -$$ - -\int_M\langle \nabla B_\lambda,\nabla\psi\rangle dvol=\left(\alpha_2R(x_2)^{-1}-\alpha_1R(x_1)^{-1}\right)Vol_{h_0}(S^2)), -$$ - where $h(0)$ is the round metric of -scalar curvature $1$ and where each of $\alpha_1$ and $\alpha_2$ -limits to $1$ as $\epsilon$ goes to zero. Since $\psi\ge 0$, -Proposition~ tells us that the left-hand side of -Equation~() must be $\ge 0$. This shows that, -provided that $\epsilon$ is sufficiently small, $R(x_2)$ is bounded -above by $2R(x_1)$. This completes the proof of the proposition. -\end{proof} - -**Corollary.** -Fix $\epsilon>0$ sufficiently small so that Lemma~ holds. Then -there is a constant $C<\infty$ depending on $\epsilon$ such that the following -holds. Suppose that $M$ is a non-compact $3$-manifold of positive sectional -curvature. Suppose that $N$ is an $\epsilon$-neck in $M$ centered at a point -$x$ and disjoint from a soul $p$ of $M$. Then for any $\epsilon$-neck $N'$ that -is separated from $p$ by $N$ with center $x'$ we have $R(x')\le CR(x)$. - -## Forward difference quotients - -Let us review quickly some standard material on forward difference quotients. - -Let $f\colon [a,b]\to \Ar$ be a continuous function on an interval. -We say that the {\em forward difference quotient of $f$ at a point -$t\in [a,b)$, denoted $\frac{df}{dt}(t)$, is less than -$c$} provided that -$$\overlinelim_{\triangle t\rightarrow -0^+}\frac{f(t+\triangle t)-f(t)}{\triangle t}\le c.$$ We say that it -is greater than or equal to $c'$ if $$c'\le \underlinelim_{\triangle t\rightarrow 0^+}\frac{f(t+\triangle -t)-f(t)}{\triangle t}.$$ Standard comparison arguments show: - -**Lemma.** -Suppose that $f\colon [a,b]\to \Ar$ is a continuous function. Suppose that -$\psi$ is a $C^1$-function on $[a,b]\times \Ar$ and suppose that -$\frac{df}{dt}(t)\le \psi(t,f(t))$ for every $t\in [a,b)$ in the sense of -forward difference quotients. Suppose also that there is a function $G(t)$ -defined on $[a,b]$ that satisfies the differential equation -$G'(t)=\psi(t,G(t))$ and has $f(a)\le G(a)$. Then $f(t)\le G(t)$ for all $t\in -[a,b]$. - -The application we shall make of these results is the following. - -**Proposition.** -Let $M$ be a smooth manifold with a smooth vector field $\chi$ and a -smooth function $**t**\colon M\to [a,b]$ with $\chi(**t**)=1$. -Suppose also that $F\colon M\to \Ar$ is a smooth function with the -properties: -\begin{enumerate} -\item for each $t_0\in [a,b]$ the restriction of $F$ to the level set $**t**^{-1}(t_0)$ achieves its maximum, and \item the subset ${\mathcal Z}$ of $M$ -consisting of all $x$ for which $F(x)\ge F(y)$ for all $y\in **t**^{-1}(**t**(x))$ is a compact set. -\end{enumerate} - Suppose also that at each $x\in {\mathcal Z}$ we have $\chi(F(x))\le \psi(**t**(x),F(x))$. Set $F_max(t)=max_{x\in **t**^{-1}(t)}F(x)$. Then $F_max(t)$ is a continuous function and -$$\frac{dF_max}{dt}(t)\le \psi(t,F_max(t))$$ -in the sense of forward difference quotients. - Suppose that $G(t)$ satisfies the -differential equation -$$G'(t)=\psi(t,G(t))$$ and has initial condition $F_max(a)\le G(a)$. Then for all $t\in [a,b]$ we have -$$F_max(t)\le G(t).$$ - -**Proof.** Under the given hypothesis it is a standard and easy exercise to -establish the statement about the forward difference quotient of -$F_max$. The second statement then is an immediate corollary -of the previous result. diff --git a/projects/poincare-conjecture/.astrolabe/docs/02-preliminaries-from-riemannian-geometry.mdx b/projects/poincare-conjecture/.astrolabe/docs/02-preliminaries-from-riemannian-geometry.mdx new file mode 100644 index 00000000..d5145946 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/docs/02-preliminaries-from-riemannian-geometry.mdx @@ -0,0 +1,887 @@ + + +# Chapter 1 -- Preliminaries from Riemannian geometry + +In this chapter we will recall some basic facts in Riemannian +geometry. For more details we refer the reader to [doCarmo] and +[Petersen]. Throughout, we always adopt Einstein's summation +convention on repeated indices and 'manifold' means a paracompact, +Hausdorff, smooth manifold. + +## Riemannian metrics and the Levi-Civita connection + +Let $M$ be a manifold and let $p$ be a point of $M$. Then $TM$ +denotes the tangent bundle of $M$ and $T_pM$ is the tangent space at +$p$. Similarly, $T^*M$ denotes the cotangent bundle of $M$ and +$T^*_pM$ is the cotangent space at $p$. For any vector bundle +${\mathcal V}$ over $M$ we denote by $\Gamma({\mathcal V})$ the +vector space of smooth sections of ${\mathcal V}$. + +\entryblock{0c6b379e7dca} + +Using a partition of unity one can easily see that any manifold +admits a Riemannian metric. A Riemannian metric on $M$ allows us to +measure lengths of smooth paths in $M$ and hence to define a +distance function by setting $d(p,q)$ equal to the infimum of the +lengths of smooth paths from $p$ to $q$. This makes $M$ a metric +space. For a point $p$ in a Riemannian manifold $(M,g)$ and for $r> +0$ we denote the metric ball of radius $r$ centered at $p$ in $M$ by +$B(p,r)$ or by $B_g(p,r)$ if the metric needs specifying or +emphasizing. It is defined by + +$$ +B(p,r)=\{q\in M\left|\right. d(p,q) 0$, such that the restriction of $\mathit{exp}_p$ to the ball +$B_{g|T_pM}(0,r_0)$ in $T_pM$ is a diffeomorphism onto +$B_g(p,r_0)$. Fix $g$-orthonormal linear coordinates on $T_pM$. +Transferring these coordinates via $\mathit{exp}_p$ to coordinates on +$B(p,r_0)$ gives us *Gaussian normal coordinates* on $B(p,r_0)\subset M$. + +Suppose now that $M$ is complete, and fix a point $p\in M$. For +every $q\in M$, there is a length-minimizing path from $p$ to $q$. +When parameterized at constant speed equal to its length, this path +is a geodesic with domain interval $[0,1]$. Consequently, $\mathit{exp}_p\colon T_pM\to M$ is onto. The differential of the exponential +mapping is given by Jacobi fields: + Let +$\gamma\colon [0,1]\to M$ be a geodesic from $p$ to $q$, and let + $X\in T_pM$ be $\gamma'(0)$. Then the exponential +mapping at $p$ is a smooth map from $T_p(M)\to M$ sending $X$ to +$q$. Fix $Z\in T_pM$. Then there is a unique Jacobi field $Y_Z$ +along $\gamma$ with $\nabla_XY_Z(0)=Z$. The association $Z\mapsto +Y_Z(1)\in T_qM$ is a linear map from $T_p(M)\to T_qM$. Under the +natural identification of $T_pM$ with the tangent plane to $T_pM$ at +the point $Z$, this linear mapping is the differential of $\mathit{exp}_p\colon T_pM\to M$ at the point $X\in T_pM$. + +\entryblock{8141585de23a} + +**Proof.** Of course, $\mathit{exp}_p(t_0X(0))=\gamma(t_0)$. According to the +previous discussion, the kernel of the differential of the +exponential mapping at $t_0X(0)$ is identified with the space of +Jacobi fields along $\gamma$ vanishing at $\gamma(t_0)$. According +to \entryref{50646105e5e0} the only such Jacobi field is the +trivial one. Hence, the differential of $\mathit{exp}_p$ at $t_0X(0)$ +is an isomorphism, completing the proof. + +\entryblock{19afab2e3115} + +It follows from \entryref{8141585de23a} that $U\subset T_pM$ is a +star-shaped open neighborhood of $0\in T_pM$. + +\entryblock{26a127323e5a} + +For a proof see p. 139 of [Petersen]. + +\entryblock{8ebd93835797} + +Suppose that $\mathit{inj}_M(p)=r$. There are two possibilities: +Either there is a broken, closed geodesic through $p$, broken only +at $p$, of length $2r$, or there is a geodesic $\gamma$ of length +$r$ emanating from $p$ whose endpoint is a conjugate point along +$\gamma$. The first case happens when the exponential mapping is not +one-to-one of the closed ball of radius $r$ in $T_pM$, and the +second happens when there is a tangent vector in $T_pM$ of length +$r$ at which $\mathit{exp}_p$ is not a local diffeomorphism. + +## Computations in Gaussian normal coordinates + +In this section we compute the metric and the Laplacian (on +functions) in local Gaussian coordinates. A direct computation shows +that in Gaussian normal coordinates on a metric ball about $p\in M$ +the metric takes the form + +$$ +\begin{aligned} +g_{ij}(x)& = & \delta_{ij} + \frac{1}{3}R_{iklj}x^kx^l + +\frac{1}{6}R_{iklj,s}x^kx^lx^s \\ +& & +(\frac{1}{20}R_{iklj,st} +\frac{2}{45}\sum_m +R_{iklm}R_{jstm})x^kx^lx^sx^t + O(r^{5}), \nonumber +\end{aligned} +$$ + +where $r$ is the distance from $p$. (See, for example Proposition +3.1 on page 41 of [Sakai], with the understanding that, with +the conventions there, the quantity $R_{ijkl}$ there differs by sign +from ours.) + +Let $\gamma$ be a geodesic in $M$ emanating from $p$ in the direction $v$. +Choose local coordinates $\theta^1,\ldots,\theta^{n-1}$ on the unit sphere in +$T_pM$ in a neighborhood of $v/|v|$. Then $(r, \theta^{1}, ... ,\theta^{n-1})$ +are local coordinates at any point of the ray emanating from the origin in the +$v$ direction (except at $p$). Transferring these via $\mathit{exp}_p$ produces +local coordinates $(r,\theta^{1}, ... ,\theta^{n-1})$ along $\gamma$. Using +Gauss's lemma (Lemma 12 of Chapter 5 on p. 133 of [Petersen]), we can +write the metric locally as + +$$ +g=dr^{2}+ +r^{2}h_{ij}(r,\theta)d\theta^{i}\otimes d\theta^{j}. +$$ + + Then the +volume form + +$$ +\begin{aligned} +dV &= \sqrt{\mathit{det}(g_{ij})}dr\wedge d\theta^{1} +\wedge\cdots\wedge +d\theta^{n-1}\\ +& = r^{n-1}\sqrt{\mathit{det}(h_{ij})}dr\wedge d\theta^{1} +\wedge\cdots\wedge d\theta^{n-1}. +\end{aligned} +$$ + +\entryblock{0cf778ecc2ec} + +**Proof.** Let us compute the derivative at a point $p$. We have + +$$ +\frac{1}{\sqrt{\mathit{det}(g)}}\partial_i +\left(g^{ij}\sqrt{\mathit{det}(g)}\partial_j\right)f=g^{ij}\partial_i\partial_jf+\partial_ig^{ij}\partial_jf+ +\frac{1}{2}g^{ij}\partial_iTr(\tilde g)\partial_jf, +$$ + + where $\tilde +g=g(p)^{-1}g$. On the other hand from the definition of the +Laplacian, Equation (1.4), and +Equation (\entryref{98af9c99c3c2}) we have + +$$ +\triangle +f=g^{ij}\mathit{Hess}(f)(\partial_i,\partial_j)=g^{ij}\left(\partial_i\partial_j(f)- +\nabla_{\partial_i}\partial_jf\right) +=g^{ij}\partial_i\partial_jf-g^{ij}\Gamma_{ij}^k\partial_kf. +$$ + + Thus, +to prove the claim it suffices to show that + +$$ +g^{ij}\Gamma_{ij}^k=-(\partial_ig^{ik}+\frac{1}{2}g^{ik}\mathit{Tr}(\partial_i\tilde g)). +$$ + +From the definition of the Christoffel symbols we have + +$$ +g^{ij}\Gamma_{ij}^k=\frac{1}{2}g^{ij}g^{kl}(\partial_ig_{jl}+\partial_jg_{il}-\partial_lg_{ij}). +$$ + +Of course, $g^{ij}\partial_ig_{jl}=-\partial_ig^{ij}g_{jl}$, so that + $g^{ij}g^{kl}\partial_ig_{jl}=-\partial_ig^{ik}$. It follows by symmetry that $g^{ij}g^{jl}\partial_jg_{il} + =-\partial_ig^{ik}$. The last term is clearly $-\frac{1}{2}g^{ik}\mathit{Tr}(\partial_i\tilde g).$ + +Using Gaussian local coordinates near $p$, we have + +$$ +\begin{aligned} +\triangle r &= +\frac{1}{r^{n-1}\sqrt{\mathit{det}(h)}}\partial_r\left(r^{n-1}\sqrt{\mathit{det}(h)}\right)\\ +&= \frac{n-1}{r} + + \partial_r\log\left(\sqrt{\mathit{det}(h)}\right). +\end{aligned} +$$ + +From this one computes directly that + +$$ +\triangle r + = \frac{n-1}{r} - \frac{r}{3}\mathit{Ric}(v,v)+ O(r^{2}), +$$ + + where $v = \dot r(0)$, cf, p.265-268 of [Petersen]. So + +$$ +\triangle r \leq + \frac{n-1}{r}\ \ \mathit{when\ \} r \ll 1\ \ \mathit{and\} \mathit{Ric} > 0. +$$ + +This local computation has the following global analogue. + +\entryblock{53758e75bb17} + +[Compare [Petersen], p. 284 Lemma 42]. + +\entryblock{9f5db77333af} + +Since $\abs{\nabla f} = 1$ and $\triangle f$ is the mean curvature +of the geodesic sphere $\partial B(x,r)$, $\mathit{Ric}(v,v)$ measures +the difference of the mean curvature between the standard Euclidean +sphere and the geodesic sphere in the direction $v$. Another +important geometric object is the shape operator associated to $f$, +denoted $S$. By definition it is the Hessian of $f$; i.e., +$S=\nabla^2f=\mathit{Hess}(f)$. + +## Basic curvature comparison results + +In this section we will recall some of the basic curvature +comparison results in Riemannian geometry. The reader can refer to +[Petersen], Section 1 of Chapter 9 for details. + +We fix a point $p\in M$. For any real number $k\ge 0$ let $H^n_k$ denote the +simply connected, complete Riemannian $n$-manifold of constant sectional +curvature $-k$. Fix a point $q_k\in H^n_k$, and consider the exponential map +$\mathit{exp}_{q_k}\colon T_{q_k}(H^n_k)\to H^n_k$. This map is a global +diffeomorphism. Let us consider the pullback, $\tilde h_k$, of the Riemannian +metric on $H^n_k$ to $T_{q_k}H^n_k$. A formula for this tensor is easily given +in polar coordinates on $T_{q_k}(H^n_k)$ in terms of the following function. + +\entryblock{b4ef6801b3e2} + +The function $\mathit{sn}_{k}(r)$ is the solution to the equation + +$$ +\begin{aligned} +\varphi'' - k\varphi &= 0,\\ +\varphi(0) &= 0,\\ +\varphi'(0) &= 1. +\end{aligned} +$$ + +We define + $\mathit{ct}_{k}(r) = \frac{\mathit{sn}_{k}^{'}(r)}{\sqrt{k}\mathit{sn}_{k}(r)}$. + +Now we can compare manifolds of varying sectional curvature with +those of constant curvature. + +\entryblock{95de178b529d} + +There is also an analogous result for a positive upper bound to the +sectional curvature, but in fact all we shall need is the local +diffeomorphism property of the exponential mapping. + +\entryblock{4a0173b99805} + +There is a crucial comparison result for volume which involves the +Ricci curvature. + +\entryblock{569be40ea88e} + +Note that the inequality in \entryref{9f5db77333af} follows from this +theorem. + +The comparison result in \entryref{569be40ea88e} holds out to +every radius, a fact that will be used repeatedly in our arguments. +This result evolved over the period 1964-1980 and now is referred to +as the Bishop-Gromov inequality; see +Proposition 4.1 of [CheegerGromovTaylor] + +\entryblock{1a1b2c4fdc9b} + +## Local volume and the injectivity radius + +As the following results show, in the presence of bounded curvature the volume +of a ball $B(p,r)$ in $M$ is bounded away from zero if and only if the +injectivity radius of $M$ at $p$ is bounded away from zero. + +\entryblock{7fec5a940413} + +**Proof.** Suppose that $|\mathit{Rm}(x)|\le r^{-2}$ for all $x\in B(p,r)$. +Replacing $g$ by $r^2 g$ allows us to assume that $r=1$. Without +loss of generality we can assume that $\epsilon\le 1$. The map $\mathit{exp}_p$ is a diffeomorphism on the ball $B(0,\epsilon)$ in the +tangent space, and by \entryref{95de178b529d} the volume of +$B(p,\epsilon)$ is at least that of the ball of radius $\epsilon$ in +the $n$-sphere of radius $1$. This gives a lower bound to the volume +of $B(p,\epsilon)$, and a fortiori to $B(p,1)$, in terms of $n$ and +$\epsilon$. + + We shall normally work with volume, which behaves nicely under Ricci +flow, but in order to take limits we need to bound the injectivity +radius away from zero. Thus, the more important, indeed crucial, +result for our purposes is the converse to the previous proposition; +see Theorem 4.3, especially Inequality (4.22), on page 46 of +[CheegerGromovTaylor], or see Theorem 5.8 on page 96 of +[CheegerEbin]. + +\entryblock{f90cce212e88} diff --git a/projects/poincare-conjecture/.astrolabe/docs/03-manifolds-of-non-negative-curvature.mdx b/projects/poincare-conjecture/.astrolabe/docs/03-manifolds-of-non-negative-curvature.mdx new file mode 100644 index 00000000..e5386fba --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/docs/03-manifolds-of-non-negative-curvature.mdx @@ -0,0 +1,508 @@ + + +# Chapter 2 -- Manifolds of non-negative curvature + +In studying singularity development in $3$-dimensional Ricci flows +one forms blow-up limits. By this we mean the following. One +considers a sequence of points $x_k$ in the flow converging to the +singularity. It will be the case that $R(x_k)$ tends to $\infty$ as +$k$ tends to $\infty$. We form a sequence of based Riemannian +manifolds labeled by $k$, where the $k^{th}$ Riemannian manifold is +obtained by taking the time-slice of $x_k$, rescaling its metric by +$R(x_k)$, and then taking $x_k$ as the base point. This creates a +sequence with the property that for each member of the sequence the +scalar curvature at the base point is one. Because of a pinching +result of Hamilton's (see Chapter 4), if there is a +geometric limit of this sequence, or of any subsequence of it, then +that limit is non-negatively curved. Hence, it is important to +understand the basic properties of Riemannian manifolds of +non-negative curvature in order to study singularity development. In +this chapter we review the properties that we shall need. We suppose +that $M$ is non-compact and of positive (resp., non-negative) +curvature. The key to understanding these manifolds is the Busemann +function associated to a minimizing geodesic ray. + +## Busemann functions + +A geodesic ray $\lambda\colon [0,\infty)\to M$ is said to be *minimizing* if the restriction of $\lambda$ to every compact +subinterval of $[0,\infty)$ is a length-minimizing geodesic arc, +i.e., a geodesic arc whose length is equal to the distance between +its endpoints. Likewise, a geodesic line $\lambda\colon +(-\infty,\infty)\to M$ is said to be *minimizing* if its +restriction to every compact sub-interval of $\Ar$ is a length +minimizing geodesic arc. + +Clearly, if a sequence of minimizing geodesic arcs $\lambda_k$ +converges to a geodesic arc, then the limiting geodesic arc is also +minimizing. More generally, if $\lambda_k$ is a sequence of length +minimizing geodesic arcs whose initial points converge and whose +lengths go to infinity, then, after passing to a subsequence, there +is a limit which is a minimizing geodesic ray. (The existence of a +limit of a subsequence is a consequence of the fact that a geodesic +ray is determined by its initial point and its initial tangent +direction.) Similarly, if $I_k$ is an sequence of compact intervals +with the property that every compact subset of $\Ar$ is contained in +$I_k$ for all sufficiently large $k$, if for each $k$ the map +$\lambda_k\colon I_k\to M$ is a minimizing geodesic arc, and if +$\mathit{lim}_{k\rightarrow \infty}\lambda_k(0)$ exists, then, after +passing to a subsequence there is a limit which is a minimizing +geodesic line. Using these facts one establishes the following +elementary lemma. + +\entryblock{89a6041c42b9} + +\entryblock{4d6e47f3ec09} + +Clearly, $B_\lambda(x)\ge -d(x,\lambda(0))$. By equicontinuity +$B_\lambda(x)$ is a continuous function of $x$ and in fact a +Lipschitz function satisfying $|B_\lambda(x)-B_\lambda(y)|\le +d(x,y)$ for all $x,y\in X$. Clearly $B_\lambda(\lambda(s))=-s$ for +all $s\ge 0$. Since $B_\lambda$ is Lipschitz, $\nabla B_\lambda$ is +well-defined as an $L^2$-vector field. + +\entryblock{7c06c4b22b84} + +**Proof.** First notice that since $B_\lambda$ is Lipschitz, $\nabla B_\lambda$ +is an $L^2$-vector field on $M$. That is to say, $B_\lambda\in +W_\mathit{loc}^{1,2}$, i.e., $B_\lambda$ locally has one derivative in +$L^2$. Hence, there is a sequence of $C^\infty$-functions $f_n$ +converging to $B_\lambda$ in $W^{1,2}_{loc}$. Let $\varphi$ be a +test function (i.e., a compactly supported $C^\infty$-function). +Integrating by parts yields + +$$ +-\int_M\langle \nabla +f_n,\nabla\varphi\rangle d\mathit{vol}=\int_Mf_n\triangle \varphi d\mathit{vol}. +$$ + + Using the fact that $f_n$ converges to $B_\lambda$ in +$W^{1,2}_{loc}$ and taking limits yields + +$$ +-\int_M\langle \nabla B_\lambda,\nabla \varphi\rangle d\mathit{vol}= \int_M +B_\lambda\triangle \varphi d\mathit{vol}. +$$ + +Thus, to prove the proposition we need only show that if $\varphi$ +is a non-negative test function, then + +$$ +-\int_M \langle\nabla B_\lambda,\nabla\varphi\rangle d\mathit{vol}\le 0. +$$ + +For a proof of this see + Proposition 1.1 and its proof on pp. 7 and 8 in [SchoenYau]. + +## Comparison results in non-negative curvature + +Let us review some elementary comparison results for manifolds of +non-negative curvature. These form the basis for Toponogov +theory, [Toponogov]. For any pair +of points $x,y$ in a complete Riemannian manifold $s_{xy}$ denotes a +minimizing geodesic from $x$ to $y$. We set $|s_{xy}|=d(x,y)$ and +call it the *length* of the side. A *triangle* in a +Riemannian manifold consists of three vertices $a,b,c$ and three +sides $s_{ab}$,$s_{ac}$,$s_{bc}$. We denote by $\angle_a$ the angle +of the triangle at $a$, i.e., the angle at $a$ between the geodesic +rays $s_{ab}$ and $s_{ac}$. + +\entryblock{04518fa1ac2f} + +See Fig. 2.1. For a proof of this result see +Theorem 4.2 on page 161 of [Sakai], or Theorem 2.2 on page 42 +of [CheegerEbin]. + +One corollary is a monotonicity result. Suppose that +$\triangle(a,b,c)$ is a triangle in a complete manifold of +non-negative curvature. Define a function $EA(u,v)$ defined for +$0\le u\le |s_{ab}|$ and $0\le v\le |s_{ac}|$ as follows. For $u$ +and $v$ in the indicated ranges, let $x(u)$ be the point on $s_{ab}$ +at distance $u$ from $a$ and let $y(v)$ be the point of $s_{ac}$ at +distance $v$ from $a$. Let $EA(u,v)$ be the angle at $a'$ of the +Euclidean triangle with side lengths $|s_{a'b'}|=u$, $|s_{a'c'}|=v$ +and $|s_{b'c'}|=d(x(u),y(v))$. + +\entryblock{c81962d31706} + +Suppose that $\alpha, \beta,\gamma$ are three geodesics +emanating from a point $p$ in a Riemannian manifold. Let $\angle_p +(\alpha,\beta)$, $\angle_p(\beta,\gamma)$ and +$\angle_p(\alpha,\gamma)$ be the angles of these geodesics at $p$ as +measured by the Riemannian metric. Then of course + +$$ +\angle_p(\alpha,\beta)+\angle_p(\beta,\gamma)+\angle_p(\alpha,\gamma)\le +2\pi +$$ + + since this inequality holds for the angles between straight +lines in Euclidean $n$-space. There is a second corollary of +\entryref{04518fa1ac2f} which gives an analogous result for the +associated Euclidean angles. + +\entryblock{dc263832416c} + +**Proof.** Consider the sum of these angles as the geodesic arcs in $M$ are +shortened without changing their direction. By the first property of +\entryref{04518fa1ac2f} the sum of the angles of these triangles +is a monotone decreasing function of the lengths. Of course, the +limit as the lengths all go to zero is the corresponding Euclidean +angle. The result is now clear. + +## The soul theorem + +A subset $X$ of a Riemannian manifold $(M,g)$ is said to be *totally convex* if every geodesic segment with endpoints in $X$ is +contained in $X$. Thus, a point $p$ in $M$ is totally convex if and +only if there is no broken geodesic arc in $M$ broken exactly at +$x$. + +\entryblock{2b5f18292e62} + +\entryblock{8e408e4f9091} + +The rest of this section is devoted to a sketch of the proof of this +result. Our discussion follows closely that in [Petersen] +starting on p. 349. We shall need more information about complete, +non-compact manifolds of non-negative curvature, so we review a +little of their theory as we sketch the proof of the soul theorem. + +\entryblock{dc7704c76958} + +See Fig. 2.2. + +**Proof.** The proof is by contradiction. Fix $0<\epsilon<1$ sufficiently small +so that $\mathit{cos}(\epsilon/2)<1-\epsilon^2/12$. Suppose that there +is a sequence of points $q_n$ tending to infinity such that for each +$n$ there are minimizing geodesics $\gamma_n$ and $\mu_n$ from $p$ +to $q_n$ making angle at least $\epsilon$ at $q_n$. For each $n$ let +$d_n=d(p,q_n)$. By passing to a subsequence we can suppose that for +all $n$ and $m$ the cosine of the angle at $p$ between $\gamma_n$ +and $\gamma_m$ at least $1-\epsilon^2/24$, and the cosine of the +angle at $p$ between $\mu_n$ and $\mu_m$ is at least +$1-\epsilon^2/24$. We can also assume that for all $n\ge 1$ we have +$d_{n+1}\ge (100/\epsilon^2) d_n$. Let $\delta_n=d(q_n,q_{n+1})$. +Applying the first Toponogov property at $p$, we see that +$\delta_n^2\le d_n^2+d_{n+1}^2-2d_nd_{n+1}(1-\epsilon^2/24)$. +Applying the same property at $q_n$ we have + +$$ +d_{n+1}^2\le d_n^2+\delta_n^2-2d_n\delta_n\mathit{cos}(\theta), +$$ + + where +$\theta\le \pi$ is the angle at $q_n$ between $\gamma_n$ and a +minimal geodesic joining $q_n$ to $q_{n+1}$. Thus, + +$$ +\mathit{cos}(\theta)\le +\frac{d_n-d_{n+1}(1-\epsilon^2/24)}{\delta_n}. +$$ + + By the triangle +inequality (and the fact that $\epsilon<1$) we have $\delta_n\ge +(99/\epsilon)d_n$ and $\delta_n\ge d_{n+1}(1-(\epsilon^2/100))$. +Thus, + +$$ +\mathit{cos}(\theta)\le +\epsilon^2/99-(1-\epsilon^2/24)/(1-(\epsilon^2/100))<-(1-\epsilon^2/12). +$$ + +This implies that $\mathit{cos}(\pi-\theta)>(1-\epsilon^2/12)$, which +implies that $\pi-\theta<\epsilon/2$. That is to say, the angle at +$q_n$ between $\gamma_n$ and a shortest geodesic from $q_n$ to +$q_{n+1}$ is between $\pi-\epsilon/2$ and $\pi$. By symmetry, the +same is true for the angle between $\mu_n$ and the same shortest +geodesic from $q_n$ to $q_{n+1}$. Thus, the angle between $\gamma_n$ +and $\mu_n$ at $q_n$ is less than $\epsilon$, contradicting our +assumption. + +\entryblock{64a0f82323d3} + +**Proof.** Given $(M,g)$ and $p\in M$ as in the statement of the corollary, choose a +constant $R<\infty$ such that any two minimal geodesics from $p$ to a point $q$ +with $d(p,q)\ge R/2$ make an angle at most $\pi/6$ at $q$. Now following +[Petersen] p. 335, it is possible to find a smooth unit vector field $X$ +on $U=M-\overline{B(p,R/2)}$ with the property that $f(\cdot)=d(p,\cdot)$ is +increasing along any integral curve for $X$ at a rate bounded below by $\mathit{cos}(\pi/3)$. In particular, for any $s\ge R$ each integral curve of $X$ +crosses the level set $f^{-1}(s)$ in a single point. Using this vector field +we see that for any $s,s'>R$, the pre-image $f^{-1}([s,s'])$ is homeomorphic to +$f^{-1}(s)\times[s,s']$ and that the end $f^{-1}\left([s,\infty)\right)$ is +homeomorphic to $f^{-1}(s)\times [s,\infty)$. + +In a complete, non-compact $n$-manifold of positive curvature any soul is a +point. While the proof of this result uses the same ideas as discussed above, +we shall not give a proof. Rather we refer the reader to Theorem 84 of +[Petersen] on p. 349. A soul has the property that if two minimal +geodesics emanate from $p$ and end at the same point $q\not= p$, then the angle +that they make at $q$ is less than $\pi/2$. Also, of course, the exponential +mapping is a diffeomorphism sufficiently close to the soul. Applying the above +lemma and a standard compactness argument, we see that in fact there is +$\epsilon>0$ such that all such pairs of minimal geodesics from $p$ ending at +the same point make angle less than $\pi/2-\epsilon$ at that point. Hence, in +this case there is a vector field $X$ on all of $M$ vanishing only at the soul, +and agreeing with the gradient of the distance function near the soul, so that +the distance function from $p$ is strictly increasing to infinity along each +flow line of $X$ (except the fixed point). Using $X$ one establishes that $M$ +is diffeomorphic to $\Ar^n$. It also follows that all the level surfaces +$f^{-1}(s)$ for $s>0$ are homeomorphic to $S^{n-1}$ and for $00$ +and for any $x\in M$ we have $d(x,\lambda(t))+d(x,\lambda(-s))\ge +s+t$, and hence $B_+(x)+B_-(x)\ge 0$. Clearly, $B_+(x)+B_-(x)=0$ for +any $x$ in the image of $\lambda$. Thus, the function $B_++B_-$ is +everywhere $\ge 0$, vanishes at at a least one point and satisfies +$\Delta (B_++B_-)\le 0$ in the weak sense. This is exactly the +set-up for the maximum principle, cf. [Petersen], p. 279. + +\entryblock{0d534e8417eb} + +Applying this result to $-(B_++B_-)$, we see that $B_++B_-=0$, so that +$B_-=-B_+$. It now follows that $\Delta B_+=0$ in the weak sense. By standard +elliptic regularity results this implies that $B_+$ is a smooth harmonic +function. + +Next, we show that for all $x\in M$ we have $|\nabla B_+(x)|=1$. +Fix $x\in M$. Take a sequence $t_n$ tending to infinity and consider +minimizing geodesics $\mu_{+,n}$ from $x$ to $\lambda_+(t_n)$. By +passing to a subsequence we can assume that there is a limit as +$n\rightarrow \infty$. This limit is a minimizing geodesic ray +$\mu_+$ from $x$, which we think of as being `asymptotic at +infinity' to $\lambda_+$. Similarly, we construct a minimizing +geodesic ray $\mu_-$ from $x$ asymptotic at infinity to $\lambda_+$. + Since +$\mu_+$ is a minimal geodesic ray, it follows that for any $t$ the +restriction $\mu_+|_{[0,t]}$ is the unique length minimizing +geodesic from $x$ to $\mu_+(t)$ and that $\mu_+(t)$ is not a +conjugate point along $\mu_+$. It follows by symmetry that $x$ is +not a conjugate point along the reversed geodesic $-\mu_+|_{[0,t]}$ +and hence that $x\in U_{\mu_+(t)}$. This means that the function +$d(\mu_+(t),\cdot)$ is smooth at $x$ with gradient equal to the unit +tangent vector in the negative direction at $x$ to $\mu_+$, and +consequently that $B_{\mu_+,t}$ is smooth at $x$. Symmetrically, for +any $t>0$ the function $B_{\mu_-,t}$ is smooth at $x$ with the +opposite gradient. Notice that these gradients have norm one. We +have + +$$ +B_{\mu_+,t}+B_+(x)\ge B_+=-B_-\ge -(B_{\mu_-,t}+B_-(x)). +$$ + +Of course, $B_{\mu_+,t}(x)=0$ and $B_{\mu_-,t}(x)=0$, so that + +$$ +B_{\mu_+,t}(x)+B_+(x)=-(B_{\mu_-,t}(x)+B_-(x)). +$$ + +This squeezes $B_+$ between two smooth functions with the same value +and same gradient at $x$ and hence shows that $B_+$ is $C^1$ at $x$ +and $|\nabla B_+(x)|$ is of norm one. + +Thus, $B$ defines a smooth Riemannian submersion from $M\to \Ar$ +which implies that $M$ is isometric to a product of the fiber over +the origin with $\Ar$. + +This result together with \entryref{89a6041c42b9} shows that if $M$ +satisfies the hypothesis of the theorem, then it can be written as +a Riemannian product $M=N\times \Ar$. Since $M$ has at least two +ends, it follows immediately that $N$ is compact. This completes the +proof of the theorem. + +## $\epsilon$-necks + +Certain types of (incomplete) Riemannian manifolds play an +especially important role in our analysis. The purpose of this +section is to introduce these manifolds and use them to prove one +essential result in Riemannian geometry. + +For all of the following definitions we fix $0<\epsilon<1/2$. Set +$k$ equal to the greatest integer less than or equal to +$\epsilon^{-1}$. In particular, $k\ge 2$. + +\entryblock{755d47f17b43} + +\entryblock{bdfb2841e15e} + +The first object of interest is one that, up to scale, is close to a +long, round cylinder. + +\entryblock{d91d54ddbd1c} + +Here is the result that will be so important in our later arguments. + +\entryblock{cb9ca6d6db9f} + +**Proof.** The result is obvious if $M$ is compact, so we assume that $M$ is +non-compact. Let $p\in M$ be a soul for $M$ (\entryref{2b5f18292e62}), +and let $f$ be the distance function from $p$. Then $f^{-1}(s)$ is +connected for all $s>0$. + +\entryblock{818772530e5a} + +**Proof.** Let $N$ be an $\epsilon$-neck disjoint from $p$. By +\entryref{04dd3c3ea1c8} for any point $z$ in the middle third of $N$, the +boundary of the metric ball $B(p,d(z,p))$ is a topological +$2$-sphere in $N$ isotopic in $N$ to the central $2$-sphere of $N$. +Hence, the central $2$-sphere separates the soul from the end of +$M$. The second statement follows immediately by applying this to +$N_1$ and $N_2$. + +Let $N_1$ and $N_2$ be disjoint $\epsilon$-necks, each disjoint from +the soul. By the previous lemma, the central $2$-spheres $S_1$ and +$S_2$ of these necks are smoothly isotopic to each other and they +are the boundary components of a region diffeomorphic to $S^2\times +I$. Reversing the indices if necessary we can assume that $N_2$ is +closer to $\infty$ than $N_1$, i.e., further from the soul. +Reversing the directions of the necks if necessary, we can arrange +that for $i=1,2$ the function $s_{N_i}$ is increasing as we go away +from the soul. We define $C^\infty$- functions $\psi_i$ on $N_i$, +functions depending only on $s_{N_i}$, as follows. The function +$\psi_1$ is zero on the negative side of the middle third of $N_1$ +and increases to be identically one on the positive side of the +middle third. The function $\psi_2$ is one on the negative side of +the middle third of $N_2$ and decreases to be zero on the positive +side. We extend $\psi_1,\psi_2$ to a function $\psi$ defined on all +of $M$ by requiring that it be identically one on the region $X$ +between $N_1$ and $N_2$ and to be identically zero on $M\setminus +(N_1\cup X\cup N_2)$. + +Let $\lambda$ be a geodesic ray from the soul of $M$ to infinity, +and $B_\lambda$ its Busemann function. Let $N$ be any +$\epsilon$-neck disjoint from the soul, with $ s_N$ direction chosen +so that it points away from the soul. At any point of the middle +third of $N$ where $B_\lambda$ is smooth, $\nabla B_\lambda$ is a +unit vector in the direction of the unique minimal geodesic ray from +the end of $\lambda$ to this point. Invoking \entryref{88a5a2233418} +from the appendix we see that at such points $\nabla B_\lambda$ is +close to $-R(x)^{1/2}\partial/\partial s_N$, where $x\in N$ is the +center of the $\epsilon$-neck. Since $\nabla B_\lambda$ is $L^2$ its +non-smooth points have measure zero and hence, the restriction of +$\nabla B_\lambda$ to the middle third of $N$ is close in the +$L^2$-sense to $-R(x)^{1/2}\partial/\partial s_N$. + +Applying this to $N_1$ and $N_2$ we see that + +$$ +\int_M\langle \nabla B_\lambda,\nabla\psi\rangle d\mathit{vol}=\left(\alpha_2R(x_2)^{-1}-\alpha_1R(x_1)^{-1}\right)\mathit{Vol}_{h_0}(S^2)), +$$ + + where $h(0)$ is the round metric of +scalar curvature $1$ and where each of $\alpha_1$ and $\alpha_2$ +limits to $1$ as $\epsilon$ goes to zero. Since $\psi\ge 0$, +\entryref{7c06c4b22b84} tells us that the left-hand side of +Equation (2.2) must be $\ge 0$. This shows that, +provided that $\epsilon$ is sufficiently small, $R(x_2)$ is bounded +above by $2R(x_1)$. This completes the proof of the proposition. + +\entryblock{554068bff2cf} + +## Forward difference quotients + +Let us review quickly some standard material on forward difference quotients. + +Let $f\colon [a,b]\to \Ar$ be a continuous function on an interval. +We say that the *forward difference quotient of $f$ at a point +$t\in [a,b)$, denoted $\frac{df}{dt}(t)$, is less than +$c$* provided that + +$$ +\overline\mathit{lim}_{\triangle t\rightarrow +0^+}\frac{f(t+\triangle t)-f(t)}{\triangle t}\le c. +$$ + + We say that it +is greater than or equal to $c'$ if + +$$ +c'\le \underline\mathit{lim}_{\triangle t\rightarrow 0^+}\frac{f(t+\triangle +t)-f(t)}{\triangle t}. +$$ + + Standard comparison arguments show: + +\entryblock{67c378f87457} + +The application we shall make of these results is the following. + +\entryblock{37d7d059a526} + +**Proof.** Under the given hypothesis it is a standard and easy exercise to +establish the statement about the forward difference quotient of +$F_\mathit{max}$. The second statement then is an immediate corollary +of the previous result. diff --git a/projects/poincare-conjecture/.astrolabe/docs/04-basics-of-ricci-flow.mdx b/projects/poincare-conjecture/.astrolabe/docs/04-basics-of-ricci-flow.mdx new file mode 100644 index 00000000..ffe11b08 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/docs/04-basics-of-ricci-flow.mdx @@ -0,0 +1,1507 @@ + + +# Chapter 3 -- Basics of Ricci flow + + In this chater we introduce the Ricci flow equation due to R. +Hamilton [Hamilton3MPRC]. For the basic material on the Ricci flow +equation see [ChowKnopf]. + +## The definition of the Ricci flow + +\entryblock{7e9caf7ba4da} + +Let us give a quick indication of what the Ricci flow equation +means. In harmonic coordinates $(x^1,\ldots,x^n)$ about $p$, + that is to say coordinates where $\triangle x^i=0$ for all $i$, +we have + +$$ +\mathit{Ric}_{ij}=\mathit{Ric}(\frac{\partial}{\partial x^i},\frac{\partial}{\partial x^j})= +-\frac{1}{2}\triangle g_{ij} + Q_{ij}(g^{-1},\partial g) +$$ + +where $Q$ is a quadratic form in $g^{-1}$ and $\partial g$, and so in particular is a lower order term +in the derivatives of $g$. See Lemma 3.32 on page 92 of [ChowKnopf]. +So, in these coordinates, the Ricci flow equation is actually a heat +equation for the Riemannian metric + +$$ +\frac{\partial}{\partial t}g = +\triangle g +2Q(g^{-1},\partial g). +$$ + +\entryblock{95eb1bef8937} + +## Some exact solutions to the Ricci flow + +### Einstein manifolds + +Let $g_0$ be an Einstein metric: $\mathit{Ric}(g_0) = \lambda g_0$, where $\lambda$ is a constant. Then for +any positive constant $c$, setting $g = cg_0$ we have $\mathit{Ric}(g) += \mathit{Ric}(g_0) = \lambda g_0= \frac{\lambda}{c} g.$ Using this we +can construct solutions to the Ricci flow equation as follows. +Consider $g(t) = u(t)g_{0}$. If this one-parameter family of +metrics is a solution of the Ricci flow, then + +$$ +\begin{aligned} +\frac{\partial g}{\partial t}& =u'(t)g_{0}\\& = -2\mathit{Ric}(u(t)g_{0})\\& += -2\mathit{Ric}(g_{0})\\& = -2\lambda g_{0}. +\end{aligned} +$$ + +So $u'(t)= -2\lambda$, and hence $u(t) = 1-2 \lambda t$. Thus +$g(t)= (1-2 \lambda t)g_{0}$ is a solution of the Ricci flow. The +cases $\lambda +>0, \lambda=0,$ and $\lambda< 0$ correspond to *shrinking*, *steady* +and *expanding* solutions. Notice that in the shrinking case the +solution exists for $t\in [0,\frac{1}{2\lambda})$ and goes singular at +$t=\frac{1}{2\lambda}$. + +\entryblock{43be10053c44} + +\entryblock{fbf53047d0c1} + +\entryblock{78d9cbe9be13} + +The standard shrinking round cylinder is a model for evolving +$\epsilon$-necks. In Chapter 1 we introduced the +notion of an $\epsilon$-neck. In the case of flows in order to take +smooth geometric limits, it is important to have a stronger version +of this notion. In this stronger notion, the neck not only exists in +one time-slice but it exists backwards in the flow for an +appropriate amount of time and is close to the standard shrinking +round cylinder on the entire time interval. The existence of +evolving necks + is exploited when we study limits of Ricci flows. + +\entryblock{6d6227668950} + +### Solitons + +A *Ricci soliton* +is a Ricci flow $(M,g(t)),\ 0\le t0$. Then, for all $t\in [0,T)$ we define a function + +$$ +\sigma(t)=1-2\lambda t, +$$ + +and a vector field + +$$ +Y_t(x)=\frac{X(x)}{\sigma(t)}. +$$ + + Then we define $\varphi_t$ as the one-parameter +family of diffeomorphisms generated by the time-dependent vector +fields $Y_t$. + +\entryblock{7fdbbb57c26b} + +**Proof.** We check that this flow satisfies the Ricci flow equation; from +that, the result follows immediately. We have + +$$ +\begin{aligned} +\frac{\partial g(t)}{\partial t} & = & +\sigma'(t)\varphi_t^*g(0)+\sigma(t)\varphi_t^*{\mathcal L}_{Y(t)}g(0)\\ +& = & \varphi_t^*(-2\lambda+{\mathcal L}_X)g(0)\\ +& = & \varphi_t^*(-2\mathit{Ric}(g(0))) =-2\mathit{Ric}(\varphi_t^*(g(0))). +\end{aligned} +$$ + +Since $\mathit{Ric}(\alpha g)=\mathit{Ric}(g)$ for any $\alpha>0$, it follows that + +$$ +\frac{\partial g(t)}{\partial t} = -2\mathit{Ric}(g(t)). +$$ + +There is one class of shrinking solitons which are of special +importance to us. These are the gradient shrinking solitons. + +\entryblock{6ace86bb0b70} + +\entryblock{8a667675febd} + +**Proof.** Since + +$$ +{\mathcal L}_{\nabla f}g(0)=2Hess(f), +$$ + + Equation (3.3) is the soliton equation, +Equation (3.2), with the vector field $X$ being the +gradient vector field $\nabla f$. It is a shrinking soliton by +assumption since $\lambda>0$. + +\entryblock{a560479acf35} + +## Local existence and uniqueness + The following is the +first basic result in the theory -- local existence and uniqueness for Ricci +flow in the case of compact manifolds. + +\entryblock{e6b1bd9fa0d7} + +We remark that the Ricci flow is a weakly parabolic system where +degeneracy comes from the gauge invariance of the equation under +diffeomorphisms. Therefore the short-time existence does not come +from general theory. R. Hamilton's original proof of the short-time +existence was involved and used the Nash-Moser inverse function +theorem, [HamiltonIFTNM]. Soon after, DeTurck [DeTurck] +substantially simplified the short-time existence proof by breaking +the diffeomorphism invariance of the equation. For the reader's +convenience, and also because in establishing the uniqueness for +Ricci flows from the standard solution in Section 12.4 +we use a version of this idea +in the non-compact case, we sketch DeTurck's argument. + +**Proof.** Let's sketch the proof due to DeTurck [DeTurck], cf, Section +3 of Chapter 3 starting on page 78 of [ChowKnopf] for more details. First, we compute the first variation +at a Riemannian metric $g$ of minus twice the Ricci curvature tensor in the direction $h$: + +$$ +\begin{aligned} +\delta_g(-2\mathit{Ric})(h)=\triangle h-\mathit{Sym}(\nabla V)+S +\end{aligned} +$$ + +where: + +1. $V$ is the one-form given by + +$$ +V_k=\frac{1}{2}g^{pq}(\nabla_ph_{qk}+\nabla_qh_{pk}-\nabla_kh_{pq}), +$$ + +1. $\mathit{Sym}(\nabla V)$ is the symmetric two-tensor obtained by +symmetrizing the covariant derivative of +$V$, and +1. $S$ is a symmetric two-tensor constructed from the inverse of the metric, +the Riemann curvature tensor and $h$, but involves no derivatives of $h$. + +Now let $g_0$ be the initial metric. For any metric $g$ we define a one-form +$\hat W$ by taking the trace, with respect to $g$, of the matrix-valued +one-form that is the difference of the connections of $g$ and $g_0$. Now we +form a second-order operator of $g$ by setting + +$$ +P(g)={\mathcal L}_Wg, +$$ + +the Lie derivative of $g$ with respect to the vector field $W$ dual to $\hat +W$. Thus, in local coordinates we have +$P(g)_{ij}=\nabla_i\hat W_j+\nabla_j\hat W_i$. +The linearization at $g$ of the second-order operator $P$ in the direction +$h$ is symmetric and is given by + +$$ +\delta_gP(h)=\mathit{Sym}(\nabla V)+T +$$ + +where $T$ is a first-order operator in $h$. +Thus, defining $Q=-2\mathit{Ric}+P$ we have + +$$ +\delta_g(Q)(h)=\triangle h+U +$$ + +where $U$ is a first-order operator in $h$. +Now we introduce the Ricci-DeTurck flow + +$$ +\begin{aligned} +\frac{\partial g}{\partial t}&=-2\mathit{Ric}(g)+P. +\end{aligned} +$$ + +The computations above show that the Ricci-DeTurck flow is strictly parabolic. + Thus, Equation (3.4) has a +short-time solution $\bar g(t)$ with $\bar g(0)=g_0$ by the standard PDE theory. +Given this solution $\bar g(t)$ we +define the time-dependent vector field $W(t)=W(\bar g(t),g_0)$ as above. + Let $\phi_t$ be a one-parameter family of diffeomorphisms, with $\phi_0=\mathit{Id}$, +generated by this time-dependent vector field, i.e., + +$$ +\frac{\partial\phi_t}{\partial t}=W(t). +$$ + +Then, direct computation shows that $g(t)=\phi_t^*\bar g(t)$ solves the Ricci flow equation. + +In performing surgery at time $T$, we will have an open submanifold $\Omega$ of +the compact manifold with the following property. As $t$ approaches $T$ from +below, the metrics $g(t)|_\Omega$ converge smoothly to a limiting metric $g(T)$ +on $\Omega$. We will 'cut away' the rest of the manifold $M\setminus \Omega$ +where the metrics are not converging and glue in a piece $E$ coming from the +standard solution to form a new compact manifold $M'$. Then we extend the +Riemannian metric $g(T)$ on $\Omega$ to one defined on $M'=\Omega\cup E$. The +resulting Riemannian manifold forms the initial manifold at time $T$ for +continuing the Ricci flow $\tilde g(t)$ on an interval $T\le tr$ so that the exponential +mapping $\mathit{exp}_p\colon B(0,r')\to U$ is a local diffeomorphism +onto $B(p,0,r')$. Pulling back by the exponential map, we replace +the Ricci flow on $U$ by a Ricci flow on $B(0,r')$ in $T_pM$. +Clearly, it suffices to establish the estimates in the statement of +the proposition for $B(0,r/2)$. This remark allows us to assume that +the exponential mapping is a diffeomorphism onto $B(p,0,r)$. Bounded +curvature then comes into play in the following crucial proposition, +which goes back to Shi. The function given in the next proposition allows us to localize +the computation in the ball $B(p,0,r)$. + +\entryblock{81f8de5102de} + +For a proof of this result see Lemma 6.62 on page 225 of +[ChowLuNi]. + +We can apply this proposition to our situation, because we are +assuming that $r\le \pi/2\sqrt{K}$ so that the exponential mapping +is a local diffeomorphism onto $B(p,0,r)$ and we have pulled the +Ricci flow back to the ball in the tangent space. + +Fix any $y\in B(p,0,r/2)$ and choose $\eta$ as in the previous +proposition for the constants $C_2(\alpha,n)$ and +$C_2'(K,\alpha,r/4,n)$. Notice that $B(y,0,r/4)\subset B(p,0,3r/4)$ +so that the conclusion of \entryref{e7fe06259f43} holds for every $(z,t)$ +with $z\in B(y,0,r/4)$ and $t\in [0,T]$. We shall show that the restriction of $\eta +F_m$ to $P(y,0,r/4,T)$ is bounded by a constant that depends only on +$K,\alpha,r,n,A_1,\ldots,A_m$. It will then follow immediately that +the restriction of $F_m$ to $P(y,0,r/8,T)$ is bounded by the same +constant. In particular, the values of $F_m(y,t)$ are bounded by the +same constant for all $y\in B(p,0,r/2)$ and $t\in [0,T]$. + + Consider a point $(x,t)\in B(y,0,r/2)\times +[0,T]$ where $\eta F_m$ achieves its maximum; such a point exists +since the ball $B(y,0,r/2)\subset B(p,0,r)$, and hence $B(y,0,r/2)$ +has compact closure in $U$. If $t=0$, then $\eta F_m$ is bounded by +$(C+K^2)K^2$ which is a constant depending only on $K$ and $A_m$. +This, of course, immediately implies the result. Thus we can assume +that the maximum is achieved at some $t>0$. When $s\left\{ +\max\left\{ m+1-l,0\right\} \right\} -0,$ according to the +\entryref{e7fe06259f43}, we have + +$$ +\left( \frac{\partial}{\partial t}-\Delta\right) +F_{m}\leq-c_1\left( F_{m}-C_0\right) ^{2}+C_1. +$$ + +We compute + +$$ +\left( \frac{\partial}{\partial t}-\Delta\right) \left( \eta +F_{m}\right) \leq\eta\left( -c_1\left( +F_{m-C_0}\right)^{2}+C_1\right) -\Delta\eta\cdot +F_m-2\nabla\eta\cdot\nabla F_m. +$$ + + Since $(x,t)$ is a maximum point for $\eta F_m$ and since +$t>0$, a simple maximum principle argument shows that + +$$ +\left( \frac{\partial}{\partial t}-\Delta\right)\eta F_{m}(x,t)\ge 0. +$$ + +Hence, in this case we conclude that + +$$ +\begin{aligned} +0& \le & \left( +\frac{\partial}{\partial t}-\Delta\right) \left( \eta(x) +F_{m}(x,t)\right) \leq\eta(x)\left( -c_1\left( +F_{m}(x,t)-C_0\right)^{2}+C_1\right) \\ +& & -\Delta\eta(x)\cdot F_m(x,t)-2\nabla\eta(x)\cdot\nabla +F_m(x,t). +\end{aligned} +$$ + + Hence, + +$$ +c_1\eta (F_m(x,t)-C_0)^2\le \eta(x) +C_1-\Delta\eta(x)\cdot F_m(x,t)-2\nabla\eta(x)\cdot \nabla F_m(x,t). +$$ + + Since we +are proving that $F_m$ is bounded, we are free to argue by contradiction and +assume that $F_m(x,t)\ge 2C_0$, implying that $F_m(x,t)-C_0\ge F_m(x,t)/2$. +Using this inequality yields + +$$ +\begin{aligned} +\eta(x) (F_m(x,t)-C_0) & \le & +\frac{2\eta C_1}{c_1F_m(x,t)}-\frac{2\Delta\eta(x)}{c_1} -\frac{4}{c_1F_m(x,t)} +\nabla\eta(x)\cdot\nabla F_m(x,t) \\ +&\le & \frac{\eta C_1}{c_1C_0}-\frac{2\Delta\eta(x)}{c_1} +-\frac{4}{c_1F_m(x,t)} \nabla\eta(x)\cdot\nabla F_m(x,t) +\end{aligned} +$$ + +Since +$(x,t)$ is a maximum for $\eta F_m$ we have + +$$ +0=\nabla(\eta(x) F_m(x,t))=\nabla\eta(x) F_m(x,t)+\eta(x)\nabla F_m(x,t), +$$ + +so that + +$$ +\frac{\nabla\eta(x)}{\eta(x)}=-\frac{\nabla F_m(x,t)}{F_m(x,t)}. +$$ + +Plugging this in gives + +$$ +\eta(x) F_m(x,t)\le +\frac{C_1}{c_1C_0}-\frac{2\Delta\eta(x)}{c_1}+4\frac{|\nabla\eta(x)|^2}{c_1\eta(x)}+\eta +C_0. +$$ + +Of course, the gradient and Laplacian of $\eta$ are taken at the +point $(x,t)$. Thus, because of the properties of $\eta$ given in +\entryref{81f8de5102de}, it immediately follows that $\eta F_m(x,t)$ is +bounded by a constant depending + only on $K,n,\alpha,r,c_1,C_0,C_1$, and as we have already seen, $c_1,C_0,C_1$ depend only on + $K,\alpha,A_1,\ldots,A_m$. + +Now suppose that $s\left\{ \max\left\{ m-l+1,0\right\} +\right\}=1.$ Again we compute the evolution inequality for $\eta +F_{m}$. The result is + +$$ +\left( \frac{\partial}{\partial +t}-\Delta\right) \left( \eta F_{m}\right) \leq\eta\left( +-\frac{c_1}{t} (F_{m}-C_0)^{2}+\frac{C_1}{t}\right) -\Delta\eta\cdot +F_m-2\nabla\eta\cdot\nabla F_m. +$$ + + Thus, using the maximum principle +as before, we have + +$$ +\left( \frac{\partial}{\partial t}-\Delta\right)\eta F_{m}(x,t)\ge 0. +$$ + +Hence, + +$$ +\frac{\eta(x) c_1(F_m(x,t)-C_0)^2}{t}\le +\frac{\eta(x)C_1}{t}-\Delta\eta(x)F_m(x,t)-2\nabla\eta(x)\cdot\nabla F_m(x,t). +$$ + +Using the assumption that $F_m(x,t)\ge 2C_0$ as before, and rewriting the last +term as before, we have + +$$ +\eta F_m(x,t)\le +\frac{\eta(x)C_1}{c_1C_0}-\frac{2t\Delta\eta(x)} +{c_1}+\frac{4t|\nabla\eta(x)|^2}{c_1\eta(x)}+\eta C_0. +$$ + + The right-hand side is +bounded by a constant depending only on $K,n,\alpha,r,c_1,C_0,C_1$. We conclude +that in all cases $\eta F_{m}$ is bounded by a constant depending only on +$K,n,\alpha,r,c_1,C_0,C_1$, and hence on $K,n,\alpha,r,A_1,\ldots,A_m$. + +This proves that for any $y\in B(p,0,r/2)$, the value + $\eta F_m(x,t)$ is bounded by a +constant $A_{m+1}$ depending only on $(m+1,l,K,n,\alpha,r)$ for all +$(x,t)\in B(y,0,r/2)\times [0,T]$. Since $\eta(y)=1$, for all $0\le t\le T$ we have + +$$ +t^{\mathit{max}\{m+1-l,0\}}|\nabla^{m+1}\mathit{Rm}(y,t)|^2\le F_m(y,t)=\eta(y)F_m(y,t)\le A_{m+1}. +$$ + +This completes the inductive proof that the result holds for +$k=m+1$ and hence establishes \entryref{e0b6a1f646a7}, modulo the proof of \entryref{e7fe06259f43}. + +Now we turn to the proof of \entryref{e7fe06259f43}. + +**Proof.** In this argument we fix +$(x,t)\in B(p,0,3r/4)\times (0,T]$ and we drop $(x,t)$ from the +notation. Recall that by Equations (7.4a) and (7.4b) on p. 229 of +[ChowKnopf] we have + +$$ +\begin{aligned} +\frac{\partial}{\partial t}\left\vert \nabla^{\ell +}\operatorname*{Rm}\right\vert ^{2} & \leq\Delta\left\vert +\nabla^{\ell }\operatorname*{Rm}\right\vert ^{2} -2\left\vert +\nabla^{\ell +1}\operatorname*{Rm}\right\vert ^{2}+\sum_{i=0} ^{\ell +}c_{\ell,j}\left\vert \nabla^{i}\operatorname*{Rm}\right\vert +\left\vert \nabla^{\ell -i}\operatorname*{Rm}\right\vert \left\vert +\nabla^{\ell } \operatorname*{Rm}\right\vert, +\end{aligned} +$$ + +where the constants $c_{\ell,j}$ depend only on $\ell$ and $j$. + +Hence, setting $m_l= \max\left\{ m+1-l,0\right\}$ and denoting +$c_{m+1,i}$ by $\tilde c_i$, we have + +$$ +\begin{aligned} +\frac{\partial}{\partial t}\left( t^{m_l}\left\vert +\nabla^{m+1}\operatorname*{Rm}\right\vert ^{2}\right) + \leq\Delta\left( t^{m_l }\left\vert \nabla +^{m+1}\operatorname*{Rm}\right\vert ^{2}\right) -2t^{m_l } +\left\vert \nabla^{m+2}\operatorname*{Rm}\right\vert ^{2}\\ +& & +t^{m_l }\sum_{i=0}^{m+1}\tilde c_{i}\left\vert \nabla +^{i}\operatorname*{Rm}\right\vert \left\vert +\nabla^{m+1-i}\operatorname*{Rm} \right\vert \left\vert +\nabla^{m+1}\operatorname*{Rm}\right\vert + +m_l t^{m_l -1}\left\vert \nabla^{m+1}\operatorname*{Rm}\right\vert ^{2}\nonumber\\ +& \leq & \Delta\left( t^{m_l }\left\vert \nabla +^{m+1}\operatorname*{Rm}\right\vert ^{2}\right) -2t^{m_l } +\left\vert \nabla^{m+2}\operatorname*{Rm}\right\vert ^{2} +(\tilde +c_{0}+\tilde +c_{m+1})t^{m_l}\left\vert\operatorname*{Rm}\right\vert\left\vert\nabla^{m+1}\operatorname*{Rm} +\right\vert^2\nonumber\\ & &+ t^{m_l }\sum_{i=1}^{m}\tilde +c_{i}\left\vert \nabla +^{i}\operatorname*{Rm}\right\vert \left\vert \nabla^{m+1-i}\operatorname*{Rm} +\right\vert \left\vert \nabla^{m+1}\operatorname*{Rm}\right\vert ++m_l t^{m_l -1}\left\vert \nabla^{m+1}\operatorname*{Rm}\right\vert +^{2}.\nonumber +\end{aligned} +$$ + +Using the inductive hypothesis, Inequality (3.14), there is a constant +$A<\infty$ depending only on $K,\alpha,A_1,\ldots,A_m$ such that + +$$ +\sum_{i=1}^{m}\tilde c_{i}\left\vert \nabla +^{i}\operatorname*{Rm}\right\vert \left\vert +\nabla^{m+1-i}\operatorname*{Rm} \right\vert \leq At^{-m_l/2}. +$$ + +Also, let $c=\tilde c_{0}+\tilde c_{m+1}$ and define a new constant +$B$ +by + +$$ +B=c(\alpha+K)+m_l. +$$ + + Then, since $t\le T\le \alpha/K$ and $m_l\ge 0$, we have + +$$ +((\tilde c_{0}+\tilde c_{m+1})t\left\vert\operatorname*{Rm}\right\vert +m_l)t^{m_l-1}\le +\frac{Bt^{m_l}}{t^{s(m_l)}}. +$$ + + Putting this together allows us to +rewrite Inequality (3.16) as + +$$ +\begin{aligned} +\frac{\partial}{\partial t}\left( t^{m_l}\left\vert +\nabla^{m+1}\operatorname*{Rm}\right\vert ^{2}\right) + \leq &\Delta\left( t^{m_l }\left\vert \nabla +^{m+1}\operatorname*{Rm}\right\vert ^{2}\right) -2t^{m_l } +\left\vert \nabla^{m+2}\operatorname*{Rm}\right\vert ^{2}\\ + & ++At^{m_l /2}\left\vert \nabla^{m+1} \operatorname*{Rm}\right\vert ++\left(ct\left\vert \operatorname*{Rm}\right\vert +m_l \right) +t^{m_l -1}\left\vert \nabla^{m+1}\operatorname*{Rm}\right\vert +^{2}\\ + \leq &\Delta\left( t^{m_l }\left\vert \nabla +^{m+1}\operatorname*{Rm}\right\vert ^{2}\right) -2t^{m_l } +\left\vert \nabla^{m+2}\operatorname*{Rm}\right\vert ^{2}\\ +& + +\frac{B}{t^{s(m_l)}}t^{m_l}\left\vert\nabla^{m+1}\operatorname*{Rm}\right\vert^2 ++ At^{m_l/2}\left\vert\nabla^{m+1}\operatorname*{Rm}\right\vert. +\end{aligned} +$$ + +Completing the square gives + +$$ +\begin{aligned} +\frac{\partial}{\partial t}\left( t^{m_l}\left\vert +\nabla^{m+1}\operatorname*{Rm}\right\vert ^{2}\right) + & +\leq\Delta\left( t^{m_l }\left\vert \nabla +^{m+1}\operatorname*{Rm}\right\vert ^{2}\right) -2t^{m_l } +\left\vert \nabla^{m+2}\operatorname*{Rm}\right\vert ^{2}\\ +& +(B+1)t^{m_l -s(m_l)}\left\vert \nabla^{m+1} +\operatorname*{Rm}\right\vert ^{2}+\frac{A^2}{4}t^{s(m_l)}. +\end{aligned} +$$ + +Let $\hat{m}_l = \max\left\{ m-l,0\right\} $. From +(3.15) and the induction hypothesis, there is a constant +$D$, depending on $K,\alpha,A_1,\ldots,A_m$ such that + +$$ +\begin{aligned} +\frac{\partial}{\partial t}\left( t^{\hat{m}_l }\left\vert +\nabla^{m}\operatorname*{Rm}\right\vert ^{2}\right) & \leq & +\Delta\left( t^{\hat{m}_l} \left\vert +\nabla^{m}\operatorname*{Rm}\right\vert ^{2}\right) -2t^{\hat{m}_l +}\left\vert +\nabla^{m+1}\operatorname*{Rm}\right\vert ^{2}\\ +& & +\hat{m}_l t^{\hat{m}_l -1}\left\vert +\nabla^{m}\operatorname*{Rm}\right\vert ^{2}+D. +\end{aligned} +$$ + +Now, defining new constants $\widetilde B=B+1$ and $\widetilde +A=A^2/4$ we have + +$$ +\begin{aligned} + \left( \frac{\partial}{\partial t}-\Delta\right)F_m= +\left( \frac{\partial}{\partial t}-\Delta\right) \left[ \left( +C+t^{\hat{m}_l }\left\vert \nabla^{m}\operatorname*{Rm} \right\vert +^{2}\right) t^{m_l }\left\vert +\nabla^{m+1}\operatorname*{Rm}\right\vert ^{2}\right] \leq \\ +& & \left( C+t^{\hat{m}_l }\left\vert \nabla^{m} +\operatorname*{Rm}\right\vert ^{2}\right) \left( -2t^{m_l +}\left\vert \nabla^{m+2} \operatorname*{Rm} \right\vert ^{2} ++\frac{\widetilde B}{t^{s\{m_l\}}} t^{m_l}\left\vert \nabla^{m+1} +\operatorname*{Rm} \right\vert ^{2}+\widetilde At^{s(m_l)} \right) \\ +& & +\left( -2t^{\hat{m}_l }\left\vert +\nabla^{m+1}\operatorname*{Rm} \right\vert ^{2} +\hat{m}_l +t^{\hat{m}_l -1}\left\vert \nabla^{m}\operatorname*{Rm}\right\vert +^{2}+D \right) t^{m_ +l} \left\vert \nabla^{m+1} \operatorname*{Rm}\right\vert ^{2}\\ +& & -2t^{\hat{m}_l +m_l } \nabla\left(\left\vert +\nabla^{m}\operatorname*{Rm}\right\vert ^{2}\right)\cdot +\nabla\left(\left\vert \nabla^{m+1}\operatorname*{Rm}\right\vert +^{2}\right). +\end{aligned} +$$ + +Since $C\ge 4t^{\hat +m_l}\left\vert\nabla^m\operatorname*{Rm}\right\vert^2$, this implies + +$$ +\begin{aligned} + \left( \frac{\partial}{\partial t}-\Delta\right) F_m \leq +-10t^{\hat{m}_l +m_l} \left\vert +\nabla^{m}\operatorname*{Rm}\right\vert ^{2}\left\vert +\nabla^{m+2}\operatorname*{Rm}\right\vert ^{2}\\ + & & -8t^{\hat{m}_l +m_l }\left\vert +\nabla^{m}\operatorname*{Rm}\right\vert \left\vert \nabla^{m+1} +\operatorname*{Rm}\right\vert ^{2}\left\vert \nabla^{m+2}\operatorname*{Rm} +\right\vert -2t^{\hat m_l +m_l }\left\vert +\nabla^{m+1}\operatorname*{Rm}\right\vert ^{4}\nonumber\\ + & & +\left( C+t^{\hat m_l}\left\vert \nabla^{m} +\operatorname*{Rm}\right\vert ^{2}\right) \left( \widetilde +Bt^{m_l-s(m_l) }\left\vert \nabla^{m+1}\operatorname*{Rm}\right\vert +^{2}+\widetilde At^{s(m_l)}\right) \nonumber\\ + & & +\left(\hat m_l t^{\hat{m}_l -1}\left\vert +\nabla^{m}\operatorname*{Rm}\right\vert ^{2}+D\right) t^{m_l +}\left\vert \nabla^{m+1}\operatorname*{Rm} \right\vert +^{2}.\nonumber +\end{aligned} +$$ + +Now we can write the first three terms on the right-hand side of +Inequality (3.17) as + +$$ +-t^{\hat +m_l+m_l}\left(\sqrt{10}\left\vert\nabla^{m+2}\operatorname*{Rm}\right\vert +\left\vert\nabla^m\operatorname*{Rm}\right\vert+\frac{4}{\sqrt{10}}\left\vert\nabla^{m+1} +\operatorname*{Rm}\right\vert^2\right)^2-\frac{2}{5}t^{\hat +m_l+m_l}\left\vert\nabla^{m+1}\operatorname*{Rm}\right\vert^4. +$$ + +In addition we have + +$$ +C+t^{\hat m_l}\left\vert \nabla^{m} \operatorname*{Rm}\right\vert +^{2} \leq C+A_m^2. +$$ + +Let us set $\widetilde D=\max(\alpha/K,1)D$. If $\hat m_l=0$, then + +$$ +\hat m_lt^{\hat +m_l-1}\left\vert\nabla^m\operatorname*{Rm}\right\vert^2+D =D\le +\frac{\widetilde D}{t^{s(m_l)}}=\hat m_lA_m^2+D\le \frac{\hat +m_lA_m^2+\widetilde D}{t^{s(m_l)}}. +$$ + + On the other +hand, if $\hat m_l>0$, then $s(\hat m_l)=s(m_l)=1$ and hence + +$$ +\hat m_lt^{\hat m_l-1}\left\vert\nabla^m\operatorname*{Rm}\right\vert^2+D\le +\frac{1}{t^{s(m_l)}}\hat m_lA_m^2+D\le \frac{\hat +m_lA_m^2+\widetilde D}{t^{s(m_l)}}. +$$ + +Since $\hat m_l =m_l-s(m_l)$, +Inequalities (3.18), (3.19), and (3.20) then +allow us rewrite Inequality (3.17) as + +$$ +\begin{aligned} + \left( \frac{\partial}{\partial t}-\Delta\right)F_m + \leq -\frac{2}{5t^{s( m_l) } + }t^{2m_l}\left\vert\nabla^{m+1}\operatorname*{Rm}\right\vert^4\\ +& & +(C+A_m^2)\left(\frac{\widetilde B}{t^{s(m_l)}} +t^{m_l}\left\vert\nabla^{m+1}\operatorname*{Rm}\right\vert^2+\widetilde +At^{s(m_l)}\right) +\frac{\hat m_lA_m^2+\widetilde +D}{t^{s(m_l)}}t^{m_l}\left\vert\nabla^{m+1}\operatorname*{Rm} +\right\vert^2. +\end{aligned} +$$ + +Setting + +$$ +B'=(C+A_m^2)\widetilde B+(\hat m_lA_m^2+\widetilde D), +$$ + +and $A'=\widetilde A(C+A_m^2)$ we have + +$$ +\begin{aligned} +\left( \frac{\partial}{\partial t}-\Delta\right) F_m & \leq & +-\frac{2}{5t^{s( m_l) } + }\left(t^{m_l}\left\vert\nabla^{m+1}\operatorname*{Rm}\right\vert^2\right)^2 \\ + & & + +\frac{B'}{t^{s(m_l)}}t^{m_l}\left\vert\nabla^{m+1}\operatorname*{Rm}\right\vert^2 ++A't^{s(m_l)}. +\end{aligned} +$$ + +We rewrite this as + +$$ +\begin{aligned} +\left( \frac{\partial}{\partial t}-\Delta\right) F_m & \leq & +-\frac{2}{5t^{s(m_l)}}\left(t^{m_l}\left\vert\nabla^{m+1}\operatorname*{Rm} +\right\vert^2 -\frac{5B'}{4}\right)^2 +\\ & & +\frac{5(B')^2}{8t^{s(m_l)}} +A't^{s(m_l)}, +\end{aligned} +$$ + +and hence + +$$ +\left( \frac{\partial}{\partial t}-\Delta\right) F_m \leq +-\frac{2}{5t^{s(m_l)}}\left(t^{m_l}\left\vert\nabla^{m+1}\operatorname*{Rm} +\right\vert^2-B''\right)^2 +\frac{A''}{t^{s(m_l)}} +$$ + +where the constants $B''$ and $A''$ are defined by $ B''=5 B'/4$ and + +$$ +A''=(\max\{\alpha/K,1\})^2+5( B')^2/8. +$$ + + (Recall that $t\le T\le +\alpha/K$.) Let + +$$ +Y=(C+t^{\hat +m_l}\left\vert\nabla^m\operatorname*{Rm}\right\vert^2). +$$ + + (Notice +that $Y$ is not a constant.) Of course, by definition + +$$ +F_m=Yt^{m_l}|\nabla^{m+1}\mathit{Rm}|^2. +$$ + + Then the previous inequality +becomes + +$$ +\left(\frac{\partial}{\partial t}-\Delta\right) F_m \leq +-\frac{2}{5t^{s(m_l)}Y^2}\left(Yt^{m_l}\left\vert\nabla^{m+1}\operatorname*{Rm} +\right\vert^2-B'' Y\right)^2 + +\frac{A''}{t^{s(m_l)}} +$$ + +Since $C\le Y\le 5C/4$ we have + +$$ +\begin{aligned} +\left( \frac{\partial}{\partial t}-\Delta\right) F_m \leq +-\frac{32}{125t^{s(m_l)}C^2}\left(F_m-B'' +Y\right)^2+\frac{A''}{t^{s(m_l)}} +\end{aligned} +$$ + +At any point where $F_m\ge 5CB''/4$, the last inequality gives + +$$ +\left( \frac{\partial}{\partial t}-\Delta\right) F_m \leq +-\frac{32}{125t^{s(m_l)}C^2}\left(F_m-5CB''/4\right)^2+\frac{A''}{t^{s(m_l)}}. +$$ + +At any point where $F_m\le 5CB''/4$, since $F_m\ge 0$ and $0\le +B''Y\le 5CB''/4$, we have $(F_m-B''Y)^2\le 25C^2(B'')^2/16$, so that + +$$ +-\frac{32}{125t^{s(m_l)}C^2}\left(F_m-5CB''/4\right)^2\ge +-2(B'')^2/5t^{s(m_l)}. +$$ + + Thus, in this case we have + +$$ +\left(\frac{\partial}{\partial t}-\Delta\right) F_m \leq +\frac{A''}{t^{s(m_l)}}\le +-\frac{32}{125t^{s(m_l)}C^2}\left(F_m-5CB''/4\right)^2+\frac{A''+2(B'')^2/5}{t^{s(m_l)}}. +$$ + +These two cases together prove \entryref{e7fe06259f43}. + +## Generalized Ricci flows + +In this section we introduce a generalization of the Ricci flow +equation. The generalization does not involve changing the PDE that +gives the flow. Rather it allows for the global topology of +space-time to be different from a product. + +### Space-time + +There are two basic ways to view an $n$-dimensional Ricci flow: (i) +as a one-parameter family of metrics $g(t)$ on a fixed smooth +$n$-dimensional manifold $M$, and (ii) as a partial metric (in the +horizontal directions) on the $(n+1)$-dimensional manifold $M\times +I$. We call the latter $(n+1)$-dimensional manifold *space-time* and the +horizontal slices are the *time-slices*. In defining the +generalized Ricci flow, it is the second approach that we +generalize. + +\entryblock{6aa2c1fdf94e} + +Notice that it follows +that $\chi(\mathbf{t})=1$. + +\entryblock{ec0b9637d440} + +Notice that a horizontal metric on space-time induces an ordinary +Riemannian metric on each time-slice. Conversely, given a Riemannian +metric on each time-slice $M_t$, the condition that they fit +together to form a horizontal metric on space-time is that they vary +smoothly on space-time. We define the curvature of a horizontal +metric $G$ to be the section of the dual of the symmetric square of +$\wedge^2{\mathcal H}T{\mathcal M}$ whose value at each point $x$ with $\mathbf{t}(x)=t$ is the usual Riemannian curvature tensor of the induced +metric on $M_t$ at the point $x$. This is a smooth section of $\mathit{Sym}^2(\wedge^2{\mathcal H}T^*{\mathcal M})$. + The Ricci curvature and the scalar curvature of a horizontal metric + are given in the usual way from its Riemannian curvature. + The Ricci curvature is a smooth section of $\mathit{Sym}^2({\mathcal + H}T^*{\mathcal M})$ while the scalar curvature is a smooth function on ${\mathcal M}$. + +### The generalized Ricci flow equation + +Because of the second condition in the definition of space-time, the +vector field $\chi$ preserves the horizontal foliation and hence the +horizontal distribution. Thus, we can form the Lie derivative of a +horizontal metric with respect to $\chi$. + +\entryblock{7048ed283c84} + +\entryblock{f4523c2b6f1e} + +Of course, any ordinary Ricci flow is a generalized Ricci flow where +space-time is a product $M\times I$ with time being the projection +to $I$ and $\chi$ being the unit vector field in the positive +$I$-direction. + +### More definitions for generalized Ricci flows + +\entryblock{faf16f1a175d} + +\entryblock{b5e6e97be37d} + +\entryblock{1aaf4983e041} + +\entryblock{0f8fc616f97c} diff --git a/projects/poincare-conjecture/.astrolabe/docs/05-the-maximum-principle.mdx b/projects/poincare-conjecture/.astrolabe/docs/05-the-maximum-principle.mdx new file mode 100644 index 00000000..6a6dac12 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/docs/05-the-maximum-principle.mdx @@ -0,0 +1,877 @@ + + +# Chapter 4 -- The maximum principle + + Recall that the maximum principle for the heat equation says that if $h$ is a solution to +the heat equation + +$$ +\frac{\partial h}{\partial t}=\Delta h +$$ + + on a +compact manifold and if $h(x,0)\ge 0$ for all $x\in M$, then +$h(x,t)\ge 0$ for all $(x,t)$. In this chapter we discuss analogues +of this result for the scalar curvature, the Ricci curvature, and +the sectional curvature under Ricci flow. Of course, in all three +cases we are working with quasi-linear versions of the heat equation +so it is important to control the lower order (non-linear) terms and +in particular show that at zero curvature they have the appropriate +sign. Also, in the latter two cases we are working with tensors +rather than with scalars and hence we require a tensor version of +the maximum principle, which was established by Hamilton in +[Hamilton4MPIC]. + +As further applications of these results beyond just establishing +non-negativity, we indicate Hamilton's result that if the initial +conditions have positive Ricci curvature then the solution becomes +singular at finite time and as it does it becomes round (pinching to +round). We also give Hamilton's result showing that at points where +the scalar curvature is sufficiently large the curvature is pinched +toward positive. This result is crucial for understanding +singularity development. As a last application, we give Hamilton's +Harnack inequality for Ricci flows of non-negative curvature. + +The maximum principle is used here in two different ways. The first +assumes non-negativity of something (e.g., a curvature) at time zero +and uses the maximum principle to establish non-negativity of this +quantity at all future times. The second assumes non-negativity of +something at all times and positivity at one point, and then uses +the maximum principle to establish positivity at all points and all +later times. In the latter application one compares the solution +with a solution to the linear heat equation where such a property is +known classically to hold. + +## Maximum principle for scalar curvature + +Let us begin with the +easiest evolution equation, that for the scalar curvature, where the +argument uses only the (non-linear) version of the maximum +principle. This result is valid in all dimensions: + +\entryblock{95cc838cfc6d} + +**Proof.** According to Equation (3.5), the evolution equation for $R$ +is + +$$ +\frac{\partial}{\partial t}R(x,t)=\Delta +R(x,t)+2|\mathit{Ric}(x,t)|^2. +$$ + + Since $M$ is compact, the function +$R_\mathit{min}(t)$ is continuous but may not be $C^1$ at points where +the minimum of the scalar curvature is achieved at more than one +point. + +The first thing to notice is the following: + +\entryblock{7f3d286772d7} + +**Proof.** This is immediate from the evolution equation for $R$, the fact +that if $R(x,t)=R_\mathit{min}(t)$, then $\Delta R(x,t)\ge 0$, and the +fact that $R$ is the trace of $\mathit{Ric}$ which implies by the +Cauchy-Schwarz inequality that $|R|^2\le n|\mathit{Ric}|^2$. + +Now it follows that: + +\entryblock{84e573c0f2b2} + +**Proof.** This is immediate from the first statement in +\entryref{37d7d059a526}. + +If follows immediately from \entryref{84e573c0f2b2} and +\entryref{67c378f87457} that $R_\mathit{min}(t)$ is a non-decreasing +function of $t$. This establishes the first item and also the second +item in the case when $R_\mathit{min}(0)=0$. + +Suppose that $R_\mathit{min}(0)\not= 0$. + Consider the function + +$$ +S(t)=\frac{-1}{R_\mathit{min}(t)}-\frac{2t}{n}+\frac{1}{R_\mathit{min}(0)}. +$$ + + Clearly, $S(0)=0$ and $S'(t)\ge 0$ (in the sense of forward difference quotients), + so that by \entryref{67c378f87457} we have $S(t)\ge 0$ for all $t$. This means that + +$$ +\frac{1}{R_\mathit{min}(t)}\le +\frac{1}{R_\mathit{min}(0)}-\frac{2t}{n} +$$ + + provided that +$R_\mathit{min}$ is not ever zero on the interval $[0,t]$. If $R_\mathit{min}(0)>0$, then by the first item, $R_\mathit{min}(t)>0$ for all $t$ +for which the flow is defined, and the inequality in the second item +of the proposition is immediate from Equation (4.1). The +third inequality in the proposition also follows easily from +Equation (4.1) when $R_\mathit{min}(t)<0$. But if $R_\mathit{min}(t)\ge 0$, then the third item is obvious. + +## The maximum principle for tensors + +For the applications to the +Ricci curvature and the curvature tensor we need a version of the +maximum principle for tensors that is due to Hamilton; see +[Hamilton4MPCO]. + +Suppose that $V$ is a finite-dimensional real vector space and $Z\subset V$ is +a closed convex set. For each $z$ in the frontier of $ Z$ we define the *tangent cone to* $Z$ at $z$, denoted $T_zZ$, to be the intersection of all +closed half-spaces $H$ of $V$ such that $z\in +\partial H$ and $Z\subset H$. For $z\in \mathit{int} Z$ we define +$T_zZ=V$. Notice that $v\notin T_zZ$ if and only if there is a +affine linear function $\ell$ vanishing at $z$ non-positive on $Z$ +and positive on $v$. + +\entryblock{d12fecfdfae3} + +It is an easy exercise to show the following; see Lemma 4.1 on page +183 of [Hamilton4MPCO]: + +\entryblock{5db278e369ae} + +### The global version + +The maximum principle for tensors generalizes this to tensor flows +evolving by parabolic equations. First we introduce a generalization +of the notion of a vector field preserving a closed convex set to +the context of vector bundles. + +\entryblock{f3441849c78b} + +The following global version of the maximum principle for tensors is +Theorem 4.2 of [Hamilton4MPCO]. + +\entryblock{c954e95b5d89} + +For a proof we refer the reader to Theorem 4.3 and its proof (and +the related Theorem 4.2 and its proof) in [Hamilton4MPCO]. + +There is a slight improvement of this result where the convex set ${\mathcal +Z}$ is allowed to vary with $t$. It is proved by the same argument; see Theorem +4.8 on page 101 of [ChowKnopf]. + +\entryblock{8f737f2b0f7d} + +### The local version + +Here is the local result. It is proved by the same argument as given +in the proof of Theorem 4.3 in [Hamilton4MPCO]. + +\entryblock{a89be7fdb431} + +## Applications of the maximum principle + +Now let us give some applications of these results to Riemann and +Ricci curvature. In order to do this we first need to specialize the +above general maximum principles for tensors to the situation of the +curvature. + +### Ricci flows with normalized initial conditions + +As we have already seen, the Ricci flow equation is invariant under multiplying +space and time by the same scale. This means that there can be no absolute +constants in the results about Ricci surgery. To break this gauge symmetry and +make the constants absolute we impose scale fixing (or rather scale bounding) +conditions on the initial metrics of the flows that we shall consider. The +following definition makes precise the exact conditions that we shall use. + +\entryblock{12a9b4e99e24} + +The evolution equation for the Riemann curvature and a standard maximum +principle argument show that if $(M,g(0))$ has an upper bound on the Riemann +curvature and a lower bound on the volume of balls of a fixed radius, then the +flow has Riemann curvature bounded above and volumes of balls bounded below on +a fixed time interval. Here is the result in the context of normalized initial +condition. + +\entryblock{c1330c5a8be3} + +**Proof.** The bound on the Riemann curvature follows directly from Lemma 6.1 +on page 207 of [ChowLuNi] and the definition of normalized +initial conditions. Once we know that the Riemann curvature is +bounded by $2$ on $[0,2^{-4}]$, there is an $00$. Let $h(x,t),\ 0\le t<\infty$, be +the solution to the heat equation on $\overline U$ + +$$ +\frac{\partial h}{\partial t}=\Delta h +$$ + +with Dirichlet boundary conditions $h(x,t)=0$ for all $x\in \partial +\overline U$ and all $t\ge 0$ and with the given initial conditions. + +Consider the tensor bundle ${\mathcal V}\oplus \Ar$ over $M$. We +define + +$$ +Z_x=\left\{({\mathcal T},h)\in V_x\oplus +\Ar\bigl|\bigr. s_x({\mathcal T})\ge h\ge 0\right\}. +$$ + + The union +over all $x\in M$ of the $Z_x$ defines a closed convex subset +${\mathcal Z}\subset {\mathcal V}\oplus \Ar$ which is invariant +under parallel translation since $s$ is. We consider the family of +sections $({\mathcal T}(x,t),h(x,t)),\ 0\le t\le T$, of ${\mathcal +V}\oplus \Ar$. These evolve by + +$$ +\frac{d\left({\mathcal T}(x,t),h(x,t)\right)}{dt}=\left(\Delta {\mathcal T}(x,t),\Delta +h(x,t)\right)+\widetilde \psi\left({\mathcal T}(x,t),h(x,t)\right) +$$ + +where $\widetilde \psi({\mathcal T},h)=\left(\psi({\mathcal +T}),0\right)$. Clearly, by our hypotheses, the vector field +$\widetilde \psi$ preserves the convex set ${\mathcal Z}$. Applying +the local version of the maximum principle (\entryref{a89be7fdb431}), +we conclude that ${\mathcal T}(x,t)\ge h(x,t)$ for all $(x,t)\in +\overline U\times [0,T]$. + +The result then follows immediately from \entryref{5b296315fdf9}. + +### Applications of the strong maximum principle + +We have the following applications of the strong maximum principle. + +\entryblock{016cf36686b2} + +**Proof.** We suppose that there is $p\in U$ with $R(p,T)=0$. Since all the +metrics in the flow are of non-negative sectional curvature, if the +flow does not consist entirely of flat manifolds then there is +$(q,t)\in U\times [0,T]$ with $R(q,t)>0$. Clearly, by continuity, we +can assume $t0$ +for all $y\in \mathit{int}(V)$. Also, from Equation (3.5) we +have + +$$ +\frac{\partial}{\partial t}(R-h)=\triangle (R-h)+2|\mathit{Ric}|^2, +$$ + +so that $(R-h)(y,0)\ge 0$ on $(V\times \{0\})\cup (\partial V\times [0,T])$. It +follows from the maximum principle that $(R-h)\ge 0$ on all of $V\times +[0,T]$. In particular, $R(p,T)\ge h(p,T)>0$. This is a contradiction, +establishing the theorem. + +\entryblock{8ec855a2aab5} + +**Proof.** First notice that it follows from \entryref{016cf36686b2} that +because $(U,g(0))$ is not flat, we have $R(y,t)>0$ for every +$(y,t)\in U\times (0,T]$. + + We consider the function $s$ on +$\mathit{Sym}^2(\wedge^2T^*_yU)$ that associates to each endomorphism +the sum of the smallest $2$ eigenvalues. Then $s_y$ is the minimum +of the traces on $2$-dimensional subsets in $\wedge^2T_yU$. Thus, +$s$ is a convex function, and the subset ${\mathcal +S}=s^{-1}([0,\infty))$ is a convex subset. Clearly, this subset is +invariant under parallel translation. By the computations in the +proof of \entryref{f8afc1c61977} it is invariant under the vector +field $\psi({\mathcal T})$. The hypothesis of the corollary tells us +that $s(p,T)=0$. Suppose that $s(q,t)>0$ for some $(q,t)\in U\times +[0,T]$. Of course, by continuity we can take $t< T$. Shift the time +parameter so that $t=0$, and fix a compact connected, +codimension-$0$ submanifold $V$ containing $p,q$ in its interior. +Then by \entryref{f696c0fa9db3} $s(y,T)>0$ for all $y\in \mathit{int}(V)$ +and in particular $s(p,T)>0$. This is a contradiction, and we +conclude that $s(q,t)=0$ for all $(q,t)\in U\times [0,T]$. + +Since we have already established that each $R(y,t)>0$ for all $(y,t)\in +U\times (0,T]$, so that $\mathit{Rm}(y,t)$ is not identically zero, this means +that for all $y\in U$ and all $t\in (0,T]$ that the null space of the operator +$\mathit{Rm}(y,t)$ is a $2$-dimensional subspace of $\wedge^2T_yU$. This +$2$-dimensional subspace is dual to a line in $T_xM$. Thus, we have a +one-dimensional distribution (a line bundle in the tangent bundle) ${\mathcal +D}$ in $U\times (0,T]$ with the property that the sectional curvature $\mathit{Rm}(y,t)$ vanishes on any $2$-plane containing the line ${\mathcal D}(y,t)$. +The fact that the sectional curvature of $g(t)$ vanishes on all two-planes in +$T_yM$ containing ${\mathcal D}(y,t)$ means that its eigenvalues are +$\{\lambda,0,0\}$ where $\lambda>0$ is the sectional curvature of the +$g(t)$-orthogonal $2$-plane to ${\mathcal D}(y,t)$. Hence ${\mathcal +R}(V(y,t),\cdot,\cdot,\cdot)=0$. + +Locally in space and time, there is a unique (up to sign) vector field $V(y,t)$ +that generates ${\mathcal D}$ and satisfies $|V(y,t)|^2_{g(t)}=1$. We wish to +show that this local vector field is invariant under parallel translation and +time translation; cf. Lemma 8.2 in [Hamilton4MPCO]. Fix a point $x\in M$, +a direction $X$ at $x$, and a time $t$. Let $\tilde V(y,t)$ be a parallel +extension of $V(x,t)$ along a curve $C$ passing through $x$ in the +$X$-direction, and let $\tilde W(y,t)$ be an arbitrary parallel vector field +along $C$. Since the sectional curvature is non-negative, we have ${\mathcal +R}(\tilde V,\tilde W,\tilde V,\tilde W)(y)\ge 0$ for all $y\in C$; furthermore, +this expression vanishes at $x$. Hence, its first variation vanishes at $x$. +That is to say + +$$ +\nabla \left({\mathcal R}(\tilde V,\tilde W,\tilde V,\tilde W)\right)(x,t)=(\nabla{\mathcal R})(\tilde +V,\tilde W,\tilde V,\tilde W) +$$ + + vanishes at $(x,t)$. Since this is true for all +$\tilde W$, it follows that the null space of the quadratic form +$\nabla{\mathcal R}(x,t)$ contains the null space of ${\mathcal R}(x,t)$, and +thus + +$$ +(\nabla {\mathcal R})(V(x,t),\cdot,\cdot,\cdot)=0. +$$ + + Now let us consider three parallel +vector fields $\tilde W_1,\tilde W_2,$ and $\tilde W_3$ along $C$. +We compute $0=\nabla_X\left({\mathcal R}( V(y,t),\tilde +W_1(y,t),\tilde W_2(y,t),\tilde W_3(y,t))\right)$. (Notice that +while the $\tilde W_i$ are parallel along $C$, $V(y,t)$ is defined +to be the vector field spanning ${\mathcal D}(y,t)$ rather than a +parallel extension of $V(x,t)$.) Given the above result we find that + +$$ +0=2{\mathcal R}(\nabla_X V(x,t),\tilde W_1(x,t),\tilde W_2(x,t), +\tilde W_3(x,t)). +$$ + + Since this is true for all triples of vector +fields $\tilde W_i(x,t)$, it follows that $\nabla_XV(x,t)$ is a real +multiple of $V(x,t)$. But since $|V(y,t)|^2_{g(t)}=1$, we see that +$\nabla_XV(x,t)$ is orthogonal to $V(x,t)$. We conclude that +$\nabla_XV(x,t)=0$. Since $x$ and $X$ are general, this shows that +the local vector field $V(x,t)$ is invariant under the parallel +translation associated to the metric $g(t)$. + +It follows that locally $(M,g(t))$ is a Riemannian product of a surface of +positive curvature with a line. Under this product decomposition, the curvature +is the pullback of the curvature of the surface. Hence, by +Equation (3.5), under Ricci flow on the $3$-manifold, the time +derivative of the curvature at time $t$ also decomposes as the pullback of the +time derivative of the curvature of the surface under Ricci flow on the +surface. In particular, $(\partial {\mathcal R}/\partial +t)(V,\cdot,\cdot,\cdot)=0$. It now follows easily that $\partial +V(x,t)/\partial t=0$. + +This completes the proof that the unit vector field in the direction ${\mathcal +D}(x,t)$ is invariant under parallel translation and under time translation. +Thus, there is a local Riemannian splitting of the $3$-manifold into a surface +and a line, and this splitting is invariant under the Ricci flow. This +completes the proof of the corollary. + +In the complete case, this local product decomposition globalizes in +some cover; see Lemma 9.1 in [Hamilton4MPCO]. + +\entryblock{d21a675e4ea8} + +\entryblock{ce85fb7318bf} + +Let $(N,g)$ be a Riemannian manifold. Recall from \entryref{d4891987758a} + that the open cone +on $(N,g)$ is the space $N\times (0,\infty)$ with the Riemannian +metric $\tilde g(x,s)=s^2g(x)+ds^2$. An extremely important result +for us is that open pieces in non-flat cones cannot arise as the +result of Ricci flow with non-negative curvature. + +\entryblock{92b5c664399e} + +**Proof.** If $(U,g(T))$ is flat, then by \entryref{016cf36686b2} + for every $t\in [0,T]$ the +Riemannian manifold $(U,g(t))$ is flat. + +We must rule out the possibility that $(U,g(T))$ is non-flat. Suppose that +$(U,g(T))$ is an open subset in a non-flat cone. According to +\entryref{1b517dfd4bd7}, for each $x\in U$ the Riemann curvature tensor of +$(U,g(T))$ at $x$ has a $2$-dimensional null space in $\wedge^2T_xU$. Since we +are assuming that $(U,g(T))$ is not flat, the third eigenvalue of the Riemann +curvature tensor is not identically zero. Restricting to a smaller open subset +if necessary, we can assume that the third eigenvalue is never zero. By the +computations in \entryref{1b517dfd4bd7} the non-zero eigenvalue is not +constant, and in fact it scales by $s^{-2}$ in the terminology of that +proposition, as we move along the cone lines. Of course, the $2$-dimensional +null-space for the Riemann curvature tensor at each point is equivalent to a +line field in the tangent bundle of the manifold. Clearly, that line field is +the line field along the cone lines. \entryref{8ec855a2aab5} says that since +the Riemann curvature of $(U,g(T))$ has a $2$-dimensional null-space in +$\wedge^2T_xU$ at every point $x\in U$, the Riemannian manifold $(U,g(T))$ +locally splits as a Riemannian product of a line with a surface of positive +curvature, and the $2$-dimensional null-space for the Riemannian curvature +tensor is equivalent to the line field in the direction of the second factor. +Along these lines the non-zero eigenvalue of the curvature is constant. This is +a contradiction and establishes the result. + +Lastly, we have Hamilton's result (Theorem 15.1 in +[Hamilton3MPRC]) that compact $3$-manifolds of non-negative +Ricci curvature become round under Ricci flow: + +\entryblock{86e5a5766bda} + +Hamilton's proof in [Hamilton3MPRC] uses the maximum principle +and Shi's derivative estimates. + +### Solitons of positive curvature + +One nice application of this pinching result is the following +theorem. + +\entryblock{4f253de31e0c} + +**Proof.** Let $(M,g(t)),\ 0\le t0$ for all $x\in M$, it +follows from \entryref{86e5a5766bda} that $T<\infty$ and that as +$t$ tends to $T$ the metrics $g(t)$ converge smoothly to a round +metric. Since all the manifolds $(M,g(t))$ are isometric up to +diffeomorphism and a constant conformal factor, this implies that +all the $g(t)$ are of constant positive curvature. + +The last statement is a standard consequence of the fact that the +manifold has constant positive curvature. + +\entryblock{142b52ba2d68} + +## Pinching toward positive curvature + +As the last application of the maximum principle for tensors we give +a theorem due to R. Hamilton (Theorem 4.1 in [HamiltonNSRF3M]) +and T. Ivey [Ivey] which shows that, in dimension three, as the +scalar curvature gets large, the sectional curvatures pinch toward +the positive. Of course, if the sectional curvatures are +non-negative, then the results in the previous section apply. Here, +we are considering the case when the sectional curvature is not +everywhere positive. The pinching result says roughly the following: +At points where the Riemann curvature tensor has a negative +eigenvalue, the smallest (thus negative) eigenvalue of the Riemann +curvature tensor divided by the largest eigenvalue limits to zero as +the scalar curvature grows. This result is central in the analysis +of singularity development in finite time for a $3$-dimensional +Ricci flow. + +\entryblock{92827a66b5dd} + +For any fixed $t$, the limit as $X$ goes to $0$ from above of +$X(\mathit{log}(X)+\mathit{log}(1+t)-3)$ is zero, so that it is natural to +interpret this expression to be zero when $X=0$. Of course, when +$X(x,t)=0$ all the eigenvalues of $\mathit{Rm}(x,t)$ are non-negative +so that $R(x,t)\ge 0$ as well. Thus, with this interpretation of the +expression in Part 2 of the theorem, it remains valid even when +$X(x,t)=0$. + +\entryblock{1160f53cc783} + +We proof we give below follows Hamilton's original proof in +[HamiltonNSRF3M] very closely. + +**Proof.** First note that by \entryref{95cc838cfc6d}, if $R_\mathit{min}(0)\ge 0$, then the same is true for $R_\mathit{min}(t)$ for every +$t>0$ and thus the first inequality stated in the theorem is clearly +true. If $R_\mathit{min}(0)<0$, the first inequality stated in the +theorem follows easily from the last inequality in +\entryref{95cc838cfc6d}. + +We turn now to the second inequality in the statement of the +theorem. Consider the tensor bundle ${\mathcal V}=\mathit{Sym}^2(\wedge^2T^*M)$. Then the curvature operator written in the +evolving frame, ${\mathcal T}(x,t)$, is a one-parameter family of +smooth sections of this bundle, evolving by + +$$ +\frac{\partial{\mathcal T}}{\partial t}= +\Delta{\mathcal T}+\psi({\mathcal T}). +$$ + + We consider two subsets of ${\mathcal +V}$. There are two solutions to $x(\mathit{log}(x)+(\mathit{log}(1+t)-3)=-3/(1+t)$. +One is $x=1/(1+t)$; let $\xi(t)>1/(1+t)$ be the other. We set $S({\mathcal +T})=\mathit{tr}({\mathcal T})$, so that $R=2S$, and we set $X({\mathcal T})=\mathit{max}(-\nu({\mathcal T}),0)$. Define + +$$ +\begin{aligned} +{\mathcal Z}_1(t) & = & \{{\mathcal T}\in {\mathcal V}\bigl|\bigr. + S({\mathcal T})\geq-\frac{3}{(1+t)}\} \\ +{\mathcal Z}_2(t) & = & \{ {\mathcal T}\in {\mathcal +V}\bigl|\bigr.S({\mathcal T})\geq f_t(X({\mathcal T})),\ \ \ +\text{if }\ \ X( {\mathcal T})\geq\xi(t)\}, +\end{aligned} +$$ + +where $f_t(x)=x(\mathit{log}x+\mathit{log}(1+t)-3)$. Then we define + +$$ +{\mathcal Z}(t)={\mathcal Z}_1(t)\cap {\mathcal Z}_2(t). +$$ + +\entryblock{0ac34e2feca6} + +**Proof.** First consider the function $f_t(x)=x(\mathit{log}(x)+\mathit{log}(1+t)-3)$ on the interval $[\xi(t),\infty)$. Direct computation +shows that $f'(x)>0$ and $f''(x)>0$ on this interval. Hence, for +every $t\ge 0$ the region ${\mathcal C}(t)$ in the $S$-$X$ plane + defined by $S\ge -3/(1+t)$ +and $S\ge f_t(X)$ when $X\ge \xi(t)$ is convex and has the property +that if $(S,X)\in {\mathcal C}(t)$ then so is $(S,X')$ for all +$X'\le X$. (See Fig. 4.1). By definition an element +${\mathcal T}\in {\mathcal V}$ is contained in ${\mathcal Z}(t)$ if +and only if $(S({\mathcal T}),X({\mathcal T})\in {\mathcal C}(t)$. +Now fix $t\ge 0$ and suppose that ${\mathcal T}_1$ and ${\mathcal +T}_2$ are elements of $\mathit{Sym}^2(\wedge^2T^*M_x)$ such that +setting $S_i=\mathit{tr}({\mathcal T}_i)$ and $X_i=X({\mathcal T}_i)$ +we have $(S_i,X_i)\in {\mathcal C}(t)$ for $i=1,2$. Then we consider +${\mathcal T}=s{\mathcal T}_1+(1-s){\mathcal T}_2$ for some $s\in +[0,1]$. Let $S=\mathit{tr}({\mathcal T})$ and $X=X({\mathcal T})$. +Since ${\mathcal C}(t)$ is convex, we know that +$(sS_1+(1-s)S_2,sX_1+(1-s)X_2)\in{\mathcal C}(t)$, so that +${\mathcal T}\in {\mathcal Z}(t)$. + Clearly, +$S=sS_1+(1-s)S_2$, so that we conclude that $(S,(sX_1+(1-s)X_2))\in{\mathcal +C}(t)$. But since $\nu$ is a convex function, $X$ is a concave function, i.e., +$X\le sX_1+(1-s)X_2$. Hence $(S,X)\in {\mathcal C}(t)$. + +\entryblock{cfd7d6954385} + +**Proof.** Note that by the +hypothesis of the theorem we have + +$$ +\nu(x,0)+\mu(x,0)+\lambda(x,0)\geq -3 +$$ + + so $(S(x,0),X(x,0))\in +{\mathcal C}(0)$ for all $x\in M$. On the other hand, if $0< +X(x,0)$, then since $X(x,0)\le 1$ we have $S(x,0)\geq-3X(x,0)\geq +X(\mathit{log}X-3)$. This completes the proof that ${\mathcal +T}(x,0)\in {\mathcal C}(0)$ for all $x\in M$. + +\entryblock{d3f49c0de6a0} + +**Proof.** Fix $x\in M$ and suppose that we have an integral curve $\gamma(t),\ +t_0\le t\le T$, for $\psi$ with $\gamma(t_0)\in Z(x,t_0)$. We wish +to show that $\gamma(t)\in Z(x,t)$ for all $t\in [t_0,T]$. The +function $S(t)=S(\gamma(t))$ satisfies + +$$ +\frac{dS}{dt}=\lambda^2+\mu^2+\nu^2+\lambda\mu+\lambda\nu+\mu\nu=\frac{1}{2}\left((\lambda+\mu)^2+ +(\lambda+\nu)^2+(\mu+\nu)^2\right). +$$ + + By Cauchy-Schwarz we have + +$$ +(\lambda+\mu)^2+ +(\lambda+\nu)^2+(\mu+\nu)^2)\ge \frac{4S^2}{3}\ge \frac{2S^2}{3}. +$$ + + Since +$\gamma(t_0)\in Z(x,t_0)$ we have $S(t_0)\ge -3/(1+t_0)$. It then follows that + +$$ +S(t)\ge -3/(1+t)\ \ \ \mathit{for\ all}\ \ \ t\ge t_0. +$$ + +Now let us consider the evolution of $X(t)=X(\gamma(t))$. Assume +that we are at a point $t$ for which $X(t)>0$. For this computation +we set $Y=-\mu$. + +$$ +\begin{aligned} +\frac{dX}{dt}&=-\frac{d\nu}{dt}=-\nu^2-\mu\lambda =-X^2+Y\lambda,\\ +\frac{dS}{dt}&=\frac{d(\nu+\mu+\lambda)}{dt} +=\nu^2+\mu^2+\lambda^2+\mu\lambda+\nu\lambda+\nu\mu\\ +&=X^2+Y^2+\lambda^2+XY-\lambda(X+Y). +\end{aligned} +$$ + +Putting this together yields + +$$ +X\frac{dS}{dt}-(S+X)\frac{dX}{dt} =X^3+I, +$$ + + where +$I=XY^2+\lambda Y(Y-X)+\lambda^2(X-Y)$. + +\entryblock{61df8ef86274} + +**Proof.** First we consider the case when + $Y\leq 0$. This means that $\mu\geq 0$ and + hence that $\lambda\geq0$. Since by definition $X\ge 0$, we have $X\ge Y$. +This immediately gives $I\geq 0$. Now let us consider the case when +$Y>0$ which means that $\nu\le \mu<0$. In this case, we have + +$$ +I=Y^3+(X-Y)(\lambda^2-\lambda Y+Y^2)>0 +$$ + +since $X\geq Y$ and $\lambda^2-\lambda +Y+Y^2=(\lambda-\frac{Y}{2})^2+\frac{3Y^2}{4}>0$. + +The above claim and Equation (4.3) immediately imply that + +$$ +X\frac{dS}{dt}-(S+X)\frac{dX}{dt}\geq +X^3. +$$ + + Set $W=\frac{S}{X}-\mathit{log} X$, then rewriting + Equation (4.4) + in terms of $W$ gives + +$$ +\frac{dW}{dt}\geq X. +$$ + +Now suppose that $\gamma(t)\not\in Z(x,t)$ for some $t\in [t_0,T]$. +Let $t_11/(1+t_1)$ and $S(t_1)=f_{t_1}(X(t_1))$. But +Equation (4.2) implies that $S(t)\ge -3/(1+t)$ for all $t$. +Hence, if the first case holds then $\gamma(t)\in Z(x,t)$ for $t$ in +some interval $[t_0,t_1']$ with $t_1'>t_1$. This contradicts the +maximality of $t_1$. Thus, it must be the case that $X(t_1)\ge +\xi(t_1)$. But then $X(t)> \frac{1}{1+t}$ for all $t$ sufficiently +close to $t_1$. Hence, by Equation (4.5) we have + +$$ +\frac{dW}{dt}(t)\geq X(t)>\frac{1}{1+t}, +$$ + +for all $t$ sufficiently close to $t_1$. Also, since +$S(t_1)=f_{t_1}(X(t_1))$, we have $W(t_1)=(\mathit{log}(1+t_1)-3)$. It +follows immediately that $W(t)\ge (\mathit{log}(1+t)-3)$ for all +$t>t_1$ sufficiently close to $t_1$. This proves that $S(t)\ge +f_t(X(t))$ for all $t\ge t_1$ sufficiently close to $t_1$, again +contradicting the maximality of $t_1$. + +This contradiction proves that $\psi$ preserves the family +${\mathcal Z}(t)$. + +By \entryref{8f737f2b0f7d}, the previous three claims imply that ${\mathcal +T}(x,t)\in {\mathcal Z}(t)$ for all $x\in M$ and all $t\in [0,T)$. That is to +say, $S(x,t)\ge -3/(1+t)$ and $S(x,t)\ge f_t(X(x,t))$ whenever $X(x,t)\ge +\xi(t)$. For $X\in [1/(1+t),\xi(t)]$ we have $f_t(X)\le -3/(1+t)$, and thus in +fact $S(x,t)\ge f_t(X(x,t))$ as long as $X(x,t)\ge 1/(1+t)$. On the other hand, +if $0< X(x,t)\le 1/(1+t)$ then $f_t(X(x,t))< -3X(x,t)\le S(x,t)$. On the other +hand, since $X(x,t)$ is the negative of the smallest eigenvalue of ${\mathcal +T}(x,t)$ and $S(x,t)$ is the trace of this matrix, we have $S(x,t)\ge +-3X(x,t)$. Thus, $S(x,t)\ge f_t(X(x,t))$ in this case as well. This completes +the proof of \entryref{92827a66b5dd}. + +Actually, the proof establishes a stronger result which we shall +need. + +\entryblock{fc44d8653f35} + +Once again it is natural to interpret the right-hand side of the +inequalities relating $R$ and $X$ to be zero when $X(x,t)=0$. With +this convention the result remains true even when $X(x,t)=0$. + +\entryblock{8b5afdc35b56} + +**Proof.** Fix $R_0\ge e^4$ sufficiently large, and suppose that $R(x,t)\le +R_0$. If $X(x,t)=0$, then $|\mathit{Rm}(x,t)|\le R(x,t)/2$. If +$X(x,t)>0$, then by \entryref{fc44d8653f35} it is bounded by $R_0$. +Thus, $\lambda(x,t)\le 3R_0$. Thus, we have an upper bound on +$\lambda(x,t)$ and a lower bound on $\nu(x,t)$ in terms of $R_0$. + +This theorem leads to a definition. + +\entryblock{151c2c13281d} + +The content of \entryref{fc44d8653f35} is that if $(M,g(t)),\ 0\le a\le +t0$. Consider $A=X(x,0)/R(x,0)$. For each $t0$ for all $(x,t)\in M\times +(0,T)$. But if the $(M,g(t))$ are all flat, then the flow is trivial +and hence the diameters of the $(M,g(t))$ do not go to zero as $t$ +approaches $T$, contradicting the hypothesis. Hence, $\mathit{Rm}(x,t)>0$ for all $(x,t)\in M\times (0,T)$. According to +\entryref{86e5a5766bda} this means that as the singularity +develops the metrics are converging to round. By the shrinking +soliton hypothesis, this implies that all the metrics $(M,g(t)),\ +0 + +# Chapter 5 -- Convergence results for Ricci flow + +The most obvious notion of smooth convergence of Riemannian +manifolds is the $C^\infty$-version of Cheeger-Gromov compactness: +We have a sequence of Riemannian metrics $g_n$ on a fixed smooth +manifold $M$ converging uniformly on compact subsets of $M$ in the +$C^\infty$-topology to a limit metric $g_\infty$. There is also a +version of this compactness for based, complete Riemannian +manifolds. The most common starts with a sequence of based complete +Riemannian manifolds $(M_n,g_n,x_n)$, typically of unbounded +diameter. Then a geometric limit is a based complete +$(M_\infty,g_\infty,x_\infty)$ so that for every $R<\infty$ the +metric balls $B(x_n,R)\subset M_n$ converge uniformly in the +$C^\infty$-topology to the metric ball $B(x_\infty,R)\subset +M_\infty$. This allows the topology to change -- even if all the +$M_n$ are diffeomorphic, $M_\infty$ can have a different topological +type; for example the $M_n$ could all be compact and $M_\infty$ +could be non-compact. + +But we also need to be able to deal with incomplete limits. In the +case of incomplete limits, the basic idea remains the same, but it +requires some care to give a definition of a geometric limit that +makes it unique up to canonical isometry. One must somehow impose +conditions that imply that the limit eventually fills up most of +each of the manifolds in the sequence. + +## Geometric convergence of Riemannian manifolds + +Above we referred to filling up 'most' of the manifold. The measure of most of +the manifold is in terms of the $\delta$-regular points as defined below. + +\entryblock{1588b6b37e8c} + +Intuitively, the $\delta$-regular points of $(U,g)$ are at distance +at least $\delta$ from the boundary on $U$. + +\entryblock{786d89b0be9f} + +**Proof.** Suppose that $p_n$ converges to $p$ as $n$ tends to $\infty$ and +suppose that $p_n\in \mathit{Reg}_\delta(U,g)$ for all $n$. Fix +$r'<\delta$ and consider the + ball $B(p,r')$. For all $n$ sufficiently large, this +ball is contained in $B(p_n,(\delta+r')/2)$, and hence has compact +closure. + +Now we are ready for the basic definition of geometric convergence +of Riemannian manifolds. + +\entryblock{467f4b920031} + +\entryblock{03c26a36f415} + +Notice that if the $(U_k,g_k)$ have uniformly bounded volume by, say, $V$, then +any geometric limit has volume $\le V$. + +\entryblock{1a98d21b3855} + +**Proof.** Suppose that we have two geometric limits +$(U_\infty,g_\infty,x_\infty)$ and +$(U'_\infty,g'_\infty,x'_\infty)$. Let $\{V_k,\varphi_k\}$ and +$\{V'_k,\varphi_k'\}$ be the sequences of open subsets and maps as +required by the definition of the limit. + +Fix $k$. Since $V_k$ is connected and has compact closure, there are +$R<\infty$ and $\delta>0$ such that $V_k\subset B(x_\infty,R)\cap +\mathit{Reg}_\delta(U_\infty,g_\infty,x_\infty)$. Let $x$ be contained +in the closure of $V_k$. Then by the triangle inequality the closed +ball $\overline{B(x,\delta/3)}$ is contained in +$B(x_\infty,R+\delta)\cap\mathit{Reg}_{\delta/2}(U_\infty,g_\infty,x_\infty)$. Since the union of +these closed balls as $x$ ranges over $\overline V_k$ is a compact +set, for all $\ell$ sufficiently large, the restriction of +$\varphi_\ell^*g_\ell$ to the union of these balls is close to the +restriction of $g_\infty$ to the same subset. In particular, for all +$\ell$ sufficiently large and any $x\in V_k$ we see that +$\varphi_\ell\left(B(x,\delta/3)\right)$ contains +$B(\varphi_\ell(x),\delta/4)$. Thus, for all $\ell$ sufficiently +large $\varphi_\ell(V_k)\subset B(x_\ell,R+2\delta)\cap \mathit{Reg}_{\delta/4}(U_\ell,g_\ell,x_\ell)$. This implies that, for given +$k$, for all $\ell$ sufficiently large $\varphi_\ell(V_k)\subset +\varphi'_\ell(V'_\ell)$. Of course, $(\varphi'_\ell)^{-1}\circ +\varphi_\ell(x_\infty)=x_\infty'$. Fix $k$ and pass to a subsequence +of $\ell$, such that as $\ell\rightarrow\infty$, the compositions +$(\varphi'_\ell)^{-1}\circ\left(\varphi_\ell|_{V_k}\right)\colon +V_k\to U'_\infty$ converge to a base-point preserving isometric +embedding of $V_k$ into $U'_\infty$. Clearly, as we pass from $k$ to +$k'>k$ and take a further subsequence of $\ell$ these limiting +isometric embeddings are compatible. Their union is then a +base-point preserving isometric embedding of $U_\infty$ into +$U'_\infty$. + +The last thing we need to see is that the embedding of $U_\infty$ into +$U'_\infty$ constructed in the previous paragraph is onto. For each $n$ we have +$\overline V'_n\subset V'_{n+1}$. Since $\overline V'_n$ is compact and +connected, it follows that there are $R<\infty$ and $\delta>0$ (depending on +$n$) such that $\overline V'_n\subset B(x'_\infty,R)\cap \mathit{Reg}_\delta(V_{n+1},g'_\infty,x_\infty')$. Since $V'_{n+1}$ has compact closure +in $U'_\infty$, as $\ell$ tends to $\infty$ the metrics +$(\varphi'_\ell)^*g_\ell$ converge uniformly on $V_{n+1}$ to +$g'_\infty|_{V_{n+1}}$. This means that there are $R'<\infty$ and $\delta'>0$ +(depending on $n)$ such that for all $\ell$ sufficiently large, +$\varphi'_\ell(V_n)\subset B(x_k,R')\cap \mathit{Reg}_{\delta'}(U_\ell,g_\ell,x_\ell)$. This implies that for all $k$ +sufficiently large and any $\ell\ge k$ the image $\varphi'_\ell(V'_n)$ is +contained in the image of $\varphi_\ell(V_k)$. Hence, for all $k$ sufficiently +large and any $\ell\ge k$ we have +$V'_n\subset(\varphi'_\ell)^{-1}(\varphi_\ell(V_k))$. Hence, the isometric +embedding $U_\infty\to U'_\infty$ constructed above contains $V'_n$. Since this +is true for every $n$, it follows that this isometric embedding is in fact an +isometry $U_\infty\to U'_\infty$. + +Here is the basic existence result. + +\entryblock{74c7682a68ed} + +Before giving the proof of this result, we begin with a standard +lemma. + +\entryblock{e5006fafe096} + +The basic point in proving this lemma is to 'find the right gauge,' which in +this case means find local coordinates so that the metric tensor is controlled +by the curvature. The correct local coordinates are the Gaussian coordinates +centered at the center of the ball. + +**Proof.** (of the theorem). +Fix $R<\infty$ and $\delta>0$. Let + +$$ +X(\delta,R)=B(x_k,R)\cap +\mathit{Reg}_{2\delta}(U_k,g_k,x_k). +$$ + + From the non-collapsing assumption +and the curvature bound assumption if follows from +\entryref{f90cce212e88} that there is a uniform positive lower bound +(independent of $k$) to the injectivity radius of every point in +$X(\delta,R)$. Fix $0<\delta'\le \mathit{min}(r_0,\delta/2)$ much less +than this injectivity radius. We also choose $\delta'>0$ +sufficiently small so that any ball of radius $2\delta'$ in +$B(x_k,R+\delta)\cap \mathit{Reg}_\delta(U_k,g_k,x_k)$ is geodesically +convex. (This is possible because of the curvature bound.) We cover +$X(\delta,R)$ by balls $B_1',\ldots,B_N'$ + of radii $\delta'/2$ centered at points of $X(\delta,R)$ with the +property that the sub-balls of radius $\delta'/4$ are disjoint. We +denote by $B_i'\subset B_i\subset \widetilde B_i$ the metric balls +with the same center and radii $\delta'/2$, $\delta'$, and +$2\delta'$ respectively. + Notice that each of the balls $\widetilde B_i$ is contained in $B(x_k,R+\delta)\cap \mathit{Reg}_{\delta}(U_k,g_k,x_k)$. Because $\delta'\le r_0$, because $\mathit{Vol} B(x_k,R+\delta)$ is bounded independent of $k$, and because the +concentric balls of radius $\delta'/4$ are disjoint, there is a uniform bound +(independent of $k$) to the number of such balls. Passing to a subsequence we +can assume that the number of balls in these coverings is the same for all $k$. +We number them $\widetilde B_1,\ldots, \widetilde B_N$. Next, using the +exponential mapping at the central point, identify each of these balls with the +ball of radius $2\delta'$ in $\Ar ^n$. By passing to a further subsequence we +can arrange that the metrics on each $\widetilde B_i$ converge uniformly. (This +uses the fact that the concentric balls of radius $2\delta\ge 4\delta'$ are +embedded in the $U_k$ by the exponential mapping.) Now we pass to a further +subsequence so that the distance between the centers of the balls converges, +and so that for any pair $\widetilde B_i$ and $\widetilde B_j$ for which the +limiting distance between their centers is less than $4\delta'$, the overlap +functions in the $U_k$ also converge. The limits of the overlap functions +defines a limiting equivalence relation on $\coprod_i\widetilde B_i$. + +This allows us to form a limit manifold $\widehat U_\infty$. It is +the quotient of the disjoint union of the $\widetilde B_i$ with the +limit metrics under the limit equivalence relation. We set +$(U_\infty(\delta,R),g_\infty(\delta,R),x_\infty(\delta,R))$ equal +to the submanifold of $\widehat U_\infty$ that is the union of the +sub-balls $B_i\subset \widetilde B_i$ of radii $\delta'$. A standard +argument using partitions of unity and the geodesic convexity of the +balls $\widetilde B_i$ shows that, for all $k$ sufficiently large, +there are smooth embeddings $\varphi_k(\delta,R)\colon +U_\infty(\delta,R)\to B(x_k,R+\delta)\cap \mathit{Reg}_\delta(U_k,g_k,x_k)$ sending $x_\infty(\delta,R)$ to $x_k$ and +converging as $k\rightarrow \infty$, uniformly in the +$C^\infty$-topology on each $B_i$, to the identity. Furthermore, the +images of each of these maps contains $B(x_k,R)\cap \mathit{Reg}_{2\delta}(U_k,g_k,x_k)$; compare [Cheeger]. Also, the pull +backs under these embeddings of the metrics $g_k$ converge uniformly +to $g_\infty(\delta,R)$. + +Repeat the process with $R$ replaced by $2R$ and $\delta=\delta_1$ +replaced by $\delta_2\le \delta_1/2$. This produces + +$$ +\left(U_\infty(\delta_2,2R),g_\infty(\delta_2,2R),x_\infty(\delta_2,2R)\right) +$$ + +and, for all $k$ sufficiently large, embeddings +$\varphi_k(\delta_2,2R)$ of this manifold into + +$$ +B(x_k,2R+\delta_2)\cap \mathit{Reg}_{\delta_2}(U_k,g_k,x_k). +$$ + + Hence, +the image of these embeddings contains the images of the original +embeddings. The compositions $(\varphi_k(\delta_2,2R))^{-1}\circ +\varphi_k(\delta,R)$ converge to an isometric embedding + +$$ +\left(U_\infty(\delta,R),g_\infty(\delta,R),x_\infty(\delta,R)\right)\to +\left(U_\infty(\delta_2,2R),g_\infty(\delta_2,2R),x_\infty(\delta_2,2R)\right). +$$ + +Repeating this construction infinitely often produces a manifold +$(U_\infty,g_\infty,x_\infty)$ which is written as an increasing union of open +subsets $V_k=U_\infty(\delta_k,2^kR)$, where the $\delta_k$ tend to zero as $k$ +tends to $\infty$. For each $k$ the open subset $V_k$ has compact closure +contained in $V_{k+1}$. By taking a subsequence of the original sequence we +have maps $\varphi_k\colon V_k\to U_k$ so that (2a) in the definition of +geometric limits holds. Condition (2b) clearly holds by construction. + +Now let us turn to complete Riemannian manifolds, where the result +is the $C^\infty$-version of the classical Cheeger-Gromov +compactness. + +\entryblock{6ed4f379cb9f} + +**Proof.** Since the balls $B(x_\infty,R)$ have compact closure in $U_\infty$ +and since + +$$ +\mathit{Reg}_\delta(U_\infty,g_\infty,x_\infty)=U_\infty +$$ + + for +every $\delta>0$, it is easy to see that the extra condition, (2b), in +\entryref{467f4b920031} is automatic in this case. + +Now as an immediate corollary of \entryref{74c7682a68ed} we have the +following. + +\entryblock{c81f82f6f8a1} + +**Proof.** By the curvature bounds, it follows from the Bishop-Gromov theorem +(\entryref{1a1b2c4fdc9b}) that for each $A<\infty$ there is a uniform bound +to the volumes of the balls $B(x_k,A)$ for all $k$ sufficiently large. It also +follows from the same result that the uniform lower bound on the injectivity +radius at the central point implies that for each $A<\infty$ there is a uniform +lower bound for the injectivity radius on the entire ball $B(x_k,A)$, again for +$k$ sufficiently large. Given these two facts, it follows immediately from +\entryref{74c7682a68ed} that there is a geometric limit. + +Since, for every $A<\infty$, the $B(x_k,A)$ have compact closure in +$M_k$ for all $k$ sufficiently large, it follows that for every +$A<\infty$ the ball $B(x_\infty,A)$ has compact closure in +$M_\infty$. This means that $(M_\infty,g_\infty)$ is complete. + +\entryblock{b45765a3070e} + +**Proof.** Let $A=\mathit{max}(\delta^{-2},C(0,\delta))$, where $C(0,\delta)$ is the constant +given in the second condition in \entryref{c81f82f6f8a1}. Rescale, replacing the +Riemannian metric $g_k$ by $Ag_k$. Of course, the first condition of +\entryref{c81f82f6f8a1} still holds as does the second with different constants, +and we have $\left|\mathit{Rm}_{Ag_k}(y_k)\right|\le 1$ for all $y_k\in +B_{Ag_k}(x_k,\sqrt{A}\delta)$. Also, $\mathit{Vol} B_{Ag_k}(x_k,\sqrt{A}\delta)\ge \kappa (\sqrt{A}\delta)^n$. Thus, by the +Bishop-Gromov inequality (\entryref{1a1b2c4fdc9b}), we have $\mathit{Vol}_{Ag_k}B(x_k,1)\ge \kappa/\Omega$ where + +$$ +\Omega=\frac{V(\sqrt{A}\delta)}{(\sqrt{A}\delta)^nV(1)}, +$$ + +where $V(a)$ is the volume of the ball of radius $a$ in hyperbolic +$n$-space (the simply connected $n$-manifold of constant curvature +$-1$). + Since +$\sqrt{A}\delta\ge 1$, this proves that for the rescaled manifolds the absolute +values of the sectional curvatures on $B_{Ag_k}(x_k,1)$ are bounded by $1$ and +the $\mathit{Vol}_{Ag_k}B(x_k,1)$ are bounded below by a positive constant +independent of $k$. + According to \entryref{f90cce212e88} the lower bound on the volume of the ball +of radius $1$ and the curvature bound on the ball of radius $1$ yield a uniform +positive lower bound $r>0$ for the injectivity radius of the rescaled manifolds +at $x_k$. Hence, the injectivity radii at the base points of the original +sequence are bounded below by $\delta/\sqrt{A}$. This means that the original +sequence of manifolds satisfies the third condition in \entryref{c81f82f6f8a1}. +Invoking this theorem gives the result. + +### Geometric convergence of manifolds in the case of Ricci flow + +As the next theorem shows, because of Shi's theorem, it is much +easier to establish the geometric convergence manifolds in the +context of Ricci flows than in general. + +\entryblock{d9f3522dec12} + +**Proof.** The first condition in \entryref{c81f82f6f8a1} holds by our first +assumption. It is immediate from Shi's theorem (\entryref{ed4b72caf9f9}) +that the second condition of \entryref{c81f82f6f8a1} holds. The result +is then immediate from \entryref{b45765a3070e}. + +## Geometric convergence of Ricci flows + + In this section we extend this notion of geometric convergence for based + Riemannian manifolds in the obvious way to geometric convergence of based + Ricci flows. Then we give Hamilton's theorem about the existence of + such geometric limits. + +\entryblock{9a27ed21e742} + +Notice that the restriction to the $0$ time-slices of a partial +geometric limit of generalized Ricci flows is a partial geometric +limit of the $0$ time-slices. + +\entryblock{59dc7d470f3e} + +Again Shi's theorem, together with a computation of Hamilton, allows us to form +geometric limits of generalized Ricci flows. We have the following result due +originally to Hamilton [Hamiltonlimits]. + +\entryblock{4673eb9f4424} + +**Proof.** Suppose that we have a partial geometric limit of the +time-zero slices as stated in the proposition. Fix a compact subset $K\subset +M_\infty$ and a compact sub-interval $I\subset (T',T)$. For all $k$ +sufficiently large we have embeddings $\widetilde \varphi_k(K,I)$ as stated. We +consider the flows $g_k(K,I)(t)$ on $K\times I$ defined by pulling back the +horizontal metrics $G_k$ under the maps $\widetilde \varphi_k(K,I)$. These of +course satisfy the Ricci flow equation on $K\times I$. Furthermore, by +assumption the flows $g_k(K,I)(t)$ have uniformly bounded curvature. Then under +these hypothesis, Shi's theorem can be used to show that the curvatures of the +$g_k(K,I)$ are uniformly bounded $C^\infty$-topology. The basic computation + done by Hamilton in [Hamiltonlimits] shows that after passing +to a further subsequence, the Ricci flows $g_k(K,I)$ converge +uniformly in the $C^\infty$-topology to a limit flow on $K\times I$. +A standard diagonalization argument allows us to pass to a further +subsequence so that the pullbacks $\widetilde\varphi_k^*G_k$ +converge uniformly in the $C^\infty$-topology on every compact +subset of $M_\infty\times (T',T)$. Of course, the limit satisfies +the Ricci flow equation. + +This 'local' result leads immediately to the following result for +complete limits. + +\entryblock{a0212d0eefcf} + +**Proof.** By \entryref{d9f3522dec12} there is a geometric limit $(M_\infty,g_\infty(0))$ +of the $0$ time-slices, and the limit is a complete Riemannian manifold. Then +by \entryref{4673eb9f4424} there is a geometric limit flow defined +on the time interval $(T',T)$. Since for every $t\in (T',T)$ there is a compact +interval $I$ containing $0$ and $t$, it follows that the Riemann curvature of +the limit is bounded on $M_\infty\times I$. This means that the metrics +$g_\infty(0)$ and $g_\infty(t)$ are commensurable with each other. Since +$g_\infty(0)$ is complete so is $g_\infty(t)$. + +\entryblock{00e5410e6aa1} + +**Proof.** Take a sequence $t_n\rightarrow T$ and consider the +sequence of Riemannian manifolds $(V,g(t_n))$. By Shi's theorem and the fact +that $V$ has compact closure in $U$, the restriction of this sequence of +metrics to $V$ has uniformly bounded curvature derivatives. Hence, this +sequence has a convergent subsequence with limit $(V,g_\infty)$, where the +convergence is uniform in the $C^\infty$-topology. Now by Hamilton's result +[Hamiltonlimits] it follows that, passing to a further subsequence, the +flows $(V,g(T+t-t_n),(p,0))$ converge to a flow $(V,g_\infty(t),(p,0))$ defined +on $(0,T]$. Clearly, for any $0 T$ +using the local existence results. The extension to $[T,T_1)$ fits together +smoothly with the flow on $[0,T]$ by \entryref{b6bc786f7f44}. + +## Gromov-Hausdorff convergence + +Let us begin with the notion of the Gromov-Hausdorff distance between based +metric spaces of finite diameter. Let $Z$ be a metric space. We define the +Hausdorff distance between subsets of $Z$ as follows: $d^Z_H(X,Y)$ is the +infimum of all $\delta\ge 0$ such that $X$ is contained in the +$\delta$-neighborhood of $Y$ and $Y$ is contained in the $\delta$-neighborhood +of $X$. For metric spaces $X$ and $Y$ we define the Gromov-Hausdorff distance +between them, denoted $D_{GH}(X,Y)$, to be the infimum over all metric spaces +$Z$ and isometric embeddings $f\colon X\to Z$ and $g\colon Y\to Z$ of the +Hausdorff distance between $f(X)$ and $g(Y)$. For pointed metric spaces $(X,x)$ +and $(Y,y)$ of finite diameter, we define the Gromov-Hausdorff +distance between them, denoted +$D_{GH}((X,x),(Y,y))$, to be the infimum of $D_H^Z(f(X),g(Y))$ over all triples +$((Z,z),f,g)$ where $(Z,z)$ is a pointed metric space and $f\colon (X,x)\to +(Z,z)$ and $g\colon (Y,y)\to (Z,z)$ are base-point preserving isometries. + +To see that $D_{GH}$ is a distance function we must establish the +triangle inequality. For this it is convenient to introduce +$\delta$-nets in metric spaces. + +\entryblock{9baa1c0fea07} + +Clearly, the Gromov-Hausdorff distance from a based +metric space $(X,x)$ to a $\delta$-net $(L,x)$ contained in it is at +most $\delta$. Furthermore, for every $\delta>0$ the based space +$(X,x)$ has a $\delta$-net: Consider subsets $L\subset X$ containing +$x$ with the property that the $\delta/2$-balls centered at the +points of $L$ are disjoint. Any maximal such subset (with respect to +the inclusion relation) is a $\delta$-net in $X$. + +\entryblock{243579230a8c} + +**Proof.** Suppose that $D_{GH}((X,x),(Y,y))=a$ and $D_{GH}((Y,y),(Z,z))=b$. +Fix any $\delta>0$. Then there is a metric $d_1$ on $X\vee Y$ such +that $d_1$ extends the metrics on $X,Y$ and the +$(a+\delta)$-neighborhood of $X$ is all of $X\vee Y$ as is the +$(a+\delta)$-neighborhood of $Y$. Similarly, there is a metric $d_2$ +on $Y\vee Z$ with the analogous properties (with $b$ replacing $a$). +Take a $\delta$-net $(L,y)\subset (Y,y)$, and define + +$$ +d(x',z')=\mathit{inf}_{\ell\in L}d(x',\ell)+d(\ell,z'). +$$ + +We claim that $d(x',z')>0$ unless $x'=z'$ is the common base point. The reason +is that if $\mathit{inf}_{\ell\in L}d(x',\ell)=0$, then by the triangle +inequality, any sequence of $\ell_n\in L$ with $d(x',\ell_n)$ converging to +zero is a Cauchy sequence, and hence is eventually constant. This means that +for all $n$ sufficiently large, $x'=\ell_n\in L\cap X$ and hence $x'$ is the +common base point. Similarly for $z'$. + +A straightforward computation shows that the function $d$ above, +together with the given metrics on $X$ and $Z$, define a metric on +$X\vee Z$ with the property that the $(a+b+3\delta)$-neighborhood of +$X$ is all of $X\vee Z$ and likewise for $Z$. Since we can do this +for any $\delta>0$, we conclude that $D_{GH}((X,x),(Z,z))\le a+b$. + +Thus, the Gromov-Hausdorff distance is a pseudo-metric. In fact, the +restriction of the Gromov-Hausdorff distance to complete metric +spaces of bounded diameter is a metric. We shall not establish this +result, though we prove below closely related results about the +uniqueness of Gromov-Hausdorff limits. + +\entryblock{d06299f3fe46} + +\entryblock{af79df2b9075} + +\entryblock{7744e53809c5} + +Notice that, with a different realization sequence for the +convergence, a sequence $\ell_k\in X_k$ can converge to a different +point of $Y$. Also notice that, given a realization sequence for the +convergence, every $y\in Y$ is the limit of some sequence $x_k\in +X_k$, a sequence $x_k\in X_k$ has at most one limit in $Y$, and if +$Y$ is compact then every sequence $x_k\in X_k$ has a subsequence +converging to a point of $Y$. Lastly, notice that under any +realization sequence for the convergence, the base points $x_k$ +converge to the base point $y$. + +\entryblock{5be0c15e901a} + +For a proof see Proposition 3.5 on page 36 of [Gromov]. + +\entryblock{cd2cfdd79bea} + +**Proof.** By the triangle inequality for Gromov-Hausdorff distance, it follows +from the hypothesis of the lemma that $D_{GH}((Y,y),(Y',y'))=0$. Fix +$\delta>0$. Since $D_{GH}((Y,y),(Y',y'))=0$, for any $n>0$ and +finite $1/n$-net $L_n\subset Y$ containing $y$ there is an embedding +$\varphi_n\colon L_n\to Y'$ sending $y$ to $y'$ such that the image +is a $2/n$-net in $Y'$ and such that the map from $L_n$ to its image +is a $(1+\delta)$-bi-Lipschitz homeomorphism. Clearly, we can +suppose that in addition the $L_n$ are nested: $L_n\subset +L_{n+1}\subset \cdots$. Since $Y'$ is compact and $L_n$ is finite, +and we can pass to a subsequence so that $\mathit{lim}_{k\rightarrow\infty}\varphi_k|_{L_n}$ converges to a map +$\psi_n\colon L_n\to Y'$ which is a $(1+\delta)$-bi-Lipschitz map +onto its image which is a $2/n$ net in $Y'$. By a standard +diagonalization argument, we can arrange that +$\psi_{n+1}|_{L_{n}}=\psi_{n}$ for all $n$. The $\{\psi_n\}$ then +define an embedding $\cup_n L_n\to Y'$ that is a +$(1+\delta)$-bi-Lipschitz map onto its image which is a dense subset +of $Y'$. Clearly, using the compactness of $Y'$ this map extends to +a $(1+\delta)$-bi-Lipschitz embedding $\psi_\delta\colon (Y,y)\to +(Y',y')$ onto a dense subset of $Y'$. Since $Y$ is also compact, +this image is in fact all of $Y'$. That is to say, $\psi_\delta$ is +a $(1+\delta)$-bi-Lipschitz homeomorphism $(Y,y)\to (Y',y')$. Now +perform this construction for a sequence of $\delta_n\rightarrow 0$ +and $(1+\delta_n)$-bi-Lipschitz homeomorphisms +$\psi_{\delta_n}\colon (Y,y)\to (Y',y')$. These form an +equicontinuous family so that by passing to a subsequence we can +extract a limit $\psi\colon (Y,y)\to (Y',y')$. Clearly, this limit +is an isometry. + +Now let us consider the more general case of spaces of not +necessarily bounded diameter. It turns out that the above definition +is too restrictive when applied to such spaces. Rather one takes: + +\entryblock{11159b600ca2} + +Thus, a sequence of cylinders $S^{n-1}\times \Ar$ with any base points and with +the radii of the cylinders going to zero has the real line as Gromov-Hausdorff +limit. + +\entryblock{beb6a354ee6a} + +**Proof.** We show that for each $r<\infty$ there is an isometry between the +closed balls $\overline{B(y,r)}$ and $\overline{B(y',r)}$. By the +local compactness and completeness, these closed balls are compact. +Each is the limit in the Gromov-Hausdorff sense of a sequence +$B(x_k,r+\delta_k)$ for some $\delta_k\rightarrow 0$ as +$k\rightarrow\infty$. Thus, invoking the previous lemma we see that +these closed balls are isometric. We take a sequence +$r_n\rightarrow\infty$ and isometries $\varphi_n\colon +(B(y,r_n),y)\to (B(y',r_n),y')$. By a standard diagonalization +argument, we pass to a subsequence such that for each $r<\infty$ the +sequence $\varphi_n|_{B(y,r)}$ of isometry converges to an isometry +$\varphi_r\colon B(y,r)\to B(y',r)$. These then fit together to +define a global isometry $\varphi\colon (Y,y)\to (Y',y')$. + +If follows from this that if a sequence of points $\ell_k\in X_k$ converges to +$\ell\in Y$ under one realization sequence for the convergence and to $\ell'\in +Y$ under another, then there is an isometry of $(Y,y)$ to itself carrying +$\ell$ to $\ell'$. + +\entryblock{4f373966523b} + +### Precompactness + +There is a fundamental compactness result due to Gromov. We begin +with a definition. + +\entryblock{9b970f13e747} + +For any based metric space $(X,x)$ and constants $\delta>0$ and +$R<\infty$ let $N(\delta,R,X)$ be the maximal number of disjoint +$\delta$-balls in $X$ that can be contained in $B(x,R)$. + +\entryblock{17352ed43755} + +For a proof of this result see Proposition 5.2 on page 63 of +[Gromov]. + +### The Tits cone + +Let $(M,g)$ be a complete, non-compact Riemannian manifold of +non-negative sectional curvature. Fix a point $p\in M$, and let +$\gamma$ and $\mu$ be minimal geodesic rays emanating from $p$. For +each $r>0$ let $\gamma(r)$ and $\mu(r)$ be the points along these +geodesic rays at distance $r$ from $p$. Then by Part 1 of +\entryref{04518fa1ac2f} we see that + +$$ +\ell(\gamma,\mu,r)=\frac{d(\gamma(r),\mu(r))}{r} +$$ + +is a non-increasing function of $r$. Hence, there is a limit +$\ell(\gamma,\mu)\ge 0$ of $\ell(\gamma,\mu,r)$ as +$r\rightarrow\infty$. We define the *angle at infinity* between +$\gamma$ and $\mu$, $0\le\theta_\infty(\gamma,\mu)\le \pi$, to be +the angle at $b$ of the Euclidean triangle $a,b,c$ with side lengths +$|ab|=|bc|=1$ and $|bc|=\ell(\gamma,\mu)$, see Fig. 5.1. If $\nu$ is a third geodesic ray +emanating from $p$, then clearly, +$\theta_\infty(\gamma,\mu)+\theta_\infty(\mu,\nu)\ge +\theta_\infty(\gamma,\nu)$. + +\entryblock{97c2fdddbf04} + +\entryblock{cb86e24eee63} + +**Proof.** Let $\{[\gamma_n]\}_n$ be a sequence of points in $S_\infty(M,p)$. We show that +there is a subsequence with a limit point. By passing to a subsequence we can +arrange that the unit tangent vectors to the $\gamma_n$ at $p$ converge to a +unit tangent vector $\tau$, say. Fix $d<\infty$, and let $x_n$ be the point of +$\gamma_n$ at distance $d$ from $p$. Then by passing to a subsequence we can +arrange that the $x_n$ converge to a point $x$. The minimizing geodesic +segments $[p,x_n]$ on $\gamma_n$ then converge to a minimizing geodesic segment +connecting $p$ to $x$. Performing this construction for a sequence of $d$ +tending to infinity and then taking a diagonal subsequence produces a +minimizing geodesic ray $\gamma$ from $p$ whose class is the limit of a +subsequence of the $\{[\gamma_n]\}$ + + We define the *Tits cone* of $M$ at $p$, denoted ${\mathcal T}(M,p)$, + to be the cone over +$S_\infty(M,p)$, i.e., the quotient of the space $S_\infty(M,p)\times +[0,\infty)$ where all points $(x,0)$ are identified together (to become the +cone point). The cone metric on this space is given as follows: Let $(x_1,a_1)$ +and $(x_2,a_2)$ be points of $S_\infty(M,p)\times [0,\infty)$. Then the +distance between their images in the cone is determined by + +$$ +d^2([x_1,a_1],[x_2,a_2])=a_1^2+a_2^2-2a_1a_2\mathit{cos}(\theta_\infty(x_1,x_2)). +$$ + + It is an easy exercise to show that the Tits cone of $M$ at $p$ is in fact +independent of the choice of $p$. From the previous claim, it follows that the +Tits cone of $M$ is locally compact and complete. + +\entryblock{88a75d6dd09e} + +**Proof.** Let $c$ be the cone point of ${\mathcal T}(M,p)$, and denote by $d$ the +distance function on ${\mathcal T}(M,p)$. Consider the ball $B(c,R)\subset +{\mathcal T}(M,p)$. Since $S_\infty(M,p)$ is the metric completion of the +quotient space of minimal geodesic rays emanating from $p$, for any $\delta>0$ +there is a $\delta$-net $L\subset B(c,R)$ consisting of the cone point together +with points of the form $([\gamma],t)$ where $\gamma$ is a minimal geodesic ray +emanating from $p$ and $t>0$. We define a map from $\psi_n\colon L \to (M,g_n)$ +by sending the cone point to $p$ and sending $([\gamma],t)$ to the point at +$g_n$-distance $t$ from $p$ along $\gamma$. Clearly, $\psi_n(L)$ is contained +in $B_{g_n}(p,R)$. From the second item of \entryref{04518fa1ac2f} and the +monotonicity of angles it follows that the map $\psi_n\colon L\to (M,g_n)$ is a +distance non-decreasing map; i.e., $\psi_n^*(g_n|\psi_n(L))\ge d|_L$. On the +other hand, by the monotonicity, $\psi_{n+1}^*(g_{n+1}|\psi_{n+1}(L))\le +\psi_n^*(g_n|\psi_n(L))$ and this non-increasing sequence of metrics converges +to $d|_L$. This proves that for any $\delta>0$ for all $n$ sufficiently large, +the embedding $\psi_n$ is a $(1+\delta)$-bi-Lipschitz homeomorphism. + + It remains to show that for any $\eta>0$ the images $\psi_n(L)$ are eventually +$\delta+\eta$-nets in $B_{g_n}(p,R)$. Suppose not. Then after +passing to a subsequence, for each $n$ we have a point $x_n\in +B_{g_n}(p,R)$ whose distance from $\psi_n(L)$ is at least +$\delta+\eta$. In particular, $d_{g_n}(x_n,p)\ge \delta$. Consider a +sequence of minimal geodesic rays $\mu_n$ connecting $p$ to the +$x_n$. Since the $g$-length of $\mu_n$ is at least $n\delta$, by +passing to a further subsequence, we can arrange that the $\mu_n$ +converge to a minimal geodesic ray $\gamma$ emanating from $p$. By +passing to a further subsequence if necessary, we arrange that +$d_{g_n}(x_n,p)$ converges to $r>0$. Now consider the points $\tilde +x_n$ on $\gamma$ at $g$-distance $\sqrt{\lambda_n}r$ from $p$. +Clearly, from the second item of \entryref{04518fa1ac2f} and the +fact that the angle at $p$ between the $\mu_n$ and $\mu$ tends to +zero as $n\rightarrow\infty$ we have $d_{g_n}(x_n,\tilde +x_n)\rightarrow 0$ as $n\rightarrow\infty$. Hence, it suffices to +show that for all $n$ sufficiently large, $\tilde x_n$ is within +$\delta$ of $\psi_n(L)$ to obtain a contradiction. Consider the +point $z=([\mu],r)\in {\mathcal T}(M,p)$. There is a point +$\ell=([\gamma],t')\in L$ within distance $\delta$ of $z$ in the +metric $d$. Let $\tilde y_n\in M$ be the point in $M$ at +$g$-distance $\sqrt{\lambda_n}t'$ along $\gamma$. Of course, $\tilde +y_n=\psi_n(\ell)$. Then $d_{g_n}(\tilde x_n,\tilde y_n)\rightarrow +d(\ell,z)<\delta$. Hence, for all $n$ sufficiently large, +$d_{g_n}(\tilde x_n,\tilde y_n)<\delta$. This proves that for all +$n$ sufficiently large $\tilde x_n$ is within $\delta$ of +$\psi_n(L)$ and hence for all $n$ sufficiently large $x_n$ is within +$\delta+\eta$ of $\psi_n(L)$. + +We have established that for every $\delta,\eta>0$ and every $R<\infty$ there +is a finite $\delta$-net $L$ in $({\mathcal T}(M,p),c)$ and for all $n$ +sufficiently large an $(1+\delta)$-bi-Lipschitz embedding $\psi_n$ of $L$ into +$(M,g_n,p)$ with image a $\delta+\eta$-net for $(M,g_n,p)$. This proves that +the sequence $(M,g_n,p)$ converges in the Gromov-Hausdorff sense to ${\mathcal +T}(M,p),c))$. + +## Blow-up limits + +Here we introduce a type of geometric limit. These were originally +introduced and studied by Hamilton in [Hamiltonsurvey], where, +among other things, he showed that $3$-dimensional blow-up limits +have non-negative sectional curvature. We shall use repeatedly +blow-up limits and the positive curvature result in the arguments in +the later sections. + +\entryblock{666538bf577e} + +The significance of the condition that the generalized Ricci flows +have curvature pinched toward positive is that, as Hamilton +originally established in [Hamiltonsurvey], the latter +condition implies that any blow-up limit has non-negative curvature. + +\entryblock{2ddfbc74b081} + +**Proof.** Let us consider the case of the geometric limit of the zero time-slice first. +Let $( M_\infty,g_\infty(0),x_\infty)$ be a blow-up limit of the zero +time-slices in the sequence. Let $V_k\subset M_\infty$ and $\varphi_k\colon +V_k\to (M_k)_0$ be as in the definition of the geometric limit. Let $y\in +{\mathcal M}_\infty$ be a point and let $\lambda(y)\ge \mu(y)\ge \nu(y)$ be the +eigenvalues of the Riemann curvature operator for $g_\infty$ at $y$. Let +$\{y_k\}$ be a sequence in $Q_k{\mathcal M}_k'$ converging to $y$, in the sense +that $y_k=\varphi_k(y)$ for all $k$ sufficiently large. Then + +$$ +\begin{aligned} +\lambda(y) & = & \mathit{lim}_{n\rightarrow\infty}Q_k^{-1}\lambda(y_k) \\ +\mu(y) & = & \mathit{lim}_{n\rightarrow\infty}Q_k^{-1}\mu(y_k) \\ +\nu(y) & = & \mathit{lim}_{n\rightarrow\infty}Q_k^{-1}\nu(y_k) +\end{aligned} +$$ + +Since by Equation (4.6) we have $R(y_k)\ge -6$ for all $k$ and since +by hypothesis $Q_k$ tends to infinity as $n$ does, it follows that $R(y)\ge 0$. +Thus if $\lambda(y)=0$, then $\mathit{Rm}(y)=0$ and the result is established at +$y$. Hence, we may assume that $\lambda(y)>0$, which means that $\lambda(y_k)$ +tends to infinity as $k$ does. If $\nu(y_k)$ remains bounded below as $k$ tends +to infinity, then $Q_k^{-1}\nu(y_k)$ converges to a limit which is $\ge 0$, and +consequently $Q_k^{-1}\mu(y_k)\ge Q_k^{-1}\nu(y_k)$ has a non-negative limit. +Thus, in this case the Riemann curvature of $g_\infty$ at $y$ is non-negative. +On the other hand, if $\nu(y_k)$ goes to $-\infty$ as $k$ does, then according +to Equation (4.6) the ratio of $X(y_k)/R(y_k)$ goes to zero. Since +$Q_k^{-1}R(y_k)$ converges to the finite limit $R(y)$, the product +$Q_k^{-1}X(y_k)$ converges to zero as $k$ goes to infinity. This means that +$\nu(y)=0$ and consequently that $\mu(y)\ge 0$. Thus, once again we have +non-negative curvature for $g_\infty$ at $y$. + +The argument in the case of a geometric limit flow is identical. + +\entryblock{e67bd2c19008} + +**Proof.** According to \entryref{92827a66b5dd} the hypotheses imply that for every $k$ the +Ricci flow $(M_k,g_k(t))$ has curvature pinched toward positive. From this, the +corollary follows immediately from the previous theorem. + +## Splitting limits at infinity + +In our later arguments we shall need a splitting result at infinity +in the non-negative curvature case. Assuming that a geometric limit +exists, the splitting result is quite elementary. For this reason we +present it here, though it will not be used until +Chapter 9. + +The main result of this section gives a condition under which a geometric limit +automatically splits off a line; see Fig. 5.2. + +\entryblock{e9a9172d16e2} + +**Proof.** Let $d_n$ be the distance from $p$ to $x_n$. Consider minimizing +geodesic arcs $\gamma_n$ from $p$ to $x_n$. By passing to a +subsequence we can assume that tangent directions at $p$ of these +arcs converge. Hence, for every $0<\delta<1$ there is $N$ such that +for all $n,m\ge N$ the angle between $\gamma_n$ and $\gamma_m$ at +$p$ is less than $\delta$. For any $n$ we can choose $m(n)$ such +that $d_{m(n))}\ge d_n(1+1/\delta)$. Let $\mu_n$ be a minimizing +geodesic from $x_n$ to $x_{m(n)}$. + + Now applying the Toponogov comparison +(first part of \entryref{04518fa1ac2f}) and the usual law of +cosines in Euclidean space, we see that the distance $d$ from $x_n$ +to $x_{m(n)}$ satisfies + +$$ +d_{m(n)}-d_n\le d\le \sqrt{d_n^2+d_{m(n)}^2-2d_nd_{m(n)}\mathit{cos}(\delta)}. +$$ + + Let $\theta_n=\angle_{x_n'}$ of the Euclidean triangle +$\triangle(x_n',p',x_{m(n)}')$ with $|s_{x_n'p'}|=d_n,|s_{x_n'x_{m(n)}'}|=d$ +and $|s_{p'x_{m(n)}'}|=d_{m(n)}$. Then for any $\alpha0$, for all $n$ sufficiently large, $1+\mathit{cos}(\theta_n)<\delta$. + +We are assuming that the based Riemannian manifolds +$\{(M,\lambda_ng,x_n)\}_{n=1}^\infty$ converge to a geometric limit +$(M_\infty,g_\infty,x_\infty)$. Also, by assumption, +$d_{\lambda_ng_n}(p,x_n)\rightarrow \infty$ as $n\rightarrow +\infty$, so that the lengths of the $\gamma_n$ tend to infinity in +the metrics $\lambda_ng_n$. This also means that the lengths of +$\mu_n$, measured in the metrics $\lambda_ng_n$, tend to infinity. +Thus, by passing to a subsequence we can assume that each of these +families, $\{\gamma_n\}$ and $\{\mu_n\}$, of minimizing geodesic +arcs converges to a minimizing geodesic arc, which we denote $\tilde +\gamma$ and $\tilde \mu$, respectively, in $M_\infty$ emanating from +$x_\infty$. The above computation shows that the angle between these +arcs is $\pi$ and hence that their union is a geodesic, say $\ell$. +The same computation shows that $\ell $ is minimizing. + +The existence of the minimizing geodesic line $\ell$ together with +the fact that the sectional curvatures of the limit are $\ge 0$ +implies by \entryref{006e42ef5bb5} that the limit manifold is a Riemannian +product $N\times \Ar$ in such a way that $\ell$ is of the form +$\{x\}\times \Ar$ for some $x\in N$. diff --git a/projects/poincare-conjecture/.astrolabe/docs/07-a-comparison-geometry-approach-to-the-ricci-flow.mdx b/projects/poincare-conjecture/.astrolabe/docs/07-a-comparison-geometry-approach-to-the-ricci-flow.mdx new file mode 100644 index 00000000..2c18f6e5 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/docs/07-a-comparison-geometry-approach-to-the-ricci-flow.mdx @@ -0,0 +1,2247 @@ + + +# Chapter 6 -- A comparison geometry approach to the Ricci flow + +In this section we discuss Perelman's notions, introduced in +[Perelman1], of the ${\mathcal L}$-length in the context of generalized Ricci flows. This is a +functional defined on paths in space-time parameterized by backward +time, denoted $\tau$. The ${\mathcal L}$-length is the analogue in +this context of the energy for paths in a Riemannian manifold. We +derive the associated Euler-Lagrange equation for the ${\mathcal +L}$-length; the critical paths are then ${\mathcal +L}$-geodesics. Using ${\mathcal +L}$-geodesics we define the ${\mathcal L}$-exponential +mapping. We derive the +${\mathcal L}$-Jacobi equation +and relate ${\mathcal L}$-Jacobi fields to the differential of the ${\mathcal L}$-exponential +mapping. There is the analogue of the interior of the cut locus. It +is the open subset, depending on the parameter $\tau$, of the +tangent space of initial vectors for ${\mathcal L}$-geodesics which +are minimizing out to time $\tau$ and at which the ${\mathcal +L}$-geodesic map is a local diffeomorphism at time $\tau$. The +difference between this situation and that of geodesics in a +Riemannian manifold is that there is such an open set in the tangent +space for each positive $\tau$. The analogue of the fact that, for +ordinary geodesics, the interior of the cut locus in the tangent +space is star-shaped from the origin is that the open set of 'good' +initial conditions at $\tau$ is contained the open subset of 'good' +initial conditions at time $\tau'$ for any $\tau'<\tau$. All of +these results are local and are established in the context of +generalized Ricci flows. In the next section we consider the case of +ordinary Ricci flows, where we are able to extend our results over +the entire manifold. + +There are two applications of this theory in our study. In +Section 8 we use the theory of ${\mathcal L}$-geodesics and the +associated notion of reduced volume to establish non-collapsing results. These +are crucial when we wish to take blow-up limits in studying singularities in +Ricci flows and Ricci flows with surgery. The second application will be in +Section 9 to $\kappa$-solutions (ancient, $\kappa$-non-collapsed +solutions of bounded non-negative curvature). Here the second-order +inequalities on the length function that we establish in this section are used +to prove the existence of an asymptotic soliton for any $\kappa$-solution. This +asymptotic soliton is important for giving qualitative results on +$\kappa$-solutions. + +## ${\mathcal L}$-length and ${\mathcal L}$-geodesics + +The running assumption throughout this section is that we have an +$n$-dimensional generalized Ricci flow $({\mathcal M},G)$. In +particular, the space-time ${\mathcal M}$ is a smooth manifold of +dimension $n+1$ whose boundary lies at the initial and final times +(if they exist). Recall that its tangent bundle naturally decomposes +as the direct sum of the sub-line bundle spanned by the vector field +$\chi$ and the horizontal tangent bundle, denoted ${\mathcal +HT}{\mathcal M}$. We also fix a time $T$ in the time interval of +definition of the flow distinct from the initial time. + +\entryblock{6a961a764f44} + +Throughout this section the paths $\gamma$ that we consider shall be +parameterized by backward time. We begin with the definition of +${\mathcal L}$-length of such a path. + +\entryblock{09af70d51aac} + +With a view toward better understanding the properties of the paths +that are critical points of this functional, the so-called +$\mathcal{L}$-geodesics, especially near $\tau=0$, it is helpful to +introduce a convenient reparameterization. We set $s=\sqrt{\tau}$. +We use the notation $A(s)$ to denote the horizontal component of the +derivative of $\gamma$ with respect to the variable $s$. One sees +immediately by the chain rule that + +$$ +A(s^2) = 2s X(s^2)\ \ \mathit{or} \ \ +A(\tau)=2\sqrt{\tau}X(\tau). +$$ + + With respect to the +variable $s$, the ${\mathcal L}$-functional is + +$$ +\mathcal{L}(\gamma)= +\int_{\sqrt{\tau_1}}^{\sqrt{\tau_2}}\left(\frac{1}{2}|A(s)|^{2} + +2R(\gamma(s))s^{2}\right)ds. +$$ + +Let's consider the simplest example. + +\entryblock{867793768688} + +### ${\mathcal L}$-geodesics + +\entryblock{75695d1b1b54} + +\entryblock{91d88796541d} + +**Proof.** First, let us suppose that the +generalized Ricci flow is an ordinary Ricci flow $(M,g(t))$. Let +$\gamma_u(\tau)=\gamma(\tau,u)$ be a family of curves parameterized +by backward time. Let + +$$ +\widetilde Y(\tau,u)=\frac{\partial\gamma}{\partial u}. +$$ + + Then $\widetilde X(\tau,u)=X_{\gamma_u}(\tau,u)$ + and $\widetilde Y(\tau,u)$ +are the coordinate vector fields along the surface obtained by taking the +projection of $\gamma(\tau,u)$ into $M$. Thus, $[\widetilde X,\widetilde Y] +=0$. We denote by $X$ and $Y$ the restrictions of $\widetilde X$ and +$\widetilde Y$, respectively to $\gamma_0$. We have + +$$ +\begin{aligned} +\frac{d}{du}\mathcal{L}(\gamma_{u})\bigl|_{u=0}\bigr. & = +\frac{d}{du}\left(\int_{\tau_{1}}^{\tau_{2}} +\sqrt{\tau}(R(\gamma_u(\tau)) + \abs{\widetilde +X(\tau,u)}^{2})d\tau\right)\Bigl|_{u=0}\Bigr. +\\&=\int_{\tau_{1}}^{\tau_{2}} \sqrt{\tau}(\langle \nabla R, Y\rangle + +2\langle (\nabla_{Y}\widetilde X)|_{u=0},X\rangle )d\tau +\end{aligned} +$$ + +On the other hand, since $\partial g/\partial \tau =2\mathit{Ric}$ and +since $[\widetilde X,\widetilde Y]=0$, we have + +$$ +\begin{aligned} +2\frac{d}{d\tau}(\sqrt{\tau}\langle Y,X\rangle_{g(T-\tau)} ) +& = & \frac{1}{\sqrt{\tau}}\langle Y,X\rangle + +2\sqrt{\tau}\langle\nabla_XY,X\rangle ++2\sqrt{\tau}\langle Y,\nabla_XX\rangle \\ & & +4\sqrt{\tau}\mathit{Ric}(Y,X) \\ +& = & \frac{1}{\sqrt{\tau}}\langle Y,X\rangle + +2\sqrt{\tau}\langle(\nabla_Y\widetilde X)|_{u=0},X\rangle ++2\sqrt{\tau}\langle Y,\nabla_XX\rangle \\ & & +4\sqrt{\tau}\mathit{Ric}(Y,X) +\end{aligned} +$$ + +Using this we obtain + +$$ +\begin{aligned} +\nonumber \frac{d}{du} +\mathcal{L}(\gamma_u)\bigl|_{u=0}\bigr. & = & \int_{\tau_1}^{\tau_2} +\Bigl(2\frac{d}{d\tau}\left[(\sqrt{\tau})\langle +Y,X\rangle\right]-\frac{1}{\sqrt{\tau}}\langle +Y,X\rangle\Bigr. \\ +& & +\sqrt{\tau}\bigl(\langle\nabla R,Y\rangle + - 2\langle Y,\nabla_{X}X\rangle - 4\mathit{Ric}(X,Y)\bigr)\Bigr)d\tau \nonumber \\ + & = & 2\sqrt{\tau}\langle Y,X\rangle |_{\tau_{1}}^{\tau_{2}} \nonumber \\ +& & + \int_{\tau_{1}}^{\tau_{2}} \sqrt{\tau}\langle Y,\bigl(\nabla R +- \frac{1}{\tau}X - 2\nabla_{X}X - 4\mathit{Ric}(X,\cdot)^*\bigr)\rangle d\tau. +\end{aligned} +$$ + +Now we drop the assumption that the generalized Ricci flow is an +ordinary Ricci flow. Still we can partition the interval +$[\tau_1,\tau_2]$ into finitely many sub-intervals with the property +that the restriction of $\gamma_0$ to each of the sub-intervals is +contained in a patch of space-time on which the generalized Ricci +flow is isomorphic to an ordinary Ricci flow. The above argument +then applies to each of the sub-intervals. Adding up +Equation (6.4) over these sub-intervals shows that the +same equation for the first variation of length for the entire +family $\gamma_u$ holds. + +We consider a variation $\gamma(\tau,u)$ with fixed endpoints, so +that $Y(\tau_{1})=Y(\tau_{2})= 0.$ Thus, the condition that +$\gamma$ be a critical path for the ${\mathcal L}$-length is that +the integral expression vanish for all variations $Y$ satisfying +$Y(\tau_1)=Y(\tau_2)=0$. Equation (6.4) holds for all +such $Y$ if and only if $\gamma$ satisfies +Equation (6.3). + +\entryblock{388e599c4bee} + +\entryblock{d23dc29f2aae} + +Written with respect to the variable $s=\sqrt{\tau}$ the +$\mathcal{L}$-geodesic equation becomes + +$$ +\nabla_{A(s)}A(s) -2s^{2}\nabla R + 4s\mathit{Ric}(A(s),\cdot)^*=0. +$$ + + Notice that in this form the ODE is +regular even at $s=0$. + +\entryblock{b2c992468118} + +**Proof.** Since the ODE in Equation (6.5) is non-singular even at +zero, it follows that $A(s)$ is a smooth function of $s$ in a +neighborhood of $s=0$. , The lemma follows easily by the change of +variables formula, $A(\tau)=2\sqrt{\tau}X_\gamma(\tau)$. + +\entryblock{f09cbffcbdce} + +### The ${\mathcal L}$-Jacobi equation + +Consider a family $\gamma(\tau,u)$ of ${\mathcal L}$-geodesics parameterized by +$u$ and defined on $[\tau_1,\tau_2]$ with $0\le \tau_1<\tau_2$. Let $Y(\tau)$ +be the horizontal vector field along $\gamma$ defined by + +$$ +Y(\tau)=\frac{\partial}{\partial u}\gamma(\tau,u)|_{u=0}. +$$ + +\entryblock{deb9b83d83f2} + +**Proof.** Given a family $\gamma(\tau,u)$ of ${\mathcal L}$-geodesics, then +from \entryref{75695d1b1b54} we have + +$$ +\nabla_{\widetilde X}\widetilde X = \frac{1}{2}\nabla R(\gamma) +-\frac{1}{2\tau}\widetilde X - 2\mathit{Ric}(\widetilde X,\cdot)^*. +$$ + +Differentiating this equation in the $u$-direction along the curve +$u=0$ yields + +$$ +\nabla_Y +\nabla_{\widetilde X}\widetilde X|_{u=0}=\frac{1}{2}\nabla_Y(\nabla +R)-\frac{1}{2\tau}\nabla_Y(\widetilde X)|_{u=0} -2\nabla_Y(\mathit{Ric}(\widetilde X,\cdot))^*|_{u=0}. +$$ + + Of course, we have + +$$ +\nabla_Y(\mathit{Ric}(\widetilde X,\cdot)^*)|_{u=0}=(\nabla_Y\mathit{Ric})(X,\cdot)^* ++\mathit{Ric}(\nabla_Y\widetilde X|_{u=0},\cdot)^*. +$$ + + Plugging this in, +interchanging the orders of differentiation on the left-hand side, using +$\nabla_{\widetilde Y}\widetilde X=\nabla_{\widetilde X}\widetilde Y$, and +restricting to $u=0$ yields the equation given in the statement of the lemma. +This equation is a regular, second-order linear equation for all $\tau>0$, and +hence is determined by specifying the value and first derivative at any +$\tau>0$. + +Equation (6.6) is obtained by applying $\nabla_Y$ to +Equation (6.3) and exchanging orders of differentiation. +The result Equation (6.6) is a second-order differential +equation for $Y$ that makes no reference to an extension of +$\gamma(\tau)$ to an entire family of curves. + +\entryblock{200f280a8e3b} + +In fact, there is a similar result even for $\tau_1=0$. + +\entryblock{6b205e5b51a2} + +**Proof.** We use the variable $s=\sqrt{\tau}$, and let $A(s)$ be the +horizontal component of $d\gamma/ds$. Then differentiating the +${\mathcal L}$-geodesic equation written with respect to this +variable we see + +$$ +\nabla_A\nabla_AY=-{\mathcal R}(Y,A)A+2s^2\nabla_Y(\nabla R)-4s(\nabla_Y\mathit{Ric})(A,\cdot) +-4s\mathit{Ric}(\nabla_AY,\cdot). +$$ + + Hence, for each tangent vector $Z$, +there is a unique solution to this equation with the two initial +conditions $Y(0)=0$ and $\nabla_AY(0)=Z$. + +On the other hand, from Equation (6.1) we have +$\nabla_X(Y)=\frac{1}{2\sqrt{\tau}}\nabla_A(Y)$, so that + +$$ +\sqrt{\tau}\nabla_X(Y)=\frac{1}{2}\nabla_A(Y). +$$ + +### Second order variation of ${\mathcal L}$ + We shall need the relationship of the ${\mathcal L}$-Jacobi +equation to the second-order variation of ${\mathcal L}$. This is +given in the next proposition. + +\entryblock{26b7f69e803e} + +Let us begin the proof of this proposition with the essential computation. + +\entryblock{f48558c60ca2} + +**Proof.** According to Equation (6.4) we have + +$$ +\begin{aligned} +\frac{\partial}{\partial u_2}{\mathcal L}(\gamma)(u_1,u_2) + & = & 2\sqrt{\tau_2}\langle \widetilde Y_2(\tau_2,u_1,u_2),\widetilde +X(\tau_2,u_1,u_2)\rangle +\\ & & -\int_{\tau_1}^{\tau_2}2\sqrt{\tau}\langle EL(\widetilde +X(\tau,u_1,u_2),\widetilde Y_2(\tau,u_1,u_2)\rangle d\tau, +\end{aligned} +$$ + + where +$EL(\widetilde X(\tau,u_1,u_2))$ is the Euler-Lagrange expression for +geodesics, i.e., the left-hand side of Equation (6.3). +Differentiating again yields: + +$$ +\begin{aligned} +\frac{\partial }{\partial u_1}\frac{\partial }{\partial u_2}{\mathcal +L}(\gamma_{u_1,u_2}\bigl)|_{u_1=u_2=0}\big. + = 2\sqrt{\tau_2}Y_1(\tau_2)\langle \widetilde +Y_2(\tau_2,u_1,0),\widetilde X(\tau_2,u_1,0)\rangle\bigl|_{u_1=0}\bigr. \nonumber\\ +& & - +\int_{\tau_1}^{\tau_2}2\sqrt{\tau}\left(\langle\nabla_{Y_1}EL(\widetilde +X), Y_2\rangle +\langle EL(X),\nabla_{Y_1}\widetilde +Y_2\rangle\right)(\tau,0,0)d\tau. +\end{aligned} +$$ + +Since $\gamma_{0,0}=\gamma$ is a geodesic, the second term in the +integrand vanishes, and since $[\widetilde X,\widetilde Y_1]=0$, we +have $\nabla_{Y_1}EL(\widetilde X(\tau,0,0))=\mathit{Jac}(Y_1)(\tau)$. +This proves the lemma. + +\entryblock{2fef81872a01} + +\entryblock{abd40a465b53} + +**Proof.** Given $Y_1$ and $Y_2$ along $\gamma$ we construct a two-parameter +family of curves parameterized by backward time as follows. Let +$\gamma(\tau,u_1)$ be the value at $u_1$ of the geodesic through +$\gamma(\tau)$ with tangent vector $Y_1(\tau)$. This defines a +family of curves parameterized by backward time, the family being +parameterized by $u_1$ sufficiently close to $0$. We extend + $Y_1$ and $X$ to vector fields on this entire family by defining them to be + $\partial/\partial u_1$ and the +horizontal projection of $\partial/\partial \tau$, respectively. + Now we +extend the vector field $Y_2$ along $\gamma$ to a vector field on +this entire one-parameter family of curves. We do this so that +$Y_2(\tau_2,u_1)=Y_1(\tau_2,u_1)$. Now given this extension +$Y_2(\tau,u_1)$ we define a two-parameter family of curves +parameterized by backward time + by setting $\gamma(\tau,u_1,u_2)$ equal to the value at $u_2$ of the geodesic + through $\gamma(\tau,u_1)$ in the direction $Y_2(\tau,u_1)$. + We then extend $Y_1$, $Y_2$, and $X$ over this entire family by letting them be + $\partial/\partial u_1$, +$\partial/\partial u_2$, and the horizontal projection of $\partial/ \partial +\tau$, respectively. Applying \entryref{f48558c60ca2} and using the fact that +$Y_i(\bar\tau)=0$ we conclude that + +$$ +\frac{\partial}{\partial u_1}\frac{\partial}{\partial u_2}{\mathcal L}(\gamma)|_{u_1=u_2=0}= -\int_{\tau_1}^{\tau_2}2\sqrt{\tau}\langle +\mathit{Jac}(Y_1),Y_2\rangle d\tau +$$ + + and symmetrically that + +$$ +\frac{\partial}{\partial u_2}\frac{\partial}{\partial u_1}{\mathcal L}(\gamma)|_{u_1=u_2=0}= -\int_{\tau_1}^{\tau_2}2\sqrt{\tau}\langle +\mathit{Jac}(Y_2),Y_1\rangle d\tau. +$$ + + Since the second cross partials +are equal, the corollary follows. + +Now we are in a position to establish \entryref{26b7f69e803e}. + +**Proof.** (Of \entryref{26b7f69e803e}) +From the equation in \entryref{f48558c60ca2}, the equality of the second variation +of ${\mathcal L}$-length at $u=0$ and the integral is immediate from the fact +that $Y(\tau_2)=0$. It follows immediately that, if $Y$ is an ${\mathcal +L}$-Jacobi field vanishing at $\tau_2$, then the second variation of the length +vanishes at $u=0$. Conversely, suppose given a family $\gamma_u$ with +$\gamma_0=\gamma$ with the property that the second variation of length +vanishes at $u=0$, and that the vector field $Y=(\partial\gamma/\partial +u)|_{u=0}$ along $\gamma$ vanishes at the end points. It follows that the +integral also vanishes. Since $\gamma$ is a minimizing ${\mathcal L}$-geodesic, +for any variation $W$, vanishing at the endpoints, the first variation of the +length vanishes and the second variation of length is non-negative. That is to +say, + +$$ +-\int_{\tau_1}^{\tau_2}2\sqrt{\tau}\langle \mathit{Jac}(W),W\rangle d\tau\ge 0 +$$ + +for all vector fields $W$ along $\gamma$ vanishing at the endpoints. +Hence, the restriction to the space of vector fields along $\gamma$ +vanishing at the endpoints of the bilinear form + +$$ +B(Y_1,Y_2)=-\int_{\tau_1}^{\tau_2}2\sqrt{\tau}\langle +\mathit{Jac}(Y_1),(Y_2)d\tau, +$$ + + which is symmetric by \entryref{abd40a465b53}, is +positive semi-definite. Since $B(Y,Y)=0$, it follows that $B(Y,\cdot)=0$; that +is to say, $\mathit{Jac}(Y)=0$. + +## The ${\mathcal L}$-exponential map and its first-order properties + +We use ${\mathcal L}$-geodesics in order to define the ${\mathcal +L}$-exponential map. + +For Section 6.2 we fix $\tau_1\ge 0$ and a point $x\in +{\mathcal M}$ with $\mathbf{t}(x)=T-\tau_1$. We suppose that $T-\tau_1$ +is greater than the initial time of the generalized Ricci flow. +Then, for every $Z \in T_{x}M_{T-\tau_1},$ there is a maximal +$\mathcal{L}$-geodesic, denoted $\gamma_{Z}$, defined on some +positive $\tau$-interval, with $\gamma_{Z}(\tau_1)=x$ and with +$\sqrt{\tau_1}X(\tau_1)=Z$. (In the case $\tau_1=0$, this equation +is interpreted to mean $\lim_{\tau\rightarrow 0} \sqrt{\tau}X(\tau) += Z$.) + +\entryblock{df842ef2ab9f} + +\entryblock{948d50598f2f} + +**Proof.** The tangent vector in space-time of the ${\mathcal L}$-geodesic +$\gamma$ is the vector field $-\chi+X_\gamma(\tau)$ along $\gamma$, +where $X_\gamma(\tau)$ satisfies Equation (6.3). As +above, in the case $\tau_1=0$, it is convenient to replace the +independent variable $\tau$ by $s=\sqrt{\tau}$, so that the ODE +becomes Equation (6.5) which is regular at $0$. With this +change, the lemma then follows immediately by the usual results on +existence, uniqueness and $C^\infty$-variation with parameters of +ODE's. + +### The differential of ${\mathcal L}\mathit{exp}$ + +Now we compute the differential of ${\mathcal L}\mathit{exp}$. + +\entryblock{d2e61ac52277} + +**Proof.** Let $Z(u)$ be a curve in ${\mathcal D}_x^{\bar\tau}$ with $Z(0)=Z$. +Let $\gamma_u$ be the ${\mathcal L}$-geodesic starting at $x$ with +$\sqrt{\tau_1}X_{\gamma_u}(\tau_1)=Z(u)$. Then, clearly, + +$$ +d_Z{\mathcal L}\mathit{exp}^{\bar\tau}_x\left(\frac{dZ}{du}(0)\right) +=\frac{\partial}{\partial u}\left(\gamma_u(\bar\tau)\right)|_{u=0}. +$$ + + On the +other hand, the vector field $Y(\tau)=\left(\partial\gamma_u(\tau)/\partial +u\right)|_{u=0} $ is an ${\mathcal L}$-Jacobi field along $\gamma_Z$. Thus, to +complete the proof in the case when $\tau_1>0$ we need only see that +$\nabla_X\widetilde Y(\tau_1)=\nabla_Y\widetilde X(\tau_1)$. This is clear +since, as we have already seen, $[\widetilde X,\widetilde Y]=0$. + + When $\tau_1=0$, we complete the +argument using the following claim. + +\entryblock{da7306fc733d} + +**Proof.** This follows immediately by changing variables, +replacing $\tau$ by $s=\sqrt{\tau}$. + +This completes the proof of \entryref{d2e61ac52277}. + +### Positivity of the second variation at a minimizing ${\mathcal L}$-geodesic + +If $\gamma$ is a minimizing ${\mathcal L}$-geodesic, then variations +of $\gamma$ fixing the endpoints give curves whose ${\mathcal +L}$-length is no less than that of $\gamma$. In fact, there is a +second-order version of this inequality which we shall need later. + +\entryblock{faca28b0bd43} + +**Proof.** According to \entryref{26b7f69e803e} the second variation in the +$Y$-direction is non-negative and vanishes if and only if $Y$ is an ${\mathcal +L}$-Jacobi field. But since $d_Z{\mathcal L}\mathit{exp}_x^{\bar\tau}$ is a +diffeomorphism, by \entryref{d2e61ac52277} there are no non-zero ${\mathcal +L}$-Jacobi fields vanishing at both endpoints of $\gamma_Z$. + +### The gradient of $\widetilde L^\tau$ + +Recall that $\widetilde +L^{\tau}$ is the map from ${\mathcal D}^{\tau}_x$ to $\Ar$ that +assigns to each $Z$ the ${\mathcal L}$-length of +$\gamma_Z|_{[\tau_1,\tau]}$. We compute its gradient. + +\entryblock{abf9cbb10c68} + +**Proof.** Since ${\mathcal D}^{\tau}_x$ is an open subset of $T_x(M_{T-\tau_1})$, it +follows that for any $\widetilde Y\in T_x(M_{T-\tau_1})$ there is a +one-parameter family $\gamma_u(\tau')=\gamma(\tau',u)$ of + ${\mathcal L}$-geodesics, defined for $\tau_1\le \tau'\le \tau$, +starting at $x$ with $\gamma(\cdot,0)=\gamma_Z$ and with +$\frac{\partial}{\partial u}\left(\sqrt{\tau_1}X(\tau_1)\right)=\widetilde Y$. +(Again, when $\tau_1=0$, this equation is interpreted to mean +$\frac{\partial}{\partial u}\mathit{lim}_{\tau'\rightarrow +0}(\sqrt{\tau'}X(\tau',u))=\widetilde Y$.) Let +$Y(\tau')=\frac{\partial}{\partial u}(\gamma(\tau',u))|_{u=0}$ be the +corresponding ${\mathcal L}$-Jacobi field along $\gamma_Z$. Since +$\gamma(\tau_1,u)=x$ for all $u$, we have $Y(\tau_1)=0$. Since +$\gamma(\cdot,u)$ is an ${\mathcal L}$-geodesic for all $u$, according to +Equation (6.4), and in the case $\tau_1=0$, using the fact that +$\sqrt{\tau}X(\tau')$ approaches a finite limit as $\tau\rightarrow 0$, we have + +$$ +\frac{d}{du}{\mathcal +L}(\gamma_u)|_{u=0}=2\sqrt{\tau}\langle X(\tau),Y(\tau)\rangle. +$$ + + By +\entryref{d2e61ac52277} we have $Y(\tau)=d_Z{\mathcal L}\mathit{exp}_x^{\tau}(\tilde +Y)$. Thus, + +$$ +\langle\nabla\widetilde L^{\tau},\widetilde Y\rangle=\frac{d}{du}{\mathcal +L}(\gamma_u)|_{u=0}=2\sqrt{\tau}\langle +X(\tau),Y(\tau)\rangle=2\sqrt{\tau}\langle X(\tau),d_Z({\mathcal +L}\mathit{exp}^{\tau}_x)(\widetilde Y)\rangle. +$$ + +### Local diffeomorphism near the initial $\tau$ + +Now let us use the nature of the ${\mathcal L}$-Jacobi equation to +study ${\mathcal L}\mathit{exp}_x$ for $\tau>\tau_1$ but $\tau$ +sufficiently close to $\tau_1$. + +\entryblock{4180719ab2c8} + +**Proof.** Fix $x$ and $Z$ as in the statement of the lemma. To establish the result it +suffices to prove that there is $\delta>0$ such that $d_Z{\mathcal L}\mathit{exp}_x^{\tau}$ is an isomorphism for all $\tau_1<\tau<\tau_1+\delta$. By +\entryref{d2e61ac52277} it is enough to find a $\delta>0$ such that any ${\mathcal +L}$-Jacobi field $Y$ along + $\gamma_Z$ with $\sqrt{\tau_1}\nabla_XY(\tau_1)\not= 0$ does + not vanish on the interval $(\tau_1,\tau_1+\delta)$. + Because the ${\mathcal L}$-Jacobi equation is linear, it suffices to consider the + case of ${\mathcal L}$-Jacobi fields with + $|\nabla_XY(\tau_1)|=1$. The space of such fields is identified with the + unit sphere in $T_xM_{T-\tau_1}$. + Let us consider first the case when $\tau_1\not= 0$. Then for any + such tangent vector $\nabla_XY(\tau_1)\not=0$. Since $Y(\tau_1)=0$, + it follows that $Y(\tau)\not=0$ in some interval + $(\tau_1,\tau_1+\delta)$, where $\delta$ can depend on $Y$. Using + compactness of the unit sphere in the tangent space, we see that there is $\delta>0$ + independent of $Y$ so that the above holds. + + In case when $\tau_1=0$, it is convenient to shift to the + $s=\sqrt{\tau}$ parameterization. Then the geodesic equation and + the ${\mathcal L}$-Jacobi equation are non-singular at the origin. Also, letting + $A=d\gamma_Z/ds$ we have $\nabla_AY=2\mathit{lim}_{\tau\rightarrow + 0}\sqrt{\tau}\nabla_XY$. In these variables, the argument for $\tau_1=0$ + is the same as the one above for $\tau_1>0$. + +\entryblock{fa3234930898} + +## Minimizing ${\mathcal L}$-geodesics and the injectivity domain + +Now we discuss the analogue of the interior of the cut locus for the +usual exponential map of a Riemannian manifold. For +Section 6.3 we keep the assumption that $x\in +{\mathcal M}$ with $\mathbf{t}(x)=T-\tau_1$ for some $\tau_1\ge 0$. + +\entryblock{90fe97584bdd} + +It is clear from the definition that $\widetilde{\mathcal +U}_x\subset {\mathcal D}_x$ is an open subset and hence +$\widetilde{\mathcal U}_x$ is an open subset of +$T_xM_{T-\tau_1}\times (\tau_1,\infty)$. Of course, this implies +that $\widetilde {\mathcal U}_x(\tau)$ is an open subset of +${\mathcal D}_x^\tau$ for every $\tau>\tau_1$. + +\entryblock{440e7d27aee8} + +By definition, for every point $q\in {\mathcal U}_x$ for any +$(Z,\tau)\in \widetilde{\mathcal U}_x$ with ${\mathcal L}\mathit{exp}_x(Z,\tau)=q$, the ${\mathcal L}$-geodesic +$\gamma_{Z}|_{[\tau_1,\tau]}$ is a minimizing + ${\mathcal L}$-geodesic to $q$. In particular, there is a minimizing + ${\mathcal L}$-geodesic from $x$ to $q$. + +\entryblock{5578a00f3308} + +This brings us to the +analogue of the fact that in Riemannian geometry the restriction to +the interior of the cut locus of the exponential mapping is a +diffeomorphism onto an open subset of the manifold. + +\entryblock{17254b50b27a} + +**Proof.** We consider the differential of ${\mathcal L}\mathit{exp}_x$ at any + $(Z,\tau)\in\widetilde {\mathcal U}_x$. By construction + the restriction of this differential to $T_xM_{T-\tau_1}$ is a local + isomorphism onto ${\mathcal H}T{\mathcal M}$ at the image point. On the other + hand, the differential of ${\mathcal L}\mathit{exp}_x$ in the $\tau$ + direction is $\gamma_Z'(\tau)$, whose 'vertical' component is + $-\chi$. By the inverse function theorem this + shows that ${\mathcal L}\mathit{exp}_x$ is a local diffeomorphism at + $(Z,\tau)$, and + its image is an open subset of ${\mathcal M}$. +The uniqueness in Condition 2, of the definition immediately implies +that the restriction of ${\mathcal L}\mathit{exp}_x$ to +$\widetilde{\mathcal U}_x$ is one-to-one, and hence that it is a +global diffeomorphism onto its image ${\mathcal U}_x$. + +Since for every $(Z,\tau)\in\widetilde{\mathcal U}_x$ the ${\mathcal +L}$-geodesic $\gamma_Z|_{[\tau_1,\tau]}$ is ${\mathcal +L}$-minimizing, we see that $L_x\circ {\mathcal L}\mathit{exp}_x|_{\widetilde{\mathcal U}_x}=\widetilde L|_{\widetilde +{\mathcal U}_x}$ and that $L_x\colon {\mathcal U}_x\to \Ar$ is a +smooth function. + +According to \entryref{abf9cbb10c68} we have: + +\entryblock{033786a89c6d} + +At the level of generality that we are working (arbitrary +generalized Ricci flows) there is no result analogous to the fact in +Riemannian geometry that the image under the exponential mapping of +the interior of the cut locus is an open dense subset of the +manifold. There is an analogue in the special case of Ricci flows on +compact manifolds or on complete manifolds of bounded curvature. +These will be discussed in Section 7. + +### Monotonicity of the $\widetilde{\mathcal U}_x(\tau)$ with respect to $\tau$ + +Next, we have the analogue of the fact in Riemannian geometry that +the cut locus is star-shaped. + +\entryblock{8b9b3403dbc1} + +**Proof.** For $Z\in \widetilde{\mathcal U}_x(\bar\tau')$, + we shall show that: +(i) the ${\mathcal L}$-geodesic $\gamma_{Z'}|_{[\tau_1,\bar\tau]}$ +is the unique minimizing ${\mathcal L}$-geodesic from $x$ to +$\gamma_{Z}(\bar\tau)$, and (ii) the differential $d_Z{\mathcal +L}\mathit{exp}_x^{\bar\tau}$ is an isomorphism. Given these two +conditions, it follows from the definition that $\widetilde{\mathcal +U}_x(\bar\tau')$ is contained in $\widetilde{\mathcal +U}_x(\bar\tau)$. + + We show that the ${\mathcal L}$-geodesic +$\gamma_{Z}|_{[\tau_1,\bar\tau]}$ is the unique minimizing +${\mathcal L}$- geodesic to its endpoint. If there is an ${\mathcal +L}$-geodesic $\gamma_1$, distinct from +$\gamma_{Z}|_{[\tau_1,\bar\tau]}$, from $x$ to $\gamma_Z(\bar\tau)$ +whose ${\mathcal L}$-length is at most that of +$\gamma_Z|_{[\tau_1,\bar\tau]}$, then we can create a broken path +$\gamma_1*\gamma_Z|_{[\bar\tau,\bar\tau']}$ parameterized by +backward time whose ${\mathcal L}$-length is at most that of +$\gamma_{Z}$. Since this latter path is not smooth, its ${\mathcal +L}$-length cannot be the minimum, which is a contradiction. + + Now suppose that $d_Z{\mathcal L}\mathit{exp}_x^{\bar\tau}$ is not an isomorphism. +The argument is similar to the one above, using a non-zero +${\mathcal L}$-Jacobi field vanishing at both endpoints rather than +another geodesic. Let $\tau'_2$ be the first $\tau$ for which +$d_Z{\mathcal L}\mathit{exp}_x^\tau$ is not an isomorphism. +According to \entryref{4180719ab2c8}, $\tau_1<\tau'_2\le \bar\tau$. Since +${\mathcal L}\mathit{exp}_x^{\tau'_2}$ is not a local diffeomorphism at +$(Z,\tau'_2)$, by \entryref{d2e61ac52277} there is a non-zero +${\mathcal L}$-Jacobi field $Y$ along $\gamma_Z|_{[\tau_1,\tau'_2]}$ +vanishing at both ends. Since $\gamma_Z|_{[\tau_1,\tau_2']}$ is +${\mathcal L}$-minimizing, according to \entryref{26b7f69e803e}, +the second variation of the length of $\gamma_Z|_{\tau_1,\tau'_2]}$ +in the $Y$-direction vanishes, in the sense that if $\gamma(u,\tau)$ +is any one-parameter family of paths parameterized by backward time +from $x$ to $\gamma_Z(\tau'_2)$ with $(\partial \gamma/\partial +u)|_{u=0}=Y$ then + +$$ +\frac{\partial^2{\mathcal L}(\gamma_u)}{\partial +u^2}\bigl|_{u=0}\bigr.=0. +$$ + + Extend $Y$ to a horizontal vector field +$\widehat Y$ along $\gamma_Z$ by setting $\widehat Y(\tau)=0$ for +all $\tau\in [\tau'_2,\bar\tau]$. Of course, the extended horizontal +vector field $\widehat Y$ is not $C^2$ at $\tau'_2$ since $Y$, being +a non-zero ${\mathcal L}$-Jacobi field, does not vanish to second +order there. This is the first-order variation of the family +$\hat\gamma(u,\tau)$ that agrees with $\gamma(u,\tau)$ for all +$\tau\le \tau'_2$ and has $\hat\gamma(u,\tau)=\gamma_Z(\tau)$ for +all $\tau\in [\tau'_2,\bar\tau]$. Of course, the second-order +variation of this extended family at $u=0$ agrees with the +second-order variation of the original family at $u=0$, and hence +vanishes. But according to \entryref{26b7f69e803e} this means that +$\widehat Y$ is an ${\mathcal L}$-Jacobi field, which is absurd +since it is not a $C^2$-vector field. + +We shall also need a closely related result. + +\entryblock{4b9af3660539} + +**Proof.** It suffices to show that the differential of ${\mathcal L}\mathit{exp}_{q_2}^\tau$ is an isomorphism for all $\tau\in +(\tau_2,\bar\tau]$. If this is not the case, then there is a +$\tau'\in (\tau_2,\bar\tau]$ and a non-zero ${\mathcal L}$-Jacobi +field $Y$ along $\gamma_Z|_{[\tau_2,\tau']}$ vanishing at both ends. +We extend $Y$ to a horizontal vector field $\widehat Y$ along all of +$\gamma_Z|_{[\tau_1,\tau']}$ by setting it equal to zero on +$[\tau_1,\tau_2]$. Since $Y$ is an ${\mathcal L}$-Jacobi field, the +second-order variation of ${\mathcal L}$-length in the direction of +$Y$ is zero, and consequently the second-order variation of the +length of $\gamma_Z|_{[\tau_1,\tau']}$ vanishes. Hence by +\entryref{26b7f69e803e} it must be the case that $\widehat Y$ is a +${\mathcal L}$-Jacobi field. This is impossible since $\widehat Y$ +is not smooth at $\tau'$. + +We finish this section with a computation of the $\tau$-derivative +of $L_x$. + +\entryblock{b3bf2f526cfc} + +**Proof.** By definition and the Fundamental Theorem of Calculus, we have + +$$ +\frac{d}{d\tau}L_x(\gamma(\tau))= \sqrt{\tau}\left(R(\gamma(\tau))+|X(\tau)|^2\right). +$$ + +On the other hand since $\gamma'(\tau)=-\partial/\partial t+X(\tau)$ +the chain rule implies + +$$ +\frac{d}{d\tau}L_x(\gamma(\tau))=\langle \nabla +L_x,X(\tau)\rangle+\frac{\partial L_x}{\partial +\tau}(\gamma(\tau)), +$$ + + so that + +$$ +\frac{\partial L_x}{\partial \tau}(\gamma(\tau))=\sqrt{\tau}\left(R(\gamma(\tau))+|X(\tau)|^2\right)-\langle\nabla +L_x,X(\tau)\rangle. +$$ + + Now using \entryref{033786a89c6d}, and rearranging +the terms gives the result. + +## Second-order differential inequalities for $\widetilde L^{\bar\tau}$ and $L_x^{\bar\tau}$ + +Throughout Section 6.4 we fix $x\in {\mathcal M}$ with $x\in +M_{T-\tau_1}$. + +### The second variation formula for $\widetilde L^{\bar\tau}$ + +Our goal here is to compute the second variation of $\widetilde +L^{\bar\tau}$ in the direction of a horizontal vector field +$Y(\tau)$ along an ${\mathcal L}$-geodesic $\gamma$. Here is the +main result of this subsection. + +\entryblock{3add96057999} + +As we shall see, this is simply a rewriting of the equation in +\entryref{f48558c60ca2} in the special case when $u_1=u_2$. + +We begin the proof of this result with the following computation. + +\entryblock{a75b1ef40db8} + +**Proof.** We can break $\frac{\partial}{\partial\tau}\langle \nabla_{X}Y,Y\rangle $ into +two parts: the first assumes that the metric is constant and the second deals +with the variation with $\tau$ of the metric. The first contribution is the +usual formula + +$$ +\frac{\partial}{\partial\tau}\langle \nabla_{X}Y,Y\rangle _{G(T-\tau_0)}= +\langle \nabla_{X}Y,\nabla_{X}Y\rangle _{G(T-\tau_0)}+ \langle +\nabla_{X}\nabla_{X}Y,Y\rangle _{G(T-\tau_0)}. +$$ + + This gives us the first two +terms of the right-hand side of the equation in the claim. + +We show that the last two terms in that equation come from differentiating the +metric with respect to $\tau$. To do this recall that in local coordinates, +writing the metric $G(T-\tau)$ as $g_{ij}$, we have + +$$ +\langle \nabla_{X}Y,Y \rangle=g_{ij}\bigl(X^k\partial_k +Y^i+\Gamma_{kl}^iX^kY^l\bigr)Y^j. +$$ + + There are two contributions coming from +differentiating the metric with respect to $\tau$. The first is when we +differentiate $g_{ij}$. This leads to + +$$ +2\mathit{Ric}_{ij}\bigl(X^k\partial_k +Y^i+\Gamma_{kl}^iX^kY^l\bigr)Y^j=2\mathit{Ric}(\nabla_XY,Y\rangle. +$$ + +The other contribution is from differentiating the Christoffel +symbols. This yields + +$$ +g_{ij}\frac{\partial\Gamma^{i}_{kl}}{\partial \tau}X^{k}Y^{l}Y^{j}. +$$ + +Differentiating the formula +$\Gamma_{kl}^i=\frac{1}{2}g^{si}(\partial_kg_{sl}+\partial_lg_{sk}-\partial_s +g_{kl})$ leads to + +$$ +\begin{aligned} +g_{ij}\frac{\partial +\Gamma_{kl}^i}{\partial \tau} & = & -2\mathit{Ric}_{ij}\Gamma_{kl}^i+g_{ij}g^{si}(\partial_k\mathit{Ric}_{sl}+\partial_l\mathit{Ric}_{sk} +-\partial_s\mathit{Ric}_{kl})\\ +& = & -2\mathit{Ric}_{ij}\Gamma_{kl}^i+ +\partial_k\mathit{Ric}_{jl}+\partial_l\mathit{Ric}_{jk} -\partial_j\mathit{Ric}_{kl}. +\end{aligned} +$$ + +Thus, we have + +$$ +\begin{aligned} +g_{ij}\frac{\partial\Gamma^{i}_{kl}}{\partial \tau}X^{k}Y^{l}Y^{j} + & = & \bigl(-2\mathit{Ric}_{ij}\Gamma^i_{kl}+\partial_k\mathit{Ric}_{jl})\bigr)X^kY^lY^j \\ +& = & (\nabla_X\mathit{Ric})(Y,Y) +\end{aligned} +$$ + +This completes the proof of the claim. + +Now we return to the proof of the second variational formula in +\entryref{3add96057999}. + +**Proof.** According to \entryref{f48558c60ca2} we have + +$$ +\frac{d^2}{du^2}{\mathcal L}_{u=0}=2\sqrt{\bar +\tau}Y(\bar\tau)(\langle \widetilde Y(\bar\tau,u),\widetilde +X(\bar\tau,u)\rangle)|_{u=0}-\int_{\tau_1}^{\tau_2}2\sqrt{\tau}\langle +\mathit{Jac}(Y),Y\rangle d\tau. +$$ + + We plug in +Equation 6.6 for $\mathit{Jac}(Y)$ and this makes the +integrand + +$$ +\begin{aligned} +\sqrt{\tau}\langle\nabla_Y(\nabla +R),Y\rangle+2\sqrt{\tau}\langle{\mathcal R}(Y,X)Y,X\rangle +-\bigl(2\sqrt{\tau}\langle\nabla_X\nabla_XY,Y\rangle+\frac{1}{\sqrt{\tau}}\langle\nabla_XY,Y\rangle\bigr)\\ +-4\sqrt{\tau}(\nabla_Y\mathit{Ric})(X,Y)-4\sqrt{\tau}\mathit{Ric}(\nabla_XY,Y) +\end{aligned} +$$ + +The first term is $\sqrt{\tau}\mathit{Hess}(R)(Y,Y)$. Let us deal with +the third and fourth terms, which are grouped together within +parentheses. According to the previous claim, we have + +$$ +\begin{aligned} +\frac{\partial}{\partial \tau}\bigl(2\sqrt{\tau}\langle +\nabla_XY,Y\rangle\bigr) & = & +\frac{1}{\sqrt{\tau}}\langle\nabla_XY,Y\rangle+2\sqrt{\tau}\langle\nabla_X\nabla_XY,Y\rangle + +2\sqrt{\tau}\langle\nabla_XY,\nabla_XY\rangle \\ & & +4\sqrt{\tau}\mathit{Ric}(\nabla_XY,Y) + +2\sqrt{\tau} (\nabla_X\mathit{Ric})(Y,Y). +\end{aligned} +$$ + +This allows us to replace the two terms under consideration by + +$$ +-\frac{\partial}{\partial t}\bigl(2\sqrt{\tau}\langle +\nabla_XY,Y\rangle\bigr)+2\sqrt{\tau}\langle\nabla_XY,\nabla_XY\rangle+4\sqrt{\tau}\mathit{Ric}(\nabla_XY,Y) +2\sqrt{\tau} (\nabla_X\mathit{Ric})(Y,Y). +$$ + +Integrating the total derivative out of the integrand and canceling +terms leaves the integrand as + +$$ +\begin{aligned} +\sqrt{\tau}\mathit{Hess}(R)(Y,Y)+2\sqrt{\tau}\langle{\mathcal R}(Y,X)Y,X\rangle ++2\sqrt{\tau}|\nabla_XY|^2 \\ +-4\sqrt{\tau}(\nabla_Y\mathit{Ric})(X,Y)+2\sqrt{\tau} (\nabla_X\mathit{Ric})(Y,Y), +\end{aligned} +$$ + +and makes the boundary term (the one in front of the integral) equal +to + +$$ +2\sqrt{\bar\tau}\bigl(Y(\bar\tau)\langle +\widetilde Y(\bar\tau,u),\widetilde +X(\bar\tau,u)\rangle|_{u=0}-\langle +\nabla_XY(\bar\tau),Y(\bar\tau)\rangle\bigr)=2\sqrt{\bar\tau}\langle +X(\bar\tau),\nabla_Y\widetilde Y(\bar\tau,u)|_{u=0}\rangle. +$$ + + This +completes the proof of the proposition. + +### Inequalities for the Hessian of $L_x^{\bar\tau}$ + +Now we shall specialize the type of vector fields along $\gamma$. +This will allow us to give an inequality for the Hessian of +${\mathcal L}$ involving the integral of the vector field along +$\gamma$. These lead to inequalities for the Hessian of +$L_x^{\bar\tau}$. The main result +of this section is \entryref{88d4ef67440f} below. In the end we +are interested in the case when the $\tau_1=0$. In this case the +formulas simplify. The reason for working here in the full +generality of all $\tau_1$ is in order to establish differential +inequalities at points not in the injectivity domain. As in the case +of the theory of geodesics, the method is to establish weak +inequalities at these points by working with smooth barrier +functions. In the geodesic case the barriers are constructed by +moving the initial point out the geodesic a small amount. Here the +analogue is to move the initial point of an ${\mathcal L}$-geodesic +from $\tau_1=0$ to a small positive $\tau_1$. Thus, the case of +general $\tau_1$ is needed so that we can establish the differential +inequalities for the barrier functions that yield the weak +inequalities at non-smooth points. + +\entryblock{1ac0cdbfd850} + +Direct computation shows the following: + +\entryblock{e1277342360a} + +Now we come to the main result of this subsection, which is an +extremely important inequality for the Hessian of $L_x^{\bar\tau}$. + +\entryblock{88d4ef67440f} + +\entryblock{a17e64a8c11e} + +We begin the proof of this proposition with three elementary lemmas. The first +is an immediate consequence of the definition of $\widetilde {\mathcal +U}_x(\bar\tau)$. + +\entryblock{2e4da429dc95} + +\entryblock{44abf07e60ec} + +**Proof.** Let $\gamma(\tau,u)$ be a one-parameter family of ${\mathcal +L}$-geodesics emanating from $x$ with $\gamma(u,0)$ being the +${\mathcal L}$-geodesic in the statement of the lemma and with +$\frac{\partial}{\partial u}\gamma(\tau,0)=Y(\tau)$. We have the +extensions of $X(\tau)$ and $Y(\tau)$ to vector fields $\widetilde +X(\tau,u)$ and $\widetilde Y(\tau,u)$ defined at $\gamma(\tau,u)$ +for all $\tau$ and $u$. Of course, + +$$ +\begin{aligned} +2\sqrt{\bar\tau}\langle \nabla_Y\widetilde +X(\bar\tau,u)|_{u=0},Y(\bar\tau)\rangle \\ +& = & Y(\langle 2\sqrt{\bar\tau}\widetilde X(\bar\tau,u),\widetilde +Y(\bar\tau,u)\rangle)|_{u=0}-\langle +2\sqrt{\bar\tau}X(\bar\tau),\nabla_Y\widetilde +Y(\bar\tau,u)|_{u=0}\rangle. +\end{aligned} +$$ + + Then by +\entryref{033786a89c6d} we have + +$$ +\begin{aligned} +2\sqrt{\bar\tau}\langle \nabla_Y\widetilde +X(\bar\tau,u)|_{u=0},Y(\bar\tau)\rangle & = Y(\bar\tau)(\langle +\nabla L_x^{\bar\tau},\widetilde +Y(\bar\tau,u)\rangle)|_{u=0}-\langle \nabla +L_x^{\bar\tau},\nabla_{Y(\bar\tau)}\widetilde +Y(\bar\tau,u)|_{u=0}\rangle +\\ +& = Y(\bar\tau)(\widetilde Y(\bar\tau,u) +L_x^{\bar\tau})|_{u=0}-\nabla_{Y(\bar\tau)}\widetilde +Y(\bar\tau,u)|_{u=0}(L_x^{\bar\tau}) \\ & = \mathit{Hess}(L_x^{\bar\tau})(Y(\bar\tau),Y(\bar\tau)). +\end{aligned} +$$ + +Now suppose that we have a horizontal vector field that is both +adapted and ${\mathcal L}$-Jacobi. We get: + +\entryblock{72949e6350d4} + +**Proof.** From the definition of an adapted vector field $Y(\tau)$ we have + +$$ +\mathit{Ric}(Y(\tau),Y(\tau)) + \langle +\nabla_{X}Y(\tau),Y(\tau)\rangle = +\frac{1}{2\sqrt{\tau}(\sqrt{\tau}-\sqrt{\tau_1})}\langle +Y(\tau),Y(\tau)\rangle. +$$ + +Since $Y(\tau)$ is an ${\mathcal L}$-Jacobi field, according to +\entryref{44abf07e60ec} we have + +$$ +\langle\nabla_XY(\bar\tau),Y(\bar\tau)\rangle = +\frac{1}{2\sqrt{\bar\tau}}\mathit{Hess}(L_x^{\bar\tau})(Y(\bar\tau),Y(\bar\tau)). +$$ + + Putting these +together gives the result. + +Now we are ready to begin the proof of \entryref{88d4ef67440f}. + +**Proof.** Let $\tilde\gamma(\tau,u)$ be a family of curves with +$\gamma(\tau,0)=\gamma_Z$ and with $\frac{\partial}{\partial +u}\gamma(\tau,u)=\widetilde Y(\tau,u)$. We denote by $Y$ the +horizontal vector field which is the restriction of $\widetilde Y$ +to $\gamma_0=\gamma_Z$. We denote by +$q(u)=\tilde\gamma(\bar\tau,u)$. By restricting to a smaller +neighborhood of $0$ in the $u$-direction, we can assume that +$q(u)\in {\mathcal U}_x(\bar\tau)$ for all $u$. Then ${\mathcal +L}(\tilde\gamma_u)\ge L_x^{\bar\tau}(q(u))$. Of course, +$L_x^{\bar\tau}(q(0))={\mathcal L}(\gamma_Z)$. This implies that + +$$ +\frac{d}{d u}L_x^{\bar\tau}(q(u))\bigl|\bigr._{u=0}= +\frac{d}{du}{\mathcal L}(\gamma_u)\bigl|\bigr._{u=0}, +$$ + + and + +$$ +Y(\bar\tau)(\widetilde +Y(\bar\tau,u)(L^{\bar\tau}_x))|_{u=0}=\frac{d^2}{du^2}L_x^{\bar\tau}(q(u))\bigl|\bigr._{u=0}\le +\frac{d^2}{du^2}{\mathcal L}(\gamma_u)\bigl|\bigr._{u=0}. +$$ + + Recall that +$\nabla L_x^{\bar\tau}(q) = 2\sqrt{\bar\tau}X(\bar\tau)$, so that + +$$ +\nabla_{Y(\bar\tau)}\widetilde Y(\bar\tau,u)|_{u=0}(L_x^{\bar\tau})= +\langle \nabla_{Y(\bar\tau)}\widetilde Y(\bar\tau,u)|_{u=0},\nabla +L^{\bar\tau}\rangle = 2\sqrt{\bar\tau}\langle +\nabla_{Y(\bar\tau)}\widetilde +Y(\bar\tau,u)|_{u=0},X(\bar\tau)\rangle . +$$ + + Thus, by +\entryref{3add96057999}, and using the fact that +$Y(\tau_1)=0$, we have + +$$ +\begin{aligned} +\mathit{Hess}(L^{\bar\tau})(Y(\bar\tau),Y(\bar\tau)) = +Y(\bar\tau)\left(\widetilde +Y(\bar\tau,u)(L_x^{\bar\tau})\right)|_{u=0}- +\nabla_{Y(\bar\tau)}\widetilde Y(\bar\tau,u)|_{u=0}(L_x^{\bar\tau}) \\ +& \le & \frac{d^2}{du^2}{\mathcal +L}(\gamma_u)-2\sqrt{\bar\tau}\langle +\nabla_{Y(\bar\tau)}\widetilde Y(\bar\tau,u)|_{u=0},X(\bar\tau)\rangle \\ + & = & \int_{\tau_1}^{\bar\tau} +\sqrt{\tau}\bigl(\mathit{Hess}(R)(Y,Y) + +2\langle {\mathcal R}(Y,X)Y,X\rangle - 4(\nabla_{Y}\mathit{Ric})(X,Y)\bigr. \\ +& & \ \ \ \ \bigl.+ 2(\nabla_{X}\mathit{Ric})(Y,Y) + +2\abs{\nabla_XY}^2\bigr)d\tau. +\end{aligned} +$$ + + Plugging in Equation (6.9), and using the fact that + $|Y(\tau)|^2= +|Y(\bar\tau)|^2\frac{(\sqrt{\tau}-\sqrt{\tau_1})^2}{(\sqrt{\bar\tau}-\sqrt{\tau_1})^2}$, +gives + +$$ +\begin{aligned} +\mathit{Hess}(L^{\bar\tau})(Y(\bar\tau),Y(\bar\tau)) \\ & +\le & \int_{\tau_1}^{\bar\tau} \sqrt{\tau}\bigl(\mathit{Hess}(R)(Y,Y) + +2\langle {\mathcal R}(Y,X)Y,X\rangle - 4(\nabla_{Y}\mathit{Ric})(X,Y)\bigr. \\ +& & \ \ \ \ \bigl. +2(\nabla_X\mathit{Ric})(Y,Y) + + 2\abs{\mathit{Ric}(Y,\cdot)}^{2}\bigr)d\tau \\ & & + +\int_{\tau_1}^{\bar\tau}\left[ +\frac{|Y(\bar\tau)|^2}{2\sqrt{\tau}(\sqrt{\bar\tau}-\sqrt{\tau_1})^2} +- \frac{2}{(\sqrt{\tau}-\sqrt{\tau_1})}\mathit{Ric}(Y,Y)\right] d\tau +\end{aligned} +$$ + + Using the definition of $H(X,Y)$ given in the statement, Equation (6.12), + allows us to write + +$$ +\begin{aligned} +\mathit{Hess}(L^{\bar\tau})(Y(\bar\tau),Y(\bar\tau)) \\ + & \le & +-\int_{\tau_1}^{\bar\tau}\sqrt{\tau}H(X,Y)d\tau \\ & & ++\int_{\tau_1}^{\bar\tau}\Bigl[\sqrt{\tau} \bigl(-2(\nabla_{X}\mathit{Ric})(Y,Y) +-2\frac{\partial\mathit{Ric}}{\partial\tau}(Y,Y) +4|\mathit{Ric}(Y,\cdot)|^2 \bigr) \Bigr.\\ +& & \Bigl.+ +\frac{|Y(\bar\tau)|^2}{2\sqrt{\tau}(\sqrt{\bar\tau}-\sqrt{\tau_1})^2} +-\left(\frac{2}{(\sqrt{\tau}- +\sqrt{\tau_1})}+\frac{1}{\sqrt{\tau}}\right) \mathit{Ric}(Y,Y)\Bigr]d\tau, +\end{aligned} +$$ + + To simplify +further, we compute, using Equation (6.9) + +$$ +\begin{aligned} +\frac{d}{d\tau}\bigl(\mathit{Ric}(Y(\tau),Y(\tau))\bigr) &= & \frac{\partial\mathit{Ric}}{\partial{\tau}}(Y,Y) + + 2\mathit{Ric}(\nabla_{X}Y,Y) + (\nabla_{X}\mathit{Ric})(Y,Y)\\ + & = & \frac{\partial\mathit{Ric}}{\partial{\tau}}(Y,Y)+ (\nabla_{X}\mathit{Ric})(Y,Y) \\ & & + +\frac{1}{\sqrt{\tau}(\sqrt{\tau}-\sqrt{\tau_1})}\mathit{Ric}(Y,Y) - +2|\mathit{Ric}(Y,\cdot)|^{2}. +\end{aligned} +$$ + + Consequently, we have + +$$ +\begin{aligned} +\frac{d\left(2\sqrt{\tau}\mathit{Ric}(Y(\tau),Y(\tau))\right)}{d\tau} & += & 2\sqrt{\tau}\left(\frac{\partial\mathit{Ric}}{\partial{\tau}}(Y,Y) ++ (\nabla_{X}\mathit{Ric})(Y,Y)- 2|\mathit{Ric}(Y,\cdot)|^{2}\right)\\ + & & + \left(\frac{2}{(\sqrt{\tau}-\sqrt{\tau_1})} +\frac{1}{\sqrt{\tau}}\right)\mathit{Ric}(Y,Y) +\end{aligned} +$$ + +Using this, and the fact that $Y(\tau_1)=0$, gives + +$$ +\begin{aligned} +\mathit{Hess}(L_x^{\bar\tau})(Y(\bar\tau),Y(\bar\tau)) \le \\ + & & +-\int_{\tau_1}^{\bar\tau}\left(\sqrt{\tau}H(X,Y) +-\frac{d}{d\tau}\left(2\sqrt{\tau}\mathit{Ric}(Y,Y)\right) +-\frac{|Y(\bar\tau)|^2}{2\sqrt{\tau}(\sqrt{\bar\tau}-\sqrt{\tau_1})^2}\right)d\tau \nonumber\\ + & = & +\frac{|Y(\bar\tau)|^2}{\sqrt{\bar\tau}-\sqrt{\tau_1}} +-2\sqrt{\bar\tau}\mathit{Ric}(Y(\bar\tau),Y(\bar\tau))-\int_{\tau_1}^{\bar\tau}\sqrt{\tau}H(X,Y)d\tau. +\nonumber +\end{aligned} +$$ + +This proves Inequality (6.11). Now we examine when equality +holds in this expression. Given an adapted vector field $Y(\tau)$ +along $\gamma$, let $\mu(v)$ be a geodesic through +$\gamma(\bar\tau,0)$ with tangent vector $Y(\bar\tau)$. Then there +is a one-parameter family $\mu(\tau,v)$ of minimizing ${\mathcal +L}$-geodesics with the property that $\mu(\bar\tau,v)=\mu(v)$. Let +$\widetilde Y'(\tau,v)$ be $\partial\mu(\tau,v)/\partial v$. It is +an ${\mathcal L}$-Jacobi field with $\widetilde +Y'(\bar\tau,0)=Y(\bar\tau)$. Since $L_x\circ {\mathcal L}\mathit{exp}_x=\widetilde L$, we see that + +$$ +\frac{d^2}{d v^2}{\mathcal +L}(\mu_v)|_{v=0}=\frac{d^2}{du^2}L^{\bar \tau}_x(\mu(u))|_{u=0}. +$$ + +Hence, the assumption that we have equality in (6.11) +implies that + +$$ +\frac{d^2}{d v^2}{\mathcal +L}(\mu_v)|_{v=0}=\frac{d^2}{d u^2}{\mathcal L}(\tilde\gamma_u)|_{u=0}. +$$ + + Now we extend this +one-parameter family to a two-parameter family $\mu(\tau,u,v)$ so +that $\partial\mu(\tau,0,0)/\partial v=\widetilde Y'$ and +$\partial\mu(\tau,0,0)/\partial u=Y(\tau)$. Let $w$ be the variable +$u-v$, and let $\widetilde W$ be the tangent vector in this +coordinate direction, so that $\widetilde W=\widetilde Y-\widetilde +Y'$. We denote by $W$ the restriction of $\widetilde W$ to +$\gamma_{0,0}=\gamma_Z$. By \entryref{2fef81872a01} the second +partial derivative of the length of this family in the $u$-direction +at $u=v=0$ agrees with the second derivative of the length of the +original family $\widetilde \gamma$ in the $u$-direction. + +\entryblock{b20ded1b2964} + +**Proof.** Of course, the second partial derivatives are equal. According to +\entryref{f48558c60ca2} we have + +$$ +\frac{\partial}{\partial v}\frac{\partial}{\partial w}{\mathcal L}(\mu)|_{v=w=0} +=2\sqrt{\bar\tau} \widetilde Y'(\bar\tau)\langle \widetilde +W(\bar\tau),X(\bar\tau)\rangle-\int_{\tau_1}^{\bar\tau}2\sqrt{\tau}\langle +\mathit{Jac}(\widetilde Y'),W\rangle d\tau. +$$ + + Since $W(\bar\tau)=0$ and +since $\nabla_{\widetilde Y'}(\widetilde W)=\nabla_W(\widetilde +Y')$, we see that the boundary term in the above expression +vanishes. The integral vanishes since $\widetilde Y'$ is an +${\mathcal L}$-Jacobi field. + +If Inequality (6.11) is an equality, then + +$$ +\frac{\partial^2}{\partial v^2}{\mathcal L}(\mu)|_{u=v=0}=\frac{\partial^2}{\partial u^2}{\mathcal +L}(\mu)|_{u=v=0}. +$$ + + We write $\partial/\partial u=\partial/\partial +v+\partial/\partial w$. Expanding out the right-hand side and +canceling the common terms gives + +$$ +0=\left(\frac{\partial}{\partial v}\frac{\partial}{\partial w}+ +\frac{\partial}{\partial w}\frac{\partial}{\partial +v}+\frac{\partial^2}{\partial w^2}\right){\mathcal +L}(\mu)|_{u=v=0}. +$$ + + The previous claim tells us that the first two +terms on the right-hand side of this equation vanish, and hence we +conclude + +$$ +\frac{\partial^2}{\partial w^2}{\mathcal L}(\mu)|_{u=v=0}=0 +$$ + +Since $W$ vanishes at both endpoints this implies, according to +\entryref{26b7f69e803e}, that $\widetilde W(\tau,0,0)=0$ for all +$\tau$, or in other words $Y(\tau)=\widetilde Y'(\tau,0,0)$ for all +$\tau$. Of course by construction $\widetilde Y'(\tau,0,0)$ is an +${\mathcal L}$-Jacobi field. This shows that equality holds only if +the adapted vector field $Y(\tau)$ is also an ${\mathcal L}$-Jacobi +field. + +Conversely, if the adapted vector field $Y(\tau)$ is also an +${\mathcal L}$-Jacobi field, then inequality between the second +variations at the beginning of the proof is an equality. In the rest +of the argument we dealt only with equalities. Hence, in this case +Inequality (6.11) is an equality. + +This shows that we have equality in (6.11) if and only if +the adapted vector field $Y(\tau)$ is also an ${\mathcal L}$-Jacobi +field. + +### Inequalities for $\triangle L_x^{\bar\tau}$ + +The inequalities for the Hessian of $L_x^{\bar\tau}$ lead to +inequalities for $\triangle L_x^{\bar\tau}$ which we establish in this section. Here is the +main result. + +\entryblock{882677eecdc1} + +**Proof.** Choose an orthonormal basis $\{Y_{\alpha}\}$ for +$T_q(M_{T-\bar\tau})$. For each $\alpha$, extend $\{Y_{\alpha}\}$ to +an adapted vector field along the $ \mathcal{L}$-geodesic $\gamma_Z$ +by solving + +$$ +\nabla_{X}Y_{\alpha} = \frac{1}{2\sqrt{\tau}(\sqrt{\tau}-\sqrt{\tau_1})}Y_{\alpha} +-\mathit{Ric}(Y_{\alpha},\cdot)^*. +$$ + + As in Equation (6.10), +we have + +$$ +\begin{aligned} +\frac{d}{d\tau}\langle Y_{\alpha},Y_{\beta}\rangle &= \langle +\nabla_{X}Y_{\alpha},Y_{\beta}\rangle + \langle +\nabla_{X}Y_{\beta}, Y_{\alpha}\rangle + 2\mathit{Ric}(Y_{\alpha},Y_{\beta}) +\\&= \frac{1}{\sqrt{\tau}(\sqrt{\tau}-\sqrt{\tau_1})}\langle Y_{\alpha},Y_{\beta}\rangle . +\end{aligned} +$$ + + By integrating we get + +$$ +\langle Y_{\alpha},Y_{\beta}\rangle (\tau) = +\frac{(\sqrt{\tau}-\sqrt{\tau_1})^2}{(\sqrt{\bar\tau}-\sqrt{\tau_1})^2}\delta_{\alpha\beta}. +$$ + +To simplify the notation we set + +$$ +I(\tau)=\frac{\sqrt{\bar\tau}-\sqrt{\tau_1}}{\sqrt{\tau}-\sqrt{\tau_1}} +$$ + +and $W_\alpha(\tau)=I(\tau)Y_\alpha(\tau)$. + Then $\{W_{\alpha}(\tau)\}_\alpha$ +form an orthonormal basis at $\tau$. Consequently, summing +Inequality (6.13) over $\alpha$ gives + +$$ +\triangle L_x^{\bar\tau}(q)\le +\frac{n}{\sqrt{\bar\tau}-\sqrt{\tau_1}} +-2\sqrt{\bar\tau}R(q)-\sum_\alpha\int_{\tau_1}^{\bar\tau}\sqrt{\tau}H(X,Y_\alpha)d\tau. +$$ + +To establish Inequality (6.14) it remains to prove the +following claim. + +\entryblock{11e3d00eb60d} + +**Proof.** To prove the claim we sum Equation (6.12) giving + +$$ +\begin{aligned} +I^2(\tau) \sum_{\alpha}H(X,Y_{\alpha}) &= & \sum_\alpha +H(X,W_\alpha) +\\ +& = & -\triangle R + 2\mathit{Ric}(X,X) - 4\langle \nabla R,X\rangle ++4\sum_{\alpha}(\nabla_{W_\alpha} \mathit{Ric})(W_{\alpha},X) \\ & & +-2\sum_{\alpha}\mathit{Ric}_{\tau}( W_{\alpha},W_{\alpha}) +2|\mathit{Ric}|^{2} -\frac{1}{\tau}R. +\end{aligned} +$$ + +Taking the trace of the second Bianchi identity, we get + +$$ +\sum_{\alpha}(\nabla_{W_\alpha} +\mathit{Ric})(W_{\alpha},X) =\frac{1}{2}\langle \nabla R,X\rangle . +$$ + + In addition +by (3.5), recalling that $\partial R/\partial\tau=-\partial R/\partial +t$, we have + +$$ +\frac{\partial R}{\partial \tau} = -\triangle R - 2|\mathit{Ric}|^{2}. +$$ + +On the other hand, + +$$ +\frac{\partial R}{\partial\tau} = +\partial(g^{ij}R_{ij})/\partial{\tau}= -2|\mathit{Ric}|^{2} + \sum_{\alpha}\frac{\partial\mathit{Ric}}{\partial\tau}(W_{\alpha}, W_{\alpha}), +$$ + + and so +$\sum_{\alpha}\frac{\partial\mathit{Ric}}{\partial\tau}(W_{\alpha},W_{\alpha}) = -\triangle R$. Putting +all this together gives + +$$ +I^2(\tau)\sum_{\alpha}H(X,Y_{\alpha})= H(X). +$$ + +Clearly, Inequality (6.14) follows immediately from +Inequality (6.16) and the claim. The last statement of +\entryref{882677eecdc1} follows directly from the last statement of +\entryref{88d4ef67440f} and \entryref{72949e6350d4}. This completes +the proof of \entryref{882677eecdc1}. + +## Reduced length + +We introduce the reduced length function both on the tangent space +and on space-time. The reason that the reduced length $l_x$ is +easier to work with is that it is scale invariant when $\tau_1=0$. +Throughout Section 6.5 we fix $x\in{\mathcal M}$ with +$\mathbf{t}(x)=T-\tau_1$. We shall always suppose that $T-\tau_1$ is +greater than the initial time of the generalized Ricci flow. + +### The reduced length function $l_x$ on space-time + +\entryblock{50447da4ee48} + +In order to understand the differential inequalities that $l_x$ +satisfies, we first need to introduce a quantity closely related to +the function ${\mathcal K}^{\bar\tau}_{\tau_1}$ defined in +\entryref{882677eecdc1}. + +\entryblock{c958eb1d5427} + +The following is immediate from the definitions. + +\entryblock{7dec33ee6a59} + +\entryblock{7a68e2160e8a} + +**Proof.** Using the ${\mathcal L}$-geodesic equation and the definition of $H$ we have + +$$ +\begin{aligned} +\frac{d}{d\tau}(R(\gamma_Z(\tau))+ \abs{X(\tau)}^{2}) \\ & += & \frac{\partial R}{\partial\tau}(\gamma_Z(\tau)) + \langle \nabla +R(\gamma_Z(\tau)), +X(\tau)\rangle + 2\langle \nabla_{X}X(\tau),X(\tau)\rangle \\ +& & + 2\mathit{Ric}(X(\tau),X(\tau)) +\\&= & \frac{\partial R}{\partial\tau}(\gamma_Z(\tau)) + 2X(\tau)(R) - \frac{1}{\tau}\abs{X(\tau)}^{2} - +2\mathit{Ric}(X(\tau),X(\tau)) \\&= &-H(X(\tau)) +-\frac{1}{\tau}(R(\gamma_Z(\tau)+ \abs{X(\tau)}^{2}). +\end{aligned} +$$ + +Thus + +$$ +\frac{d}{d\tau}(\tau^{\frac{3}{2}}(R(\gamma_Z(\tau)+\abs{X(\tau)}^{2}) )= +\frac{1}{2}\sqrt{\tau}(R(\gamma_Z(\tau)+\abs{X(\tau)}^{2}) - +\tau^{\frac{3}{2}}H(X(\tau)). +$$ + + Integration from $\tau_1$ to $\bar\tau$ gives + +$$ +\bar\tau^{3/2}\left(R(q))+|X(\bar\tau)|^2\right)-\tau_1^{3/2}(R(x)+|X(\tau_1)|^2) +=\frac{L_x^{\bar\tau}(q)}{2}-K_{\tau_1}^{\bar\tau}(\gamma_Z), +$$ + +which is equivalent to Equation (6.17). In the case when +$\tau_1=0$, the last term on the right-hand side vanishes since + +$$ +\mathit{lim}_{\tau\rightarrow 0}\tau^{3/2}|X(\tau)|^2=0. +$$ + +Now we come to the most general of the differential inequalities for +$l_x$ that will be so important in what follows. Whenever the +expression +$\left(\frac{\tau_1}{\bar\tau}\right)^{3/2}\left(R(x)+|X(\tau_1)|^2\right)$ +appears in a formula, it is interpreted to be zero in the case when +$\tau_1=0$. + +\entryblock{2b8ab39364e7} + +**Proof.** It follows immediately from Equation (6.8) that + +$$ +\frac{\partial l_x}{\partial \tau}=R-\frac{1}{2}(R+|X|^2)-\frac{l_x}{2\tau}. +$$ + +Using Equation (6.17) this gives the first equality stated +in the lemma. + It follows immediately from \entryref{033786a89c6d} that +$\nabla l_x^\tau=X(\tau)$ and hence $|\nabla +l_x^\tau|^2=|X(\tau)|^2$. From this and Equation (6.17) the +second equation follows. The last inequality is immediate from +\entryref{882677eecdc1}. + +When $\tau_1=0$, the last terms on the right-hand sides of the first +two equations vanish, since the last term on the right-hand side of +Equation (6.17) vanishes in this case. + +When $\tau_1=0$, which is the case of main interest, all these +formulas simplify and we get: + +\entryblock{4917e81d7684} + +**Proof.** This is immediate from the formulas in the previous lemma. + +Now let us reformulate the differential inequalities in +\entryref{4917e81d7684} in a way that will be useful later. + +\entryblock{967c0f8bb774} + +### The tangential version $\widetilde l$ of the reduced length function + +For any path $\gamma\colon [\tau_1,\bar\tau]\to ({\mathcal M},G)$ +parameterized by backward time we define + +$$ +l(\gamma)=\frac{1}{2\sqrt{\bar\tau}}{\mathcal L}(\gamma). +$$ + +This leads immediately to a reduced length on $\widetilde {\mathcal +U}_x$. + +\entryblock{77ed5048ae37} + +At first glance it may appear that the computations of the gradient +and $\tau$-derivatives for $l_x$ pass immediately to those for +$\widetilde l$. For the spatial derivative this is correct, but for +the $\tau$-derivative it is not true. As the computation below +shows, the $\tau$-derivatives of $\widetilde l$ and $l_x$ do not +agree under the identification ${\mathcal L}\mathit{exp}_x$. The reason +is that this identification does not line up the $\tau$-vector field +in the domain with $-\partial/\partial t$ in the range. So it is an +entirely different computation with a different answer. + +\entryblock{5e142c14df9d} + +**Proof.** By the Fundamental Theorem of Calculus + +$$ +\frac{\partial}{\partial\tau}\widetilde L(Z,\tau)=\sqrt{\tau}\left(R(\gamma_z(\tau))+|X(\tau)|^2\right). +$$ + +Thus, + +$$ +\frac{\partial}{\partial\tau}\widetilde l(Z,\tau)=\frac{1}{2}\left(R(\gamma_z(\tau))+|X(\tau)|^2\right) +-\frac{\widetilde l(Z,\tau)}{2\tau}. +$$ + +\entryblock{451b4cc590c1} + +**Proof.** This is immediate from \entryref{5e142c14df9d} and \entryref{7a68e2160e8a} (after the +latter is rewritten using $\widetilde L$ instead of $L_x$). + +## Local Lipschitz estimates for $l_x$ + +It is important for the applications to have results on the +Lipschitz properties of $l_x$, or equivalently $L_x$. Of course, +these are the analogues of the fact that in Riemannian geometry the +distance function from a point is Lipschitz. The proof of the +Lipschitz property given here is based on the exposition in +[Ye]. In Section 6.6, we fix $x\in +M_{T-\tau_1}\subset {\mathcal M}$. + +### Statement and corollaries + +\entryblock{27c072f30ccf} + +Here is the main result of this subsection. + +\entryblock{db4ced24b05f} + +\entryblock{a89a833cf7cb} + +### The proof of Proposition \entryref{db4ced24b05f} + +\entryref{db4ced24b05f} follows from a much more precise, though more +complicated to state, result. In order to state this more technical +result we introduce the following definition. + +\entryblock{0fc5cfacef3f} + +Now we are ready for the more precise, technical result. + +\entryblock{96ced8c484df} + +Before proving \entryref{96ced8c484df}, let us show how it implies +\entryref{db4ced24b05f}. + +**Proof.** (that \entryref{96ced8c484df} implies +\entryref{db4ced24b05f}) Suppose given $\epsilon>0$, $A$, $\nu(A)$ and $F$ as in +the statement of \entryref{db4ced24b05f}. For each $y\in A$ there is +$0<\epsilon'<\epsilon$ and a neighborhood $\nu'(y)$ with (i) the closure +$\overline{\nu}'(y)$ of $\nu'(y)$ being a compact subset of $\nu(A)$ and (ii) +for each $z\in \overline{\nu}'(y)$ the parabolic neighborhood $\widetilde +P(z,\epsilon',\epsilon')$ exists and has compact closure in $\nu(A)$. It +follows that for every $z\in \overline{\nu}'(A)$, $\mathit{Rm}_G$ is bounded on +$\widetilde P(z,\epsilon',\epsilon')$ and every point of $\widetilde +P(z,\epsilon',\epsilon')$ is connected to $x$ by a minimizing ${\mathcal +L}$-geodesic in $F$. Thus, \entryref{96ced8c484df}, with $\epsilon$ replaced +by $\epsilon'$, applies to $z$. In particular, $l_x$ is continuous at $z$, and +hence is continuous on all of $\overline{\nu}'(y)$. Thus, $l_x$ is bounded on +$\overline{\nu}'(y)$. Since we have uniform bounds for the curvature on +$\widetilde P(z,\epsilon',\epsilon')$ according to \entryref{96ced8c484df} +there are constants $C<\infty$ and $0<\delta<\epsilon'$ such that for any $z\in +\overline{\nu}'(y)$ and any $z'\in \widetilde P(z,\delta,\delta)$, we have + +$$ +|l_x(z)-l_x(z')|\le C|z-z'|_{G(\mathbf{t}(z)+dt^2}. +$$ + +Since we have a uniform bound for the curvature on $\widetilde +P(z,\epsilon',\epsilon')$ independent of $z\in \nu'(y)$, the metrics $\widehat +G=G+dt^2$ and $G(\mathbf{t}(z))+dt^2$ are uniformly comparable on all of +$\widetilde P(z,\delta,\delta)$. It follows that there is a constant +$C'<\infty$ such that for all $z\in \nu'(y)$ and all $z'\in \widetilde +P(z,\delta,\delta)$ we have + +$$ +|l_x(z)-l_x(z')|\le C'|Z-z'|_{\widehat G}. +$$ + +We set $\nu_0(A)=\cup_{y\in A}\nu'(y)$. This is an open neighborhood of $A$ +contained in $\nu(A)$ on which $l_x$ is locally Lipschitz with respect to the +metric $\widehat G$. + +Now we turn to the proof of \entryref{96ced8c484df}. We begin with +several preliminary results. + +\entryblock{333be2e58780} + +**Proof.** The geodesic equation in terms of the variable $s$, +Equation (6.5), gives + +$$ +\begin{aligned} +\frac{d|\gamma'(s)|^2}{ds} & = & 2\langle +\nabla_{\gamma'(s)}\gamma'(s),\gamma'(s)\rangle+4s\mathit{Ric}(\gamma'(s),\gamma'(s))\nonumber \\ +& = & 4s^2\langle\nabla R,\gamma'(s)\rangle -4s\mathit{Ric}(\gamma'(s),\gamma'(s)). +\end{aligned} +$$ + +Thus,, by our assumption on $|\nabla R|$ and $|\mathit{Ric}|$ along +$\gamma$, we have + +$$ +\left|\frac{d|\gamma'(s)|^2}{ds}\right| \le +4C_0s^2|\gamma'(s)|+4C_0s|\gamma'(s)|^2. +$$ + + It follows that + +$$ +\left|\frac{d|\gamma'(s)|}{ds}\right|\le 2C_0s^2+2C_0s|\gamma'(s)|\le 2C_0\bar\tau+2 +C_0\sqrt{\bar\tau}|\gamma'(s)|, +$$ + + and hence that + +$$ +-2C_0\sqrt{\bar\tau}ds\le \frac{d|\gamma'(s)|}{\sqrt{\bar\tau}+|\gamma'(s)|} +\le 2C_0\sqrt{\bar\tau}ds. +$$ + + Suppose that $s_00$. We rescale to form $(Q{\mathcal M},QG)$ and then we shift +the time by $T-QT$ so that the time-slice $M_T$ in the original flow +is the $T$ time-slice of the new flow. We call the result +$({\mathcal M}',G')$. Recall that $\tau=T-\mathbf{t}$ is the parameter +for ${\mathcal L}$-geodesics in $({\mathcal M},G)$ The corresponding +parameter in the rescaled flow $({\mathcal M}',G')$ is $\tau'=T-\mathbf{t'}=Q\tau$. We denote by ${\mathcal L}'\mathit{exp}_x$ the ${\mathcal +L}$-exponential map from $x$ in $({\mathcal M}',G')$, and by $l_x'$ +the reduced length function for this Ricci flow. The associated +function on the tangent space is denoted $\tilde l'$. + +\entryblock{ad48b42bb0a0} + +\entryblock{7237f6ec73b4} + +**Proof.** It is clear that $\beta(0)=x$ and that $\beta$ is parameterized by +backward time in $({\mathcal M}',G')$. Because of the scaling of +space and time by $Q$, we have $R_{G'}=R_G/Q$ and +$X_\beta(\tau')=d\iota(X_\gamma(\tau))/Q$, and hence +$|X_\beta(\tau')|^2_{G'}=\frac{1}{Q}|X_\gamma(\tau)|^2_{G}$. A +direct change of variables in the integral then shows that + +$$ +{\mathcal L}(\beta)=\sqrt{Q}{\mathcal L}(\gamma). +$$ + +It follows that $\beta$ is an ${\mathcal L}$-geodesic if and only if +$\gamma$ is. The last statement follows directly. + +Immediately from the definitions we see the following: + +\entryblock{541f71387f84} + +### The integrand in the reduced volume integral + +Now we turn our attention to the integrand (over $\widetilde {\mathcal U}_x(\tau)$) in the +reduced volume integral. Namely, set + +$$ +f(\tau)=\tau^{-n/2}e^{-\tilde l(Z,\tau)}{\mathcal J}(Z,\tau), +$$ + +where ${\mathcal J}(Z,\tau)$ is the Jacobian +determinant of ${\mathcal L}\mathit{exp}_x^\tau$ at the point $Z\in \widetilde +U_x(\tau)\subset T_xM_T$. We wish to see that this quantity is invariant under +the rescaling. + +\entryblock{76c5f88ad2b4} + +**Proof.** It follows from the first equation in \entryref{541f71387f84} that + +$$ +J(\iota){\mathcal J}(Z,\tau)=J(\sqrt{Q^{-1}}){\mathcal +J}'(\sqrt{Q^{-1}}Z,\tau'), +$$ + + where $J(\iota)$ is the Jacobian +determinant of $\iota$ at ${\mathcal L}\mathit{exp}_x(Z,\tau)$ and +$J(\sqrt{Q^{-1}})$ is the Jacobian determinant of multiplication by +$\sqrt{Q^{-1}}$ as a map from $T_xM_T$ to itself, where the domain +has the metric $G$ and the range has metric $G'=QG$. Clearly, with +these conventions, we have $J(\iota)=Q^{n/2}$ and +$J(\sqrt{Q^{-1}})=1$. Hence, we conclude + +$$ +Q^{n/2}{\mathcal J}(Z,\tau)={\mathcal J}'(\sqrt{Q^{-1}}Z,\tau'). +$$ + +Letting $\gamma$ be the ${\mathcal L}$-geodesic in $({\mathcal +M},G)$ with initial condition $Z$ and $\beta$ the ${\mathcal +L}$-geodesic in $({\mathcal M}',G')$ with initial condition +$\sqrt{Q^{-1}}Z$, by \entryref{ad48b42bb0a0} we have +$\gamma(\tau)=\beta(\tau')$. From \entryref{541f71387f84} and the +definition of the reduced length, we get + +$$ +\tilde +l'(\sqrt{Q^{-1}}Z,\tau')=\tilde l(\gamma,\tau). +$$ + + Plugging these in +gives the result. + +Let us evaluate $f(\tau)$ in the case of $\Ar^n$ with the Ricci flow being the +constant family of Euclidean metrics. + +\entryblock{a6ea3cf8fb92} + +Putting these computations together gives the following. + +\entryblock{9d4d83df4874} + +This computation has consequences for all Ricci flows. + +\entryblock{4e6330b2aa46} + +**Proof.** First notice that since $T$ is greater than the initial time of ${\mathcal M}$, +there is $\epsilon>0$, and an embedding $\rho\colon B(x,T,\epsilon)\times +[T-\epsilon,T]\to {\mathcal M}$ compatible with time and the vector field. By +taking $\epsilon>0$ smaller if necessary, we can assume that the image of +$\rho$ has compact closure in ${\mathcal M}$. By compactness every higher +partial derivative (both spatial and temporal) of the metric is bounded on the +image of $\rho$. + +Now take a sequence of positive constants $\tau_k$ tending to $0$ as +$k\rightarrow\infty$, and set $Q_k=\tau_k^{-1}$. We let $({\mathcal +M}_k,G_k)$ be the $Q_k$-rescaling and shifting of $({\mathcal M},G)$ +as described at the beginning of this section. The rescaled version +of $\rho$ is an embedding + +$$ +\rho_k\colon B_{G_k}(x,T,\sqrt{Q_k}\epsilon)\times [T-Q_k\epsilon,T]\to +{\mathcal M}_k +$$ + + compatible with the time function $\mathbf{t}_k$ and the vector +field. Furthermore, uniformly on the image of $\rho_k$, every higher partial +derivative of the metric is bounded by a constant that goes to zero with $k$. +Thus, the generalized Ricci flows $({\mathcal M}_k,G_k)$ based at $x$ converge +geometrically to the constant family of Euclidean metrics on $\Ar^n$. Since the +ODE given in Equation (6.5) is regular even at $0$, this implies that +the ${\mathcal L}$-exponential maps for these flows converge uniformly on the +balls of finite radius centered at the origin of the tangent spaces at $x$ to +the ${\mathcal L}$-exponential map of $\Ar^n$ at the origin. Of course, if +$Z\in T_xM_T$ is an initial condition for an ${\mathcal L}$-geodesic in +$({\mathcal M},G)$, then $\sqrt{Q_k^{-1}}Z$ is the initial condition for the +corresponding ${\mathcal L}$-geodesic in $({\mathcal M}_k,G_k)$. But +$|Z|_G=|\sqrt{Q_k^{-1}}Z|_{G_k}$, so that if $Z\in B_G(0,A)$ then +$\sqrt{Q_k^{-1}}Z\in B_{G_k}(0,A)$. + In particular, we see +that for any $A<\infty$, for all $k$ sufficiently large, the +${\mathcal L}$-geodesics are defined on $B_{G_k}(0,A)\times (0,1]$ +and the image is contained in the image of $\rho_k$. Rescaling shows +that for any $A<\infty$ there is $k$ for which the ${\mathcal +L}$-exponential map is defined on $B_G(0,A)\times (0,\tau_k]$ and +has image contained in $\rho$. + +Let $Z\in B_Q(0,A)\subset T_xM_T$, and let $\gamma$ be the +${\mathcal L}$-geodesic with $\mathit{lim}_{\tau\rightarrow +0}\sqrt{\tau}X_\gamma(\tau)=Z$. Let $\gamma_k$ be the corresponding +${\mathcal L}$-geodesic in $({\mathcal M}_k,G_k)$. Then $\mathit{lim}_{\tau\rightarrow +0}\sqrt{\tau}X_{\gamma_k}(\tau)=\sqrt{\tau_k}Z=Z_k$. Of course, +$|Z_k|^2_{G_k}=|Z|^2_G$, meaning that $Z_k$ is contained in the ball +$B_{G_k}(0,A)\subset T_xM_T$ for all $k$. Hence, by passing to a +subsequence we can assume that, in the geometric limit, the +$\sqrt{\tau_k}Z$ converge to a tangent vector $Z'$ in the ball of +radius $A$ centered at the origin in the tangent space to Euclidean +space. Of course $|Z'|^2=|Z|_G^2$. By \entryref{9d4d83df4874}, this +means that we have + +$$ +\mathit{lim}_{k\rightarrow \infty}1^{-n/2}e^{-\tilde l_k(\sqrt{Q_k^{-1}}Z,1)}{\mathcal +J}_k(\sqrt{Q_k^{-1}}Z,1)=2^ne^{-|Z|^2}, +$$ + + where ${\mathcal J}_k$ is +the Jacobian determinant of the ${\mathcal L}$-exponential map for +$({\mathcal M}_k,G_k)$. + Of course, since +$\tau_k=Q_k^{-1}$, by \entryref{76c5f88ad2b4} we have + +$$ +1^{-n/2}e^{-\tilde l_k(\sqrt{Q_k^{-1}}Z,1)}{\mathcal +J}_k(\sqrt{Q_k^{-1}}Z,1)=\tau_k^{-n/2}e^{-\tilde +l(Z,\tau_k)}{\mathcal J}(Z,\tau_k). +$$ + + This establishes the limiting +result. + +Since the geometric limits are uniform on balls of finite radius +centered at the origin in the tangent space, the above limit also is +uniform over each of these balls. + +\entryblock{f1123507a2ea} + +**Proof.** According to the last result, given $R<\infty$, for all $\delta>0$ sufficiently +small the ball of radius $R$ centered at the origin in $T_xM_T$ is contained in +${\mathcal D}_x^\delta$, in the domain of definition of ${\mathcal L}\mathit{exp}_x^\delta$ as given in \entryref{df842ef2ab9f}, and ${\mathcal L}\mathit{exp}_x$ is a diffeomorphism on this subset. We shall show that if $\delta>0$ is +sufficiently small, then the resulting ${\mathcal L}$-geodesic $\gamma$ is the +unique minimizing ${\mathcal L}$-geodesic. If not then there must be another, +distinct ${\mathcal L}$-geodesic to this point whose ${\mathcal L}$-length is +no greater than that of $\gamma$. According to \entryref{333be2e58780} there is a +constant $C_1$ depending on the curvature bound and on $\delta$ such that if +$Z$ is an initial condition for an ${\mathcal L}$-geodesic then for all +$\tau\in (0,\delta)$ we have + +$$ +C_1^{-1}\left(|Z|-\frac{(C_1-1)}{2}\sqrt{\delta}\right)\le \sqrt{\tau}|X(\tau)|\le +C_1|Z|+\frac{(C_1-1)}{2}\sqrt{\delta}. +$$ + + From the formula given in +\entryref{333be2e58780} for $C_1$, it follows that, fixing the bound of +the curvature and its derivatives, + $C_1\rightarrow 1$ as $\delta\rightarrow 0$. Thus, with +a given curvature bound, for $\delta$ sufficiently small, +$\sqrt{\tau}|X(\tau)|$ is almost a constant along ${\mathcal +L}$-geodesics. Hence, the integral of $\sqrt{\tau}|X(\tau)|^2$ is +approximately $2\sqrt{\delta}|Z|^2$. On the other hand, the absolute +value of the integral of $\sqrt{\tau}R(\gamma(\tau))$ is at most +$2C_0\delta^{3/2}/3$ where $C_0$ is an upper bound for the absolute +value of the scalar curvature. + +Given $R<\infty$, choose $\delta>0$ sufficiently small such that ${\mathcal +L}\mathit{exp}_x$ is a diffeomorphism on the ball of radius $9R$ centered at the +origin and such that the following estimate holds: The ${\mathcal L}$-length of +an ${\mathcal L}$-geodesic defined on $[0,\delta]$ with initial condition $Z$ +is between $\sqrt{\delta}|Z|^2$ and $3\sqrt{\delta}|Z|^2$. To ensure the latter +estimate we need only take $\delta$ sufficiently small given the curvature +bounds and the dimension. Hence, for these $\delta$ no ${\mathcal L}$-geodesic +with initial condition outside the ball of radius $9R$ centered at the origin +in $T_xM_T$ can be as short as any ${\mathcal L}$-geodesic with initial +condition in the ball of radius $R$ centered at the same point. This means that +the ${\mathcal L}$-geodesics defined on $[0,\delta]$ with initial condition +$|Z|$ with $|Z| + +# Chapter 7 -- Complete Ricci flows of bounded curvature + + In this chapter we establish strong results for ${\mathcal L}\mathit{exp}_x$ in the case of ordinary Ricci flow on complete $n$-manifolds +with appropriate curvature bounds. In particular, for these flows we +show that there is a minimizing ${\mathcal L}$-geodesic to every +point. This means that $l_x$ is everywhere defined. +We extend the differential inequalities for $l_x$ established in +Section 6.4 at the 'smooth points' to weak +inequalities (i.e., inequalities in the distributional sense) valid +on the whole manifold. Using this we prove an upper bound for the +minimum of $l_x^\tau$. + +Let us begin with a definition that captures the necessary curvature +bound for these results. + +\entryblock{2336b554a054} + +## The functions $L_{x}$ and $l_{x}$ + +Throughout Chapter 7 we have a Ricci flow $(M,g(t)),\ +0\le t\le T<\infty$, and we set $\tau=T-t$. All the results of the +last chapter apply in this context, + but in fact in this context there are much stronger results, which +we develop here. + +### Existence of ${\mathcal L}$-geodesics + +We assume here that $(M,g(t)),\ 0\le t\le T<\infty$, is a Ricci flow +which is complete of bounded curvature. In Shi's Theorem +(\entryref{ed4b72caf9f9}) we take $K$ equal to the bound of the norm of +the Riemannian curvature on $M\times [0,T]$, we take $\alpha=1$, and +we take $t_0=T$. It follows from \entryref{ed4b72caf9f9} that there is a +constant $C(K,T)$ such that $|\nabla R(x,t)|\le C/t^{1/2}$. Thus, +for any $\epsilon>0$ we have a uniform bound for $|\nabla R|$ on +$M\times [\epsilon,T]$. Also, because of the uniform bound for the +Riemann curvature and the fact that $T<\infty$, there is a constant +$C$, depending on the curvature bound and $T$ such that + +$$ +C^{-1}g(x,t)\le g(x,0)\le Cg(x,t) +$$ + +for all $(x,t)\in M\times [0,T]$. + +\entryblock{cd062526b91a} + +**Proof.** For any curve $\gamma$ parameterized by backward time, we set +$\bar \gamma$ equal to the path in $M$ that is the image under +projection of $\gamma$. We set $A(s)=\bar\gamma'(s)$. Define + +$$ +c((p_1,\tau_1),(p_2,\tau_{2})) = \mathit{inf} +\{\mathcal{L}(\gamma)|\gamma\colon [\tau_{1},\tau_{2}] \rightarrow +M\times [0,T], \bar\gamma(\tau_{1}) = p_1, \bar\gamma(\tau_{2})= +p_2\}. +$$ + + From Equation (6.2) we see that the infimum exists +since, by assumption, the curvature is uniformly bounded (below). +Furthermore, for a minimizing sequence $\gamma_{i}$, we have +$\int_{s_1}^{s_{2}}\abs{A_i(s)}^{2}ds \leq C_0$, for some constant +$C_0$, where $s_i=\sqrt{\tau_i}$ for $i=1,2$. It follows from this +and the inequality in Equation (7.1) that there is a +constant $C_1<\infty$ such that for all $i$ we have + +$$ +\int_{s_{1}}^{s_{2}} \abs{ +A_{i}}^{2}_{g(0)}d\tau \leq C_1. +$$ + + Therefore the sequence $\{\gamma_{i}\}$ is +uniformly continuous with respect to the metric $g(0)$; by Cauchy-Schwarz we +have + +$$ +\abs{\bar\gamma_{i}(s) - \bar\gamma_{i}(s')}_{g(0)} \leq +\int_{s'}^{s} \abs{A_{i}}_{g(0)}ds \leq \sqrt{C_1}\sqrt{s-s'}. +$$ + + By +the uniform continuity, we see that a subsequence of the $\gamma_i$ +converges uniformly pointwise to a continuous curve $\gamma$ +parameterized by $s$, the square root backward time. By passing to a +subsequence we can arrange that the $\gamma_i$ converge weakly in +$H^{2,1}$. Of course, the limit in $H^{2,1}$ is represented by the +continuous limit $\gamma$. That is to say, after passing to a +subsequence, the $\gamma_i$ converge uniformly and weakly in +$H^{2,1}$ to a continuous curve $\gamma$. Let $A(s)$ be the +$L^2$-derivative of $\gamma$. Weak convergence in $H^{2,1}$ implies +that $\int_{s'}^s|A(s)|^2ds\le \mathit{lim}_{i\rightarrow\infty}\int_{s'}^s|A_i(s)|^2ds$, so that +${\mathcal L}(\gamma)\le \mathit{lim}_{i\rightarrow\infty}{\mathcal +L}(\gamma_i)$. This means that $\gamma$ minimizes the ${\mathcal +L}$-length. Being a minimizer of ${\mathcal L}$-length, $\gamma$ +satisfies the Euler-Lagrange equation and is smooth by the +regularity theorem of differential equations. This then is the +required minimizing ${\mathcal L}$-geodesic from $(p_1,\tau_1)$ to +$(p_2,\tau_2)$. + +Let us now show that it is always possible to uniquely extend +${\mathcal L}$-geodesics up to time $T$. + +\entryblock{44032fc43e5d} + +**Proof.** We work with the parameter $s=\sqrt{\tau}$. According to +Equation (6.5), we have + +$$ +\nabla_{\gamma'(s)}\gamma'(s)=2s^2\nabla +R-4s\mathit{Ric}(\gamma'(s),\cdot). +$$ + + This is an everywhere +non-singular ODE. Since the manifolds $(M,g(t))$ are complete and +their metrics are uniformly related as in +Inequality (7.1), to show that the solution is defined +on the entire interval $s\in [0,\sqrt{T})$ we need only show that +there is a uniform bound to the length, or equivalently the energy +of $\gamma$ of any compact subinterval of $[0,T)$ on which it is +defined. Fix $\epsilon>0$. It follows immediately from +\entryref{333be2e58780}, and the fact that the quantities $R$, $|\nabla +R|$ and $|\mathit{Rm}|$ are bounded on $M\times [\epsilon,T]$, that +there is a bound on $\mathit{max}|\gamma'(s)|$ in terms of +$|\gamma'(\tau_1)|$, for all $s\in [0,\sqrt{T-\epsilon}]$ for which +$\gamma$ is defined. Since $(M,g(0))$ is complete, this, together +with a standard extension result for second-order ODE's, implies +that $\gamma$ extends uniquely to the entire interval +$[0,\sqrt{T-\epsilon}]$. Changing the variable from $s$ to +$\tau=s^2$ shows that the ${\mathcal L}$-geodesic extends uniquely +to the entire interval $[0,T-\epsilon]$. Since this is true for +every $\epsilon>0$, this completes the proof. + +Let $p\in M$ and set $x=(p,T)\in M\times [0,T]$. Recall that from +\entryref{90fe97584bdd} for every $\tau>0$, the injectivity set +$\widetilde{\mathcal U}_{x}(\tau)\subset T_pM$ consists of all $Z\in +T_pM$ for which (i) the ${\mathcal L}$-geodesic +$\gamma_Z|_{[0,\tau]}$ is the unique minimizing ${\mathcal +L}$-geodesic from $x$ to its endpoint, (ii) the differential of +${\mathcal L}\mathit{exp}_{x}^\tau$ is an isomorphism at $Z$, and (iii) +for all $Z'$ sufficiently close to + $Z$ the ${\mathcal L}$-geodesic $\gamma_{Z'}|_{[0,\tau]}$ is the unique minimizing + ${\mathcal L}$-geodesic to its endpoint. The image +of $\widetilde{\mathcal U}_{x}(\tau)$ is denoted ${\mathcal U}_{x}(\tau)\subset +M$ + +The existence of minimizing ${\mathcal L}$-geodesics from $x$ to +every point of $M\times (0,T)$ means that the functions $L_{x}$ and +$l_{x}$ are defined on all of $M\times (0,T)$. This leads to: + +\entryblock{0bfc692d99b1} + +### Results about $l_{x}$ and ${\mathcal U}_{x}(\tau)$ + +Now we come to our main result about the nature of ${\mathcal +U}_{x}(\tau)$ and the function $l_{x}$ +in the context of Ricci flows which are complete and of bounded +curvature. + +\entryblock{16fbe263a7c8} + +**Proof.** By Shi's Theorem (\entryref{ed4b72caf9f9}) the curvature bound on $M\times +[0,T]$ implies that for each $\epsilon>0$ there is a bound for +$|\nabla R|$ on $M\times (\epsilon,T]$. Thus, +\entryref{96ced8c484df} shows that $L_{x}$ is a + locally Lipschitz function on $M\times (\epsilon,T)$. Since this is +true for every $\epsilon>0$, $L_x$ is a locally Lipschitz function +on $M\times (0,T)$. Of course, the same is true for $l_{x}$. The +second statement is contained in \entryref{17254b50b27a}, and the +last one is contained in \entryref{8b9b3403dbc1}. It remains to +prove the third statement, namely that the complement of ${\mathcal +U}_{x}(\tau)$ is closed nowhere dense. This follows immediately from +\entryref{7aad50e49af2} since $|\mathit{Ric}|$ and $|\nabla R|$ are +bounded on $F=M\times [T-\tau,T]$. + +\entryblock{2e14dd57a334} + +## A bound for $\mathit{min} l^\tau_x$ + +We continue to assume that we have a Ricci flow $(M,g(t)),\ 0\le +t\le T<\infty$, complete and of bounded curvature and a point +$x=(p,T)\in M\times [0,T]$. Our purpose here is to extend the first +differential inequality given in + \entryref{967c0f8bb774} to a differential +inequality in the weak or distributional sense for $l_{x}$ valid on +all of $M\times (0,T)$. We then use this to establish that $\mathit{min}_{q\in M} l^\tau_{x}(q)\le n/2$ for all $0<\tau0$ there is an upper barrier +$\varphi$ for $l_x$ at $(q,\tau)$ defined on an open subset $U$ of +$(q,\tau)\in M\times (0,T)$ and satisfying + +$$ +\frac{d\varphi}{d\tau}(q,\tau)+\triangle \varphi(q,\tau)\le \frac{(n/2)-l_x(q,t)}{\tau}+\epsilon. +$$ + +Since $l_x(q,\tau)=l_\mathit{min}(\tau)$, it follows that +$\varphi(q',\tau)\ge \varphi(q,\tau)$ for all $(q',\tau)\in U\cap +M_{T-\tau}$. This means that $\triangle \varphi(q,\tau)\ge 0$, and +we conclude that + +$$ +\frac{d\varphi}{d\tau}(q,\tau)\le \frac{(n/2)-l_\mathit{min}(\tau)}{\tau}+\epsilon. +$$ + +Since $\varphi$ is an upper barrier for $l_x$ at $(q,\tau)$ it +follows immediately that + +$$ +\mathit{limsup}_{\tau'\rightarrow +\tau^+}\frac{l_x(q,\tau')-l_x(q,\tau)}{\tau'-\tau}\le +\frac{(n/2)-l_x(q,\tau)}{\tau}+\epsilon. +$$ + + Since this is true for every +$\epsilon>0$, we see that + +$$ +\mathit{limsup}_{\tau'\rightarrow +\tau^+}\frac{l_x(q,\tau')-l_x(q,\tau)}{\tau'-\tau}\le +\frac{(n/2)-l_x(q,\tau)}{\tau}. +$$ + + Since $l_\mathit{min}(\tau)=l_x(q,\tau)$, the same inequality holds for the forward +difference quotient of $l_\mathit{min}$ at $\tau$. That is to say, we +have + +$$ +\mathit{limsup}_{\tau'\rightarrow \tau^+}\frac{l_\mathit{min}(\tau')-l_\mathit{min}(\tau)}{\tau'-\tau}\le \frac{(n/2)-l_\mathit{min}(\tau)}{\tau}. +$$ + +The preceding equation implies that if $l_\mathit{min}(\tau)\le n/2$ +then $l_\mathit{min}(\tau')\le n/2$ for every $\tau'\ge \tau$. On the +other hand $\mathit{lim}_{\tau\rightarrow 0}l_\mathit{min}(\tau)=0$. Then +reason for this is that the path $\tau'\mapsto (P,T-\tau')$ for +$\tau'\in [0,\tau]$ has ${\mathcal L}$-length $ O(\tau^{3/2})$ as +$\tau\rightarrow 0$. It follows that $l_\mathit{min}(\tau)0$, with $\tau_1<(T-t)/2$, let +$t_1=T-\tau_1$, and let $q_1=\gamma(\tau_1)$. Consider +$\varphi_{(q,t)}={\mathcal L}(\gamma_{[0,\tau_1]})+ +L^\tau_{(q_1,t_1)}$. This is an upper barrier for $L_x^\tau$ at $q$ +defined in some neighborhood $V\subset M$ of $q$. Clearly, $\nabla +\varphi_{(q,t)}=\nabla L^\tau_{(q_1,t_1)}$ and $\mathit{Hess}(\varphi_{(q,t)}=\mathit{Hess}( L^\tau_{(q_1,t_1)})$. + +According to \entryref{033786a89c6d} we have $\nabla +L^\tau_{(q_1,t_1)}(q)=2\sqrt{\tau}X_{\gamma}(\tau)$. On the other +hand, by \entryref{a7c2ebee07a3} there is a bound on +$\sqrt{\tau}|X_{\gamma}(\tau)|$ depending only on the bounds on +curvature and its first derivatives, on $\tau$ and $\tau_1$ and on +$l_x(q,\tau)$. Of course, by Shi's theorem (\entryref{dbe4fabd0076}) +for every $\epsilon>0$ the norms of the first derivatives of +curvature on $M\times [\epsilon,T]$ are bounded in terms of +$\epsilon$ and the bounds on curvature. + This +proves that $|\nabla \varphi_{(q,t)}(q)|$ is bounded by a continuous +function $C(q,t)$ defined on all of $M\times (0,T)$. + + Now consider Inequality (6.11) for $\gamma$ at $\bar\tau=\tau$. + It is clear that the first + two terms on the right-hand side are bounded by $C|Y(\tau)|^2$, + where $C$ depends on the curvature bound and on $T-t$. We consider + the last term, $\int_{\tau_1}^{\tau}\sqrt{\tau'}H(X,Y)d\tau'$. + We claim that this integral is also bounded by $C'|Y(\tau)|^2$ + where $C'$ depends on the bounds on curvature and its first and + second derivatives along $\gamma_1$ and on $T-t$. +We consider $\tau'\in [\tau_1,\tau]$. Of course, + $\sqrt{\tau'}|X(\tau')|$ is bounded on this interval. Also, + +$$ +|Y(\tau')|=\left(\frac{\sqrt{\tau'}-\sqrt{\tau_1}}{\sqrt{\tau} + -\sqrt{\tau_1}}\right)|Y(\tau)|\le \frac{\sqrt{\tau'}}{\sqrt{\tau}}|Y(\tau)|. +$$ + +Hence $|Y(\tau')|/\sqrt{\tau'}$ and $|Y(\tau')||X(\tau')|$ are +bounded in terms of $T-t$, $|Y(\tau)|$, and the bound on +$\sqrt{\tau'}|X(\tau')|$ along the ${\mathcal L}$-geodesic. From +this it follows immediately from Equation (6.12) that $H(X,Y)$ +is bounded along the ${\mathcal L}$-geodesic by $C|Y(\bar\tau)|^2$ + where the constant $C$ depends on $T-t$ and the + bounds on curvature and its first two derivatives. + +Of course, if $(q,t)\in {\mathcal U}_{x}(\tau)$, then this argument +shows that the Hessian of $L^\tau_{x}$ is bounded near $(q,t)$. + +At this point in the proof of \entryref{7d5d94efe4e9} we wish to employ +arguments using convexity. To carry these out we find it convenient +to work with a Euclidean metric and usual convexity rather than the +given metric $g(t)$ and convexity measured using $g(t)$-geodesics. +In order to switch to a Euclidean metric we must find one that well +approximates $g(t)$. The following is straightforward to prove. + +\entryblock{d427ad10ceee} + +Instead of working in the given metric $g(t)$ on $B_{(q,t)}$ we +shall use the Euclidean metric $h$ as in the above claim. For any +function $f$ on $B_{(q,t)}$ we denote by $\mathit{Hess}(f)$ the Hessian +of $f$ with respect to the metric $g(t)$ and by $\mathit{Hess}^h(f)$ +the Hessian of $f$ with respect to the metric $h$. By +Formula (1.2), for any $z\in B_{(q,t)}$ and any $v\in +T_zM$, we have + +$$ +\mathit{Hess}(\varphi_{(z,t)})(v,v)=\mathit{Hess}^h(\varphi_{(z,t)})(v,v) +-\sum_{i,j,k}v^iv^j\Gamma_{ij}^k\frac{\partial +\varphi_{(z,t)}}{\partial x^k}. +$$ + + Thus, it follows from the above +assumptions on the $\Gamma_{ij}^k$ and the bound on $|\nabla +\varphi_{(z,t)}|$ that for all $z\in B_{(q,t)}$ we have + +$$ +\left|\mathit{Hess}(\varphi_{(z,t)})(v,v)-\mathit{Hess}^h(\varphi_{(z,t)})(v,v)\right|\le +\frac{1}{4}|v|^2_h, +$$ + + and hence for every $z\in +B_{(q,t)}$ we have + +$$ +\mathit{Hess}^h(\varphi_{(z,t)})(v,v)\le 2C(q,t)|v|^2_g+\frac{|v|_h^2}{4}\le +\left(4C(q,t)+\frac{1}{4}\right)|v|_h^2. +$$ + +This means: + +\entryblock{b58a798d5eea} + +**Proof.** Set + +$$ +\psi_{(q,t)}=-(2C(q,t)+1)d^2_h(q,\cdot). +$$ + +Then for any $z\in B_{(q,t)}$ the function +$b_{(z,t)}=\varphi_{(z,t)}+\psi_{(q,t)}$ is an upper barrier for +$L^\tau+\psi_{(q,t)}$ at $z$. Clearly, for all $v\in T_zM$ we have + +$$ +\mathit{Hess}^h(b_{(z,t)})(v,v)=\mathit{Hess}^h(\varphi_{(z,t)})(v,v)+\mathit{Hess}^h(\psi_{(q,t)})(v,v)\le -3|v|_h^2/2. +$$ + +This implies that if $\alpha\colon [a,b]\to B_{(q,t)}$ is any +Euclidean straight-line segment in $B_{(q,t)}$ parameterized by +Euclidean arc length and if $z= \alpha(s)$ for some $s\in (a,b)$, +then + +$$ +(b_{(z,t)}\circ\alpha)''(s)\le -3/2. +$$ + +\entryblock{49c665aedacc} + +**Proof.** Fix $c<3/4$ and define a continuous function + +$$ +A(s)=\frac{(\beta(-s)+\beta(s))}{2}+cs^2-\beta(0) +$$ + + for $s\in [0,a]$. +Clearly, $A(0)=0$. Also, using the upper barrier at $0$ we see that +for $s>0$ sufficiently small $A(s)<0$. For any $s\in (0,a)$ there is +an upper barrier $c_s=(\hat b_{s}+\hat b_{-s})/2+cs^2-\beta(0)$ for +$A(s)$ at $s$, and $c_s''(t)\le 2c-3/2<0$. By the maximum principle +this implies that $A$ has no local minimum in $(0,a)$, and +consequently that it is a non-increasing function of $s$ on this +interval. That is to say, $A(s)<0$ for all $s\in (0,a)$ and hence +$A(a)\le 0$, i.e., $(\beta(a)+\beta(-a))/2+ca^2\le \beta(0)$. Since +this is true for every $c<3/4$, the result follows. + +Now applying this to Euclidean intervals in $B_{(q,t)}$ we conclude: + +\entryblock{528f2ee171eb} + +What follows is a simple interpolation result (see [GreeneWu]). +For each $q\in M$ we let $B_{(q,t)}'\subset B_{(q,t)}$ be a smaller +ball centered at $q$, so that $B'_{(q,t)}$ has compact closure in +$B_{(q,t)}$. + +\entryblock{675c76833e83} + +**Proof.** Fix $\epsilon>0$ sufficiently small so that for any $z\in +B_{(q,t)}'$ the Euclidean $\epsilon$-ball centered at $z$ is +contained in $B_{(q,t)}$. Let $B_0$ be the ball of radius $\epsilon$ +centered at the origin in $\Ar^n$ and let $\xi\colon B_0\to \Ar$ be +a non-negative $C^\infty$-function with compact support and with +$\int_{B_0}\xi d\mathit{vol}_h=1$. We define + +$$ +\beta_{(q,t)}^\epsilon(z)=\int_{B_0}\xi(y)\beta_{(q,t)}(z+y)dy, +$$ + +for all $z\in B_{(q,t)}'$. It is clear that for each $\epsilon>0$ +sufficiently small, the function $\beta_{(q,t)}^\epsilon\colon +B_{(q,t)}'\to \Ar$ is $C^\infty$ and that as $\epsilon\rightarrow 0$ +the $\beta_{(q,t)}^\epsilon$ converge uniformly on $B_{(q,t)}'$ to +$\beta_{(q,t)}$. It is also clear that for every $\epsilon>0$ +sufficiently small, the conclusion of \entryref{528f2ee171eb} holds +for $\beta_{(q,t)}^\epsilon$ and for each Euclidean straight-line +segment $\alpha$ in $B_{(q,t)}'$. This implies that $\mathit{Hess}^h(\beta_{(q,t)}^\epsilon)(v,v)\le -3|v|_h^2/2$, and hence that +by Inequality (7.2) that + +$$ +\mathit{Hess}(\beta_{(q,t)}^\epsilon)(v,v)\le -|v|^2_h=-|v|^2_{g(t)}/2. +$$ + +This means that $\beta_{(q,t)}^\epsilon$ is convex with respect to +$g(t)$. Now take a sequence $\epsilon_k\rightarrow 0$ and let +$f_k=\beta_{(q,t)}^{\epsilon_k}$. + +Lastly, it is a standard fact that $f_k$ converge uniformly in the +$C^\infty$-topology to $\beta_{(q,t)}$ on any subset of $B_{(q,t)}'$ +whose closure is disjoint from $S_{(q,t)}$. + +\entryblock{06ac7beab852} + +We now have: + +\entryblock{de785906ce83} + +\entryblock{f93a1c4b1057} + +**Proof.** Since $f_k\rightarrow \beta_{(q,t)}$ uniformly on $B_{(q,t)}'$ we +have + +$$ +\int_{B_{(q,t)}'}\beta_{(q,t)}\triangle \phi d\mathit{vol}(g(t))=\mathit{lim}_{k\rightarrow +\infty}\int_{B_{(q,t)}'}f_k\triangle \phi d\mathit{vol}(g(t)). +$$ + + Since +$f_k$ is strictly convex with respect to the metric $g(t)$, +$\triangle f_k\le 0$ on all of $B_{(q,t)}'$. Since $\phi\ge 0$, for +every $\epsilon$ and $k$ we have + +$$ +\int_{\nu_\epsilon(S_{(q,t)})\cap B_{(q,t)}'}\phi\triangle f_k d\mathit{vol}(g(t))\le +0. +$$ + + Hence, for every $k$ and for every $\epsilon$ we have + +$$ +\begin{aligned} +\int_{B_{(q,t)}'}f_k\triangle \phi d\mathit{vol}(g(t)) & = & +\int_{B_{(q,t)}'}\phi\triangle f_k d\mathit{vol}(g(t)) \\ +& \le & \int_{B_{(q,t)}'\setminus \left(B_{(q,t)}'\cap +\nu_\epsilon(S_{(q,t)})\right)}\phi\triangle f_k d\mathit{vol}(g(t)). +\end{aligned} +$$ + + Taking the limit as $k\rightarrow\infty$, +using the fact that $f_k\rightarrow \beta_{(q,t)}$ uniformly on +$B_{(q,t)}'$ and that restricted to +$B_{(q,t)}'\setminus(B_{(q,t)}'\cap \nu_\epsilon (S_{(q,t)}))$ the +$f_k$ converge uniformly in the $C^\infty$-topology to +$\beta_{(q,t)}$ yields + +$$ +\int_{B_{(q,t)}'}\beta_{(q,t)}\triangle \phi d\mathit{vol}(g(t)) +\le \int_{B_{(q,t)}'\setminus \left(B_{(q,t)}'\cap +\nu_\epsilon(S_{(q,t)})\right)}\phi\triangle \beta_q d\mathit{vol}(g(t)). +$$ + + Now taking the limit as $\epsilon\rightarrow 0$ +establishes the claim. + +\entryblock{490290f4c322} + +**Proof.** Recall that $\beta_{(q,t)}=L^\tau+\psi_{(q,t)}$ and that +$\psi_{(q,t)}$ is a $C^\infty$-function. Hence, + +$$ +\int_{B_{(q,t)}'}\psi_{(q,t)}\triangle \phi d\mathit{vol}(g(t))= \int_{(B_{(q,t)}')^*}\phi +\triangle \psi_{(q,t)} d\mathit{vol}(g(t)). +$$ + + Subtracting this equality +from the inequality in the previous claim and dividing by +$2\sqrt{\tau}$ gives the result. + +Now we turn to the proof proper of \entryref{7d5d94efe4e9}. + +**Proof.** Let $\phi\colon M\to \Ar$ be a non-negative, smooth function of +compact support. Cover $M$ by open subsets of the form $B_{(q,t)}'$ +as above. Using a partition of unity we can write +$\phi=\sum_i\phi_i$ where each $\phi_i$ is a non-negative smooth +function supported in some $B_{(q_i,t)}'$. Since the inequalities we +are trying to establish are linear in $\phi$, it suffices to prove +the result for each $\phi_i$. This allows us to assume (and we shall +assume) that $\phi$ is supported in $B_{(q,t)}'$ for some $q\in M$. + +Since $l_x^\tau$ is a locally Lipschitz function, the restriction of +$|\nabla l_x^\tau|^2$ to $B_{(q,t)}'$ is an $L_\mathit{loc}^\infty$-function. Similarly, $\partial l_x/\partial \tau$ is an +$L_\mathit{loc}^\infty$-function. Hence + +$$ +\begin{aligned} +\int_{B_{(q,t)}'} +\phi\cdot\left(\frac{\partial l_x}{\partial \tau}+|\nabla +l_x^\tau|^2-R+\frac{n}{2\tau}\right) d\mathit{vol}(g(t)) \\ & = & +\int_{(B_{(q,t)}')^*}\phi\left(\frac{\partial l_x}{\partial +\tau}+|\nabla l_x^\tau|^2-R+\frac{n}{2\tau}\right) d\mathit{vol}(g(t)). +\end{aligned} +$$ + + On the other hand, by +\entryref{490290f4c322} we have + +$$ +\int_{B_{(q,t)}'}l_x^\tau\triangle \phi d\mathit{vol}(g(t))\le \int_{(B_{(q,t)}')^*}\phi\triangle l_x^\tau +d\mathit{vol}(g(t)). +$$ + + Putting these together we see + +$$ +\begin{aligned} +\int_{B_{(q,t)}'}\phi\left(\frac{\partial l_x}{\partial +\tau}+|\nabla +l_x^\tau|^2-R+\frac{n}{2\tau}\right) -l_x^\tau\triangle \phi d\mathit{vol}(g(t)) \\ +& \ge & \int_{(B_{(q,t)}')^*}\phi\left(\frac{\partial l_x}{\partial +\tau}+|\nabla l_x^\tau|^2-R+\frac{n}{2\tau} -\triangle +l_x^\tau\right) d\mathit{vol}(g(t)). +\end{aligned} +$$ + +It follows immediately from the second inequality in +\entryref{967c0f8bb774} that, since $\phi\ge 0$ and +$\left(B'_{(q,t)}\right)^*\subset {\mathcal U}_x(\tau)$, we have + +$$ +\int_{(B_{(q,t)}')^*}\phi\left(\frac{\partial l_x}{\partial \tau}+|\nabla l_x^\tau|^2-R+ +\frac{n}{2\tau}-\triangle l_x^\tau\right) d\mathit{vol}(g(t))\ge 0. +$$ + +This proves the first inequality in the statement of the theorem. + +The second inequality in the statement of the theorem is proved in +exactly the same way using the third inequality in +\entryref{967c0f8bb774}. + +Now let us consider the distributions + +$$ +D_1=\frac{\partial l_x}{\partial \tau}+|\nabla l_x^\tau|^2-R+\frac{n}{2\tau}- \triangle +l_x^\tau +$$ + + and + +$$ +D_2=-|\nabla +l_x^\tau|^2+R+\frac{l_x^\tau-n}{\tau}+2\triangle l_x^\tau +$$ + + on $M\times\{\tau\}$. According to \entryref{967c0f8bb774} the following equality +holdes on ${\mathcal U}_x(\tau)$: + +$$ +2\frac{\partial l_x}{\partial \tau}+|\nabla l_x^\tau|^2-R+\frac{l_x^\tau}{\tau}=0. +$$ + +By \entryref{16fbe263a7c8} the open set ${\mathcal U}_x(\tau)$ +has full measure in $M$ and $|\nabla l_x^\tau|^2$ and $\partial +l_x/\partial\tau$ are locally essentially bounded. Thus, this +equality is an equality of locally essentially bounded, measurable +functions, i.e., elements of $L_\mathit{loc}^\infty(M)$, and hence is +an equality of distributions on $M$. Subtracting $2D_1$ from this +equality yields $D_2$. Thus, + +$$ +D_2=-2D_1, +$$ + +as distributions on $M$. This shows that $D_2$ vanishes as a +distribution if and only if $D_1$ does. But if $D_2=0$ as a +distribution for some $\tau$, then by elliptic regularity $l_x^\tau$ +is smooth on $M\times\{\tau\}$ and the equality is the naïve one +for smooth functions. Thus, if $D_2=0$ for all $\tau$, then +$l_x^\tau$ and $\partial l/\partial \tau$ are $C^\infty$ functions +on each slice $M\times\{\tau\}$ and both $D_1$ and $D_2$ hold in the +naïve sense on each slice $M\times \{\tau\}$. It follows from a +standard bootstrap argument that in this case $l_x^\tau$ is smooth +on all of space-time. + +## Reduced volume + +We have established that for a Ricci flow $(M,g(t)),\ 0\le t\le T$, +and a point $x=(p,T)\in M\times [0,T]$ the reduced length function +$l_x$ is defined on all of $M\times (0,T)$. This allows us to +defined the reduced volume of $M\times \{\tau\}$ for any $\tau\in +(0,T)$ Recall that the *reduced volume* of +$M$ is defined to be + +$$ +\widetilde V_x(M,\tau) = \int_{M} \tau^{-\frac{n}{2}}exp(-l_x(q,\tau))dq. +$$ + +This function is defined for $0<\tau0$ and any $X\in T_0\Ar^n$ we have ${\mathcal U}(\tau)=T_pM$, +and ${\mathcal J}(X,\tau)=2^n\tau^{n/2}$. Also, + $L_x(X,\tau)=2\sqrt{\tau}|X|^2$, so that $l_x(X,\tau)=|X|^2$. + Thus, for any $\tau>0$ + +$$ +\widetilde + V_x(\Ar^n,\tau)=\int_{\Ar^n}\tau^{-n/2}e^{-|X|^2}2^n\tau^{n/2}dX=(4\pi)^{n/2}. +$$ + +In the case when $M$ is non-compact, it is not clear *a priori* +that the integral defining the reduced volume is finite in general. +In fact, as the next proposition shows, it is always finite and +indeed, it is bounded above by the integral for $\Ar^n$. + +\entryblock{ae6f1c6d00d9} + +**Proof.** By \entryref{16fbe263a7c8} ${\mathcal U}_x(\tau)$ is an open +subset of full measure in $M$. Hence, + +$$ +\widetilde +V_x(M,\tau)=\int_{{\mathcal U}_x(\tau)}\tau^{-\frac{n}{2}}\mathit{exp}(-l_x(q,\tau))dq. +$$ + + Take +linear orthonormal coordinates $(z^1,\ldots,z^n)$ on $T_pM$. It +follows from the previous equality and \entryref{c3d4eaed00fd} that + +$$ +\widetilde V_x(M,\tau)=\int_{\widetilde {\mathcal U}_{x}(\tau)}f(Z,\tau)dz^1\cdots dz^n, +$$ + +where $f(Z,\tau)=\tau^{-\frac{n}{2}}e^{-\widetilde +l(Z,\tau)}{\mathcal J}(Z,\tau)$. By \entryref{4dc40f37cd6d} for +each $Z$ the integrand, $f(Z,\tau)$, is a non-increasing function of +$\tau$ and the function converges uniformly on compact sets as +$\tau\rightarrow 0$ to $2^ne^{-|Z|^2}$. This implies that +$f(Z,\tau)\le 2^ne^{-|Z|^2}$ for all $\tau>0$, and hence that + +$$ +\int_{\widetilde{\mathcal U}_x(\tau)}f(Z,\tau)dz^1\ldots dz^n +$$ + +converges absolutely for each $\tau>0$, and the integral has value +at most $(4\pi)^{n/2}$. + +Fix $0<\tau_00$ sufficiently small + $\widetilde {\mathcal U}_x(\tau)$ contains the ball of radius $A$ centered at the origin + in $T_pM$. Since the curvature is bounded, this is exactly + the content of \entryref{f1123507a2ea}. + +### Converse to Lemma \entryref{4a4c8aae3a8d} + +In \entryref{4a4c8aae3a8d} we showed that for the trivial flow on flat +Euclidean $n$-space and for any point $x\in \Ar^n\times \{T\}$ the +reduced volume $\widetilde V_x(\Ar^n,\tau)$ is independent of +$\tau>0$ and is equal to $(4\pi)^{n/2}$. In this subsection we use +the monotonicity results of the last subsection to establish the +converse to \entryref{4a4c8aae3a8d}, namely to show that if $(M,g(t)),0\le +t\le T$, is a Ricci flow complete with bounded curvature and if +$\widetilde V_x(M,\bar\tau)=(4\pi)^{n/2}$ for some $\bar\tau>0$ and +some $x\in M\times\{T\}$, then the flow on the interval +$[T-\bar\tau,T]$ is the trivial flow on flat Euclidean $n$-space. + +\entryblock{613fb3994956} + +**Proof.** If $\widetilde V_x(M,\bar\tau)=(4\pi)^{n/2}$, then by +\entryref{4a4c8aae3a8d}, $\widetilde V_x(M,\tau)$ is constant on the +interval $(0,\bar\tau]$. Hence, it follows from the proof of +\entryref{ae6f1c6d00d9} that the closure of $\widetilde{\mathcal U}(\tau)$ +is all of $T_pM$ for all $\tau\in (0,\bar\tau]$ and that +$f(Z,\tau)=e^{-|Z|^2}2^n$ for all $Z\in T_pM$ and all $\tau\le +\bar\tau$. In particular, + +$$ +\frac{\partial \mathit{ln}(f(Z,\tau))}{\partial \tau}=0. +$$ + +From the proof of + \entryref{4dc40f37cd6d} this means that +Inequality (6.20) is an equality and consequently, so is +Inequality (6.19). Thus, by \entryref{88d4ef67440f} + (with $\tau_1=0$) each of the vector fields $Y_\alpha(\tau)=\tilde +Y_\alpha(\tau)$ is both a Jacobi field and adapted. By +\entryref{882677eecdc1} we then have + +$$ +\mathit{Ric}+\mathit{Hess}(l^\tau_x)=\frac{g}{2\tau}. +$$ + +In particular, $l_x$ is smooth. Let $\varphi_\tau\colon M\to M,\ +0<\tau\le \bar\tau$, be the one-parameter family of diffeomorphisms +obtained by solving + +$$ +\frac{d\varphi_\tau}{d\tau}=\nabla +l_x(\cdot,\tau)\ \ \ \ \mathit{and} \ \ \ \varphi_{\bar\tau}=\mathit{Id}. +$$ + + We now consider + +$$ +h(\tau)=\frac{\bar\tau}{\tau}\varphi_\tau^*g(\tau). +$$ + +We compute + +$$ +\begin{aligned} +\frac{\partial h}{\partial \tau} & = & +-\frac{\bar\tau}{\tau^2}\varphi_\tau^*g(\tau) ++\frac{\bar\tau}{\tau}\varphi_\tau^*{\mathcal +L}_{\frac{d\varphi_\tau}{d\tau}}(g(\tau)) ++\frac{\bar\tau}{\tau}\varphi_t^*2\mathit{Ric}(g(\tau)) \\ +& = & -\frac{\bar\tau}{\tau^2}\varphi_\tau^*g(\tau) + +\frac{\bar\tau}{\tau}\varphi_\tau^*2\mathit{Hess}(l_x^\tau)+\frac{\bar\tau}{\tau} +\varphi_\tau^*\left(\frac{1}{\tau}g(\tau)-2\mathit{Hess}(l_x^\tau)\right)=0. +\end{aligned} +$$ + + That is to say the family +of metrics $h(\tau)$ is constant in $\tau$: for all $\tau\in +(0,\bar\tau]$ we have $h(\tau)=h(\bar\tau)=g(\bar\tau)$. It then +follows that + +$$ +g(\tau)=\frac{\tau}{\bar\tau}(\varphi_\tau^{-1})^*g(\bar\tau), +$$ + +which means that the entire flow in the interval $(0,\bar\tau]$ +differs by diffeomorphism and scaling from $g(\bar\tau)$. Suppose +that $g(\bar\tau)$ is not flat, i.e., suppose that there is some +$(x,\bar\tau)$ with $|\mathit{Rm}(x,\bar\tau)|=K>0$. Then from the flow +equation we see that $|\mathit{Rm}(\varphi^{-1}_\tau(x),\tau)|=K\bar\tau^2/\tau^2$, and these +curvatures are not bounded as $\tau\rightarrow 0$. This is a +contradiction. We conclude that $g(\bar\tau)$ is flat, and hence, +again by the flow equation so are all the $g(\tau)$ for $0<\tau\le +\bar\tau$, and by continuity, so is $g(0)$. Thus, $(M,g(\tau))$ is +isometric to a quotient of $\Ar^n$ by a free, properly discontinuous +group action. Lastly, since $\widetilde V_x(M,\tau)=(4\pi)^{n/2}$, +it follows that $(M,g(\tau))$ is isometric to $\Ar^n$ for every +$\tau\in [0,\bar\tau]$. Of course, it then follows that the flow is +the constant flow. diff --git a/projects/poincare-conjecture/.astrolabe/docs/09-non-collapsed-results.mdx b/projects/poincare-conjecture/.astrolabe/docs/09-non-collapsed-results.mdx new file mode 100644 index 00000000..333795d0 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/docs/09-non-collapsed-results.mdx @@ -0,0 +1,455 @@ + + +# Chapter 8 -- Non-collapsed results + +In this chapter we apply the results for the reduced length function +and reduced volume established in the last two sections to prove +non-collapsing results. In the first section we give a general +result that applies to generalized Ricci flows and will eventually +be applied to Ricci flows with surgery to prove the requisite +non-collapsing. In the second section we give a non-collapsing +result for Ricci flows on compact $3$-manifolds with normalized +initial metrics. + +## A non-collapsing result for generalized Ricci flows + +The main result of this chapter is a +$\kappa$-non-collapsed result. + +\entryblock{3defa9b2209c} + +See Fig. 8.1. + +In this section we denote by $g(\tau), \ 0\le \tau\le r^2$, the +family of metrics on $B(x,T,r)$ induced from pulling back $G$ under +the embedding $B(x,T,r)\times [T-r^2,T]\to {\mathcal M}$. Of course, +this family of metrics satisfies the backward Ricci flow equation. + +**Proof.** Clearly from the definition of the reduced volume, we have + +$$ +\widetilde V_x(W(\tau_0))\ge \tau_0^{-n/2}Ve^{-l_0}\ge +\bar\tau_0^{-n/2}Ve^{-l_0}. +$$ + + By the +monotonicity result (\entryref{6a52537bef2f}) it follows that for any +$\tau\le \tau_0$, and in particular for any $\tau\le r^2$, we have + +$$ +\widetilde V_x(W(\tau))\ge +\bar\tau_0^{-n/2}Ve^{-l_0}. +$$ + +Let $\varepsilon=\sqrt[n]{\mathit{Vol}(B(x,T,r))}/r$, so that $\mathit{Vol} B(x,T,r)=\varepsilon^nr^n$. The basic result we need to +establish in order to prove this theorem is the following: + +\entryblock{12d93b430400} + +Given this proposition, it follows immediately that either +$\varepsilon>\varepsilon_0$ or + +$$ +\varepsilon\ge \left(\frac{\widetilde V_x(W(\tau_1))}{3}\right)^{2/n}\ge +\frac{1}{3^{2/n}\bar\tau_0}V^{2/n}e^{-2l_0/n}. +$$ + + Since +$\kappa=\varepsilon^n$, this proves the theorem. + +**Proof.** We divide $\widetilde W$ +into + +$$ +\widetilde W_\mathit{sm}=\widetilde W\cap \left\{Z\in +T_xM_T\bigl|\bigr. |Z|\le \frac{1}{8}\varepsilon^{-1/2}\right\} +$$ + +and + +$$ +\widetilde W_\mathit{lg}=\widetilde W\setminus \widetilde W_\mathit{sm}, +$$ + +(see Fig. 8.2). + +We set $W_\mathit{sm}(\tau_1)={\mathcal L}\mathit{exp}_x^{\tau_1}(\widetilde W_\mathit{sm})$ and $W_\mathit{lg}(\tau_1)={\mathcal L}\mathit{exp}_x^{\tau_1}(\widetilde W_\mathit{lg}).$ Clearly, since $W(\tau_1)$ is the union of $W_\mathit{sm}(\tau_1)$ and $W_\mathit{lg}(\tau_1)$ and since these subsets are +disjoint measurable subsets, we have + +$$ +\widetilde V_x(W(\tau_1))=\widetilde V_x(W_\mathit{sm}(\tau_1))+\widetilde V_x(W_\mathit{lg}(\tau_1)). +$$ + + We shall show that there is $\varepsilon_0$ such that either +$\varepsilon>\varepsilon_0$ or $\widetilde V_x(W_\mathit{sm}(\tau_1))\le +2\varepsilon^{n/2}$ and $\widetilde V_x(W_\mathit{lg}(\tau_1))\le +\varepsilon^{n/2}$. This will establish \entryref{12d93b430400} and hence +\entryref{3defa9b2209c}. + +### Upper bound for $\widetilde V_x(W_\mathit{sm}(\tau_1))$ + +The idea here is that ${\mathcal L}$-geodesics with initial vector +in $\widetilde W_\mathit{sm}$ remain in the parabolic neighborhood +$P=B(x,T,r)\times [T-r^2,T]$ for $\tau\in [0,r^2]$. Once we know +this it is easy to see that their ${\mathcal L}$-lengths are bounded +from below. Then if the volume of $B(x,T,r)$ was arbitrarily small, +the reduced volume of $W_\mathit{sm}(\tau_1)$ would be arbitrarily +small. + +\entryblock{26e1a6292fef} + +Of course, we have + +$$ +\widetilde V_x(W_\mathit{sm}(\tau_1))=\int_{W_\mathit{sm}(\tau_1)}\tau_1^{-n/2}e^{-l(q,\tau_1)}d\mathit{vol}_{g(\tau_1)}=\int_{\widetilde W_\mathit{sm}}\tau_1^{-n/2}e^{-\tilde +l(Z,\tau_1)}{\mathcal J}(Z,\tau_1)dZ, +$$ + + so that it will follow +immediately from the lemma that: + +\entryblock{b49cbb9f66c6} + +**Proof.** *(Of \entryref{26e1a6292fef})* +In order to establish \entryref{26e1a6292fef} we need two preliminary +estimates: + +\entryblock{bf8f356cfd7a} + +**Proof.** Recall that by hypothesis $|\mathit{Rm}(y,t)|\le 1/r^2$ on +$B(x,T,r)\times [T-r^2,T]$. Rescale the flow by multiplying the +metric and time by $r^{-2}$ resulting in a ball $\widetilde B$ of +radius one and a flow defined for a time interval of length one with +$|\mathit{Rm}|\le 1$ on the entire parabolic neighborhood +$B(x,T,1)\times [T-1,T]$. Then according to \entryref{ed4b72caf9f9} there +is a universal constant $C_1$ such that $|\nabla R(y,t)|\le C_1$ for +all $(y,t)\in B(x,T,1/2)\times [T-1/2,T]$. Rescaling back by $r^2$ +to the original flow, we see that on this flow $|\nabla R(y,t)|\le +C_1/r^3$ for all $(y,t)\in B(x,T,r/2)\times [T-r^2/2,T]$. Taking +$\varepsilon'_0\le 1/2$ gives the first item in the claim. + + Since $|\mathit{Ric}|\le (n-1)/r^2$ for all +$(y,t)\in B\times[T-r^2,T]$ it follows by integrating that + +$$ +e^{-2(n-1)(T-t)/r^2}\le \frac{g(x,t)}{g(x,T)}\le e^{2(n-1)(T-t)/r^2}. +$$ + +Thus, for $t\in [T-\tau_1,T]$ we have + +$$ +e^{-2(n-1)\varepsilon}\le \frac{g(x,t)}{g(x,T)}\le e^{2(n-1)\varepsilon}. +$$ + +From this the second item in the claim is immediate. + +At this point we view the ${\mathcal L}$-geodesics as paths +$\gamma\colon [0,\tau_1]\to B(x,T,r)$ (with the understanding that +the path in space-time is given by the composition of the path $ +(\gamma(\tau),T-\tau)$ in $B(x,T,r)\times [T-r^2,T]$ followed by the +given inclusion of this product into ${\mathcal M}$. + +The next step in the proof is to show that for any $Z\in \widetilde +W_\mathit{sm}$ the ${\mathcal L}$-geodesic $\gamma_Z$ (the one having +$\mathit{lim}_{\tau\rightarrow 0}\sqrt\tau X_{\gamma_Z}(\tau)=Z$) +remains in $B(x,T,r/2)$ up to time $\tau_1$. Because of this, as we +shall see, these paths contribute a small amount to the reduced +volume since $B(x,T,r/2)$ has small volume. We set +$X(\tau)=X_{\gamma_Z}(\tau)$ + +\entryblock{59189adac6be} + +**Proof.** First we make sure that $\varepsilon_0$ is less than or equal to the +universal constant $\varepsilon_0'$ of the last claim. For all +$(y,t)\in B(x,T,r)\times [T-r^2,T]$ we have $|\mathit{Rm}(y,t)|\le +r^{-2}$ and $|\nabla R(y,t)|\le C_1/r^3$ for some universal constant +$C_1$. Of course, $r^2\le \bar\tau$. Thus, at the expense of +replacing $C_1$ by a larger constant, we can (and shall) assume that +$C_1/r^3>(n-1)r^{-2}\ge |\mathit{Ric}(y,t)|$ for all $(y,t)\in +B(x,T,r)\times [T-r^2,T]$. Thus, we can take the constant $C_0$ in +the hypothesis of \entryref{333be2e58780} to be $C_1/r^3$. We take the +constant $\bar\tau$ in the hypothesis of that lemma to be +$\varepsilon r^2$. Then, we have that + +$$ +\mathit{max}_{0\le \tau\le \tau_1'}\sqrt{\tau}|X(\tau)|\le +e^{2C_1\varepsilon^2}|Z|+\frac{e^{2C_1\varepsilon^2}-1}{2}\sqrt{\varepsilon} +r, +$$ + + and + +$$ +|Z|\le e^{2C_1\varepsilon^2}\mathit{min}_{0\le \tau\le +\tau_1'}\sqrt{\tau}|X(\tau)|+\frac{e^{2C_1\varepsilon^2}-1}{2}\sqrt{\varepsilon}r. +$$ + +By choosing $\varepsilon_0>0$ sufficiently small (as determined by +the universal constant $C_1$ and by $\bar\tau_0$), we have + +$$ +\mathit{max}_{0\le \tau\le \tau_1'}\sqrt{\tau}|X(\tau)|_{g(T-\tau)}\le +(1+\frac{\varepsilon}{2})|Z|+\frac{\varepsilon}{2}, +$$ + + and + +$$ +|Z|\le (1+\frac{\varepsilon}{2})\mathit{min}_{0\le \tau\le +\tau_1'}\sqrt{\tau}|X(\tau)|_{g(T-\tau)}+\frac{\varepsilon}{2}. +$$ + + It +is now immediate that + +$$ +\abs{|\sqrt{\tau}X(\tau)|_{g(T-\tau)} - |Z|} \leq \varepsilon (1+|Z|). +$$ + +Again choosing $\varepsilon_0$ sufficiently small the result now +follows from the second inequality in \entryref{bf8f356cfd7a} + +Now we are ready to establish that the ${\mathcal L}$-geodesics +whose initial conditions are elements of $\widetilde W_\mathit{sm}$ do +not leave $B(x,T,r/2)\times [T-r^2,T]$ for any $\tau\le\tau_1$. + +\entryblock{0a1eac346e21} + +**Proof.** Since $\varepsilon\le \varepsilon_0\le 1/4n(n-1)\le 1/8$, by the +last claim we have + +$$ +|\sqrt{\tau}X(\tau)|_{g(T)}\le (1+2\varepsilon )|Z|+2\varepsilon\le +\frac{5}{4}|Z|+\frac{3}{32\sqrt\varepsilon}, +$$ + + provided that +$\gamma|_{[0,\tau)}$ is contained in $B(x,T,r/2)\times [T-\tau,T]$. +Since $|Z|\le (8\sqrt\varepsilon)^{-1}$ we conclude that + +$$ +|\sqrt{\tau}X(\tau)|_{g(T)}\le \frac{1}{4\sqrt\varepsilon}, +$$ + +as long as $\gamma([0,\tau))$ is contained in $B(x,T,r/2)\times +[T-\tau,T]$. + + Suppose that there is +$\tau'<\tau_1=\varepsilon r^2$ for which $\gamma_Z$ exits +$B(x,T,r/2)\times [T-r^2,T]$. We take $\tau'$ to be the first such +time. Then we have + +$$ +|\gamma_Z(\tau')- x|_{g(T)} \leq +\int_{0}^{\tau'}|X(\tau)|_{g(T)}d\tau \le +\frac{1}{4\varepsilon^{\frac{1}{2}}}\int_{0}^{\tau'}\frac{d\tau}{\sqrt{\tau}} +=\frac{1}{2\varepsilon^{\frac{1}{2}}}\sqrt{\tau'} < r/2. +$$ + +This contradiction implies that $\gamma_Z(\tau) \in B(x,T,r/2)$ for +all $\tau< \tau_1=\varepsilon r^2$. + +Now we assume that $\varepsilon_0>0$ depending on $\bar\tau_0$ is as +above and that $\varepsilon\le \varepsilon_0$, and we shall estimate + +$$ +\widetilde V_x(W_\mathit{sm}(\tau_1)) = \int_{W_\mathit{sm}(\tau_1)} +(\tau_1)^{-\frac{n}{2}}e^{- l(q,\tau_1)}d\mathit{vol}_{g(\tau_1)}. +$$ + + In +order to do this we estimate $ l_x(q,\tau_1)$ on $ W_\mathit{sm}(\tau_1)$. By hypothesis $|\mathit{Rm}|\le 1/r^2$ on +$B(x,T,r/2)\times [0,\tau_1]$ and by \entryref{0a1eac346e21} every +${\mathcal L}$-geodesic $\gamma_Z$, defined on $[0,\tau_1]$, with +initial conditions $Z$ satisfying $|Z|\leq +\frac{1}{8}\varepsilon^{-\frac{1}{2}}$ remains in $B(x,T,r/2)$. +Thus, for such $\gamma_Z$ we have $R(\gamma_Z(\tau))\ge +-n(n-1)/r^2$. Thus, for any $q\in W_\mathit{sm}(\tau_1)$ we have + +$$ +L_x(q,\tau_1) = +\int_{0}^{\tau_1}\sqrt{\tau}(R+|X(\tau)|^{2})d \tau \geq +-\frac{2n(n-1)}{3r^{2}}(\tau_1)^{\frac{3}{2}}= +-\frac{2n(n-1)}{3}\varepsilon^{\frac{3}{2}}r, +$$ + + and hence + +$$ +l+x(q,\tau_1)=\frac{L_x(q,\tau_1)}{2\sqrt{\tau_1}}\ge -\frac{n(n-1)}{3}\varepsilon. +$$ + + Since $W_\mathit{sm}(\tau)\subset B(x,T,r/2)\subset B(x,T,r)$, we have: + +$$ +\begin{aligned} +\widetilde V_x(W_\mathit{sm}(\tau_1)) & \leq & +\varepsilon^{-\frac{n}{2}}r^{-n}e^{n(n-1)\varepsilon/3}\mathit{Vol}_{g(T-\tau_1)} W_\mathit{sm}(\tau) \\ +& \le & \varepsilon^{-\frac{n}{2}}r^{-n}e^{n(n-1)\varepsilon/3}\mathit{Vol}_{g(T-\tau_1)} B(x,T,r).\nonumber +\end{aligned} +$$ + +\entryblock{35010eaa585e} + +**Proof.** This is immediate from the second item in \entryref{bf8f356cfd7a}. + +Now assume that $\varepsilon_0$ also satisfies this claim. Plugging +this into Equation (8.2), and using the fact that +$\varepsilon\le \varepsilon_0\le 1/4n(n-1)$, , so that +$n(n-1)\varepsilon/3\le 1/12$, and the fact that from the definition +we have $\mathit{Vol}_{g(T)} B(x,T,r)=\varepsilon^nr^n$, gives + +$$ +\widetilde V_x(W_\mathit{sm}(\tau_1))\leq \varepsilon^{-\frac{n}{2}}r^{-n}e^{n(n-1)\varepsilon/3} +(1.1)\mathit{Vol}_{g(T)}B(x,T,r)\le(1.1) +\varepsilon^{\frac{n}{2}}e^{\frac{1}{12}}. +$$ + + Thus, + +$$ +\widetilde V_x(W_\mathit{sm}(\tau_1))\le +2\varepsilon^{\frac{n}{2}}. +$$ + +This completes the proof of \entryref{26e1a6292fef}. + +### Upper bound for $\widetilde V_x( W_\mathit{lg}(\tau_1))$ + +Here the basic point is to approximate the reduced volume integrand +by the heat kernel, which drops off exponentially fast as we go away +from the origin. + +Recall that $\mathit{Vol} B(x,T,r)=\varepsilon^nr^n$ and +$\tau_1=\varepsilon r^2$. + +\entryblock{4b41e7ff8317} + +**Proof.** By the monotonicity result (\entryref{4dc40f37cd6d}), we see +that the restriction of the function +$\tau_1^{-\frac{n}{2}}e^{-\tilde l(Z,\tau_1)}{\mathcal J}(Z,\tau_1)$ +to ${\widetilde U}(\tau_1)$ is less than or equal to the restriction +of the function $2^ne^{-|Z|^2}$ to the same subset. This means that + +$$ +\widetilde V_x(W_\mathit{lg}(\tau_1))\le \int_{{\widetilde U}(\tau_1)\setminus +\widetilde U(\tau_1)\cap B(0,\frac{1}{8}\varepsilon^{-1/2})}2^n +e^{-|Z|^2}dZ\le \int_{T_p{\mathcal M}_T\setminus +B(0,\frac{1}{8}\varepsilon^{-1/2})}2^ne^{-|Z|^2}dZ. +$$ + + So it suffices +to estimate this latter integral. + +Fix some $a>0$ and let $I(a)=\int_{B(0,a)}2^ne^{-|Z|^2}dZ$. Let +$R(a/\sqrt{n})$ be the $n$-cube centered at the origin with side +lengths $2a/\sqrt{n}$. Then $R(a/\sqrt{n})\subset B(0,a)$, so that + +$$ +\begin{aligned} +I(a) & \ge & \int_{R(a/\sqrt{n})}2^ne^{-|Z|^2}dZ \\ +& = & \prod_{i=1}^n +\left(\int_{-a/\sqrt{n}}^{a/\sqrt{n}}2e^{-z_i^2}dz_i\right) +\\ +& = & +\left(\int_0^{2\pi}\int_0^{a/\sqrt{n}}4e^{-r^2}rdrd\theta\right)^{n/2}. +\end{aligned} +$$ + +Now + +$$ +\int_0^{2\pi}\int_0^{a/\sqrt{n}}4e^{-r^2}rdrd\theta=4\pi(1-e^{-\frac{a^2}{n}}). +$$ + +Applying this with $a=(8\sqrt\varepsilon)^{-1}$ we have + +$$ +\widetilde V_x(W_\mathit{lg}(\tau_1))\le \int_{\Ar^n}2^ne^{-|Z|^2}dZ-I(1/8\sqrt\varepsilon) +\le(4\pi)^{n/2}\left(1-\left(1-e^{-1/(64n\varepsilon)}\right)^{n/2}\right). +$$ + +Thus, + +$$ +\widetilde V_x(W_\mathit{lg}(\tau_1))\le +(4\pi)^{n/2}\frac{n}{2}e^{-1/(64n\varepsilon)}. +$$ + + There is $\varepsilon_0>0$ such that the expression on the +right-hand side is less than $\varepsilon^{n/2}$ if $\varepsilon\le +\varepsilon_0$. This completes the proof of \entryref{4b41e7ff8317}. + +Putting Lemmas \entryref{26e1a6292fef} and \entryref{4b41e7ff8317} together, establishes +\entryref{12d93b430400}. + +As we have already remarked, \entryref{12d93b430400} immediately +implies \entryref{3defa9b2209c}. This completes the proof of +\entryref{3defa9b2209c}. + +## Application to compact Ricci flows + +Now let us apply this result to Ricci flows with normalized initial +metrics to show that they are universally $\kappa$-non-collapsed on +any fixed, finite time interval. In this section we specialize to +$3$-dimensional Ricci flows. We do not need this result in what +follows for we shall prove a more delicate result in the context of +Ricci flows with surgery. Still, this result is much simpler and +serves as a paradigm of what will come. + +\entryblock{198e7475ad2f} + +**Proof.** Fix any $x=(p,t_0)\in M\times [0,T]$. First, we claim that we can +suppose that $t_0\ge 1$. For if not, then rescale the flow by +$Q=1/t_0$. This does not affect the curvature inequality at time +zero. Furthermore, there is $\omega'>0$ depending only on $\omega$ +such that for any ball $B$ at time zero and of radius one in the +rescaled flow we have $\mathit{Vol} B\ge \omega'$. The reason for the +latter fact is the following: By the Bishop-Gromov inequality +(\entryref{1a1b2c4fdc9b}) there is $\omega'>0$ depending only on +$\omega$ such that for any $q\in M$ and any $r\le 1$ we have $\mathit{Vol} B(q,0,r)\ge \omega'r^3$. Of course, the rescaling increases +$T$, but simply restrict to the rescaled flow on $[0,1]$. + +Next, we claim that we can assume that $r\le \sqrt{t_0}/2$. If $r$ +does not satisfy this inequality, then we replace $r$ with +$r'=\sqrt{t_0}/2$. Of course, the curvature inequalities hold for +$r'$ if they hold for $r$. Suppose that we have established the +result for $r'$. Then + +$$ +\mathit{Vol} B(p,T,r)\ge \mathit{Vol} B(p,T,r')\ge \kappa(r')^3\ge +\kappa\left(\frac{r}{2}\right)^3=\frac{\kappa}{8}r^3. +$$ + +From now on we assume that $t_0\ge 1$ and $r\le\sqrt{t_0}/2$. +According to \entryref{c1330c5a8be3} for any $(p,t)\in M\times +[0,2^{-4}]$ we have $|\mathit{Rm}(p,t)|\le 2$ and $\mathit{Vol} B(p,t,r)\ge \kappa_0 r^3$ for all $r\le 1$. + +Once we know that $|\mathit{Rm}|$ is universally bounded on $M\times +[0,2^{-4}]$ it follows that there is a universal constant $C_1$ such +that $C_1^{-1}g(q,0)\le g(q,t) \le C_1g(q,0)$ for all $q\in M$ and +all $t\in [0,2^{-4}]$. This means that there is a universal constant +$C<\infty $ such that the following holds. For any points $q_0,q\in +M$ with $d_0(q_0,q)\le 1$ let $\gamma_{q_0,q}$ be the path in +$M\times [2^{-5},2^{-4}]$ given by + +$$ +\gamma_{q_0,q}(\tau)=(A_{q_0,q}(\tau),2^{-4}-\tau),\ 0\le \tau\le 2^{-5}, +$$ + +where $A_{q_0,q}$ is a shortest $g(0)$-geodesic from $q_0$ to $q$. +Then ${\mathcal L}(\gamma_{q_0,q})\le C$. + +By \entryref{aebc3ff2bd3b} there is a point $q_0\in M$ and an ${\mathcal +L}$-geodesic $\gamma_0$ from $x=(p,t_0)$ to $(q_0,2^{-4})$ with +$l(\gamma_0)\le 3/2$. Since $t_0\ge 1$, this means that there is a +universal constant $C'<\infty$ such that for each point $q\in +B(q_0,0,1)$ the path which is the composite of $\gamma_0$ followed +by $\gamma_{q_0,q}$ has $\tilde l$-length at most $C'$. Setting +$\tau_0=t_0-2^{-5}$, this implies that $l_x(q,\tau_0)\le C'$ for +every $q\in B(q_0,0,1)$. This ball has volume at least $\kappa_0$. +By \entryref{16fbe263a7c8}, the open subset ${\mathcal +U}_x(\tau_0)$ is of full measure in $M\times\{2^{-5}\}$. Hence, +$W(\tau_0)=\left(B(q_0,0,1)\times \{2^{-5}\}\right)\cap {\mathcal +U}_x(\tau_0)$ also has volume at least $\kappa_0$. Since $r^2\le +t_0/4<\tau_0$, \entryref{3defa9b2209c} now gives the result. (See Fig. 8.3.) diff --git a/projects/poincare-conjecture/.astrolabe/docs/10-kappa-non-collapsed-ancient-solutions.mdx b/projects/poincare-conjecture/.astrolabe/docs/10-kappa-non-collapsed-ancient-solutions.mdx new file mode 100644 index 00000000..fbe932f2 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/docs/10-kappa-non-collapsed-ancient-solutions.mdx @@ -0,0 +1,3482 @@ + + +# Chapter 9 -- $\kappa$-non-collapsed ancient solutions + + In this chapter we discuss the qualitative properties of +$\kappa$-non-collapsed, ancient solutions. One of the most important is the +existence of a gradient shrinking soliton that is asymptotic at $-\infty$ to +the solution. The other main qualitative result is the compactness result (up +to scaling) for these solutions. Also extremely important for us is +classification of $3$-dimensional gradient shrinking solitons -- up to finite +covers there are only two: a shrinking family of round $S^3$'s and a shrinking +family of products of round $S^2$'s with $\Ar$. This leads to a rough +classification of all $3$-dimensional $\kappa$-non-collapsed, ancient +solutions. The $\kappa$-solutions are in turn the models for singularity +development in $3$-dimensional Ricci flows on compact manifolds, and eventually +for singularity development in $3$-dimensional Ricci flows with surgery. + +## Preliminaries + +Our objects of study are Ricci flows $(M,g(t)),\ -\infty0$ and we consider an $n$-dimensional +$\kappa$-solution $(M,g(t)),\ -\infty 0$ there is a point $q(\tau)\in +M$ at which the function $l_x(\cdot,^\tau)$ achieves its minimum, +and furthermore, we have + +$$ +l_x(q(\tau),\tau) \leq \frac{n}{2}. +$$ + + For $\bar\tau>0,$ define + +$$ +g_{\bar\tau}(t) = \frac{1}{\bar\tau}g(\bar\tau t),\ \ -\infty0$ for all $\tau$. +Hence, $\mathit{lim}_{\tau\rightarrow\infty}\widetilde V_x(\tau)$ +exists. By \entryref{613fb3994956} either this limit as $\tau$ goes +to infinity is less than $(4\pi)^{n/2}$ or the flow is the constant +flow on flat Euclidean space. The latter is ruled out by our +assumption that the manifolds are non-flat. It follows immediately +from this and \entryref{dbf6a3c0bae7} that: + +\entryblock{e08895f2b0d8} + +Now let us turn to the length functions $l_{x_k}$. + +\entryblock{88571a22a2bc} + +**Proof.** By the choice of $q_k$ we have $l_{x_k}(q_k,\tau_k)\le \frac{n}{2}$. + By the scale invariance of $l$ (\entryref{541f71387f84}) we have + $l_{x_k}(q_k,-1)\le n/2$ for all $k$. Fix $0<\tau<1$. Integrating the inequality + +$$ +\frac{-2l_x(q_k,\tau)}{\tau}\le \frac{\partial l_{x_k}(q_k,\tau)}{\partial \tau} +$$ + + from $\tau$ to $1$ yields + +$$ +l_{x_k}(q_k,\tau)\le \frac{n}{2\tau^2}. +$$ + +If $\tau>1$, then + integrating the second inequality in the second displayed line of \entryref{8b8e3a0f3e36} gives + $l_{x_k}(q_k,\tau)\le \frac{n\tau}{2}$. + +\entryblock{611a73900deb} + +**Proof.** By \entryref{8b8e3a0f3e36}, for any $q\in M_k$ we have $|\nabla +l_{x_k}(q,\tau)|^2\le 3l_{x_k}(q,\tau)/\tau$. Since +$l_{x_k}(q_k,\tau)\le \frac {n}{2\tau_0^2}+\frac{n\tau}{2}$, +integrating yields + +$$ +l_{x_k}(q,\tau)\le +\left(\sqrt{\frac{3}{\tau}}d_{g_k(-\tau)}(q_k,q)+C_1(\tau)\right)^2, +$$ + +with $C_1(\tau)$ being $\sqrt{(n/2\tau^2)+(n\tau/2)}$. The second +statement follows from this and \entryref{9387b1b04824}. + +It follows immediately from \entryref{611a73900deb} that for each +$A<\infty$ and $\tau_0>0$, the functions $l_{x_k}$ are uniformly +bounded (by a bound that is independent of $k$ but depends on +$\tau_0$ and $A$) on the balls $B(q_k,-\tau_0,A)$. Once we know that +the $l_{x_k}$ are uniformly bounded on $B(q_k,-\tau_0,A)$, it +follows from \entryref{8b8e3a0f3e36} that $R_{g_k}$ are also +uniformly bounded on the $B(q_k,-\tau_0,A)$. Invoking +\entryref{36dd2f6fa9c2}, + we see +that for any $A<\infty$ the scalar curvatures of the metrics $g_k$ +are uniformly bounded on $B_{g_k}(q_k,-\tau_0,A)\times (-\infty, +-\tau_0]$. Since the metrics have non-negative curvature operator, +this implies that the eigenvalues of this operator are uniformly +bounded on these regions. + Since we are assuming +that the original Ricci flows are $\kappa$-non-collapsed on all +scales, it follows from \entryref{a0212d0eefcf} that after passing +to a subsequence there is a geometric limit +$(M_\infty,g_\infty(t),(q_\infty,-1)),\ -\infty0$, by a standard +diagonalization argument passing to a further subsequence we get a +geometric limit flow $(M_\infty,g_\infty(t),(q_\infty,-1)),\ +-\infty0$ on the support of +$\varphi$. We can rewrite + +$$ +\begin{aligned} +\mathit{lim}_{k\rightarrow\infty}\int_{M\times\{t\}}\langle\left(\nabla^{h_k} +l_k-\nabla l_\infty\right),\varphi\nabla^{h_k} l_k\rangle_{h_k} +d\mathit{vol}(h_k) = \\ +& & \mathit{lim}_{k\rightarrow\infty}\int_{M\times\{t\}}\langle\nabla^{h_k}( +l_k-l_\infty-\epsilon_k),\varphi\nabla^{h_k}l_k\rangle_{h_k} d\mathit{vol}(h_k). +\end{aligned} +$$ + +\entryblock{598f6059280c} + +**Proof.** Since $\varphi$ is a compactly supported, non-negative smooth +function, it follows from \entryref{7d5d94efe4e9} that we have the +following inequality of distributions: + +$$ +\varphi\triangle^{h_k} l_k\le +\frac{\varphi}{2}\left(|\nabla^{h_k} l_k|_{h_k}^2-R_{h_k}-\frac{ +l_k-n}{\tau}\right). +$$ + + (Here $R_{h_k}$ is the scalar curvature of +$h_k$.) + That is to say, for any non-negative +$C^\infty$-function $f$ we have + +$$ +\begin{aligned} +\int_{M\times\{t\}}-\langle\nabla^{h_k} +l_k,\nabla^{h_k}(\varphi \cdot f)\rangle_{h_k} d\mathit{vol}(h_k) \le +\\ +& & \int_{M\times\{t\}}\frac{\varphi f}{2}\left(|\nabla^{h_k} +l_k|_{h_k}^2-R_{h_k}-\frac{l_k-n}{\tau}\right)d\mathit{vol}(h_k). +\end{aligned} +$$ + +We claim that the same inequality holds as long as $f$ is a +non-negative, locally Lipschitz function. The point is that given +such a function $f$, we can find a sequence of non-negative +$C^\infty$-functions $f_k$ on the support of $\varphi$ (by say +mollifying $f$) that converge to $f$ strongly in the $W^{1,2}$-norm +on the support of $\varphi$. The sought-after inequality holds for +every $f_k$. Since both sides of the inequality are continuous in +the $W^{1,2}$-norm of the function, the result holds for the limit +function $f$ as well. + +Now we apply this with $f$ being the non-negative locally Lipschitz +function $l_\infty-l_k+\epsilon_k$. We conclude that + +$$ +\begin{aligned} +\int_{M\times\{t\}}\langle \nabla^{h_k}(\varphi( +l_k-l_\infty-\epsilon_k)),\nabla^{h_k} l_k\rangle_{h_k}d\mathit{vol}(h_k) \le \\ +& & \int_{M\times\{t\}}\frac{\varphi(l_\infty- +l_k+\epsilon_k)}{2}\left(|\nabla^{h_k} +l_k|_{h_k}^2-R_{h_k}-\frac{l_k-n}{\tau}\right)d\mathit{vol}(h_k). +\end{aligned} +$$ + +Now taking the limit as $k\rightarrow\infty$, we see that the +right-hand side of this inequality tends to zero since $(l_\infty- +l_k+\epsilon_k)$ tends uniformly to zero on the support of $\varphi$ +and $|\nabla^{h_k}l_k|_{h_k}^2$, $R_k$ and $l_k$ are all uniformly +essentially bounded on the support of $\varphi$. Thus, the term + +$$ +\int_{M\times\{t\}}\langle\nabla^{h_k}(\varphi( +l_k-l_\infty-\epsilon_k)),\nabla^{h_k} l_k\rangle_{h_k}d\mathit{vol}(h_k) +$$ + + has a limsup $\le 0$ as $k$ tends to $\infty$. Now we +expand + +$$ +\nabla^{h_k}(\varphi(l_k-l_\infty-\epsilon_k))=\nabla^{h_k}(\varphi)(l_k-l_\infty-\epsilon_k)+ +\varphi\nabla^{h_k}(l_k-l_\infty-\epsilon_k). +$$ + + The first term on +the right-hand side converges to zero as $k\rightarrow\infty$ since +$l_k-l_\infty-\epsilon_k$ tends uniformly to zero on the support of +$\varphi$. This completes the proof of the claim. + +We have now established the inequalities in both directions and +hence completed the proof of \entryref{366e3fdadf6f}. + +\entryblock{3776c086ec3b} + +**Proof.** Clearly, since the $l_k$ converge uniformly on compact subsets of +$M_{\infty}\times (-\infty,0)$ to $l_\infty$ and the metrics $h_k$ +converge smoothly to $g_\infty$, uniformly on compact sets, it +follows that the $\triangle ^{h_k}l_k $ converge in the weak sense +to $\triangle l_\infty$ and similarly, the $\partial l_k/\partial +\tau$ converge in the weak sense to $\partial l_\infty/\partial +\tau$. Hence, by taking limits from \entryref{7d5d94efe4e9}, using +\entryref{366e3fdadf6f}, we see that + +$$ +{\mathcal D}=\frac{\partial l_\infty}{\partial \tau}+|\nabla +l_\infty|^2-R+\frac{n}{2\tau}-\triangle l_\infty\ge 0 +$$ + + in the weak sense on $M\times (-\infty,0)$. + +Since $R$ and $\frac{n}{2\tau}$ are $C^\infty$-functions, it is +clear that the distributions given by these terms extend to +continuous linear functionals on the space of compactly supported +$W^{1,2}$-functions. Similarly, since $|\nabla l_\infty|^2$ is an +element of $L^\infty_\mathit{loc}$, it also extends to a continuous +linear functional on compactly supported $W^{1,2}$-functions. Since +$|\partial l_\infty/\partial \tau|$ is an locally essentially +bounded function, $\partial l_\infty/\partial \tau$ extends to a +continuous functional on the space of compactly supported $W^{1,2}$ +functions. Lastly, we consider $\triangle l_\infty$. As we have +seen, the value of the associated distribution on $\varphi$ is given +by + +$$ +\int_{M\times (-\infty,,0)}-\langle \nabla \varphi,\nabla +l_\infty\rangle_{g_\infty} d\mathit{vol}(g_\infty)d\tau. +$$ + + Since +$|\nabla l_\infty|$ is a locally essentially bounded function, this +expression also extends to a continuous linear functional on +compactly supported $W^{1,2}$-functions. + +Lastly, if $\psi$ is an element of $W^{1,2}$ with compact support +and hence can be approximated in the $W^{1,2}$-norm by non-negative +smooth functions. The last statement is now immediate from +Equation (9.9). + +This leads immediately to: + +\entryblock{a8e10db32f90} + +**Proof.** If $\varphi$ is a compactly supported non-negative +$C^\infty$-function, then $e^{-l_\infty}\varphi$ is a compactly +supported non-negative Lipschitz function. Hence, this result is an +immediate consequence of the previous corollary. + +### Extension to non-compactly supported functions + +The next step in this proof is to estimate the $l_{x_k}$ uniformly from below +in order to show that the integrals involved in the distributions in +\entryref{2e9e6c89f40c} are absolutely convergent so that they extend to +continuous functionals on a certain space of functions that includes +non-compactly supported functions. + +\entryblock{a6ee3b2ab615} + +**Proof.** Since both sides of this inequality and also Ricci flow are +invariant if the metric and time are simultaneously rescaled, it +suffices to consider the case when $\tau=1$. Also, since ${\mathcal +U}_x(1)$ is a dense subset, it suffices to assume that $p,q\in +{\mathcal U}_x(1)$. Also, by symmetry, we can suppose that +$l_{x_k}(q,1)\le l_{x_k}(p,1)$. + + Let +$\gamma_1$ and $\gamma_2$ be the minimizing ${\mathcal L}$-geodesics +from $x$ to $(p,1)$ and $(q,1)$ respectively. We define a function +$f\colon M_k\times M_k\times [0,\infty)\to \Ar$ by + +$$ +f(a,b,\tau)=d_{g_k(-\tau)}(a,b). +$$ + +Since $\gamma_1(0)=\gamma_2(0)$ we have + +$$ +\begin{aligned} +\nonumber +d_{g_k(-1)}(p,q) & = & f(p,q,1) \\ \nonumber & =& +\int_0^1\frac{d}{d\tau}f(\gamma_1(\tau),\gamma_2(\tau),\tau)d\tau \\ +\nonumber & = & \int_0^1\bigl(\frac{\partial f}{\partial +\tau}(\gamma_1(\tau),\gamma_2(\tau),\tau)+\langle \nabla +f_a,\gamma_1'(\tau)\rangle\bigr. \\ + & & \hskip.2in+\bigl.\langle \nabla +f_b,\gamma_2'(\tau)\rangle\bigr)d\tau, +\end{aligned} +$$ + +where $\nabla_af$ and $\nabla_bf$ refer respectively to the +gradient of $f$ with respect to the first copy of $M_k$ in the +domain and the second copy of $M_k$ in the domain. Of course, +$|\nabla f_a|=1$ and $|\nabla f_b|=1$. + +By \entryref{033786a89c6d}, we have $\gamma_1'(\tau)=\nabla +l_{x_k}(\gamma_1(\tau),\tau)$ and $\gamma_2'(\tau)=\nabla +l_{x_k}(\gamma_2(\tau),\tau)$. Since $R\ge 0$ we have + +$$ +l_{x_k}(\gamma_1(\tau),\tau)=\frac{1}{2\sqrt{\tau}}{\mathcal +L}_{x_k}(\gamma_1|_{[0,\tau]})\le \frac{1}{2\sqrt{\tau}}{\mathcal +L}_{x_k}(\gamma_1)=\frac{1}{\sqrt{\tau}}l_{x_k}(p,1). +$$ + +Symmetrically, we have + +$$ +l_{x_k}(\gamma_2(\tau),\tau)\le \frac{1}{\sqrt{\tau}}l_{x_k}(q,1). +$$ + +From this inequality, \entryref{8b8e3a0f3e36}, and the fact that +$R\ge 0$, we have + +$$ +\begin{aligned} +\nonumber \bigl|\langle \nabla +f_a(\gamma_1(\tau),\gamma_2(\tau),\tau),\gamma_1'(\tau)\rangle +\bigr|& \le & +|\gamma_1'(\tau)| =|\nabla l_{x_k}(\gamma_1(\tau),\tau) | \nonumber \\ +& \le & +\frac{\sqrt{3}}{\tau^{3/4}}\sqrt{l_{x_k}(p,1)} \nonumber \\ +& \le & +\frac{\sqrt{3}}{\tau^{3/4}}\sqrt{l_{x_k}(p,1)+1}. +\end{aligned} +$$ + +Symmetrically, we have + +$$ +\left|\langle +\nabla +f_b(\gamma_1(\tau),\gamma_2(\tau),\tau),\gamma'_2(\tau)\rangle\right|\le +\frac{\sqrt{3}}{\tau^{3/4}}\sqrt{l_{x_k}(q,1)}\le +\frac{\sqrt{3}}{\tau^{3/4}}\sqrt{l_{x_k}(q,1)+1}. +$$ + + It follows from +\entryref{8b8e3a0f3e36} that for any $p$ + +$$ +|\nabla (\sqrt{l_{x_k}(p,\tau)})|\le +\frac{\sqrt{3}}{2\sqrt{\tau}}. +$$ + + Set +$r_0(\tau)=\tau^{3/4}(l_{x_k}(q,1)+1)^{-1/2}$. For any $p'\in +B_{g_k}(\gamma_1(\tau),\tau,r_0(\tau))$ integrating gives + +$$ +l_{x_k}^{1/2}(p',\tau)\le +l_{x_k}^{1/2}(\gamma_1(\tau),\tau)+\frac{\sqrt{3}}{2\sqrt{\tau}}r_0(\tau)\le +\left(\tau^{-1/4}+\frac{\sqrt{3}}{2}\tau^{1/4}\right)\sqrt{l_{x_k}(p,1)+1}, +$$ + +where in the last inequality we have used the fact that $1\le +l_{x_k}(q,1)+1\le l_{x_k}(p,1)+1$. Again using +\entryref{8b8e3a0f3e36} we have + +$$ +R(p',\tau)\le +\frac{3}{\tau}\left(\tau^{-1/4}+\frac{\sqrt{3}}{2}\tau^{1/4}\right)^2(l_{x_k}(p,1)+1). +$$ + +Now consider $q'\in B_{g_k(\tau)}(\gamma_2(\tau),\tau,r_0(\tau))$. +Similarly to the above computations, we have + +$$ +l^{1/2}_{x_k}(q',\tau)\le +l_{x_k}^{1/2}(q,1)+\frac{\sqrt{3}}{2\sqrt{\tau}}r_0(\tau), +$$ + + so that + +$$ +l_{x_k}^{1/2}(q',\tau)\le \left(\tau^{-1/4} ++\frac{\sqrt{3}}{2}\tau^{1/4}\right)\sqrt{l_{x_k}(q,1)+1}, +$$ + + and + +$$ +|\mathit{Ric}(q',\tau)\le R(q',\tau)\le +\frac{3}{\tau}\left(\tau^{-1/4}+\frac{\sqrt{3}}{2}\tau^{1/4}\right)^2(l_{x_k}(q,1)+1). +$$ + +We set + +$$ +K=\frac{3}{\tau}\left(\tau^{-1/4}+\frac{\sqrt{3}}{2}\tau^{1/4}\right)^2(l_{x_k}(q,1)+1). +$$ + +Now, noting that $\partial/\partial \tau$ here is +$-\partial/\partial t$ of \entryref{5d895f9c3918}, we apply +\entryref{5d895f9c3918} to see that + +$$ +\begin{aligned} +\left|\frac{\partial}{\partial +\tau}f(\gamma_1(\tau),\gamma_2(\tau),\tau)\right| +& \le & 2(n-1)\left(\frac{2}{3(n-1)}Kr_0(\tau)+r_0(\tau)^{-1}\right) \\ +& \le & +\left(C_1\tau^{-3/4}+C_2\tau^{-1/4}+C_3\tau^{1/4}\right)\sqrt{l_{x_k}(q,1)+1}, +\end{aligned} +$$ + +where $C_1,C_2,C_3$ are constants depending only on the dimension +$n$. + +Now plugging Equation (9.11) and (9.12) and the above +inequality into Equation (9.10) we see that + +$$ +\begin{aligned} +d_{g(-1)}(p,q) & \le & +\int_0^1\left(\left(C_1\tau^{-3/4}+C_2\tau^{-1/4}+C_3\tau^{1/4}\right)\sqrt{l_{x_k}(q,1)+1}\right. +\\ & & + +\left.\sqrt{3}\tau^{-3/4}\sqrt{l_{x_k}(q,1)+1}+\sqrt{3}\tau^{-3/4}\sqrt{l_{x_k}(p,1)+1}\right)d\tau. +\end{aligned} +$$ + +This implies that + +$$ +d_{g(-1)}(p,q)\le +C\left(\sqrt{l_{x_k}(q,1)+1}+\sqrt{l_{x_k}(p,1)+1}\right), +$$ + + for +some constant depending only on the dimension. Thus, since we are +assuming that $l_{x_k}(p,1)\ge l_{x_k}(q,1)$ we have + +$$ +d^2_{g(-1)}(p,q)\le +C^2\left(3(l_{x_k}(p,1)+1)+(l_{x_k}(q,1)+1)\right)\le +4C^2(l_{x_k}(p,1)+1+l_{x_k}(q,1)), +$$ + + for some constant $C<\infty$ +depending only on the dimension. The result now follows immediately. + +\entryblock{d16f43d74c54} + +**Proof.** By \entryref{88571a22a2bc} + +$$ +l_{x_k}(q_k,\tau')\le \frac{n}{2(\tau')^2}+\frac{n\tau'}{2}. +$$ + +Now applying \entryref{a6ee3b2ab615} we see that for any $0<\tau'$ and any $q'\in +M_k$ we have + +$$ +\begin{aligned} +l_{x_k}(q',\tau') & \ge & +-\frac{n}{2(\tau')^2}-\frac{n\tau'}{2}-1+c_1\frac{d^2_{g_k(-\tau')}(q_k,q')}{\tau'} +\\ +& \ge & +-\frac{n}{2(\tau')^2}-\frac{n\tau'}{2}-1+c_1\frac{d^2_{g_k(-\tau_0)}(q_k,q')}{\tau'}. +\end{aligned} +$$ + + In the last inequality, we use the fact that the +Ricci curvature is positive so that the metric is decreasing under +the Ricci flow. + +Since the time slices of all the flows in question have non-negative +curvature, by \entryref{1a1b2c4fdc9b} the volume of the ball of +radius $s$ is at most $\omega s^{n}$ where $\omega$ is the volume of +the ball of radius one in $\Ar^n$. Since the $l_k$ converge +uniformly to $l_\infty$ on compact sets and since the metrics $h_k$ +converge uniformly in the $C^\infty$-topology on compact sets to +$g_\infty$, it follows that for any $\epsilon>0$, for any +$0<\tau_0\le \tau'<\infty$ there is a radius $r$ such that for every +$k$ and any $\tau\in [\tau_0,\tau']$ the integral + +$$ +\int_{M_\infty\setminus B_{h_k(-\tau_0)}(q_k,r)}e^{-l_k(q,\tau)}dq<\epsilon. +$$ + +It follows by Lebesgue dominated convergence that + +$$ +\int_{M_\infty\setminus +B_{g_\infty(-\tau_0)}(q_\infty,r)}e^{-l_\infty(q,\tau)}dq\le +\epsilon. +$$ + +\entryblock{1766f37a13cf} + +**Proof.** It follows from the above discussion that there are constants $c>0$ + and a ball $B\subset M_\infty$ centered at $q_\infty$ such that on +$M_\infty\times [-\tau,-\tau_0]\setminus B\times [\tau,-\tau_0]$ the +function $l_\infty$ is greater than +$cd^2_{g_\infty(-\tau_0)}(q_\infty,\cdot)-C'$. Thus, +$fe^{-l_\infty}$ has fixed exponential decay at infinity. Since the +Riemann curvature of $M_\infty\times\{\tau'\}$ is non-negative for +every $\tau'$, the flow is distance decreasing, and there is a fixed +polynomial upper bound to the growth rate of volume at infinity. +This leads to the claimed convergence property. + +\entryblock{0e9703c15036} + +**Proof.** By +\entryref{8b8e3a0f3e36}, each of the Lipschitz functions $|\nabla +l_\infty|^2$, $|\partial l_\infty/\partial \tau|$ and $R$ + is at most a constant multiple of $l_\infty$. +Hence, the corollary follows from the previous claim. + +There is a slightly weaker statement that is true for $\triangle +e^{-l_\infty}$. + +\entryblock{5b4c94102f41} + +**Proof.** This time the value of the distribution on a compactly supported +smooth function $\rho$ is given by the integral of + +$$ +-\langle +\nabla\rho,\nabla e^{-l_\infty}\rangle=\langle \nabla \rho,\nabla +l_\infty\rangle e^{-l_\infty}. +$$ + + Since $|\nabla l_\infty|$ is less +than or equal to the maximum of $1$ and $|\nabla l_\infty|^2$, it +follows immediately, that if $|\nabla \rho|$ is bounded, then the +integral is absolutely convergent. From this the claim follows +easily. + +\entryblock{42d9f91e59f3} + +**Proof.** For the interval $[\tau_0,\tau']$ we construct a sequence of +uniformly Lipschitz functions $\psi_k$ on $M_\infty\times +[\tau_0,\tau']$ that are non-negative, bounded above by one and +eventually one on every compact set. Let $\rho(x)$ be a smooth bump +function which is one for $x$ less than $1/4$ and is zero from $x\ge +3/4$ and is everywhere between $0$ and $1$. For any $k$ sufficiently +large let $\psi_k$ be the composition of +$\rho(d_{g_\infty(-\tau_0)}(q_\infty,\cdot)-k)$. Being compositions +of $\rho$ with Lipschitz functions with Lipschitz constant $1$, the +$\psi_k$ are a uniformly Lipschitz family of functions on +$M_\infty\times\{-\tau_0\}$. Clearly then they form a uniformly +Lipschitz family on $M_\infty\times [\tau_0,\tau']$ as required. +This allows us to define any of the above distributions on Lipschitz +functions on $M_\infty\times [\tau_0,\tau']$. + +Take a family $\psi_k$ of uniformly Lipschitz functions, each +bounded between $0$ and $1$ and eventually one of every compact +subset of $M_\infty\times [\tau_0,\tau_1]$. Then the family +$f\psi_k$ is a uniformly Lipschitz family of compactly supported +functions. Hence, we can apply Claims \entryref{1766f37a13cf} +and \entryref{5b4c94102f41} to establish that the integral in question is the +limit of an absolutely convergence sequence. By +\entryref{a8e10db32f90} each term in the sequence is non-positive. + +### Completion of the proof of Proposition \entryref{2e9e6c89f40c} + +Lebesgue dominated convergence implies that the following limit +exists + +$$ +\mathit{lim}_{k\rightarrow\infty}\widetilde V_k(\tau)\equiv\widetilde V_\infty(\tau)=\int_{M_\infty\times\{-\tau\}}\tau^{-n/2}e^{- +l_\infty(q,\tau)}d\mathit{vol}_{g_\infty(\tau)}. +$$ + + By \entryref{e08895f2b0d8}, the function $\tau\rightarrow +\widetilde V_\infty(\tau)$ is constant. On the other hand, note that +for any $0< \tau_0<\tau_1<\infty$, we have + +$$ +\begin{aligned} +\widetilde V_\infty(\tau_1)-\widetilde V_\infty(\tau_0) & = & +\int_{\tau_0}^{\tau_1}\frac{d\widetilde +V_\infty}{d\tau}d\tau \\ +& = & \int_{\tau_0}^{\tau_1}\int_{M_\infty}\left(\frac{\partial +l_\infty}{\partial \tau}-R+\frac{n}{2\tau}\right) +\left(\tau^{-n/2}e^{-l_\infty(q,\tau)}d\mathit{vol}_{g_\infty(\tau)}\right). +\end{aligned} +$$ + + According to +\entryref{0e9703c15036} this is an absolutely convergent integral, +and so this integral is zero. + +\entryblock{a8ab8a19cc61} + +**Proof.** Since we are integrating against the constant function $1$, this +result is clear, given the convergence result, +\entryref{0e9703c15036}, necessary to show that this integral is +well defined. + +Adding these two results together gives us the following + +$$ +\int_{\tau_0}^{\tau_1}\int_{M_\infty\times\{-\tau\}}\left(\frac{\partial +l_\infty}{\partial \tau}+|\nabla l_\infty|^2 +-R+\frac{n}{2\tau}-\triangle +l_\infty\right)\tau^{-n/2}e^{-l_\infty}d\mathit{vol}_{g_\infty}=0. +$$ + +Now let $\varphi$ be any compactly supported, non-negative smooth +function. By scaling by a positive constant, we can assume that +$\varphi\le 1$ everywhere. Let $\widetilde{\mathcal D}$ denote the +distribution given by + +$$ +\widetilde{\mathcal D}(\varphi)=\int_{\tau_0}^{\tau_1}\int_{M_\infty\times\{-\tau\}} +\varphi\left(\frac{\partial l_\infty}{\partial \tau}+|\nabla +l_\infty|^2 -R+\frac{n}{2\tau}-\triangle +l_\infty\right)\tau^{-n/2}e^{-l_\infty}d\mathit{vol}_{g_\infty}. +$$ + + Then +we have seen that $\widetilde{\mathcal D}$ extends to a functional +on bounded smooth functions of bounded gradient. Furthermore, +according to Equation (9.13), we have $\widetilde {\mathcal +D}(1)=0$. Thus, + +$$ +0={\mathcal D}(1)={\mathcal D}(\varphi)+{\mathcal D}(1-\varphi). +$$ + +Since both $\varphi$ and $1-\varphi$ are non-negative, it follows from +\entryref{42d9f91e59f3}, that ${\mathcal D}(\varphi)$ and ${\mathcal +D}(1-\varphi)$ are each $\ge 0$. Since their sum is zero, it must be the case +that each is individually zero. + +This proves that the Inequality (9.9) is actually an +equality in the weak sense, i.e., an equality of distributions on +$M_\infty\times [\tau_0,\tau')$. Taking limits we see: + +$$ +\widetilde{\mathcal D}=\left(\frac{\partial l_\infty}{\partial +\tau}+|\nabla l_\infty|^2-R+\frac{n}{2\tau}-\triangle +l_\infty\right)\tau^{-n/2}e^{-l_\infty}=0, +$$ + + in the weak +sense on all of $M\times(-\infty,0)$. Of course, this implies that + +$$ +\frac{\partial l_\infty}{\partial \tau}+|\nabla +l_\infty|^2-R+\frac{n}{2\tau}-\triangle l_\infty=0 +$$ + + in the weak +sense. + +It now follows by parabolic regularity that $l_\infty$ is a smooth +function on $M_\infty\times(-\infty,0)$ and that +Equation (9.14) holds in the usual sense. + +Now from the last two equations in \entryref{967c0f8bb774} and the +convergence of the $l_{x_k}$ to $l_\infty$, we conclude that the +following equation also holds: + +$$ +2\triangle l_\infty-|\nabla +l_\infty|^2+R+\frac{l_\infty-n}{\tau}=0. +$$ + +This completes the proof of \entryref{2e9e6c89f40c}. + +### The gradient shrinking soliton equation + +Now we return to the proof of \entryref{671402eb454b}. We have shown that the +limiting Ricci flow referred to in that result exists, and we have +established that the limit $l_\infty$ of the length functions $l_{x_k}$ is a +smooth function and satisfies the differential equalities given in +\entryref{2e9e6c89f40c}. We shall use these to establish the gradient +shrinking soliton equation, +Equation (9.2), for the limit for $f=l_\infty$. + +\entryblock{07232fc42f3a} + +**Proof.** This result will follow immediately +from: + +\entryblock{ba1adbac0a75} + +Let us assume the lemma for a moment and use it to complete the +proof of the proposition. + +We apply the lemma to the limiting Ricci flow +$(M_\infty,g_\infty(t))$ with the function $f=l_\infty$. According +to \entryref{2e9e6c89f40c} and the first statement in +\entryref{ba1adbac0a75}, the function $u$ satisfies the conjugate heat +equation. Thus, according to the second statement in +\entryref{ba1adbac0a75}, setting + +$$ +v=\left[\tau\left(2\triangle f-|\nabla +f|^2+R\right)+f-n\right]u, +$$ + + we have + +$$ +\frac{\partial v}{\partial\tau}-\triangle v+Rv=-2\tau +\bigl|\mathit{Ric}_g+\mathit{Hess}(f)-\frac{1}{2\tau}g\bigr|^2u. +$$ + + On the +other hand, the second equality in \entryref{2e9e6c89f40c} shows +that $v=0$. Since $u$ is nowhere zero, this implies that + +$$ +\mathit{Ric}_{g_\infty}+\mathit{Hess}^{g_\infty}(f)-\frac{1}{2\tau}g_\infty=0. +$$ + +This completes the proof of the proposition assuming the lemma. + +Now we turn to the proof of the lemma. + +**Proof.** *(of \entryref{ba1adbac0a75})* +Direct computation shows that + +$$ +-\frac{\partial u}{\partial +t}-\triangle u+Ru= \left(-\frac{n}{2\tau}+\frac{\partial f}{\partial +t}+\triangle f-|\nabla f|^2+R\right)u. +$$ + + From this, the first +statement of the lemma is clear. + Let + +$$ +H=\bigl[\tau(2\triangle f-|\nabla f|^2+R)+f-n\bigr] +$$ + + so that $v=Hu$. +Then, of course, + +$$ +\frac{\partial v}{\partial t}=\frac{\partial H}{\partial t}u+H\frac{\partial u}{\partial t}\ \ \ \mathit{and} +$$ + +$$ +\triangle v=\triangle H\cdot u+2\langle\nabla H,\nabla u\rangle+H\triangle u. +$$ + +Since $u$ satisfies the conjugate heat equation, we have + +$$ +-\frac{\partial v}{\partial t}-\triangle v+Rv=\left(-\frac{\partial H}{\partial t}-\triangle +H\right)u-2\langle\nabla H,\nabla u\rangle. +$$ + + Differentiating the +definition of $H$ yields + +$$ +\frac{\partial H}{\partial t} = -(2\triangle f-|\nabla +f|^2+R)+\frac{\partial f}{\partial t}+ \tau \left(2\frac{\partial +}{\partial t}\triangle f -\frac{\partial }{\partial t}(|\nabla +f|^2)+\frac{\partial R}{\partial t}\right) +$$ + +\entryblock{8283cc66c217} + +**Proof.** We work in local coordinates. We have + +$$ +\triangle +f=g^{ij}\nabla_i\nabla_j +f=g^{ij}(\partial_i\partial_jf-\Gamma_{ij}^k\partial_kf), +$$ + + so that +from the Ricci flow equation we have + +$$ +\begin{aligned} +\frac{\partial }{\partial t}\triangle f & = & 2\mathit{Ric}^{ij}\mathit{Hess}(f)_{ij}+g^{ij}\frac{\partial}{\partial t}\left(\mathit{Hess}(f)_{ij}\right) +\\ +& = & 2\mathit{Ric}^{ij}\mathit{Hess}(f)_{ij}+g^{ij}\mathit{Hess}\left({\frac{\partial f}{\partial +t}}\right)_{ij}-g^{ij}\frac{\partial \Gamma_{ij}^k}{\partial +t}\partial_kf. +\end{aligned} +$$ + +Since the first term is $2\langle \mathit{Ric},\mathit{Hess}(f)\rangle$ +and the second is $\triangle(\frac{\partial f}{\partial t})$, to +complete the proof of the claim, we must show that the last term of +this equation vanishes. In order to simplify the computations, we +assume that the metric is standard to second order at the point and +time under consideration. Then, using the Ricci flow equation, the +definition of the Christoffel symbols in terms of the metric, and +the fact that $g_{ij}$ is the identity matrix at the given point and +time and that its covariant derivatives in all spatial directions +vanish at this point and time, we get + +$$ +g^{ij}\frac{\partial \Gamma_{ij}^k}{\partial t}=g^{kl}g^{ij}\left(-(\nabla_j \mathit{Ric})_{li}-(\nabla_i + \mathit{Ric})_{lj}+(\nabla_l\mathit{Ric})_{ij}\right). +$$ + + This expression vanishes by the second Bianchi identity (\entryref{c4848a1b6f80}). + This completes the proof of the claim. + +We also have + +$$ +\frac{\partial }{\partial t}(|\nabla f|^2)=2\mathit{Ric}(\nabla f,\nabla f)+2\langle +\nabla\frac{\partial f}{\partial t},\nabla f\rangle. +$$ + + (Here $\nabla +f$ is a one-form, which explains the positive sign in the Ricci +term.) + +Plugging this and \entryref{8283cc66c217} into Equation (9.16) +yields + +$$ +\begin{aligned} +\frac{\partial H}{\partial t} & = & -2\triangle f+|\nabla f|^2-R+\frac{\partial f}{\partial t} \\ +& & + \tau \left(4\langle \mathit{Ric},\mathit{Hess}(f)\rangle +2\triangle +\frac{\partial f}{\partial t} -2\mathit{Ric}(\nabla f,\nabla f) +-2\langle \nabla\frac{\partial f}{\partial t},\nabla +f\rangle+\frac{\partial R}{\partial t}\right). +\end{aligned} +$$ + +Also, + +$$ +\triangle H = \triangle +f+\tau\left(2\triangle^2f-\triangle( |\nabla f|^2)+\triangle +R\right). +$$ + +Since $u$ satisfies the conjugate heat equation, from the first part +of the lemma we have + +$$ +\frac{\partial f}{\partial t}=-\triangle f+|\nabla +f|^2-R+\frac{n}{2\tau}. +$$ + + Putting all this together and +using the Equation (3.5) for $\partial R/\partial t$ yields + +$$ +\begin{aligned} +\frac{\partial H}{\partial t}+\triangle H & = & -\triangle f+|\nabla +f|^2+\frac{\partial f}{\partial t}-R \\ & & +\tau\bigl(4\langle \mathit{Ric},\mathit{Hess}(f)\rangle+2\triangle \frac{\partial f}{\partial t} ++2\triangle^2f-2\mathit{Ric}(\nabla f,\nabla f)\bigr. \\ +& & \bigl.-\triangle(|\nabla f|^2) -2\langle \nabla\frac{\partial +f}{\partial t},\nabla f\rangle+2\triangle +R +2|\mathit{Ric}|^2\bigr) \\ +& = & -\triangle f+|\nabla f|^2+\frac{\partial f}{\partial t}-R ++\tau\bigl(4\langle \mathit{Ric},\mathit{Hess}(f)\rangle+2\triangle(|\nabla f|^2-R)\bigr. +\\ +& & \bigl.-2\mathit{Ric}(\nabla f,\nabla f)-\triangle(|\nabla +f|^2)-2\langle\nabla\frac{\partial f}{\partial t},\nabla +f\rangle+2\triangle R+2|\mathit{Ric}|^2\bigr) +\\ +& = & -\triangle f+|\nabla f|^2+\frac{\partial f}{\partial +t}-R+\tau\Bigl[4\langle \mathit{Ric},\mathit{Hess}(f)\rangle+\triangle(|\nabla f|^2)\Bigr. +\\ +& & -2\mathit{Ric}(\nabla f,\nabla f) +2\langle\nabla(\triangle +f),\nabla f\rangle -2\langle \nabla(|\nabla f|^2),\nabla +f\rangle \\ +& & \Bigl. +2\langle\nabla R,\nabla f\rangle+2|\mathit{Ric}|^2\Bigr]. +\end{aligned} +$$ + +Similarly, we have + +$$ +\begin{aligned} +\frac{2\langle\nabla u,\nabla H\rangle}{u} & = & -2\langle\nabla f,\nabla H\rangle \\ +& = & -2|\nabla f|^2-2\tau\langle\nabla f,\left(\nabla(2\triangle +f)-|\nabla f|^2+R\right)\rangle \\ +& = & -2|\nabla f|^2-\tau\left(4\langle\nabla f,\nabla(\triangle +f)\rangle-2\langle\nabla f,\nabla(|\nabla f|^2)\rangle+\langle\nabla +f,\nabla R\rangle\right). +\end{aligned} +$$ + +Thus, + +$$ +\begin{aligned} +\frac{\partial H}{\partial t} +\triangle H+\frac{2\langle\nabla +u,\nabla H\rangle}{u} & = & -\triangle f-|\nabla f|^2+\frac{\partial +f}{\partial t}-R + \tau\Bigl[4\langle \mathit{Ric},\mathit{Hess}(f)\rangle +\Bigr. +\\ +& & +\triangle(|\nabla f|^2)-2\mathit{Ric}(\nabla f,\nabla f)+2|\mathit{Ric}|^2 +\\ +& & \Bigl.-2\langle\nabla f,\nabla(\triangle f)\rangle\Bigr]. +\end{aligned} +$$ + +\entryblock{e8f087a9cbc6} + +**Proof.** We have + +$$ +\triangle(|\nabla f|^2)=\triangle \langle \nabla f,\nabla f\rangle =\triangle \langle df,df\rangle=2\langle \triangle +df,df\rangle+2\langle \nabla df,\nabla df\rangle. +$$ + + The last term is +$|\mathit{Hess}(f)|^2$. According to \entryref{8d03f3ca0aea} we have +$\triangle df=d(\triangle f)+\mathit{Ric}(\nabla f,\cdot)$. Plugging +this in gives + +$$ +\triangle(|\nabla f|^2)=2\langle d(\triangle f),df\rangle+2\langle +\mathit{Ric}(\nabla f,\cdot),df\rangle+2|\mathit{Hess}(f)|^2, +$$ + + which is +clearly another way of writing the claimed result. + +Using this we can simplify the above to + +$$ +\begin{aligned} +\frac{\partial H}{\partial t} +\triangle H+\frac{2\langle\nabla +u,\nabla H\rangle}{u} & = &-\triangle f-|\nabla f|^2+\frac{\partial +f}{\partial t}-R +\\ +& & +\tau\left(4\langle \mathit{Ric},\mathit{Hess}(f)\rangle +2|\mathit{Hess}(f)|^2+2|\mathit{Ric}|^2\right). +\end{aligned} +$$ + +Now using Equation (9.17) we have + +$$ +\begin{aligned} +\frac{\partial H}{\partial t} +\triangle H+\frac{2\langle\nabla +u,\nabla H\rangle}{u} & = & +-2\triangle f-2R+\frac{n}{2\tau} \\ +& & +\tau\left(4\langle \mathit{Ric},\mathit{Hess}(f)\rangle+2|\mathit{Ric}|^2+2|\mathit{Hess}(f)|^2\right) \\ +& = & 2\tau\bigl(2\langle \mathit{Ric},\mathit{Hess}(f)\rangle+|\mathit{Ric}|^2+|\mathit{Hess}(f)|^2\bigr. \\ +& & -\frac{\triangle f}{\tau}-\frac{R}{\tau}+\frac{n}{4\tau^2}\bigr) \\ +& = & 2\tau\bigl|\mathit{Ric}+\mathit{Hess}(f)-\frac{1}{2\tau}g_\infty\bigr|^2 +\end{aligned} +$$ + +Since + +$$ +-\frac{\partial v}{\partial t}-\triangle v+Rv=-u\left(\frac{\partial H}{\partial t} + +\triangle H+\frac{2\langle\nabla u,\nabla H\rangle}{u}\right), +$$ + + this proves the lemma. + +At this point, setting $f=l_\infty$, we have established all the +results claimed in \entryref{671402eb454b} except for the fact that +the limit is not flat. This we establish in the next chapter. + +### Completion of the proof of Theorem \entryref{671402eb454b} + + To complete the +proof of \entryref{671402eb454b} it remains to show that for no $t\in +(-\infty,0)$ is $(M_\infty,g_\infty(t))$ flat. + +\entryblock{64100c8689ff} + +**Proof.** We know that $f=l_\infty(\cdot,\tau)$ solves the equation given in +\entryref{ba1adbac0a75} and hence by the above argument, $f$ also satisfies +the equation given in \entryref{07232fc42f3a}. If the limit is +flat, then the equation becomes + +$$ +\mathit{Hess}(f)=\frac{1}{2\tau}g. +$$ + + The universal covering of $(M_\infty,g_\infty(t))$ +is isometric to $\Ar^n$. Choose an identification with $\Ar^n$, and lift $f$ to +the universal cover. Call the result $\tilde f$. Then $\tilde f$ satisfies +$\mathit{Hess}(\tilde f)=\frac{1}{2\tau}\tilde g$, where $\tilde g$ is the usual +Euclidean metric on $\Ar^n$. This means that $\tilde f-|x|^2/4\tau$ is an +affine linear function. Clearly, then $\tilde f$ is not invariant under any +free action of a non-trivial group, so that the universal covering in question +is the trivial cover. This completes the proof of the claim. + +If $(M_\infty,g_\infty(t))$ is flat for some $t\in (-\infty,0)$, then by the +above $(M_\infty,g_\infty(t))$ is isometric to $\Ar^n$. According to +\entryref{613fb3994956} this implies that $\widetilde +V_\infty(\tau)=(4\pi)^{n/2}$. This contradicts \entryref{e08895f2b0d8}, and +the contradiction establishes that $(M_\infty,g_\infty(t))$ is not flat for any +$t<0$. Together with \entryref{07232fc42f3a}, this completes the proof of +\entryref{671402eb454b}, namely of the fact that $(M_\infty,g_\infty(t)),\ +-\infty0$ and we classify all +$\kappa$-solutions $(M,g_\infty(t)),\ -\infty>1$. Since $\gamma$ is length-minimizing and parameterized +at unit speed, it follows that it is a local minimum for the energy +functional $E(\gamma)=\frac{1}{2}\int_0^{\bar s}|\gamma'(s)|^2ds$ +among all paths with the same end points. Thus, letting +$\gamma_u(s)=\gamma(s,u)$ be a one-parameter family of variations +(fixed at the endpoints) with $\gamma_0=\gamma$ and with +$d\gamma/du|_{u=0}=Y$, we see + +$$ +0\le \delta_{Y}^2E(\gamma_u)=\int_0^{\bar +s}|\nabla_XY|^2+\langle {\mathcal R}(Y,X)Y,X\rangle ds. +$$ + + We +conclude that + +$$ +\int_0^{\bar s}\langle -{\mathcal R}(Y,X)Y,X\rangle ds\le +\int_0^{\bar s}|\nabla_XY|^2ds. +$$ + + Fix an orthonormal basis $\{E_i\}_{i=1}^n$ at $p$ with $E_n=X$, and let +$\widetilde E_i$ denote the parallel translation of $E_i$ along $\gamma$. (Of +course, $\widetilde E_n=X$.) Then, for $i\le n-1$, we define + +$$ +Y_i=\left\{\begin{array}{ll} s\widetilde E_i & \text{if }0\le s\le +1 \\ \widetilde E_i & \text{if }1\le s\le \bar s-1 \\ +(\bar s-s)E_i & \text{if }\bar s-1\le s\le \bar s\text{.} \end{array} +\right. +$$ + + Adding up Equation (9.22) for each $i$ gives + +$$ +-\sum_{i=1}^{n-1}\int_0^{\bar s}\langle{\mathcal R}(Y_i,X)Y_i,X\rangle ds\le +\sum_{i=1}^{n-1}\int_0^{\bar s}|\nabla_XY_i|^2ds. +$$ + + Of course, since the +$\widetilde E_i$ are parallel along $\gamma$, we have + +$$ +|\nabla_XY_i|^2=\left\{\begin{array}{ll} 1 & \text{if }0\le s\le 1 +\\ 0 & \text{if }1\le s\le \bar s-1 \\ 1 & \text{if }\bar s-1\le s\le +\bar s\end{array}\right. , +$$ + + so that + +$$ +\sum_{i=1}^{n-1}\int_0^{\bar s}|\nabla_XY_i|^2=2(n-1). +$$ + +On the other hand, + +$$ +-\sum_{i=1}^{n-1}\langle{\mathcal R}(Y_i,X)(Y_i),X\rangle =\left\{ +\begin{array}{ll} s^2\mathit{Ric}(X,X) & \text{if }0\le s\le 1 \\ +\mathit{Ric}(X,X) & \text{if }1\le s\le \bar s-1 \\ (\bar s-s)^2\mathit{Ric}(X,X) & \text{if }\bar s-1\le s\le \bar s\text{.} \end{array} \right. +$$ + + Since the curvature is bounded and $|X|=1$, we see that +$\int_0^{\bar s}(1-s^2)\mathit{Ric}(X,X)ds+\int_{\bar s-1}^{\bar +s}(\bar s-s)^2\mathit{Ric}(X,X)$ is bounded independent of $\gamma$ and +of $\bar s$. This concludes the proof of the claim. + +\entryblock{6060c7910950} + +**Proof.** This is obvious if $n=2$, so we may as well assume that $n=3$. We +diagonalize $\mathit{Ric}$ in an orthonormal basis $\{e_i\}$. Let +$\lambda_i\ge 0$ + be the eigenvalues. Write $X=X^ie_i$ with $\sum_i(X^i)^2=1$. Then + +$$ +\mathit{Ric}(X,\cdot)=X^i\lambda_i(e_i)^*, +$$ + + so that +$|\mathit{Ric}(X,\cdot)|^2=\sum_i(X^i)^2\lambda_i^2$. Of course, since +the $\lambda_i\ge 0$, this gives + +$$ +R\cdot \mathit{Ric}(X,X)=(\sum_i\lambda_i)\sum_i\lambda_i(X^i)^2\ge +\sum_i\lambda_i^2(X^i)^2, +$$ + + establishing the claim. + +Now we compute, using Cauchy-Schwarz, + +$$ +\begin{aligned} +\left(\int_0^{\bar s}|\mathit{Ric}(X,\widetilde E_i)|ds\right)^2 & \le +&\bar s\int_0^{\bar s}|\mathit{Ric}(X,\widetilde E_i)|^2ds\le \bar +s\int_0^{\bar s}|\mathit{Ric}(x,\cdot)|^2ds\\ & \le & \bar +s\int_0^{\bar s}R\cdot \mathit{Ric}(X,X)ds. +\end{aligned} +$$ + + Since $R$ is +bounded, it follows from the first claim that there is a constant +$C'$ independent of $\gamma$ and $\bar s$ with + +$$ +\int_0^{\bar s}|\mathit{Ric}(X,\widetilde E_i)|ds\le C'\sqrt{\bar +s}. +$$ + + Since $\gamma$ is a geodesic in the metric $g$, we have +$\nabla_XX=0$. Hence, + +$$ +\frac{d^2f(\gamma(s))}{ds^2}=X(X(f))=\mathit{Hess}(f)(X,X). +$$ + +Applying the gradient shrinking soliton equation to the pair +$(X,X)$ gives + +$$ +\frac{d^2f(\gamma(s))}{ds^2}=\frac{1}{2}-\mathit{Ric}_{g}(X,X). +$$ + +Integrating we see + +$$ +\frac{df(\gamma(s))}{ds}|_{s=\bar +s}=\frac{df(\gamma(s))}{ds}|_{s=0}+\frac{\bar s}{2}-\int_0^{\bar +s}\mathit{Ric}(X,X)ds. +$$ + + It follows that + +$$ +X(f)(\gamma(\bar s))\ge \frac{\bar s}{2}-C'', +$$ + + for some +constant $C''$ depending only on $(M,g)$ and $f$. Similarly, +applying the gradient shrinking soliton equation to the pair +$(X,\widetilde E_i)$, using Equation (9.23) and the fact that +$\nabla_X\widetilde E_i=0$ gives + +$$ +|\widetilde E_i(f)(\gamma(\bar s))|\le C''(\sqrt{\bar +s}+1). +$$ + +These two inequalities imply that for $\bar s$ sufficiently large, +$f$ has no critical points and that $\nabla f$ makes a small angle +with the gradient of the distance function from $p$, and $|\nabla +f|$ goes to infinity as the distance from $p$ increases. In +particular, $f$ is a proper function going off to $+\infty$ as we +approach infinity in $M$. + +Now apply Equation (9.21) to see that $R$ is increasing along the gradient +curves of $f$. Hence, there is a sequence $p_k$ tending to infinity in $M$ with +$\mathit{lim}_kR(p_k)=\mathit{limsup}_{q\in M} R_g(q)>0$. + +The Ricci flow $(M,G(t)),\ -\infty0. +$$ + +(On a manifold with non-negative curvature $\mathit{Ric}(Y,Y)\le R/2$ for any unit +tangent vector $Y$.) This means that for $u$ sufficiently large the level +surfaces of $N_u=f^{-1}(u)$ are convex and hence have increasing area as $u$ +increases. + +According to Equations (9.24) and (9.25) the angle between $\nabla +f$ and the gradient of the distance function from $p$ goes to zero as we go to +infinity. According to \entryref{e9a9172d16e2} the gradient of the distance +function from $p$ converges to the unit vector field in the $\Ar$-direction of +the product structure. It follows that the unit vector in the $\nabla +f$-direction converges to the unit vector in the $\Ar$-direction. Hence, as $u$ +tends to $\infty$ the level surfaces $f^{-1}(u)$ converge in the $C^1$-sense to +$\Sigma\times \{0\}$. Thus, the areas of these level surfaces converge to the +area of $(\Sigma,h(-1))$ which is $8\pi$ since the scalar curvature of this +limiting surface is $\mathit{limsup}_{p\in M} R(p,-1)= 1$. It follows that the +area of $f^{-1}(u)$ is less than $8\pi$ for all $u$ sufficiently large. + +Now let us estimate the intrinsic curvature of $N=N_u=f^{-1}(u)$. +Let $K_N$ denote the sectional curvature of the induced metric on +$N$, whereas $K_M$ is the sectional curvature of $M$. We also denote +by $R_N$ the scalar curvature of the induced metric on $N$. Fix an +orthonormal basis $\{e_1,e_2,e_3\}$ at a point of $N$, where +$e_3=\nabla f/|\nabla f|$. Then by the Gauss-Codazzi formula we have + +$$ +R_N=2K_N(e_1,e_2)=2(K_M(e_1,e_2)+\mathit{det} S) +$$ + +where $S$ is the shape operator + +$$ +S=\frac{\mathit{Hess}(f|TN)}{|\nabla f|}. +$$ + +Clearly, we have $R-2\mathit{Ric}(e_3,e_3)=2K_M(e_1,e_2)$, so that + +$$ +R_N=R-2\mathit{Ric}(e_3,e_3)+2\mathit{det} S. +$$ + +We can assume that the basis is chosen so that $\mathit{Ric}|_{TN}$ is +diagonal; i.e., in the given basis we have + +$$ +\mathit{Ric}=\begin{pmatrix} r_1 & 0 & c_1 \\ 0 & r_2 & c_2 \\ c_1 & c_2 & +r_3\end{pmatrix}. +$$ + + From the gradient shrinking soliton equation we +have $\mathit{Hess}(f)=(1/2)g-\mathit{Ric}$ so that + +$$ +\begin{aligned} +\mathit{det}(\mathit{Hess}(f|TN)) & = & \left(\frac{1}{2}-r_1\right)\left(\frac{1}{2}- r_2\right) \\ +& \le & +\frac{1}{4}(1-r_1-r_2)^2 \\ +& = & \frac{1}{4} (1-R+\mathit{Ric}(e_3,e_3))^2. +\end{aligned} +$$ + +Thus, it follows that + +$$ +R_N\le R-2\mathit{Ric}(e_3,e_3)+\frac{(1-R+\mathit{Ric}(e_3,e_3))^2}{2|\nabla f|^2}. +$$ + + It follows from Equation (9.24) that $|\nabla +f(x)|\rightarrow\infty$ as $x$ goes to infinity in $M$. Thus, since +the curvature of $(M,g(-1))$ is bounded, provided that $u$ is +sufficiently large, we have $1-R+\mathit{Ric}(e_3,e_3)<2|\nabla f|^2$. +Since the left-hand side of this inequality is positive (since $R< +1$), it follows that + +$$ +(1-R+\mathit{Ric}(e_3,e_3))^2 < 2(1-R+\mathit{Ric}(e_3,e_3))|\nabla +f|^2. +$$ + +Plugging this into Equation (9.26) gives that + +$$ +R_N<1-\mathit{Ric}(e_3,e_3)\le 1, +$$ + +assuming that $u$ is sufficiently large. + +This contradicts the Gauss-Bonnet theorem for the surface $N$: Its +area is less than $8\pi$, and the scalar curvature of the induced +metric is less than $1$, meaning that its Gaussian curvature is less +than $1/2$; yet $N$ is diffeomorphic to a $2$-sphere. This completes +the proof of \entryref{03957890f08f}, that is to say this shows +that there are no non-compact positive curved examples satisfying +the hypotheses of \entryref{2d68880771ff}. + +### Case of non-positive curvature revisited + +We return now to the second case of \entryref{2d68880771ff}. We +extend $(M,g)$ to a Ricci flow $(M,G(t))$ defined for $-\infty0$ depending only on +$\kappa$. Since the limit is not flat and has non-negative +curvature, the same is true for $(N,h(t))$. Since $\partial +R/\partial t\ge 0$ for the limit, the same is true for $(N,h(t))$. +That is to say $(N,h(t))$ satisfies all the hypotheses of +\entryref{703d5ca9da05} except possibly non-compactness. It now +follows from \entryref{da767c71996b} that $(N,h(t))$ is a shrinking +family of compact, round surfaces. + +\entryblock{351e4ac14e4e} + +**Proof.** If an asymptotic gradient shrinking soliton $(M,g(t))$ of a three-dimensional +$\kappa$-solution does not have strictly positive curvature, then according to +\entryref{d21a675e4ea8}, $(M,g(t))$ has a covering that splits as a product +of a a two-dimensional Ricci flow and a line. The two-dimensional Ricci flow +satisfies all the hypotheses of \entryref{703d5ca9da05} except possibly +compactness, and hence by \entryref{da767c71996b} it is a shrinking family of +round surfaces. In this case, it is clear that each time-slice of $(M,g(t))$ +has bounded curvature. + +Now we consider the remaining case when $(M,g(t))$ has strictly positive +curvature. Assume that $(M,g(t))$ has unbounded curvature. Then there is a +sequence of points $p_i$ tending to infinity in $M$ such that $R(p_i,t)$ tends +to infinity. By \entryref{f8ff249efb81} we can replace the points $p_i$ by +points $q_i$ with $Q_i=R(q_i,t)\ge R(p_i,t)$ so that the based Riemannian +manifolds $(M,Q_ig(t),q_i)$ converge to a product of a round surface $(N,h(t))$ +with $\Ar$. The surface $N$ is either diffeomorphic to $S^2$ or $\Ar P^2$. +Since $(M,g(t))$ has positive curvature, by \entryref{2b5f18292e62}, it is +diffeomorphic to $\Ar ^3$, and hence it contains no embedded $\Ar P^2$. It +follows that $(N,h(t))$ is a round two-sphere. + +Fix $\epsilon>0$ sufficiently small as in \entryref{cb9ca6d6db9f}. Then the +limiting statement means that, for every $i$ sufficiently large, there is an +$\epsilon$-neck in $(M,g(t))$ centered at $q_i$ with scale $Q_i^{-1/2}$. This +contradicts \entryref{cb9ca6d6db9f}, establishing that for each $t<0$ the +curvature of $(M,g(t))$ is bounded. + +\entryblock{92ee3ee82b58} + +**Proof.** Let $(M_\infty,g_\infty(t)),\ -\infty0$ depending only on $\kappa$. On the other hand, according +to \entryref{671402eb454b} there is a function $f(\cdot, -1)$ from +$M_\infty$ to $\Ar$ satisfying the gradient shrinking soliton +equation at the time-slice $-1$. + Thus, +\entryref{2d68880771ff} applies to $(M_\infty,g_\infty(-1))$ to +produce a Ricci flow $G(t),\ -\infty0$ such +that for all $n$ sufficiently large, the annulus $A_n$ contains +$B_{h_n}(x_n,0,\alpha)$. Consequently, we have a bound, independent +of $n$, for the scalar curvature of $h_n(0)$ on these balls. By the +hypothesis that $\partial R/\partial t\ge 0$, there is a bound, +independent of $n$, for the scalar curvature of $h_n$ on +$B_{h_n}(x_n,0,\alpha)\times (-\infty,0]$. Using the fact that the +flows have non-negative curvature, this means that there is a bound, +independent of $n$, for $|\mathit{Rm}_{h_n}(y,0)|$ on +$B_{h_n}(x_n,0,\alpha)\times (-\infty,0]$. + This means +that by Shi's theorem (\entryref{ed4b72caf9f9}), there are bounds, +independent of $n$, for every covariant derivative of the curvature +on $B_{h_n}(x_n,0,\alpha/2)\times (-\infty,0]$. + +Since the original flow is $\kappa$-non-collapsed on all scales, it +follows that the rescaled flows are also $\kappa$ non-collapsed on +all scales. Since the curvature is bounded, independent of $n$, on +$B_{h_n}(x_n,0,\alpha)$, this implies that there is $\delta>0$, +independent of $n$, such that for all $n$ sufficiently large, every +ball of radius $\delta$ centered at any point of +$B_{h_n}(x_n,0,\alpha/2)$ has volume at least $\kappa \delta^3$, Now +applying \entryref{74c7682a68ed} we see that a subsequence +converges geometrically to a limit which will automatically be a +metric ball $B_{g_\infty}(x_\infty,0,\alpha/2)$. In fact, by +Hamilton's result (\entryref{4673eb9f4424}) there is a +limiting flow on $B_{g_\infty}(x_\infty,0,\alpha/4)\times +(-\infty,0]$. Notice that the limiting flow is not flat since +$R(x_\infty,0)=1$. + +On the other hand, according to \entryref{88a75d6dd09e} the +Gromov-Hausdorff limit of a +subsequence $(M,\lambda_n^{-1}g_n(0),x_n)$ is the Tits +cone, i.e., the cone over $S_\infty(M,p)$. Since +$Q_n={\mathcal R}(M,g(t))\lambda_n^{-1}$, the rescalings +$(M,Q_ng_n(0),x_n)$ also converge to a cone, say $(C,h,y_\infty)$, +which is in fact simply a rescaling of the Tits cone by a factor +${\mathcal R}(M,g(t))$. Pass to a subsequence so that both the +geometric limit on the ball of radius $\alpha/2$ and the +Gromov-Hausdorff limit exist. Then the geometric limit +$B_{g_\infty}(x_\infty,0,\alpha/2)$ is isometric to an open ball in +the cone. Since we have a limiting Ricci flow + +$$ +(B_{g_\infty}(x_\infty,0,\alpha/2),g_\infty(t)),\ -\infty0$ the volume of the ball of radius $r$ centered at the cone point + is exactly $\omega_3 r^3$, where +$\omega_3$ is the volume of the unit ball in $\Ar^3$. Consequently, + +$$ +\mathit{lim}_{n\rightarrow\infty}\mathit{Vol}\left(B_{g}(p,0,\sqrt{\lambda_n}r)\setminus +B_g(p,0,1)\right)\rightarrow \omega_3\lambda_n^{3/2}r^3. +$$ + + By +\entryref{1a1b2c4fdc9b} and the fact that the Ricci curvature is +non-negative, this implies that + +$$ +\mathit{Vol} B_g(p,0,R)=\omega_3R^3 +$$ + +for all $R<\infty$. Since the Ricci curvature is non-negative, this +means that $(M,g(t))$ is Ricci-flat, and hence flat. But this +contradicts the fact that $(M,g(t))$ is a $\kappa$-solution and +hence is not flat. + +Having ruled out the other two cases, we are left with only one possibility: +${\mathcal R}(M,g(t))=\infty$. + +## Universal $\kappa$ + +The first consequence of the existence of an asymptotic gradient +shrinking soliton is that there is a universal $\kappa$ for all +$3$-dimensional $\kappa$-solutions, except those of constant +positive curvature. + +\entryblock{9c54e0d0c7d8} + +**Proof.** Let $(M,g(t))$ be a non-round $3$-dimensional $\kappa$-solution. By +\entryref{01028c93347d} since $(M,g(t))$ is not a family of +round manifolds, the asymptotic soliton for the $\kappa$-solution +cannot be compact. Thus, according to \entryref{2d68880771ff} there +are only two possibilities for the asymptotic soliton +$(M_\infty,g_\infty(t))$ -- either $(M_\infty,g_\infty(t))$ is the +product of a round $2$-sphere of Gaussian curvature $1/2|t|$ with a +line or has a two-sheeted covering by such a product. In fact, there +are three possibilities: $S^2\times \Ar$, $\Ar P^2\times \Ar$ or the +twisted $\Ar$-bundle over $\Ar P^2$ whose total space is +diffeomorphic to the complement of a point in $\Ar P^3$. + +Fix a point $x=(p,0)\in M\times\{0\}$. Let $\bar\tau_k$ be a +sequence converging to $\infty$, and $q_k\in M$ a point with +$l_{x}(q_k,\bar\tau_k)\le 3/2$. The existence of an asymptotic +soliton means that, possibly after passing to a subsequence, there +is a gradient shrinking soliton $(M_\infty,g_\infty(t))$ and a ball +$B$ of radius $1$ in $(M_\infty,g_\infty(-1))$ centered at a point +$q_\infty\in M_\infty$ and a sequence of embeddings $\psi_k\colon +B\to M$ such that $\psi_k(q_\infty)=q_k$ and such that the map + +$$ +B\times [-2,-1]\to M\times [-2\bar\tau_k,-\bar\tau_k] +$$ + +given by $(b,t)\mapsto (\psi_k(b),\bar\tau_kt)$ has the property +that the pullback of $\bar\tau^{-1}_kg(\bar\tau_k t)$ converges +smoothly and uniformly as $k\rightarrow\infty$ to the restriction of +$g_\infty(t),\ -2\le t\le -1$, to $B$. Let $(M_k,g_k(t))$ be this +rescaling of the the $\kappa$-solution by $\tau_k$. Then the +embeddings $\psi_k\times \mathit{id}\colon B\times (-2,-1]\to +(M_k\times [-2,-1]$ converge as $k\rightarrow \infty$ to a +one-parameter family of isometries. That is to say, the image +$\psi_k(B\times [-2,-1])\subset M_k\times [-2,-1]$ is an almost +isometric embedding. Since the reduced length function from $x$ to +$(\psi_k(a),-1)$ is at most $3/2$ (from the invariance of reduced +length under rescalings, see \entryref{541f71387f84}), it follows easily +that the reduced length function on $\psi_k(B\times \{-2\})$ is +bounded independent of $k$. Similarly, the volume of +$\psi_k(B\times\{-2\})$ is bounded independent of $k$. This means +the reduced volume of $\psi_k(B\times\{-2\})$ in $(M_k,g_k(t))$ is +bounded independent of $k$. Now by \entryref{3defa9b2209c} this implies +that $(M_k,g_k(t))$ is $\kappa_0$-non-collapsed at $(p,0)$ on scales +$\le \sqrt{2}$ for some $\kappa_0$ depending only on the geometry of +the three possibilities for $(M_\infty,g_\infty(t)),\ -2\le t\le +-1$. Being $\kappa_0$-non-collapsed is invariant under rescalings, +so that it follows immediately that $(M,g(t))$ is +$\kappa_0$-non-collapsed on scales $\le \sqrt{2\bar\tau_k}$. Since +this is true for all $k$, it follows that $(M,g(t))$ is +$\kappa_0$-non-collapsed on all scales at $(p,0)$. + +This result holds of course for every $p\in M$, showing that at +$t=0$ the flow is $\kappa_0$-non-collapsed. To prove this result at +points of the form $(p,t)\in M\times(-\infty,0]$ we simply shift the +original $\kappa$-solution upward by $|t|$ and remove the part of +the flow at positive time. This produces a new $\kappa$-solution and +the point in question has been shifted to the time-zero slice, so +that we can apply the previous results. + +## Asymptotic volume + +Let $(M,g(t))$ be an $n$-dimensional $\kappa$-solution. For any +$t\le 0$ and any point $p\in M$ we consider $(\mathit{Vol} B_{g(t)}(p,r))/r^n$. According to the Bishop-Gromov Theorem +(\entryref{1a1b2c4fdc9b}), this is a non-increasing function of +$r$. We define the *asymptotic volume* ${\mathcal V}_\infty(M,g(t))$, or ${\mathcal +V}_\infty(t)$ if the flow is clear from the context, to be the limit +as $r\rightarrow\infty$ of this function. Clearly, this limit is +independent of $p\in M$. + +\entryblock{7468282bd017} + +**Proof.** The proof is by induction on the dimension $n$ of the solution. For $n=2$ by +\entryref{da767c71996b} there are only compact $\kappa$-solutions, which clearly +have zero asymptotic volume. Suppose that we have established the result for +$n-1\ge 2$ and let us prove it for $n$. + +According to \entryref{703d5ca9da05} there is a sequence of points $p_n\in +M$ tending to infinity such that setting $Q_n=R(p_n,0)$ the sequence of Ricci +flows + +$$ +(M,Q_ng(Q_n^{-1}t),(q_n,0)) +$$ + + converges geometrically to a limit +$(M_\infty,g_\infty(t),(q_\infty,0))$, and this limit splits off a +line: $(M_\infty,g_\infty(t))= (N,h(t))\times \Ar$. Since the ball +of radius $R$ about a point $(x,t)\in N\times \Ar$ is contained in +the product of the ball of radius $R$ about $x$ in $N$ and an +interval of length $2R$, it follows that $(N,h(t))$ is a +$\kappa/2$-ancient solution. Hence, by induction, for every $t$, the +asymptotic volume of $(N,h(t))$ is zero, and hence so is that of +$(M,g(t))$. + +### Volume comparison + +One important consequence of the asymptotic volume result is a +volume comparison result. + +\entryblock{5cfd7c3aee29} + +**Proof.** Suppose that the result fails for some $\nu>0$. Then there is a +sequence $(M_k,g_k(t)),\ -t_k\le t\le 0$, of $n$-dimensional Ricci +flows, points $p_k\in M_k$, and radii $r_k$ as in the statement of +the lemma such that for every $A<\infty$ there is an arbitrarily +large $k$ with $\mathit{Vol} B(p_k,0,A/\sqrt{Q_k})\ge +\nu(A/\sqrt{Q_k})^n$. Pass to a subsequence so that for each +$A<\infty$ we have + +$$ +\mathit{Vol} B(p_k,0,A/\sqrt{Q_k})\ge\nu(A/\sqrt{Q_k})^n +$$ + + for all $k$ +sufficiently large. Consider now the flows +$h_k(t)=Q_kg_k(Q_k^{-1}t)$, defined for $-Q_kt_k\le t\le 0$. Then +for every $A<\infty$ for all $k$ sufficiently large we have +$R_{h_k}(q,t)\le 4$ for all $q\in B_{h_k}(p_k,0,A)$ and all $t\in +(-t_kQ_k,0]$. Also, for every $A<\infty$ for all $k$ sufficiently +large we have $\mathit{Vol} B(p_k,0,A)\ge\nu A^n$. According to +\entryref{a0212d0eefcf} we can then pass to a subsequence that has a +geometric limit which is an ancient flow of complete Riemannian +manifolds. Clearly, the time-slices of the limit have non-negative +curvature operator, and the scalar curvature is bounded (by $4$) and +is equal to $1$ at the base point of the limit. Also, the asymptotic +volume ${\mathcal V}(0)\ge \nu$. + +\entryblock{d2259da4d67d} + +**Proof.** We begin with the proof of the first item. Fix $a0$ we have + +$$ +B(p_0,b,r)\subset B(p_0,a,r+A(b-a)). +$$ + +On the other hand, since $d\mathit{Vol}/dt=-Rd\mathit{Vol}$, it follows +that in the case of non-negative curvature that the volume of any +open set is non-increasing in time. Consequently, + +$$ +\mathit{Vol}_{g(b)}B(p_0,b,r)\le \mathit{Vol}_{g(a)}B(p_0,a,r+A(b-a)), +$$ + + and hence + +$$ +\frac{\mathit{Vol}_{g(b)}B(p_0,b,r)}{r^n}\le\frac{\mathit{Vol}_{g(a)}B(p_0,a,r+A(b-a))}{(r+A(b-a))^n}\frac{(r+A(b-a))^n}{r^n}. +$$ + +Taking the limit as $r\rightarrow \infty$ gives + +$$ +{\mathcal V}(b)\le {\mathcal V}(a). +$$ + +The second item of the claim is immediate from the Bishop-Gromov +inequality (\entryref{1a1b2c4fdc9b}). + +Now we return to the proof of the proposition. Under the assumption +that there is a counterexample to the proposition for some $\nu>0$, +we have constructed a limit that is an ancient Ricci flow with +bounded, non-negative curvature with ${\mathcal V}(0)\ge \nu$. Since +${\mathcal V}(0)\ge \nu$, it follows from the claim that ${\mathcal +V}(t)\ge \nu$ for all $t\le 0$ and hence, also by the claim, we see +that $(M,g(t))$ is $\nu$-non-collapsed for all $t$. This completes +the proof that the limit is a $\nu$-solution. This contradicts +\entryref{7468282bd017} applied with $\kappa=\nu$, and proves the +proposition. + +This proposition has two useful corollaries about balls in +$\kappa$-solutions with volumes bounded away from zero. The first +says that the normalized curvature is bounded on such balls. + +\entryblock{3227c39e1892} + +**Proof.** Suppose that the result fails for some $\nu>0$. Then there is a +sequence $(M_k,g_k(t))$ of $n$-dimensional Ricci flows, complete, +with bounded non-negative curvature operator and points $p_k\in +M_k$, constants $r_k>0$, and points $q_k\in B(p_k,0,r_k)$ such that: + +- **(1)** $\mathit{Vol} B(p_k,0,r_k)\ge \nu r_k^n$, and +- **(2)** setting $Q_k=R(q_k,0)$ we have $r_k^2Q_k\rightarrow\infty$ as $k\rightarrow\infty$. + +Using \entryref{351728d82639} we can find points $q_k'\in B(p_k,0,2r_k)$ +and constants $s_k\le r_k$, such that setting $Q_k'=R(q_k',0)$ we +have $Q_k's_k^2=Q_kr_k^2$ and $R(q,0)<4Q_k'$ for all $q\in +B(q_k',0,s_k)$. Of course, $Q'_ks_k^2\rightarrow\infty$ as +$k\rightarrow\infty$. Since $d_0(p_k,q'_k)<2r_k$, we have +$B(p_k,0,r_k)\subset B(q_k',0,3r_k)$ so that + +$$ +\mathit{Vol} B(q_k',0,3r_k)\ge \mathit{Vol} B(p_k,0,r_k)\ge \nu +r_k^n=(\nu/3^n)(3r_k)^n. +$$ + + Since the sectional curvatures of +$(M,g_k(0))$ are non-negative, it follows from the Bishop-Gromov +inequality (\entryref{1a1b2c4fdc9b}) that $\mathit{Vol} B(q_k',0,s)\ge (\nu/3^n)s^n$ for any $s\le s_k$. + +Of course, by \entryref{36dd2f6fa9c2}, we have $R(q,t)<4Q_k'$ for +all $t\le 0$ and all $q\in B(q'_k,0,s_k)$. Now consider the sequence +of based, rescaled flows + +$$ +(M_k,Q'_kg(Q_k'^{-1}t),(q_k',0)). +$$ + + In +these manifolds all balls centered at $(q_k',0)$ of radii at most +$\sqrt{Q_k}s_k$ are + $(\nu/3^n)$ non-collapsed. Also, the curvatures of these manifolds + are non-negative and the scalar curvature is bounded by $4$. + It follows that by passing to a subsequence we can extract a +geometric limit. + Since $Q_k's_k^2\rightarrow\infty$ +as $k\rightarrow\infty$ the asymptotic volume of this limit is at +least $\nu/3^n$. But this geometric limit is a +$\nu/3^n$-non-collapsed ancient solution with non-negative curvature +operator with scalar curvature bounded by $4$. This contradicts +\entryref{7468282bd017}. + +The second corollary gives curvature bounds at all points in terms +of the distance to the center of the ball. + +\entryblock{329d0175aed8} + +**Proof.** Fix $q\in M$ and let $d=d_0(p,q)$. We have + +$$ +\mathit{Vol} B(q,0,r+d)\ge \mathit{Vol} B(p,0,r)\ge \nu r^n=\frac{\nu}{(1+(d/r))^n}(r+d)^n. +$$ + + Let +$K(A)=C(\nu/^n)$, where $C$ is the constant provided by the previous +corollary. The result is immediate from the previous corollary. + +## Compactness of the space of $3$-dimensional $\kappa$-solutions + +This section is devoted to proving the following result. + +\entryblock{49e865228b19} + +The main point in proving this theorem is to establish the uniform +curvature bounds given in the next lemma. + +\entryblock{ee314aacd207} + +**Proof.** Fix a based $3$-dimensional $\kappa$-solution $(M,g(t),(p,0))$. By +\entryref{5d9f1b75170d} we have + +$$ +\mathit{sup}_{q\in M}d_{0}(p,q)^2R(q,0)=\infty. +$$ + +Let $q$ be a closest point to $p$ satisfying + +$$ +d_0(p,q)^2R(q,0)=1. +$$ + +We set $d=d_{0}(p,q)$, and we set $Q=R(q,0)$. Of course, $d^2Q=1$. +We carry this notation and these assumptions through the next five +claims. The goal of these claims is to show that $R(q',0)$ is +uniformly bounded for $q'$ near $(p,0)$ so that in fact the distance +$d$ from the point $q$ to $p$ is uniformly bounded from below by a +positive constant (see \entryref{7c6182738b78} for a more precise +statement). Once we have this the lemma will follow easily. To +establish this uniform bound requires a sequence of claims. + +\entryblock{ea9b285e40cb} + +**Proof.** Suppose not. Then there is a sequence $(M_k,g_k(t),(p_k,0))$ of +$3$-dimensional $\kappa$-solutions with $R(p_k,0)=1$, points $q_k$ +in $(M_k,g_k(0))$ closest to $p_k$ satisfying $d_k^2R(q_k,0)=1$, +where $d_k=d_0(p_k,q_k)$, and points $q_k'\in B(q_k,0,2d_k)$ with + +$$ +\mathit{lim}_{k\rightarrow\infty}(2d_k)^2R(q_k',0)=\infty. +$$ + + Then according +to \entryref{3227c39e1892} for every $\nu>0$ for all $k$ +sufficiently large, we have + +$$ +\mathit{Vol} B(q_k,0,2d_k)<\nu (2d_k)^3. +$$ + + Therefore, by +passing to a subsequence, we can assume that for each $\nu>0$ + +$$ +\mathit{Vol} B(q_k,0,2d_k)< \nu (2d_k)^3 +$$ + + for all $k$ +sufficiently large. Let $\omega_3$ be the volume of the unit ball in +$\Ar^3$. Then for all $k$ sufficiently large, $\mathit{Vol} B(q_k,0,2d_k)<[\omega_3/2](2d_k)^3$. Since the sectional +curvatures of $(M_k,g_k(0))$ are non-negative, by the Bishop-Gromov +inequality (\entryref{1a1b2c4fdc9b}), it follows that for every +$k$ sufficiently large there is $r_k<2d_k$ such that + +$$ +\mathit{Vol} B(q_k,0,r_k)=[\omega_3/2]r_k^3. +$$ + + Of course, +because of Equation (9.28) we see that $\mathit{lim}_{k\rightarrow\infty}r_k/d_k=0$. Then, according to +\entryref{329d0175aed8}, we have for all $q\in M_k$ + +$$ +(r_k+d_{g_k(0)}(q_k,q))^2R(q,0)\le K(d_{g_k(0)}(q_k,q)/r_k), +$$ + +where $K$ is as given in \entryref{329d0175aed8}. Form the sequence +$(M_k,g'_k(t),(q_k,0))$, where $g'_k(t)=r_k^{-2}g_k(r_k^2t)$. This +is a sequence of based Ricci flows. For each $A<\infty$ we have + +$$ +(1+A)^2R_{g'_k}(q,0)\le K(A) +$$ + + for all +$q\in B_{g_k'(0)}(q_k,0,A)$. Hence, by the consequence of Hamilton's +Harnack inequality (\entryref{36dd2f6fa9c2}) + +$$ +R_{g'_k}(q,t)\le K(A), +$$ + +for all $(q,t)\in B_{g_k'(0)}(q_k,0,A)\times(-\infty,0]$. Using +this and the fact that all the flows are $\kappa$-non-collapsed, +\entryref{a0212d0eefcf} implies that, after passing to a +subsequence, the sequence $(M_k,g_k'(t),(q_k,0))$ converges +geometrically to a limiting Ricci flow +$(M_\infty,g_\infty(t),(q_\infty,0))$ consisting of non-negatively +curved, complete manifolds $\kappa$-non-collapsed on all scales +(though possibly with unbounded curvature). + +Furthermore, Equation (9.29) passes to the limit to give + +$$ +\mathit{Vol} B_{g_\infty}(q_\infty,0,1)=\omega_3/2. +$$ + + Since $r_k/d_k\rightarrow 0$ as $k\rightarrow\infty$ and since $R_{g_k}(q_k,0)=d_k^{-2}$, we see that +$R_{g_\infty}(q_\infty,0)=0$. By the strong maximum principle for +scalar curvature (\entryref{016cf36686b2}), this implies that +the limit $(M_\infty,g_\infty(0))$ is flat. But +Equation (9.30) tells us that this limit is not $\Ar^3$. Since +it is complete and flat, it must be a quotient of $\Ar^3$ by an +action of a non-trivial group of isometries acting freely and +properly discontinuously. But the quotient of $\Ar^3$ by any +non-trivial group of isometries acting freely and properly +discontinuously has zero asymptotic volume. [Proof: It suffices to +prove the claim in the special case when the group is infinite +cyclic. The generator of this group has an axis $\alpha$ on which it +acts by translation and on the orthogonal subspace its acts by an +isometry. Consider the circle in the quotient that is the image of +$\alpha$, and let $L_\alpha$ be its length. The volume of the ball +of radius $r$ about $L_\alpha$ is $\pi r^{2}L_\alpha$. Clearly then, +for any $p\in \alpha$, the volume of the ball of radius $r$ about +$p$ is at most $\pi L_\alpha r^2$. This proves that the asymptotic +volume of the quotient is zero.] + +We have now shown that $(M_\infty,g_\infty(0))$ has zero curvature and zero +asymptotic volume. But this implies that it is not $\kappa$-non-collapsed on +all scales, which is a contradiction. This contradiction completes the proof of +\entryref{ea9b285e40cb}. + +This claim establishes the existence of a universal constant $C<\infty$ +(universal in the sense that it is independent of the $3$-dimensional +$\kappa$-solution) such that $R(q',0)\le CQ$ for all $q'\in B(q,0,2d)$. Since +the curvature of $(M,g(t))$ is non-negative and bounded, we know from the +Harnack inequality (\entryref{36dd2f6fa9c2}) that $R(q',t)\le CQ$ for all +$q'\in B(q,0,2d)$ and all $t\le 0$. Hence, the Ricci curvature $\mathit{Ric}(q',t)\le CQ$ for all $q'\in B(q,0,2d)$ and all $t\le 0$. + +\entryblock{b8f989614649} + +**Proof.** Let $\gamma\colon[0,d]\to M$ be a $g(0)$-geodesic from $p$ to $q$, +parameterized at unit speed. Denote by $\ell_t(\gamma)$ the length +of $\gamma$ under the metric $g(t)$. We have $d_t(p,q)\le +\ell_t(\gamma)$. We estimate $\ell_t(\gamma)$ using the fact that +$|\mathit{Ric}|\le CQ$ on the image of $\gamma$ at all times. + +$$ +\begin{aligned} +\frac{d}{dt}\ell_{t_0}(\gamma) & = & +\frac{d}{dt}\left(\int_0^d\sqrt{\langle\gamma'(s),\gamma'(s)\rangle_{g(t)}}ds\right) +\Bigl|_{t=t_0}\Bigr. +\\ +& = & \int_0^{d}\frac{-\mathit{Ric}_{g(t_0)}(\gamma'(s),\gamma'(s))} +{\sqrt{\langle\gamma'(s),\gamma'(s)\rangle_{g(t_0)}}}ds \\ +& \ge & -CQ\int_0^d|\gamma'(s)|_{g(t_0)}ds=-CQ\ell_{t_0}(\gamma). +\end{aligned} +$$ + +Integrating yields + +$$ +\ell_{-t}(\gamma)\le e^{CQt}\ell_0(\gamma)=e^{CQt}Q^{-1/2}. +$$ + +(Recall $d^2Q=1$.) Plugging in $t=cQ^{-1}$ gives us + +$$ +d_{-cQ^{-1}}(p,q)\le \ell_{-cQ^{-1}}(\gamma)\le e^{cC}Q^{-1/2}. +$$ + +Setting $\widetilde C=e^{cC}$ completes the proof of the claim. + +The integrated form of Hamilton's Harnack inequality, +\entryref{6b21448ff214}, tells us that + +$$ +\mathit{log}\left(\frac{R(p,0)}{R(q,-cQ^{-1})}\right)\ge +-\frac{d_{-cQ^{-1}}^2(p,q)}{2cQ^{-1}}. +$$ + + According to the above +claim, this in turn tells us + +$$ +\mathit{log}\left(\frac{R(p,0)}{R(q,-cQ^{-1})}\right)\ge -\widetilde +C^2/2c. +$$ + + Since $R(p,0)=1$, it immediately follows that +$R(q,-cQ^{-1})\le \mathit{exp}(\widetilde C^2/(2c))$. + +\entryblock{f61f72020102} + +**Proof.** Let $G'=QG$ and $t'=Qt$. Then $R_{G'}(q',0)\le C$ for all $q'\in +B_{G'}(q,0,2)$. Consequently, $R_{G'}(q',t')\le C$ for all $q'\in +B_{G'}(q,0,2)$ and all $t'\le 0$. Thus, by Shi's derivative +estimates (\entryref{ed4b72caf9f9})applied with $T=2$ and $r=2$, there is +a universal constant $C_1$ such that for all $-1\le t'\le 0$ + +$$ +|\triangle \mathit{Rm}_{G'}(q,t')|_{G'}\le C_1, +$$ + +(where the Laplacian is taken with respect to the metric $G'$). +Rescaling by $Q^{-1}$ we see that for all $-Q^{-1}\le t\le 0$ we +have + +$$ +|\triangle \mathit{Rm}_{G}(q,t)|\le C_1Q^2, +$$ + +where the Laplacian is taken with respect to the metric $G$. Since +the metric is non-negatively curved, by \entryref{36dd2f6fa9c2} we +have $2|\mathit{Ric}(q,t)|^2\le 2Q^2$ for all $t\le 0$. + From these two facts we conclude from +the flow equation (3.5) that there is a constant $10$ as required is +clear. Since $B(p,0,d)\subset B(q,0,2d)$, since $R(q',0)/R(q,0)$ is +universally bounded on $B(q,0,2d)$ by \entryref{ea9b285e40cb}, and since +$R(q,0)$ is universally bounded by \entryref{f61f72020102}, the second +statement is clear for all $(q',0)\in B(p,0,d)\subset B(q,0,2d)$. +Given this, the fact that the second statement holds for all +$(q',t)\in B(p,0,d)\times(-\infty,0]$ then follows immediately from +the derivative inequality for $\partial R(q,t)/\partial t$, +\entryref{36dd2f6fa9c2}. + +This, in turn, leads immediately to: + +\entryblock{4bff7f1e7bc4} + +Now we return to the proof of \entryref{ee314aacd207}. Since +$(M,g(t))$ is $\kappa$-non-collapsed, it follows from the previous +corollary that $\mathit{Vol} B(p,0,\delta)\ge \kappa \delta^3$. Hence, +according to \entryref{329d0175aed8} for each $A<\infty$ there is a +constant $K(A)$ such that $R(q',0)\le K(A/\delta)/(\delta+A)^2$ for +all $q'\in B(p,0,A)$. Since $\delta$ is a universal positive +constant, this completes the proof of \entryref{ee314aacd207}. + +Now let us turn to the proof of \entryref{49e865228b19}, the +compactness result for $\kappa$-solutions. + +**Proof.** Let +$(M_k,g_k(t),(p_k,0))$ be a sequence of based $3$-dimensional +$\kappa$-solutions with $R(p_k,0)=1$ for all $k$. + The immediate +consequence of \entryref{ee314aacd207} and \entryref{36dd2f6fa9c2} +is the following. For every $r<\infty$ there is a constant +$C(r)<\infty$ such that $R(q,t)\le C(r)$ for all $q\in B(p_k,0,r)$ +and for all $t\le 0$. Of course, since, in addition, the elements in +the sequence are $\kappa$-non-collapsed, by \entryref{a0212d0eefcf} +this implies that there is a subsequence of the +$(M_k,g_k(t),(p_k,0))$ that converges geometrically to an ancient +flow $(M_\infty,g_\infty(t),(p_\infty,0))$. Being a geometric limit +of $\kappa$-solutions, this limit is complete and +$\kappa$-non-collapsed, and each time-slice is of non-negative +curvature. Also, it is not flat since, by construction, +$R(p_\infty,0)=1$. Of course, it also follows from the limiting +procedure that $\partial R(q,t)/\partial t\ge 0$ for every $(q,t)\in +M_\infty\times(-\infty,0]$. Thus, according to +\entryref{351e4ac14e4e} the limit + $(M_\infty,g_\infty(t))$ has bounded curvature for each $t\le 0$. +Hence, the limit is a $\kappa$-solution. This completes the proof of +\entryref{49e865228b19}. + +\entryblock{8c090fbd21a0} + +**Proof.** Notice that the two inequalities are scale invariant. Thus, this +result is immediate from the compactness theorem, +\entryref{49e865228b19}. + +Because of \entryref{9c54e0d0c7d8}, and the fact that the previous +corollary obviously holds for any shrinking family of round metrics, +we can take the constant $C$ in the above corollary to be +independent of $\kappa>0$. + +Notice that, using Equation (3.5), we can rewrite the second +inequality in the above corollary as + +$$ +\begin{aligned} +\mathit{sup}_{(x,t)}\frac{|\triangle R+2|\mathit{Ric}|^2|}{R(x,t)^2}4$, if these inequalities hold for +all horizontal metrics $G'$ sufficiently close to $G$ in the +$C^{[1/\epsilon]}$-topology. This completes the examination of all +cases and proves the first statement. + +The second statement is proved in the same way using the fact that +if $g'(t)$ is sufficiently close to $g$ in the +$C^{[1/\epsilon]}$-topology and if $x'$ is sufficiently close to $x$ +then $R_{g'(x')}^{-1}0$ sufficiently +small, any metric $g'$ within $\delta$ of $g$ will satisfy the +diameter, volume and curvature and the derivative of the curvature +inequalities with $C'$ replacing $C$. Let $N$ be the $\epsilon$-neck +in $(U,g)$ containing the end of $U$. Assuming that $\delta$ is +sufficiently small, let $N'$ be the image of $S^2\times +\left(-\epsilon^{-1},2(\epsilon')^{-1}-\epsilon^{-1}\right)$. Then +$(N',g')$ becomes an $\epsilon'$-neck structure once we shift the +parameter in the $s$-direction by $\epsilon^{-1}-(\epsilon')^{-1}$. +We let $U'=\overline Y\cup N'$. Clearly, the $\epsilon$-neck with +central $2$-sphere $\partial \overline Y$ will also determine an +$\epsilon'$-neck with the same central $2$-sphere provided that +$\delta>0$ is sufficiently small. Thus, for $\delta>0$ sufficiently +small, for any $(C,\epsilon)$ the result of this operation is a +$(C',\epsilon')$-cap with the same core. + +The fourth statement is immediate. + +\entryblock{75b4c67c9141} + +**Proof.** Let $T$ be the final time of the flow. Suppose that $(x,t)$ is the +center of a strong $\epsilon$-neck $U\times (t-R(x,t)^{-1},t]\subset +M\times (-\infty,0]$. This neck extends backwards for all time and +forwards until the final time $T$ giving an embedding $U\times +(-\infty,T]\to M\times (-\infty,T]$. There is $a>1$ such that for +all $t'$ sufficiently close to $t$ the restriction of this embedding +determines an evolving $\epsilon$-neck centered at $(x,t')$ defined +for an interval of normalized time of length $a$. Composing this +neck structure with a self-diffeomorphism of $U$ moving $x'$ to $x$, +as described above, shows that all $(x',t')$ sufficiently close to +$(x,t)$ are centers of strong $\epsilon$-necks. + +\entryblock{0425e2f3cfe1} + +See Fig. 9.2. + +\entryblock{1138b028cc69} + +### Canonical neighborhoods for $\kappa$-solutions + +\entryblock{25e0946e7222} + +**Proof.** Suppose that $(M,g(t))$ does not have positive curvature for some +$t$. Then, by the application of the strong maximum principle given +in \entryref{d21a675e4ea8}, there is a covering $\widetilde M$ of +$M$, with either one or $2$-sheets, such that $(\widetilde M,\tilde +g(t))$ is the product of an evolving family of round surfaces with a +flat one-manifold (either a circle or the real line). Of course, +the covering must be $\kappa$-solution. In the case in which +$(\widetilde M,\tilde g(t))$ is isometric to the product of an +evolving family of round surfaces and a circle, that circle has a +fixed length, say $L<\infty$. Since the curvature of the surface in +the $t$ time-slice goes to zero as $t\rightarrow -\infty$, we see +that the flow is not $\kappa$-non-collapsed on all scales for any +$\kappa>0$. Thus, $(M,g(t))$ has either a trivial cover or a double +cover isometric to the product of a shrinking family of round +surfaces with $\Ar$. If the round surface is $S^2$, then we have +established the result. If the round surface is $\Ar P^2$ a further +double covering is a product of round two-spheres with $\Ar$. + This proves the proposition. + +\entryblock{6bda3981dbaa} + +**Proof.** Given $(M,g(t))$ and $p$, suppose that no such $D'<\infty$ exists. +Because the statement is scale invariant, we can arrange that +$R(p,0)=1$. + Then we +can find a sequence of points $p_k\in M$ with $d_0(p,p_k)\rightarrow +\infty$ as $k\rightarrow \infty$ such that no $p_k$ is the center of +an evolving $\epsilon$-neck in $(M,g(0))$ defined for an interval of +normalized time of length $2$. By passing to a subsequence we can +assume that one of two possibilities holds: either +$d_0^2(p,p_k)R(p_k,0)\rightarrow\infty$ as $k\rightarrow\infty$ or +$\mathit{lim}_{k\rightarrow\infty}d_0^2(p,p_k)R(p_k,0)=\ell<\infty$. In +the first case, set $\lambda_k=R(p_k,0)$ and consider the based +flows $(M,\lambda_kg(\lambda_k^{-1}t),(p_k,0))$. According to +\entryref{49e865228b19}, after passing to a subsequence there is +a geometric limit. Thus, by \entryref{e9a9172d16e2} and +\entryref{8ec855a2aab5} the limit splits as a product of a +$2$-dimensional $\kappa$-solution and $\Ar$. By +\entryref{da767c71996b} it follows that the limit is the standard +round evolving cylinder. This implies that for all $k$ sufficiently +large $(p_k,0)$ is the center of an evolving $\epsilon$-neck in +$(M,g(t))$ defined for an interval of normalized time of length +$2$. This contradiction establishes the existence of $D'$ as +required in this case. + +Now suppose that $\mathit{lim}_{k\rightarrow\infty}d_0^2(p,p_k)R(p_k,0)=\ell<\infty$. Of +course, since $d_0(p,p_k)\rightarrow\infty$, it must be the case +that $R(p_k,0)\rightarrow 0$ as $k\rightarrow \infty$. Set +$Q_k=R(p_k,0)$. By passing to a subsequence we can arrange that +$d_0^2(p,p_k)Q_k< \ell+1$ for all $k$. Consider the +$\kappa$-solutions $(M_k,g_k(t))=(M,Q_kg(Q_k^{-1}t))$. For each $k$ +we have $p\in B_{g_k}(p_k,0,\ell+1)$, and +$R_{g_k}(p,0)=Q_k^{-1}\rightarrow\infty$ as $k\rightarrow\infty$. +This contradicts \entryref{ee314aacd207}, and completes the proof of +the existence of $D$ as required in this case as well. + + The existence of $D'_1$ is immediate since the +closure of the ball is compact and the manifold has positive +curvature. + +In fact a much stronger result is true. The constants $D'$ and +$D_1'$ in the above lemma can be chosen independent of the +non-compact $\kappa$-solutions. + +\entryblock{c2c339703eb1} + +\entryblock{51da06702a9f} + +**Proof.** First suppose that no $D$ exists so that the first statement holds. +Then there is a sequence of such solutions $(M_k,g_k(t))$, with +$p_k\in M_k$ being a soul of $(M_k,g_k(0))$ and points $q_k\in M_k$ +with $d^2_0(p_k,q_k)R(p_k,0)\rightarrow\infty$ as +$k\rightarrow\infty$ such that $q_k$ is not the center of a strong +$\epsilon$-neck in $(M_k,g_k(0))$. By rescaling we can assume that +$R(p_k,0)=1$ for all $k$, and hence that +$d_0(p_k,q_k)\rightarrow\infty$. Then, according to +\entryref{49e865228b19}, by passing to a subsequence we can +assume that there is a geometric limit +$(M_\infty,g_\infty(t),(p_\infty,0))$ with $R(p_\infty,0)=1$. + By +\entryref{04dd3c3ea1c8}, provided that $\epsilon$ is sufficiently small +for all $k$ the soul $(p_k,0)$ is not the center of a strong +$2\epsilon$-neck in $(M_k,g_k(t))$. Hence, invoking Part 4 of +\entryref{7c2758f0e2b3} and using the fact that $R(p_k,0)=1$ for +all $k$ and hence $R(p_\infty,0)=1$, we see that $(p_\infty,0)$ is +not the center of a strong $\epsilon$-neck in +$(M_\infty,g_\infty(t))$. Since the manifolds $M_k$ are non-compact +and have metrics of positive curvature they are diffeomorphic to +$\Ar^3$ and in particular, do not contain embedded copies of $\Ar +P^2$. Thus, the limit $(M_\infty,g_\infty(t))$ is a non-compact +$\kappa$-solution containing no embedded copy of $\Ar P^2$. Thus, by +\entryref{25e0946e7222} either it is positively curved or it is a +Riemannian product $S^2$ times $\Ar$. In the second case every point +is the center of a strong $\epsilon$-neck. Since we have seen that +the point $(p_\infty,0)$ is not the center of a strong +$\epsilon$-neck, it follows that the limit is a positively curved +$\kappa$-solution. + + Then according to the previous lemma there is $D'$, depending on +$(M_\infty,g_\infty(0))$ and $p_\infty$, such that every point +outside $B(p_\infty,0,D')$ is the center of an evolving +$\epsilon/2$-neck defined for an interval of normalized time of +length $2$. + +Now since $(M_k,g_k(t),(p_k,0))$ converge geometrically to +$(M_\infty,g_\infty(t),(p_\infty,0))$, by Part 2 of +\entryref{7c2758f0e2b3} for any $L<\infty$, for all $k$ +sufficiently large, all points of $B(p_k,0,L)\setminus B(p_k,0,2D')$ +are centers of strong $\epsilon$-necks in $(M_k,g_k(t))$. In +particular, for all $k$ sufficiently large, $d_0(p_k,q_k)>L$. Let +$L_k$ be a sequence tending to infinity as $k\rightarrow \infty$. +Passing to a subsequence, we can suppose that every point of +$\left(B(p_k,0,L_k)\setminus B(p_k,0,2D')\right)\subset M_k$ is the +center of a strong $\epsilon$-neck in $(M_k,g_k(0))$. Of course, +for all $k$ sufficiently large, $q_k\in M_k\setminus B(p_k,0,2D'))$. +By \entryref{75b4c67c9141} the subset of points in $M_k\times +\{0\}$ that are centers of strong $\epsilon$-necks is an open set. +Thus, replacing $q_k$ with another point if necessary we can suppose +that it $q_k$ is a closest point to $p_k$ contained in $M_k\setminus +B(p_k,0,2D')$ with the property $q_k$ is not the center of a strong +$\epsilon$-neck. Then $q_k\in M_k\setminus B(p_k,0,L_k)$ and +$(q_k,0)$ is in the closure of the set of points in $M_k$ that are +centers of strong $\epsilon$-necks in $(M_k,g_k(t))$, and hence by +Part 3 of \entryref{7c2758f0e2b3} each $(q_k,0)$ is the center +of a $2\epsilon$-neck in $(M_k,g_k(t))$. + +Let $\gamma_k$ be a minimizing geodesic connecting $(p_k,0)$ to +$(q_k,0)$, and let $\mu_k$ be a minimizing geodesic ray from +$(q_k,0)$ to infinity. Set $Q_k=R(q_k,0)$. Since $(q_k,0)$ is the +center of a $2\epsilon$-neck, from \entryref{818772530e5a} we see +that, provided that $\epsilon$ is sufficiently small, the +$2\epsilon$-neck centered at $q_k$ separates $p$ from $\infty$, so +that $\gamma_k$ and $\mu_k$ exit this $2\epsilon$-neck at opposite +ends. According to \entryref{49e865228b19}, after passing to a +subsequence, the based, rescaled flows + +$$ +(M_k,Q_kg(Q_k^{-1}t),(q_k,0)) +$$ + +converge geometrically to a limit. Let $(q_\infty,0)$ be the base +point of the resulting limit. By Part 3 of +\entryref{7c2758f0e2b3}, it is the center of a $4\epsilon$-neck +in the limit. + +\entryblock{b9f671d590f3} + +**Proof.** Suppose not. Then by passing to a subsequence we can suppose that +these products are bounded independent of $k$. Then since +$d_0(p_k,q_k)\rightarrow\infty$, we see that $Q_k\rightarrow 0$. +Thus, in the rescaled flows $(M_k,Q_kg_k(Q_k^{-1}t))$ the curvature +at $(p_k,0)$ goes to infinity. But this is impossible since the +$Q_kg_k$-distance from $(p_k,0)$ to $(q_k,0)$ is +$\sqrt{Q_k}d_0(p_k,q_k)$ which is bounded independent of $k$ and the +scalar curvature of $(p,0)$ in the metric $Q_kg_k(0)$ is +$R(p_k,0)Q_k^{-1}=Q_k^{-1}$ tends to $\infty$. Unbounded curvature +at bounded distance contradicts \entryref{ee314aacd207}, and this +establishes the claim. + +A subsequence of the based flows $(M_k,Q_kg_k(Q_k^{-1}t),(q_k,0))$ +converge geometrically to a $\kappa$-solution. According +\entryref{e9a9172d16e2} and \entryref{8ec855a2aab5}, this limiting +flow is the product of a $2$-dimensional $\kappa$-solution with a +line. Since $M$ is orientable, this $2$-dimensional +$\kappa$-solution is an evolving family of round $2$-spheres. This +implies that for all $k$ sufficiently large, $(q_k,0)$ is the center +of a strong $\epsilon$-neck in $(M_k,g_k(t))$. This is a +contradiction and proves the existence of $D<\infty$ as stated in +the proposition. + +Let $(M_k,g_k(t),(p_k,0))$ be a sequence of non-compact Ricci flows +based at a soul $p_k$ of $(M_k,g_k(0))$. We rescale so that +$R(p_k,0)=1$. By \entryref{04dd3c3ea1c8}, if $\epsilon$ is sufficiently +small, then $p_k$ cannot be the center of an $\epsilon$-neck. It +follows from \entryref{7c2758f0e2b3} that for any limit of a +subsequence the point $p_\infty$, which is the limit of the $p_k$, +is not the center of an $2\epsilon$-neck in the limit. Since the +limit manifold is orientable, it is either contractible with +strictly positive curvature or is a metric product of a round +$2$-sphere and the line. It follows that the limit manifold has +strictly positive curvature at $(p_\infty,0)$, and hence positive +curvature everywhere. The existence of $D_1<\infty$ as required is +now immediate from \entryref{49e865228b19}. + +The fact that any soul is disjoint from the middle two-thirds of any +$\epsilon$-neck and the fact that the central $2$-spheres of all +$\epsilon$-necks are isotopic in $M\setminus\{p\}$ are contained in +\entryref{04dd3c3ea1c8} and \entryref{818772530e5a}. + +\entryblock{274cda9be677} + +**Proof.** For $\epsilon>0$ sufficiently small let $D(\epsilon)$ and +$D_1(\epsilon)$ be as in the previous corollary. At the expenses of +increasing these, we can assume that they are at least the constant +$C$ in \entryref{8c090fbd21a0}. We set + +$$ +C_0(\epsilon)=\mathit{max}(D(\epsilon),D_1(\epsilon)). +$$ + + If the non-compact $\kappa$-solution has +positive curvature, then the corollary follows immediately from +\entryref{c2c339703eb1} and \entryref{8c090fbd21a0}. If the +$\kappa$-solution is the product of an evolving round $S^2$ with the +line, then every point of the zero time-slice is the center of a +strong $\epsilon$-neck for every $\epsilon>0$ so that the zero +time-slice of the solution is a strong $\epsilon$-tube. Suppose the +solution is double covered by the product of an evolving round +$2$-sphere and the line. Let $\iota$ be the involution and take the +product coordinates so that $S^2\times\{0\}$ is the invariant +$2$-sphere of $\iota$ in the zero time-slice. Then any point in the +zero time-slice at distance at least $3\epsilon^{-1}$ from +$P=(S^2\times \{0\})/\iota$ is the center of a strong +$\epsilon$-neck. Furthermore, an appropriate neighborhood of $P$ in +the time zero slice is a $(C,\epsilon)$-cap whose core contains the +$3\epsilon^{-1}$ neighborhood of $P$. The derivative bounds in this +case come from the fact that the metric is close in the +$C^{[1/\epsilon]}$-topology to the standard evolving flow. This +proves the corollary in this case and hence completes the proof. + +Now let us consider compact $\kappa$-solutions. + +\entryblock{a4ea411d0765} + +**Proof.** First notice that if $(M,g(t))$ is not of strictly positive +curvature, then the universal covering of $(M,g(0))$ is a Riemannian +product $S^2\times \Ar$, and hence $(M,g(0))$ is either non-compact +or finitely covered by the product flow on $S^2\times S^1$. The +former case is ruled out since we are assuming that $M$ is compact +and the latter is ruled out because such flows are not +$\kappa$-non-collapsed for any $\kappa>0$. We conclude that +$(M,g(t))$ is of positive curvature. This implies that the +fundamental group of $M$ is finite. If there were an embedded $\Ar +P^2$ in $M$ with trivial normal bundle, that $\Ar P^2$ cannot +separate (since the Euler characteristic of $\Ar P^2$ is one, it is +not the boundary of a compact $3$-manifold). But a non-separating +surface in $M$ induces a surjective homomorphism of $H_1(M)$ onto +$\Zee$. We conclude from this that $M$ does not contain an embedded +$\Ar P^2$ with trivial normal bundle. + + We assume that $(M,g(0))$ is not round so that by \entryref{9c54e0d0c7d8} + there is a universal $\kappa_0>0$ such that $(M,g(0))$ is a +$\kappa_0$-solution. Let $C_0(\epsilon)$ be the constant from + \entryref{274cda9be677}. + +\entryblock{494bc110f8ab} + +**Proof.** Suppose that for some $\epsilon>0$ there is no such $C_1$. Then we +take a sequence of constants $C'_k$ that diverges to $+\infty$ as +$k\rightarrow \infty$ and a sequence $(M_k,g_k(t), (p_k,0))$ of +based $\kappa_0$-solutions such that the diameter of $(M_k,0)$ is +greater than $C'_kR^{-1/2}(p_k,0)$ and yet $(p_k,0)$ is not +contained in the core of a $(C_0(\epsilon),\epsilon)$-cap nor is the +center of a strong $\epsilon$-neck. We scale $(M_k,g_k(t))$ by +$R(p_k,0)$. This allows us to assume that $R(p_k,0)=1$ for all $k$. +According to \entryref{49e865228b19}, after passing to a +subsequence we can assume these based $\kappa$-solutions converge to +a based $\kappa$-solution $(M_\infty,g_\infty(t),(p_\infty,0))$. +Since the diameters of the $(M_k,g_k(0))$ go to infinity, $M_\infty$ +is non-compact. According to \entryref{274cda9be677} the point +$p_\infty$ is either the center of a strong $\epsilon$-neck, or is +contained in the core of a $(C_0(\epsilon),\epsilon)$-cap. Since +$R(p_k,0)=1$ for all $k$, it follows from Parts 1 and 4 of +\entryref{7c2758f0e2b3} that for all $k$ sufficiently large, +$(p_k,0)$ is either the center of a strong $\epsilon$-neck in +$(M_k,g_k(t))$ or is contained in the core of a +$(C_0(\epsilon),\epsilon)$-cap. This is a contradiction, proving the +claim. + +Now it follows from \entryref{fa8d9a43f34a} that if the +diameter of $(M,g(0))$ is greater than $C_1(\mathit{max}_{x\in +M}R(x,0))^{-1/2}$ and if it is not of constant positive curvature, +then $M$ is diffeomorphic to either $S^3$, $\Ar P^3$, $\Ar P^3\#\Ar +P^3$ or is a $S^2$-fibration over $S^1$. On the other hand, since +$M$ is compact of positive curvature its fundamental group is +finite, see Theorem 4.1 on p. 154 of [Petersen]. This rules out +the last two cases. This implies that when $(M,g(0))$ has diameter +greater than $C_1(\mathit{max}_{x\in M}R(x,0))^{-1/2}$ and is not of +constant positive curvature, it is a double $C_0$-capped +$\epsilon$-tube. + +We must consider the case when $(M,g(0))$ is not of constant +positive curvature and its diameter is less than or equal to +$C_1(\mathit{max}_{x\in M}R(x,0))^{-1/2}$. Since $(M,g(0))$ is not +round, by \entryref{01028c93347d} its asymptotic soliton is not +compact. Thus, by \entryref{2d68880771ff} its asymptotic soliton is +either $S^2\times \Ar$ or is double covered by this product. This +means that for $t$ sufficiently negative the diameter of $(M,g(t))$ +is greater than $C_1(\mathit{max}_{x\in M}R(x,0))^{-1/2}$. Invoking the +previous result for this negative time tells us that $M$ is +diffeomorphic to $S^3$ or $\Ar P^3$. + +\entryblock{7744a698d8da} + +**Proof.** The result is immediate from \entryref{9c54e0d0c7d8} and +\entryref{49e865228b19}. + +\entryblock{744167bbcef7} + +Let us summarize our results. + +\entryblock{c2be59f6090c} + +An immediate consequence of this result is: + +\entryblock{fc2d2036a96c} + +\entryblock{e724298e5bc3} + +**Proof.** The limiting manifold $M_\infty$ cannot contain an embedded $\Ar +P^2$ with trivial normal bundle. Hence, by the previous corollary, +the point $(x_\infty,0)$ has a strong $(C,\epsilon)$-canonical +neighborhood in the limiting flow. If the limiting $\kappa$-solution +is round, then for all $n$ sufficiently large $x_n$ is contained in +a component of the zero time-slice that is $\epsilon$-round. If +$(x_\infty,0)$ is contained in a $C$-component of the zero +time-slice of the limiting $\kappa$-solution, then for all $n$ +sufficiently large $x_n$ is contained in a $C$-component of the zero +time-slice of ${\mathcal M}_n$. Suppose that $(x_\infty,0)$ is the +center of a strong $\epsilon$-neck in the limiting flow. This neck +extends backwards in the limiting solution some amount past an +interval of normalized time of length $1$, where by continuity it +is an evolving $\epsilon$-neck defined backwards for an interval of +normalized time of length greater than $1$. Then by Part 2 of +\entryref{7c2758f0e2b3}, any family of metrics on this neck +sufficiently close to the limiting metric will determine an strong +$\epsilon$-neck. This implies that for all $n$ sufficiently large +$x_n$ is the center of a strong $\epsilon$-neck in $({\mathcal +M}_n,G_n)$. Lastly, if $(x_\infty,0)$ is contained in the core of a +$(C,\epsilon)$-cap in the limiting flow, then by Part 1 of +\entryref{7c2758f0e2b3} for all $n$ sufficiently large $x_n$ is +contained in the core of a $(C,\epsilon)$-cap in $({\mathcal +M}_n,G_n)$. diff --git a/projects/poincare-conjecture/.astrolabe/docs/11-bounded-curvature-at-bounded-distance.mdx b/projects/poincare-conjecture/.astrolabe/docs/11-bounded-curvature-at-bounded-distance.mdx new file mode 100644 index 00000000..8eae2f15 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/docs/11-bounded-curvature-at-bounded-distance.mdx @@ -0,0 +1,1104 @@ + + +# Chapter 10 -- Bounded curvature at bounded distance + This chapter is +devoted to Perelman's result about bounded curvature at bounded +distance for blow-up limits. Crucial to the argument is that each +member of the sequence of generalized Ricci flows has curvature +pinched toward positive and also has strong canonical neighborhoods. + +## Pinching toward positive: the definitions + +In this section we give the definition of what it means for a +generalized Ricci flow to have curvature pinched toward +positive. This is the +obvious generalization of the corresponding notion for Ricci flows. + +\entryblock{7464980c4085} + +According to \entryref{fc44d8653f35} if $(M,g(t)),\ 0\le a\le t0$ sufficiently small such that for any +$0<\epsilon\le \epsilon_0$ all the results of the Appendix hold for +$2\epsilon$ and $\alpha=10^{-2}$, and \entryref{cb9ca6d6db9f} +holds for $2\epsilon$. + +## The statement of the theorem + +Here is the statement of the main theorem of this chapter, the +theorem that establishes bounded curvature at bounded distance for +blow-up limits. + +\entryblock{ecd62fc2ea2b} + +This chapter is devoted to the proof of this theorem. The proof is +by contradiction. Suppose that there is some $A_0<\infty$ for which +the result fails. Then there are a sequence of generalized +three-dimensional Ricci flows $({\mathcal M}_n,G_n)$ whose +intervals of definition are contained in $[0,\infty)$ and whose +curvatures are pinched toward positive. Also, there are points +$x_n\in {\mathcal M}_n$ satisfying the second condition given in the +theorem and points $y_n\in {\mathcal M}_n$ such that for all $n$ we +have: + +- **(1)** $\mathit{lim}_{n\rightarrow \infty}R(x_n)=\infty$. +- **(2)** $\mathbf{t}(y_n)=\mathbf{t}(x_n)$, +- **(3)** $d(x_n,y_n)< A_0R(x_n)^{-1/2}$, +- **(4)** + +$$ +\mathit{lim}_{n\rightarrow\infty}\frac{R(y_n)}{R(x_n)}=\infty. +$$ + +For the rest of this chapter we assume that such a sequence of +generalized Ricci flows exists. We shall eventually derive a +contradiction. + +Let us sketch how the argument goes. We show that there is a +(partial) geometric blow-up limit of the sequence $({\mathcal +M}_n,G_n)$ based at the $x_n$. We shall see that the following hold +for this limit. It is an incomplete manifold $U_\infty$ +diffeomorphic to $S^2\times (0,1)$ with the property that the +diameter of $U_\infty$ is finite and the curvature goes to infinity +at one end of $U_\infty$, an end denoted ${\mathcal E}$, while +remaining bounded at the other end. (The non-compact manifold in +question is diffeomorphic to $S^2\times (0,1)$ and, consequently, it +has two ends.) + Every point of $U_\infty$ +sufficiently close to ${\mathcal E}$ is the center of a +$2\epsilon$-neck in $U_\infty$. In fact, there is a partial +geometric limiting flow on $U_\infty$ so that these points are +centers of evolving $2\epsilon$-necks. Having constructed this +incomplete blow-up limit of the original sequence we then consider +further blow-up limits about the end ${\mathcal E}$, the end where +the scalar curvature goes to infinity. On the one hand, a direct +argument shows that a sequence of rescalings of $U_\infty$ around +points converging to the end ${\mathcal E}$ converge in the +Gromov-Hausdorff sense to a cone. On the other hand, a slightly +different sequence of rescalings at the same points converges +geometrically to a limiting non-flat Ricci flow. Since both limits +are non-degenerate three-dimensional spaces, we show that the ratio +of the rescaling factors used to construct them converges to a +finite, non-zero limit. This means that the two limits differ only +by an overall constant factor. That is to say the geometric blow-up +limit is isometric to an open subset of a non-flat cone. This +contradicts Hamilton's result (\entryref{92b5c664399e}) which says +that it is not possible to flow under the Ricci flow to an open +subset of a non-flat cone. Now we carry out all the steps in this +argument. + +## The incomplete geometric limit + +We fix a sequence $({\mathcal M}_n,G_n,x_n)$ of generalized Ricci +flows as above. The first step is to shift and rescale this +sequence of generalized Ricci flows so that we can form an +(incomplete) geometric limit which will be a tube of finite length +with scalar curvature going to infinity at one end. + +We shift the time parameter of $({\mathcal M}_n,G_n)$ by $-\mathbf{t}(x_n)$. We change notation and denote these shifted flows by +$({\mathcal M}_n,G_n)$. This allows us to arrange that $\mathbf{t}(x_n)=0$ for all $n$. Since shifting leaves the curvature +unchanged, the shifted flows satisfy a weaker version of curvature +pinched toward positive. Namely, for the shifted flows we have + +$$ +\begin{aligned} +R(x) & \ge & -6 \nonumber \\ +R(x) & \ge & 2X(x)\left(\mathit{log}(X(x))-3\right). +\end{aligned} +$$ + + We set $Q_n=R(x_n)$, and we denote by $M_n$ the $0$-time-slice of +${\mathcal M}_n$. We rescale $({\mathcal M}_n,G_n)$ by $Q_n$. Denote +by $({\mathcal M}'_n,G_n')$ the rescaled (and shifted) generalized +flows. For the rest of this argument we implicity use the metrics +$G'_n$. If we are referring to $G_n$ we mention it explicitly. + +### The sequence of tubes + + Let +$\gamma_n$ be a smooth path in $B_{G_n}(x_n,0,A_0Q_n^{-1/2})$ from +$x_n$ to $y_n$. For all $n$ sufficiently large we have +$R_{G'_n}(y_n) \gg 1$. Thus, there is a point $z_n\in \gamma_n$ such +that $R_{G'_n}(z_n)=4$ and such that on the sub-path +$\gamma_n|_{[z_n,y_n]}$ we have $R_{G'_n}\ge 4$. We replace +$\gamma_n$ by this sub-path. Now, with this replacement, according +to the second condition in the statement of the theorem, every point +of $\gamma_n$ has a strong $(C,\epsilon)$ canonical +neighborhood. As $n$ tends to infinity +the ratio of $R(y_n)/R(z_n)$ tends to infinity. This means that for +all $n$ sufficiently large, no point of $\gamma_n$ can be contained +in an $\epsilon$-round component +or a $C$-component, because if it were then all +of $\gamma_n$ would be contained in that component, contradicting +the fact that the curvature ratio is arbitrarily large for large +$n$. Hence, for $n$ sufficiently large, every point of $\gamma_n$ is +either contained in the core of a +$(C,\epsilon)$-cap or is the center of a +strong $\epsilon$-neck. According to +\entryref{0e9a1a167caa}, for all $n$ sufficiently large +$\gamma_n$ is contained an open submanifold $X_n$ of the zero +time-slice of ${\mathcal M}'_n$ that is one of the following: + +- **(1)** an $\epsilon$-tube and both endpoints of $\gamma_n$ are centers +of $\epsilon$-necks +contained in $X_n$, +- **(2)** a $C$-capped $\epsilon$-tube with cap ${\mathcal C}$, +and each endpoint of $\gamma_n$ either is contained in the core $Y$ +of ${\mathcal C}$ or is the center of an $\epsilon$-neck contained +in $X_n$, +- **(3)** a double $C$-capped $\epsilon$-tube, or finally +- **(4)** the union of two $(C,\epsilon)$-caps. + +The fourth possibility is incompatible with the fact that the ratio +of the curvatures at the endpoints of $\gamma_n$ grows arbitrarily +large as $n$ tends to infinity. Hence, this fourth possibility +cannot occur for $n$ sufficiently large. Thus, for all $n$ +sufficiently large $X_n$ is one of the first three types listed +above. + +\entryblock{31216ebde119} + +**Proof.** This is clear in the third case since $X_n$ is a closed manifold. + +Let us consider the first case. There are $\epsilon$-necks $N(z_n)$ +and $N(y_n)$ centered at $z_n$ and $y_n$ and contained in $X_n$. +Suppose first that the central $2$-spheres $S(z_n)$ and $S(y_n)$ of +these necks are disjoint. Then they are the boundary of a compact +sumanifold $X_n'$ of $X_n$. It follows easily from +\entryref{fc6c6ae605d2} that any sequence of minimizing paths from +$z_n$ to $y_n$ is contained in the union of $X_n'$ with the middle +halves of $N(z_n)$ and $N(y_n)$. Since this manifold has compact +closure in $X_n$, the usual arguments show that one can extract a +limit of a subsequence which is a minimizing geodesic in $X_n$ from +$z_n$ to $y_n$. If $S(z_n)\cap S(y_n)\not=\emptyset$, then $y_n$ is +contained in the middle half of $N(z_n)$, and again it follows +immediately from \entryref{fc6c6ae605d2} that there is a minimizing +geodesic in $N(z_n)$ between these points. + +Now let us consider the second case. If each of $z_n$ and $y_n$ is +the center of an $\epsilon$-neck in $X_n$, the argument as in the +first case applies. If both points are contained in the core of +${\mathcal C}$ then, since that core has compact closure in $X_n$, +the result is again immediate. Lastly, suppose that one of the +points, we can assume by the symmetry of the roles of the points +that it is $z_n$, is the center of an $\epsilon$-neck $N(z_n)$ in +$X_n$ and the other is contained in the core of ${\mathcal C}$. +Suppose that the central $2$-sphere $S(z_n)$ of $N(z_n)$ meets the +core $Y$ of ${\mathcal C}$. Then $z_n$ lies in the half of the neck +$N={\mathcal C}\setminus \overline Y$ whose closure contains the +frontier of $Y$. Orient $s_N$ so that this half is the positive +half. Thus, by \entryref{fc6c6ae605d2} any minimizing sequence of +paths from $z_n$ to $y_n$ is eventually contained in the union of +the core of ${\mathcal C}$ and the the positive three-quarters of +this neck. Hence, as before we can pass to a limit and construct a +minimizing geodesic in $X_n$ connecting $z_n$ to $y_n$. On the other +hand, if $S(z_n)$ is disjoint from $Y$, then $S(z_n)$ separates +$X_n$ into a compact complementary component and a non-compact +complementary component and the compact complementary component +contains $Y$. Orient the $s_N$-direction so that the compact +complementary component lies on the positive side of $S(z_n)$. Then +any minimizing sequence of paths in $X_n$ from $z_n$ to $y_n$ is +eventually contained in the union of the compact complementary +component of $N(z_n)$ and the positive $3/4$'s of $N(z_n)$. As +before, this allows us to pass to a limit to obtain a minimizing +geodesic in $X_n$. + +This claim allows us to assume (as we now shall) that $\gamma_n$ is +a minimizing geodesic in $X_n$ from $z_n$ to $y_n$. + +\entryblock{96519f84d54e} + +**Proof.** The first item is clear since, for all $n$, the geodesic $\gamma_n$ +has $G_n$-length at most $A_0Q_n^{-1/2}$ and hence $G'_n$-length at +most $A_0$. Suppose that we have a $(C,\epsilon)$-cap ${\mathcal C}$ +whose core $Y$ contains a point of $\gamma_n$. Let $N$ be the +$\epsilon$-neck that is the complement of the closure of $Y$ in +${\mathcal C}$, and let $\widehat Y$ be the union of $Y$ and the +closed negative half of $N$. We claim that $\widehat Y$ contains +either $z_n$ or $y_n$. By \entryref{5ef5906bb370}, since $Y$ +contains a point of $\gamma_n$, the intersection of $\widehat Y$ +with $\gamma_n$ is a subinterval containing one of the end points of +$\gamma_n$, i.e., either $z_n$ or $y_n$. This means that any point +$w$ which is contained in a $(C,\epsilon)$-cap whose core contains +a point of $\gamma_n$ must satisfy one of the following: + +$$ +R(w)< CR(z'_n)\ \ \ \mathit{or}\ \ \ R(w)> C^{-1}R(y'_n). +$$ + +We pass to a subsequence so that $R(y_n)/R(z_n)>4C^2$ for all $n$, +and we pass to a subinterval $\gamma'_n$ of $\gamma_n$ with +endpoints $z_n'$ and $y_n'$ such that: + +- **(1)** $R(z'_n)=2CR(z_n)$ +- **(2)** $R(y'_n)=(2C)^{-1}R(y_n)$ +- **(3)** $R(z'_n)\le R(w)\le R(y'_n)$ for all $w\in \gamma'_n$. + +Clearly, with these choices $R(z'_n)$ is bounded independent of $n$ +and $R(y'_n)$ tends to infinity as $n$ tends to infinity. Also, no +point of $\gamma'_n$ is contained in the core of a +$(C,\epsilon)$-cap. Since every point of $\gamma'_n$ has a strong +$(C,\epsilon)$-canonical neighborhood, it follows that every point +of $\gamma'_n$ is the center of a strong $\epsilon$-neck. It now +follows from \entryref{ba681e15e1a6} that there is a balanced +$\epsilon$-chain consisting of strong $\epsilon$-necks centered at +points of $\gamma'_n$ whose union contains $\gamma'_n$. (Even if the +$2$-spheres of these necks do not separate the zero time-slice of +${\mathcal M}'_n$, as we build the balanced $\epsilon$-chain as +described in \entryref{ba681e15e1a6} the new necks we add can +not meet the negative end of $N(z'_n)$ since the geodesic +$\gamma'_n$ is minimal.) We can take the first element in the +balanced chain to be a strong $\epsilon$-neck $N(z'_n)$ centered at +$z'_n$, and the last element to be a strong $\epsilon$-neck $N^+_n$ +containing $y'_n$. The union of this chain is $T_n$. (See Fig. 10.1.) + +Next, we show that every point of $T_n$ is the center of a strong +$\epsilon$-neck in $({\mathcal M}_n,G_n)$. We must rule out the +possibility that there is a + point of $T_n$ that is contained in the core of a $(C,\epsilon)$-cap. +Since $T_n$ is a union of $\epsilon$-necks centered at points of +$\gamma_n'$ we see that every point $w\in T_n$ has + +$$ +(3C/2)R(z_n)3$ for every $x\in T_n$. We have +just seen that every $x\in T_n$ is the center of an $\epsilon$-neck. +If $x$ is contained in the $\epsilon$-neck centered at $z'_n$ or +$y'_n$, then since $R(z'_n)\ge 4$ and $R(y'_n)\ge 4$, clearly +$R(x)>3$. We must consider the case when $x$ is not contained in +either of these $\epsilon$-necks. In this case the central +$2$-sphere $S_x$of the $\epsilon$-neck centered at $x$ is contained +in the compact submanifold of $T_n'$ bounded by the central +$2$-spheres of the necks centered at $z'_n$ and $y'_n$. These +$2$-spheres are disjoint and by Condition 4 in +\entryref{a46d72b87e07} each is a homotopically non-trivial +$2$-sphere in $T_n'$. Hence, the compact manifold with their +disjoint union as boundary is diffeomorphic to $S^2\times [0,1]$ +and, again according to Condition 4 of +\entryref{a46d72b87e07}, $S_x$ is isotopic to the +$2$-sphere factor in this product decomposition. Since the +intersection of $\gamma'_n$ with this submanifold is an arc spanning +from one boundary component to the other, $S_x$ must meet +$\gamma'_n$, in say $w$. + By construction, since $w\in \gamma'_n$ we have $R(w)\ge 4$. + This implies that $R(x)>3$. + This completes the proof of the claim. + +### Extracting a limit of a subsequence of the tubes + +Passing to a subsequence we arrange that the $R(z_n')$ converge. Now +consider the subset ${\mathcal A}\subset \Ar$ consisting of all +$A>0$ such that there is a uniform bound, independent of $n$, for +the curvature on $B(z'_n,A)\cap T_n$. The set ${\mathcal A}$ is +non-empty since $R(z'_n)$ is bounded independent of $n$ and for +every $n$ there is a strong $\epsilon$-neck $N(z'_n)$ centered at +$z'_n$ contained in $T_n$. On the other hand, since +$d_{G_n'}(z_n',y_n')$ is uniformly bounded and $R(y_n')\rightarrow +\infty$, there is a finite upper bound for ${\mathcal A}$. Let +$A_1$ be the least upper bound of ${\mathcal A}$. We set +$U_n=T_n\cap B(z'_n,A_1)$. This is an open subset of $T_n$ +containing $z'_n$. We let $g_n'=G_n'|U_n$. + +\entryblock{67090b75df54} + +**Proof.** The curvature on $N(z'_n)$ is bounded independent of $n$. Consider +a point $w$ near the end of $N(z'_n)$ that separates $y_n'$ from +$z_n'$. It is also the center of a strong $\epsilon$-neck $N(w)$. By +\entryref{a46d72b87e07} and our assumption that +$\epsilon\le \bar\epsilon(10^{-2})$, the scalar curvature on +$N(z'_n)\cup N(w)$ is between $(0.9)R(z'_n)$ and $(1.1)R(z'_n)$. +Since, by construction, the negative end of $N(z'_n)$ contains an +end of $T_n$, this implies that + +$$ +N(z'_n)\cup N(w)\supset B(z'_n,7R(z'_n)^{-1/2}\epsilon^{-1}/4)\cap T_n, +$$ + + so that we see that +$A_1\ge 7\epsilon^{-1}\mathit{lim}_{n\rightarrow\infty} +R(z'_n)^{-1/2}/4$. Thus, $A_1>3R(z'_n)^{-1}\epsilon^{-1}/2$ for all +$n$ sufficiently large. Obviously then $U_n$ contains $N(z'_n)$. + +The next claim uses terminology from \entryref{1588b6b37e8c}. + +\entryblock{6a13c64a90e4} + +**Proof.** To prove this it suffices to show that given $\delta>0$ there is +$A0$ sufficiently small, then it follows for all +$\delta>0$. First of all, by \entryref{b74cd360f72f} and +\entryref{dc5709757703}, the fact that $\epsilon\le +\bar\epsilon(10^{-2})$ implies that any point $w$ with the property +that the strong $\epsilon$-neighborhood centered at $w$ contains +$z_n'$ is contained in the ball of radius +$(1.1)R(z'_n)^{-1/2}\epsilon^{-1}\delta/4$. + Since $w_n'\in U_n$ and +consequently that $d(z_n',w_n')0$ and any $\ell\ge +0$ there is a uniform bound for $|\nabla^\ell \mathit{Rm}|$ on $\mathit{Reg}_\delta(U_n,g'_n)$. Clearly, since the base point $z_n'$ has +bounded curvature it lies in $\mathit{Reg}_\delta(U_n,g'_n)$ for +sufficiently small $\delta$ (how small being independent of $n$). +Lastly, the fact that every point in $U_n$ is the center of an +$\epsilon$-neighborhood implies that $(U_n,g'_n)$ is $\kappa$ +non-collapsed on scales $\le r_0$ where both $\kappa$ and $r_0$ are +universal. Since the $\gamma'_n$ have uniformly bounded lengths, the +$\epsilon$-tubes $T_n'$ have uniformly bounded diameter. Also, we +have seen that their have curvatures are bounded from below by $3$. +It follows that their volumes are uniformly bounded. Now invoking +\entryref{74c7682a68ed} we see that after passing to a subsequence +we have a geometric limit $(U_\infty,g_\infty,z_\infty)$ of a +subsequence of $(U_n,g'_n,z'_n)$. + +### Properties of the limiting tube + +Now we come to a result establishing all the properties we need for +the limiting manifold. + +\entryblock{e2e64cdb0b75} + +The proof of this proposition occupies the rest of +Chapter 10.3. + +**Proof.** Let $V_1\subset V_2 \subset \cdots \subset U_\infty$ be the open +subsets and $\varphi_n\colon V_n\to U_n$ be the maps having all the +properties stated in \entryref{467f4b920031} so as to exhibit +$(U_\infty,g_\infty,z_\infty)$ as the geometric limit of the +$(U_n,g_n',z_n')$. + +Since the $U_n$ are all contained in $B(z_n',A_1)$, it follows that any point +of $U_\infty$ is within $A_1$ of the limiting base point $z_\infty$. This +proves that the diameter of $U_\infty$ is bounded. + +For each $n$ there is the $\epsilon$-neck $N(z_n')$ centered at +$z_n'$ contained in $U_n$. The middle two-thirds, $N_n'$, of this +neck has closure contained in $\mathit{Reg}_\delta(U_n,g_n)$ for some +$\delta>0$ independent of $n$ (in fact, restricting to $n$ +sufficiently large, $\delta$ can be taken to be approximately equal +to $R(z_\infty)^{-1/2}\epsilon^{-1}/3$). This means that for some +$n$ sufficiently large and for all $m\ge n$ the image +$\varphi_m(V_n)\subset U_m$ contains $N'_m$. For any fixed $n$ as +$m$ tends to infinity the metrics $\varphi_m^*g_m|_{V_n}$ converge +uniformly in the $C^\infty$-topology to $g_\infty|_{V_n}$. Thus, it +follows from \entryref{7c2758f0e2b3} that for all $m$ +sufficiently large, $\varphi^{-1}_m(N'_m)$ is a $3\epsilon/2$-neck +centered at $z_\infty$. We fix such a neck $N'(z_\infty)\subset +U_\infty$. Let $S(z_\infty)$ be the central $2$-sphere of +$N'(z_\infty)$. For each $n$ sufficiently large, +$\varphi_n(S(z_\infty))$ separates $U_n$ into two components, one, +say $W^-_n$ contained in $N(z'_n)$ and the other, $W^+_n$ containing +all of $U_n\setminus N(z'_n)$. It follows that $S(z_\infty)$ +separates $U_\infty$ into two components, one, denoted $W^-_\infty$, +where the curvature is bounded (and where, in fact, the curvature is +close to $R(z_\infty)$) and the other, denoted $W^+_\infty$, where +it is unbounded. + +\entryblock{2b1f8205acbe} + +**Proof.** Fix a point $q\in W^+_\infty$. + For all $n$ sufficiently large denote by $q_n=\varphi_n(q)$. + Then for all $n$ sufficiently large, $q_n\in W^+_n$ and +$\mathit{lim}_{n\rightarrow\infty}R(q_n)=R(q)$. This means that for all +$n$ sufficiently large $R(y_n')>>R(q_n))$, and hence + the $3\epsilon/2$-neck centered at + $q_n\in U_n$ is disjoint from $N(y'_n)$. Thus, by the rescaled version of + \entryref{b74cd360f72f}, we see that the distance from + the $3\epsilon/2$-neck centered at $q_n$ to $N(y'_n)$ is bounded below by + $(0.99)\epsilon^{-1}R(q_n)^{-1/2}/4\ge \epsilon^{-1}R(q_\infty)^{-1/2}/12$. + Also, since $q_n\in W_n$, this $3\epsilon/2$-neck $N'(q_n)$ centered at + $q_n$ does not extend past the $2$-sphere at $s^{-1}(-3\epsilon^{-1}/4)$ in the + $ \epsilon$-neck $N(z'_n)$. It follows that for all $n$ sufficiently large that + this $3\epsilon/2$-neck has + compact closure contained in $\mathit{Reg}_\delta(U_n,g_n)$ for some $\delta$ + independent of $n$, and hence there is $m$ such that for all $n$ sufficiently large $N'(q_n)$ is + contained in the image $\varphi_n(V_m)$. Again using the fact that + $\varphi_n^*(g_n|_{V_m})$ converges in the $C^\infty$-topology to + $g_\infty|_{V_m}$ as $n$ tends to infinity, we see, by \entryref{7c2758f0e2b3} + that for all $n$ + sufficiently large $\varphi_n^{-1}(N_m)$ contains a $2\epsilon$-neck in $U_\infty$ centered + at $q$. + +It now follows from \entryref{0e9a1a167caa} that $W^+_\infty$ +is contained in an $2\epsilon$-tube $T_\infty$ that is contained in +$U_\infty$. Furthermore, the frontier of $W^+_\infty$ in $T_\infty$ +is the $2$-sphere $S(z_\infty)$ which is isotopic to the central +$2$-spheres of the $2\epsilon$-necks making up $T_\infty$. Hence, +the closure $\overline W^+_\infty$ of $W^+_\infty$ is a +$2\epsilon$-tube with boundary $S(z_\infty)$. In particular, +$\overline W^+_\infty$ is diffeomorphic to $S^2\times [0,1)$. + +Now we consider the closure $\overline W^-_\infty$ of $W^-_\infty$. +Since the closure of each $W^-_n$ is the closed negative half of the +$\epsilon$-neck $N(z_n')$ and the curvatures of the $z_n'$ have a +finite, positive limit, the limit $\overline W^-_\infty$ is +diffeomorphic to a product $S^2\times (-1,0]$. Hence, $U_\infty$ is +the union of $\overline W^+_\infty$ and $\overline W^-_\infty$ +along their common boundary. It follows immediately that $U_\infty$ +is diffeomorphic to $S^2\times (0,1)$. + +\entryblock{d9ac883eac42} + +**Proof.** A neighborhood of one end of $U_\infty$, the end $\overline W^-_\infty$, + is the limit of the + negative halves of $\epsilon$-necks centered at $z_n'$. + Thus, the curvature is bounded on this neighborhood, and in fact is +approximately equal to $R(z_\infty)$. Let $x_k$ be any sequence of +points in $U_\infty$ tending to the other end. We show that $R(x_k)$ +tends to $\infty$ as $k$ does. The point is that since the sequence +is tending to the end, the distance from $x_k$ to the end of +$U_\infty$ is going to zero. Yet, each $x_k$ is the center of an +$\epsilon$-neck in $U_\infty$. The only way this is possible is if +the scales of these $\epsilon$-necks are converging to zero as $k$ +goes to infinity. This is equivalent to the statement that $R(x_k)$ +tends to $\infty$ as $k$ goes to infinity. + +The next step in the proof of \entryref{e2e64cdb0b75} is to extend +the flow backwards a certain amount. As stated in the proposition, +the amount of backward time that we can extend the flow is not +uniform over all of $U_\infty$, but rather depends on the curvature +of the point at time zero. + +\entryblock{1418d3c286c7} + +**Proof.** Since $x\in U_n\subset T_n$, there is a strong $\epsilon$-neck in +${\mathcal M}_n$ centered at $x$. Both statements follow immediately +from that. + +Let $X\subset U_\infty$ be an open submanifold with compact closure +and set + +$$ +t_0(X)= \mathit{sup}_{x\in X}(-R_{g_\infty}(x)^{-1}). +$$ + + Then for +all $n$ sufficiently large $\varphi_n$ is defined on $X$ and the +scalar curvature of the flow $g_n(t)$ on $\varphi_n(X)\times +(t_0,0]$ is uniformly bounded independent of $n$. Thus, according to +\entryref{4673eb9f4424} by passing to a subsequence we +can arrange that there is a limiting flow defined on $X\times +(t_0,0]$. Let ${\mathcal U}_\infty\subset U_\infty\times +(-\infty,0]$ consist of all pairs $(x,t)$ with the property that +$-R_{g_\infty}(x,0)^{-1}0$ and a shortest geodesic ray $\gamma'$ +from $x$ to the end ${\mathcal E}$ with $|\gamma'|=|\gamma|-\delta$. +As we have just seen, $\gamma'$ is a minimizing geodesic ray. Take a +sequence of points $q_i$ tending to the end ${\mathcal E}$ and let +$S^2_i$ be the central $2$-sphere in the $2\epsilon$-neck centered +at $q_i$. Of course, for all $i$ sufficiently large, both $\gamma'$ +and $\gamma$ must cross $S^2_i$. Since the scalar curvature tends to +infinity at the end ${\mathcal E}$, it follows from +\entryref{88a5a2233418} for all $i$ sufficiently large, the extrinsic +diameter of $S^2_i$ is less than $\delta/3$. Let $p_i$ be a point of +intersection of $\gamma$ with $S^2_i$. For all $i$ sufficiently +large the length $d_i$ of the sub-ray in $\gamma$ from $p_i$ to the +end ${\mathcal E}$ is at most $\delta/3$. Let $p'_i$ be a point of +intersection of $\gamma'$ with $S^2_i$ and let $d'_i$ be the length +of the ray in $\gamma'$ from $p'_i$ to the end ${\mathcal E}$. Let +$\lambda$ be the sub-geodesic of $\gamma$ from $x$ to $p_i$ and +$\lambda'$ the sub-geodesic of $\gamma'$ from $x$ to $p'_i$. Let +$\beta$ be a minimizing geodesic from $p'_i$ to $p_i$. Of course, +$|\beta|<\delta/3$ so that by the minimality of $\lambda$ and +$\lambda'$ we have + +$$ +-\delta/3<|\lambda|-|\lambda'|<\delta/3. +$$ + + Since +$|\lambda'|+d'_i=|\lambda|+d_i-\delta$, we have + +$$ +2\delta/3\le d_i-d'_i. +$$ + +This is absurd since $d'_i>0$ and $d_i<\delta/3$. + +(2) follows immediately from (1) and the definition. + +Given this result, the usual arguments show: + +\entryblock{7ef7dfbbe151} + +Also, we have a version of the triangle inequality for distances to +${\mathcal E}$. + +\entryblock{f83f945f80e7} + +**Proof.** From the definitions it is clear that $d(x,y)+d(y,{\mathcal E})\ge +d(x,{\mathcal E})$, and symmetrically, reversing the roles of $x$ +and $y$. The remaining inequality that we must establish is the +following: $d(x,{\mathcal E})+d(y,{\mathcal E})\ge d(x,y)$. Let +$q_n$ be any sequence of points converging to ${\mathcal E}$. Since +the end is at finite distance, it is clear that $d(x,{\mathcal E})= +\mathit{lim}_{n\rightarrow\infty}d(x,q_n)$. The remaining inequality +follows from this and the usual triangle inequality applied to +$d(x,q_n)$, $d(y,q_n)$ and $d(x,y)$. + +\entryblock{e98dee3d7485} + +\entryblock{dda8a7c7acb6} + +**Proof.** Take a minimal geodesic ray $\gamma$ from a point $x$ limiting to +the end and let $y$ be a point closer to ${\mathcal E}$ than $x$ and +not lying on $\gamma$. Then a minimal geodesic ray from $y$ to +${\mathcal E}$ gives a direction at ${\mathcal E}$ distinct from the +direction determined by $\gamma$. + +\entryblock{f640a0334396} + +### The Metric on the space of directions at ${\mathcal E}$ + +\entryblock{f4ee075394ef} + +\entryblock{5b34d0c84cbb} + +**Proof.** By restricting $\gamma$ and $\mu$ to slightly smaller rays, we can +assume that each is the unique shortest ray from its endpoint to the +end ${\mathcal E}$. Let $x$, resp., $y$ be the endpoint of $\gamma$, +resp., $\mu$. Now let $q_n$ be any sequence of points in $U_\infty$ +limiting to the end ${\mathcal E}$, and consider minimizing geodesic +rays $\gamma_n$ from $q_n$ to $x$ and $\mu_n$ from $q_n$ to $y$, +each parameterized by the distance from $q_n$. By passing to a +subsequence we can assume that each of the sequences $\{\gamma_n\}$ +and $\{\mu_n\}$ converge to a minimizing geodesic ray, which by +uniqueness, implies that the first sequence limits to $\gamma$ and +the second to $\mu$. For $s,s'$ sufficiently small, let +$\theta_n(s,s')$ be the angle at $\tilde q_n$ of the Euclidean +triangle $\alpha_n\tilde q_n\beta_n$, where $|\alpha_n\tilde +q_n|=d(\gamma_n(s),q_n)$, $|\beta_n\tilde q_n|=d(\mu_n(s'),q_n)$ and +$|\alpha_n\beta_n|=d(\gamma_n(s),\mu_n(s'))$. Clearly, for fixed $s$ +and $s'$ sufficiently small, $\theta_n(s,s')$ converges as +$n\rightarrow\infty$ to $\theta(\gamma,s,\mu,s')$. By the Toponogov +property (\entryref{04518fa1ac2f}) for manifolds with +non-negative curvature, for each $n$ the function $\theta_n(s,s')$ +is a non-increasing function of each variable, when the other is +held fixed. This property then passes to the limit, giving the first +statement in the lemma. + +By the monotonicity, $\theta(\gamma,\mu)=0$ if and only if for all +$s,s'$ sufficiently small we have $\theta(\gamma,s,\mu,s')=0$, which +means one of $\gamma$ and $\mu$ is contained in the other. + +It is obvious that the last statement holds. + +It follows that $\theta(\gamma,\mu)$ yields a well-defined function +on the set of pairs of directions at ${\mathcal E}$. It is clearly a +symmetric, non-negative function which is positive off of the +diagonal. The next lemma shows that it is a metric by establishing +the triangle inequality for $\theta$. + +\entryblock{ddf1aed4b3bb} + +**Proof.** By \entryref{7ef7dfbbe151}, after replacing $\gamma, \mu,\nu$ by +equivalent, shorter geodesic arcs, we can assume that they are the +unique minimizing geodesics from their end points, say $x,y,z$ +respectively, to ${\mathcal E}$. Let $q_n$ be a sequence of points +limiting to ${\mathcal E}$, and let $\gamma_n,\mu_n,\nu_n$ be +minimizing geodesics from $x,y,z$ to $q_n$. Denote by +$\theta_n(x,y), \theta_n(y,z)$, and $\theta_n(x,z)$, respectively, +the angles at $\tilde q_n$ of the Euclidean triangles with the +following edge lengths: $\{d(x,y), d(x,q_n), d(y,q_n)\}$, +$\{d(y,z),d(y,q_n),d(z,q_n)\}$, and $\{d(z,x),d(z,q_n),d(x,q_n)\}$. +According to \entryref{dc263832416c} we have +$\theta_n(x,y)+\theta_n(y,z)\ge \theta_n(x,z)$. Passing to the limit +as $n$ goes to $\infty$ and then the limit as $x$, $y$ and $z$ tend +to ${\mathcal E}$, gives the result. + +\entryblock{7bb59bcd7682} + +\entryblock{72be1154f65a} + +**Proof.** By construction $(C_{\mathcal E},g_{\mathcal E})$ is a metric cone. +That it is not homeomorphic to a ray follows immediately from +\entryref{dda8a7c7acb6}. + +### Comparison results for distances + +\entryblock{317c62593611} + +**Proof.** By the definition of the cone metric we have + +$$ +d_{g_{\mathcal E}}(x',y')=a^2+b^2-2ab \mathit{cos}(\theta(\gamma,\mu)). +$$ + +On the other hand by \entryref{f4ee075394ef} and the law of +cosines for Euclidean triangles, we have + +$$ +d_{g_\infty}(x,y)=a^2+b^2-2ab \mathit{cos}(\theta(\gamma,a,\mu,b)). +$$ + +The result is now immediate from the fact, proved in +\entryref{f4ee075394ef} that + +$$ +0\le \theta(\gamma,a,\mu,b)\le \theta(\gamma,\mu)\le \pi, +$$ + +and the fact that the cosine is a monotone decreasing function on +the interval $[0,\pi]$. + +\entryblock{45d33631de13} + +**Proof.** This is immediate by applying the previous lemma to the rescaled +manifold $(U_\infty,\lambda g_\infty)$, and noticing that rescaling +does not affect the cone $C_{\mathcal E}$ nor its metric. + +\entryblock{aedf30fd9f7e} + +**Proof.** Let $K$ be such that, given $K$ points in the central $2$-sphere of +any $2\epsilon$-tube of scale $1$, at least two are within distance +$\delta/2$ of each other. Now suppose that we have $K$ directions +$\ell_1,\ldots,\ell_K$ at ${\mathcal E}$. Let +$\gamma_1,\ldots,\gamma_K$ be minimizing geodesic rays limiting to +${\mathcal E}$ that represent these directions. Choose a point $x$ +sufficiently close to the end ${\mathcal E}$ so that all the +$\gamma_j$ cross the central $2$-sphere $S^2$ of the +$2\epsilon$-neck centered at $x$. By replacing the $\gamma_j$ with +sub-rays we can assume that for each $j$ the endpoint $x_j$ of +$\gamma_j$ lies in $S^2$. Let $d_j$ be the length of $\gamma_j$. By +taking $x$ sufficiently close to ${\mathcal E}$ we can also assume +the following. For each $j$ and $j'$, the angle at $e$ of the +Euclidean triangle $\alpha_je\alpha_{j'}$, where $|\alpha_je|=d_j; +|\alpha_{j'}e|=d_{j'}$ and $|\alpha_j\alpha_{j'}|=d(x_j,x_{j'})$ is +within $\delta/2$ of $\theta(\ell_j,\ell_{j'})$. Now there must be +$j\not= j'$ with $d(x_i,x_j)<(\delta/2)r_i$ where $r_i$ is the scale +of $N_i$. Since $d_j,d_{j'}>\epsilon^{-1}r_i/2$, it follows that the +angle at $e$ of $\alpha_je\alpha_{j'}$ is less than $\delta/2$. +Consequently, $\theta(\ell_j,\ell_{j'})<\delta$. + +Recall that a $\delta$-net in a metric space $X$ is a finite set of +points such that $X$ is contained in the union of the +$\delta$-neighborhoods of these points. The above lemma immediately +yields: + +\entryblock{f7e1d840a49f} + +### Completion of the proof of a cone limit at ${\mathcal E}$ + +Now we are ready to prove \entryref{9cf6ad839dbb}. In fact, we +prove a version of the proposition that identifies the sequence of +points $x_n$ and also identifies the cone to which the +rescaled manifolds converge. + +\entryblock{b34d555cd7b5} + +**Proof.** It suffices to prove that given any subsequence of the original +sequence, the result holds for a further subsequence. So let us +replace the given sequence by a subsequence. Recall that for each +$00$ there is a $\delta$-net +$({\mathcal N},h)$ in $A_{\mathcal E}(r,R)$, with $([\gamma],1)\in +{\mathcal N}$ such that for all $n$ sufficiently large there are +embeddings $\varphi_n$ of ${\mathcal N}$ into $A_n(r,R)=\bar +B_{\lambda_ng_\infty}({\mathcal E},R)\setminus +B_{\lambda_ng_\infty}({\mathcal E},r)$ with the following four +properties: + +- **(1)** $\varphi_n^*(\lambda_ng_\infty)$ converge to $h$ as $n\rightarrow\infty$, +- **(2)** $\varphi_n([\gamma],1)=x_n$, +and +- **(3)** $\varphi_n({\mathcal N})$ is a $\delta$-net in $A_n(r,R)$, and +- **(4)** denoting the cone point by $c\in C_{\mathcal E}$, +if $d(p,c)=r$ then $d(\varphi_n(p),{\mathcal E})=r$. + +According to \entryref{f7e1d840a49f} there is a $\delta$-net +${\mathcal N}\subset A_{\mathcal E}(r,R)$ consisting of points +$(\ell_i,s_i)$ where the $\ell_i$ are realizable directions and +$r + +# Chapter 11 -- Geometric limits of generalized Ricci flows + +In this chapter we apply the main result of the last section, +bounded curvature at bounded distance, to blow-up limits in order to +establish the existence of a smooth limit for sequences of +generalized Ricci flows. In the first section we establish a blow-up +limit that is defined for some interval of time of positive length, +where the length of the interval of time is allowed to depend on the +limit. In the second section we give conditions under which this +blow-up limit can be extended backwards to make an ancient Ricci +flow. In the third section we construct limits at the singular time +of a generalized Ricci flow satisfying appropriate conditions. We +characterize the ends of the components of these limits. We show +that they are $\epsilon$-horns -- the ends are diffeomorphic to +$S^2\times [0,1)$ and the scalar curvature goes to infinity at the +end. In the fourth section we prove for any $\delta>0$ that there +are $\delta$-necks sufficiently deep in any $\epsilon$-horn, +provided that the curvature at the other end of the horn is not too +large. Throughout this chapter we fix $\epsilon>0$ sufficiently +small such that all the results of the Appendix hold for $2\epsilon$ +and $\alpha=10^{-2}$, and \entryref{cb9ca6d6db9f} holds for +$2\epsilon$. + +## A smooth blow-up limit defined for a small time + +We begin with a theorem that produces a blow-up limit flow defined on some small time interval. + +\entryblock{d51aab8a8065} + +Before beginning the proof of this theorem we establish a lemma that +we shall need both in its proof and also for later applications. + +\entryblock{0ddfe578961e} + +**Proof.** We first prove that for any $y\in B(z,t_0,r)$ we have + +$$ +R(y)\le \frac{16}{9}(R(z)+r_0^{-2}). +$$ + + Let $\gamma\colon +[0,s_0]\to B(z,t_0,r)$ be a path of length $s_00$ and, for each $n$ sufficiently large, an embedding of +$B_{Q_nG_n}(x_n,0,A)\times [-t_0(A),0]$ into ${\mathcal M}_n$ +compatible with time and with the vector field. In fact, we can +choose $t_0(A)$ so that more is true. + +\entryblock{3989b6b34e08} + +**Proof.** This is immediate from \entryref{0ddfe578961e} and Assumption (5) +in the hypothesis of the theorem. + +Now since the curvatures of the $Q_nG_n$ are pinched toward positive +or are non-negative, bounding the scalar curvature above gives a +bound on $|\mathit{Rm}_{Q_nG_n}|$ on the product +$B_{Q_nG_n}(x_n,0,A)\times (-t'_0(A),0]$. Now we invoke Shi's +theorem (\entryref{ed4b72caf9f9}): + +\entryblock{57c8cc8a5611} + +Also, by the curvature bound and the $\kappa$-non-collapsed +hypothesis we have the following: + +\entryblock{61143f134fcf} + +Now we are in a position to apply \entryref{b45765a3070e}. This +implies that, after passing to a subsequence, there is a geometric +limit $(M_\infty,g_\infty,x_\infty)$ of the sequence of based +Riemannian manifolds $(M_n,Q_nG_n(0),x_n)$. The geometric limit is a +complete Riemannian manifold. If the $({\mathcal M}_n,G_n)$ satisfy +the curvature pinched toward positive hypothesis, by +\entryref{2ddfbc74b081}, the limit Riemannian manifold +$(M_\infty,g_\infty)$ has non-negative curvature. If the $({\mathcal +M}_n,G_n)$ have non-negative curvature, then it is obvious that the +limit has non-negative curvature. By construction $R(x_\infty)=1$. + +In fact, by \entryref{4673eb9f4424} for each $A<\infty$, +there is $t(A)>0$ and, after passing to a subsequence, geometrically +limit flow defined on $B(x_\infty,0,A)\times (-t(A),0]$. + +\entryblock{9f8eade55fcd} + +**Proof.** The fact that $(M_\infty,g_\infty,x_\infty)$ is the geometric limit +of the $(M_n,Q_nG_n(0),x_n)$ means that we have the following. There +is an exhausting sequence $V_1\subset V_2\subset\cdots \subset +M_\infty$ of open subsets of $M_\infty$, with compact closure, each +containing $x_\infty$, and for each $n$ an embedding $\varphi_n$ of +$V_n$ into the zero time-slice of ${\mathcal M}_n$ such that +$\varphi_n(x_\infty)=x_n$ and such that the Riemannian metrics +$\varphi_n^*G_n$ converge uniformly on compact sets to $g_\infty$. +Let $q\in M_\infty$ be a point with $R_{g_\infty}(q)>4$. Then for +all $n$ sufficiently large, $q\in V_n$, so that $q_n=\varphi_n(q)$ +is defined, and $R_{Q_nG_n}(q_n)>4$. Thus, $q_n$ has an +$(C,\epsilon)$-canonical neighborhood, $U_n$, in ${\mathcal M}_n$; +and, since $R(q_n)>4$ for all $n$, there is a uniform bound to the +distance from any point of $U_n$ to $q_n$. Thus, there exists $m$ +such that for all $n$ sufficiently large $\varphi_n(V_m)$ contains +$U_n$. Clearly as $n$ goes to infinity the Riemannian metrics +$\varphi_n^*(G_n)|_{\varphi_n^{-1}(U_m)}$ converge smoothly to +$g_\infty|_{\varphi_n^{-1}(U_n)}$. Thus, by +\entryref{7c2758f0e2b3} for all $n$ sufficiently large the +restriction of $g_\infty$ to $\varphi_n^{-1}(U_n)$ contains a +$(2C,2\epsilon)$-canonical neighborhood of $q$. + +\entryblock{620ff6c41ef4} + +**Proof.** First, suppose that $(M_\infty,g_\infty)$ does not have strictly +positive curvature. Suppose that $y\in M_\infty$ has the property +that $\mathit{Rm}(y)$ has a zero eigenvalue. Fix $A<\infty$ greater +than $d_{g_\infty}(x_\infty,y)$. Then applying +\entryref{8ec855a2aab5} to the limit flow on +$B(x_\infty,0,A)\times (-t(A),0]$, we see that the Riemannian +manifold $(B(x_\infty,0,A),g_\infty)$ is locally a Riemannian +product of a compact surface of positive curvature with a +one-manifold. Since this is true for every $A<\infty$ sufficiently +large, the same is true for $(M_\infty,g_\infty)$. Hence +$(M_\infty,g_\infty)$ has a one- or two-sheeted covering that is a +global Riemannian product of a compact surface and one-manifold. +Clearly, in this case the curvature of $(M_\infty,g_\infty)$ is +bounded. + +If $M_\infty$ is compact, then it is clear that the curvature is +bounded. + +It remains to consider the case where $(M_\infty,g_\infty)$ is +non-compact and of strictly positive curvature. Since any point of +curvature greater than $4$ has a $(2C,2\epsilon)$-canonical +neighborhood, and since $M_\infty$ is non-compact, it follows that +the only possible canonical neighborhoods for $x\in M_\infty$ are a +$2\epsilon$-neck centered at $x$ or $(2C,2\epsilon)$-cap whose core +contains $x$. Each of these canonical neighborhoods contains a +$2\epsilon$-neck. Thus, if $(M_\infty,g_\infty)$ has unbounded and +positive Riemann curvature or equivalently, it has unbounded scalar +curvature, then it has $(2C,2\epsilon)$-canonical neighborhoods of +arbitrarily small scale, and hence $2\epsilon$-necks of arbitrarily +small scale. But this contradicts \entryref{cb9ca6d6db9f}. It +follows from this contradiction that the curvature of +$(M_\infty,g_\infty)$ is bounded. + +To complete the proof of \entryref{d51aab8a8065} it remains to +extend the limit for the $0$ time-slices of the $({\mathcal +M}_n,G_n)$ that we have just constructed to a limit flow defined for +some positive amount of time backward. Since the curvature of +$(M_\infty,g_\infty)$ is bounded, this implies that there is a +bound, $Q$, such that for any $A<\infty$ + the curvature of the restriction of +$Q_nG_n$ to $B_{Q_nG_n}(x_n,0,A)$ is bounded by $Q$ for all $n$ +sufficiently large. Thus, we can take the constant $Q(A)$ in +\entryref{3989b6b34e08} to be independent of $A$. According to that +corollary this implies that there is a $t'_0>0$ and $Q'<\infty$ such +that for every $A$ there is an embedding $B_{Q_nG_n}(x_N,0,A)\times +(-t'_0,0]\to {\mathcal M}_n$ compatible with time and with the +vector field so that the scalar curvature of the restriction of +$Q_nG_n$ to the image is bounded by $Q'$ for all $n$ sufficiently +large. This uniform bound on the scalar curvature yields a uniform +bound, uniform in the sense of being independent of $n$, on $|\mathit{Rm}_{Q_nG_n}|$ on the image of the embedding +$B_{Q_nG_n}(x_N,0,A)\times (-t'_0,0]$. + + Then by Hamilton's result, +\entryref{4673eb9f4424}, we see that, after passing to a +further subsequence, there is a limit flow defined on $(-t_0',0]$. +Of course, the zero time-slice of this limit flow is the limit +$(M_\infty,g_\infty)$. This completes the proof of +\entryref{d51aab8a8065}. + +## Long-time blow-up limits + +Now we wish to establish conditions under which we can, after +passing to a further subsequence, establish the existence of a +geometric limit flow defined on $-\infty0$. Our next step is to +extend the limit flow all the way back to time $-T_0$. + +\entryblock{d6919616448d} + +**Proof.** The +argument is by contradiction, so we suppose that there is a $T 4$ has a $(2C,2\epsilon)$-canonical +neighborhood. Hence, applying \entryref{0ddfe578961e} as in the +argument in the proof of + \entryref{3989b6b34e08} shows that for all $n$ sufficiently large, +every point in $B_{Q_nG_n}(x_n,-T',A)$ has a uniform size parabolic +neighborhood on which the Riemann curvature is uniformly bounded, +where both the time interval in the parabolic neighborhood and the +curvature bound on this neighborhood depend only on $C$ and the +curvature bound on $Q$ for the limit flow. According to Hamilton's +result (\entryref{4673eb9f4424}) this implies that, by +passing to a further subsequence, we can extend the limit flow +backward + beyond $-T'$ a uniform amount of time, say $2\delta$. Taking $T'>T-\delta$ + then gives the desired extension under the condition that the +scalar curvature is bounded on $(-T,0]$. + +It remains to show that, provided that $T0$ sufficiently small, +by the Harnack inequality +(the second result in \entryref{aa9d836895d8}) + we have + +$$ +\frac{\partial R}{\partial t}(x,t)\ge -\frac{R(x,t)}{t+T-\epsilon}. +$$ + +Taking the limit as $\epsilon\rightarrow 0$ gives + +$$ +\frac{\partial R}{\partial t}(x,t)\ge -\frac{R(x,t)}{t+T}, +$$ + +and hence, fixing $x$, + +$$ +\frac{dR(x,t)}{R(x,t)}\ge \frac{-dt}{(t+T)}. +$$ + +Integrating from $-t_0$ to $0$ shows that + +$$ +\mathit{log}(R(x,0))-\mathit{log}(R(x,-t_0))\ge \mathit{log}(T-t_0)-\mathit{log}(T), +$$ + +and since $R(x,0)\le Q$, this implies + +$$ +R(x,-t_0)\le Q\frac{T}{T-t_0}. +$$ + +Recalling that $n=3$ and that the curvature is non-negative we see that + +$$ +\mathit{Ric}(x,-t_0)\le (n-1)\frac{QT}{2}\frac{1}{T-t_0}. +$$ + +Hence by \entryref{01a829e22cf6}, for all $-t_0\in (-T,0]$ we have +that + +$$ +\mathit{dist}_{-t_0}(x,y)\le \mathit{dist}_0(x,y)+8\int_{-t_0}^0\sqrt{\frac{QT}{3(T+t)}} +\le \mathit{dist}_0(x,y)+16\sqrt{\frac{Q}{3}}T. +$$ + +It follows immediately from this claim that any compact subset +$X\subset M_\infty$ has uniformly bounded diameter under all the +metrics $g_\infty(t);\ -T3d_n/2$. This +contradiction proves the lemma. + +\entryblock{9a7e3fe2e5c4} + +**Proof.** Suppose this does not hold for some $y\in M_\infty\setminus +B(x_\infty,0,D)$. Let $d=d_0(x_\infty,y)$. Of course, $d\ge D$. +Thus, by the lemma there is $z\in M_\infty$ with $d_0(y,z)=d$ and +$d_0(x_\infty,z)>3d/2$. Since the scalar curvature $R(y,t)$ is not +uniformly bounded for all $t\in (-T,0]$, there is $t$ for which +$R(y,t)$ is arbitrarily large and hence $(y,t)$ has an +$(2C,2\epsilon)$-canonical neighborhood of arbitrarily small scale. +By \entryref{002f3496c54c} we have $d_{t}(x_\infty,y)\le +d+8\sqrt{\frac{Q}{3}}T$ and $d_{t}(y,z)\le d+8\sqrt{\frac{Q}{3}}T$. +Of course, since $\mathit{Ric}\ge 0$ the metric is non-increasing in +time and hence $d\le \mathit{min}(d_{t}(y,z),d_{t}(x_\infty,y))$ and +$3d/2\le d_{t}(x_0,z)$. Since $y$ has a $(2C,2\epsilon)$-canonical +neighborhood in $(M_\infty,g_\infty(t))$, either $y$ is the center +of an $2\epsilon$-neck in $(M_\infty,g_\infty(t))$ or $y$ is +contained in the core of a $(2C,2\epsilon)$-cap in +$(M_\infty,g_\infty(t))$. (The other two possibilities for canonical +neighborhoods require that $M_\infty$ be compact.) + +\entryblock{b3fae89d3264} + +**Proof.** Let ${\mathcal C}$ be a $(2C,2\epsilon)$-canonical +neighborhood of $y$ in +$(M_\infty,g_\infty(t))$. Since $R(y,t)$ can be arbitrarily large, +we can assume that $d \gg 2CR(y)^{-1/2}$, which is a bound on the +diameter of ${\mathcal C}$. This means that minimal $g(t)$-geodesics +$\gamma_{x_\infty}$ and $\gamma_z$ connecting $y$ to $x_\infty$ and +to $z$, respectively, must exit from ${\mathcal C}$. Let $a$ be a +point on $\gamma_{x_\infty}\cap{\mathcal C}$ close to the complement +of ${\mathcal C}$. Let $b$ be a point at the same $g(t)$-distance +from $y$ on $\gamma_z$. In the case that ${\mathcal C}$ is a cap +or that it is a $2\epsilon$-neck and $\gamma_{x_\infty}$ and +$\gamma_z$ exit from the same end, then +$d_t(b,y)/d_t(a,y)<4\pi\epsilon$. This means that the angle $\theta$ +of the Euclidean triangle with these side lengths at the point +corresponding to $y$ satisfies + +$$ +\mathit{cos}(\theta)\ge 1-\frac{(4\pi\epsilon)^2}{2}. +$$ + +Recall that $Q$ is the maximum value of $R(x,0)$, and that by +\entryref{002f3496c54c} we have + +$$ +d\le d_t(x_\infty,y)\le d+16\sqrt{\frac{Q}{3}}T, +$$ + + with the same inequalities holding with $d_t(z,y)$ replacing $d_t(x_\infty,y)$. + Also, by construction $d\ge 32\sqrt{\frac{Q}{3}}T$. +We set $a_0=d_t(x_\infty,y)$ and $a_1=d_t(z,y)$. Then by the +Toponogov property we have + +$$ +d_t(x,z)^2\le a_0^2+a_1^2-2a_0a_1\left( +1-\frac{(4\pi\epsilon)^2}{2}\right)=(a_0-a_1)^2+(4\pi\epsilon)^2a_0a_1. +$$ + +Since $|a_0-a_1|\le d/2$ and $a_0,a_1\le 3d/2$ and +$\epsilon<1/8\pi$, it follows that $d_t(x,z)4\pi R(y,t)^{-1/2}$, neither $z$ nor $x_\infty$ lies +in this ball.) Thus, if $R(y,t)$ is unbounded as $t\rightarrow -T$ +then arbitrarily small $g(0)$-balls centered at $y$ separate $z$ and +$x_\infty$. Since $y$ is distinct from $x_\infty$ and $z$, this is +clearly impossible. + +Next we establish that the curvature near the base point $x_\infty$ +is bounded for all $t\in (-T,0]$. + +\entryblock{a66fa0ea8720} + +**Proof.** First we pass to a subsequence so that a geometric limit flow + +$$ +(M_\infty,g_\infty(t),(x_\infty,0)) +$$ + + exists on $(-T,0]$. We let +$Q$ be the upper bound for $R(x,0)$ for all $x\in M_\infty$. We now +divide the argument into two cases: (i) $M_\infty$ is compact, and +(ii) $M_\infty$ is non-compact. + +Suppose that $M_\infty$ is compact. By \entryref{95cc838cfc6d} +we know that + +$$ +\mathit{min}_{x\in M_\infty}(R_{g_\infty}(x,t)) +$$ + +is a non-decreasing function of $t$. Since +$R_{g_\infty}(x_\infty,0)=1$, it follows that for each $t\in +(-T,0]$, we have $\mathit{min}_{x\in M_\infty}R(x,t)\le 1$, and hence +there is a point $x_{t}\in M_\infty$ with $R(x_{t},t)\le 1$. Now we +can apply \entryref{cc3a26ca3809} to see that the scalar curvature of +$g_\infty$ is bounded on all of $M_\infty\times (-T,0]$. + +If $M_\infty$ is non-compact, choose $D$ as in \entryref{9a7e3fe2e5c4}. +According to that lemma every point in the boundary of +$B(x_\infty,0,D)$ has bounded curvature under $g_\infty(t)$ for all +$t\in (-T,0]$. In particular, for each $t\in (-T,0]$ the minimum of +$R(x,t)$ over $\overline B(x_\infty,0,D)$ is bounded independent of +$t$. Now apply \entryref{cc3a26ca3809} to the closure of +$B(x_\infty,0,D)$. We conclude that the curvature of +$B(x_\infty,0,D)$ is uniformly bounded for all $g_\infty(t)$ for all +$t\in (-T,0]$. In particular, $R(x_\infty,t)$ is uniformly bounded +for all $t\in (-T,0]$. + +Now for any $A<\infty$ we apply \entryref{cc3a26ca3809} to the compact +subset $\overline B(x_\infty,0,A)$ to conclude that the curvature is +uniformly bounded on $B(x_\infty,0,A)\times (-T,0]$. This completes +the proof of the corollary. + +Now let us return to the proof of \entryref{d6919616448d}. + +\entryblock{b3b96a664796} + +**Proof.** Fix $A<\infty$ and let $K$ be the bound for the scalar curvature of +$g_\infty$ on $B(x_\infty,0,2A)\times (-T,0]$ from +\entryref{a66fa0ea8720}. \entryref{0ddfe578961e} shows that +there are $\delta>0$ and a bound in terms of $K$ and $C$ on the +scalar curvature of the restriction of $Q_nG_n$ to +$B_{Q_nG_n}(x_n,0,A)\times [-(T+\delta),0]$. + +Since the scalar curvature is bounded, by the assumption that either +the curvature is pinched toward positive or the Riemann curvature is +non-negative, this implies that the sectional curvatures of $Q_nG_n$ +are also uniformly bounded on the products +$B_{Q_nG_n}(x_n,0,A)\times [-(T+\delta),0]$ for all $n$ sufficiently +large. Consequently, it follows that by passing to a further +subsequence we can arrange that the $-T$ time-slices of the +$({\mathcal M}_n,G_n,x_n)$ converge to a limit +$(M_\infty,g_\infty(-T))$. + This limit manifold +satisfies the hypothesis of \entryref{cb9ca6d6db9f} and hence, by +that proposition, it has bounded sectional curvature. This means +that there is a uniform $\delta>0$ such that for all $n$ +sufficiently large and for any $A<\infty$ the scalar curvatures (and +hence the Riemann curvatures) of the restriction of $Q_nG_n$ to +$B_{Q_nG_n}(x_n,0,A)\times [-(T+\delta),0]$ are uniformly bounded. +This allows us to pass to a further subsequence for which there is a +geometric limit defined on $(-(T+\delta/2),-T]$. This geometric +limit is complete of bounded, non-negative curvature. Hence, we +have now constructed a limit flow on $(-(T+\delta/2),0]$ with the +property that for each $t\in (-(T+\delta/2),0]$ the Riemannian +manifold $(M,g(t))$ is complete and of bounded non-negative +curvature. (We still don't know whether the entire flow is of +bounded curvature.) But now invoking Hamilton's Harnack inequality +(\entryref{aa9d836895d8}), we see that the curvature is bounded on +$[-T,0]$. Since we already know it is bounded in $(-T+\delta/2,-T]$, +this completes the proof of the proposition. + + It follows immediately from \entryref{d6919616448d} that there is a +geometric limit flow defined on $(-T_0,0]$. The geometric limit +flow on $(-T_0,0]$ is complete of non-negative curvature, locally +bounded in time. + +It remains to prove the last statement in the theorem. So let us +suppose that $T_0=\infty$. We have just established the existence of +a geometric limit flow defined for $t\in (-\infty,0]$. Since the +$({\mathcal M}_n,G_n)$ either have curvature pinched toward positive +or are of non-negative curvature, it follows from +\entryref{2ddfbc74b081} that all time-slices of the limit flow +are complete manifolds of non-negative curvature. Since points of +scalar curvature greater than $4$ have $(2C,2\epsilon)$-canonical +neighborhoods, it follows from \entryref{cb9ca6d6db9f} that the +curvature is bounded on each time-slice, and hence universally +bounded by the Harnack inequality (\entryref{aa9d836895d8}). Since for +any $A<\infty$ and every $T<\infty$ the parabolic neighborhoods +$B_{Q_nG_n}(x_n,0,A)\times [-T,0]$ are $\kappa$-non-collapsed on +scales $Q_nr$ for every $n$ sufficiently large, the limit is +$\kappa$-non-collapsed on scales $\le \mathit{lim}_{n\rightarrow +\infty}Q_nr$. Since $r>0$ and $\mathit{lim}_{n\rightarrow +\infty}Q_n=\infty$, it follows that the limit flow is +$\kappa$-non-collapsed on all scales. Since $R_{Q_n G_n}(x_n)=1$, +$R_{g_\infty}(x_\infty,0)=1$ and the limit flow is non-flat. This +establishes all the properties need to show that the limit is a +$\kappa$-solution. This completes the proof of +\entryref{a9e023d082dc}. + +## Incomplete smooth limits at singular times + +Now we wish to consider smooth limits where we do not blow up, i.e., +do not rescale the metric. In this case the limits that occur can be +incomplete, but we have strong control over their ends. + +### Assumptions + +We shall assume the following about the generalized Ricci flow +$({\mathcal M},G)$: + +\entryblock{3009be0a37fc} + +With these assumptions we can say quite a bit about the limit metric +at time $T$. + +\entryblock{af00774dbaa9} + +\entryblock{d6585ba392cf} + +In order to prove this result we establish a sequence of lemmas. The +first in the series establishes that $\Omega$ is an open subset and +also establishes the first two of the above five conclusions. + +\entryblock{a99536ac8110} + +**Proof.** We pull back $G$ to $M\times [T^-,T)$ to define a Ricci flow +$(M,g(t)),\ T^{-}\le t0$ such that $R(y,t)$ is +uniformly bounded for $y\in B(x,t_n,r)\times [t_n,T)$. This means +that $B(x,t_n,r)\subset \Omega$, proving that $\Omega$ is open in +$M$. + +Furthermore, since $R(y,t)$ is bounded on $B(x,t_n,r)\times +[t_n,T)$, it follows from the curvature pinching toward positive +hypothesis that $|\mathit{Rm}(y,t)|$ is bounded on +$B(x,t_n,r)\times[t_n,T)$. Now applying \entryref{ed4b72caf9f9} we see +that in fact $\mathit{Rm}$ is bounded in the $C^\infty$-topology on +$B(x,t_n,r)\times[(t_n+T)/2 ,T)$. The same is of course also true +for $\mathit{Ric}$ and hence for $\frac{\partial g}{\partial t}$ in the +$C^\infty$-topology. It then follows that there is a continuous +extension of $g$ to $B(x,t_n,r)\times [t_n,T]$. Since this is true +for every $x\in \Omega$ we see that $g(t)$ converges in the +$C^\infty$-topology, uniformly on compact subsets of $\Omega$, to +$g(T)$. + +Lastly, let us consider the function $R(g(T))$ on $\Omega$. Since +the metric $g(T)$ is a smooth metric on $\Omega(T)$, this is a +smooth function. Clearly, by the curvature pinching toward positive +hypothesis, this function is bounded below. We must show that it is +proper. Since $M$ is compact, it suffices to show that if $x_n$ is a +sequence in $\Omega\subset M$ converging to a point $x\in M\setminus +\Omega$ then $R(x_n,T)$ is unbounded. Suppose that $R(x_n,T)$ is +bounded independent of $n$. It follows from \entryref{0ddfe578961e} +that there is a positive constant $\Delta t$ such that $R(x_n,t)$ is +uniformly bounded for all $n$ and all $t\in [T-\Delta t,T)$, and +hence, by the same result, there is $r>0$ such that $R(y_n, t)$ is +bounded for all $n$, all $y_n\in B(x_n,T-\Delta t,r)$, and all +$t\in [T-\Delta t,T)$. Since the $x_n\rightarrow x\in M$, it follows +that for all $n$ sufficiently large that $x\in B(x_n,T-\Delta t,r)$, +and hence $R(x,t)$ is uniformly bounded as $t\rightarrow T$. This +contradicts the fact that $x\not\in \Omega$. + +\entryblock{8651cf20bd2d} + +At this point we have established the first three of the five +conclusions stated in \entryref{af00774dbaa9}. Let us turn to the last +two. + +### Canonical neighborhoods for $(\widehat{\mathcal M},\widehat G)$ + +We continue with the notation and assumptions of the previous +subsection. Here we establish the fifth conclusion in +\entryref{af00774dbaa9}, namely the existence of strong canonical +neighborhoods for +$(\widehat{\mathcal M},\widehat G)$ + +\entryblock{0760630d3990} + +**Proof.** We fix $x\in \Omega(T)$ with $R(x,T)>r_0^{-2}$. First notice that + for all $tr_0^{-2}$. Thus, for all such $t$ the point $(x,t)$ has +a strong $(C,\epsilon)$-canonical neighborhood in $({\mathcal M},G)\subset (\widehat{\mathcal +M},\widehat G)$. Furthermore, since $\mathit{lim}_{t\rightarrow +T}R(x,t)=R(x,T)<\infty$, for all $tr_0^{-2}$ has a strong $(2C,2\epsilon)$-canonical +neighborhood. Since, by assumption, points at time before $T$ with +scalar curvature at least $r_0^{-2}$ have strong +$(C,\epsilon)$-canonical neighborhoods, this completes the proof of +the fifth conclusion of \entryref{af00774dbaa9}. It remains to establish +the fourth conclusion of that theorem. + +### The ends of $(\Omega,g(T))$ + +\entryblock{da99c0859788} + +\entryblock{08c3ff9d0a1e} + +See Fig. 11.2. + +\entryblock{c2ac821436de} + +\entryblock{2a16041cf547} + +**Proof.** Let $\Omega^0$ be a component of $\Omega$ containing no point of +$\Omega_\rho$. Then for every $x\in \Omega^0$, we have +$R(x,T)>r_0^{-2}$. Therefore, by \entryref{0760630d3990} $(x,T)$ +has a $(2C,2\epsilon)$-canonical neighborhood. Of course, this +entire neighborhood is contained in $\widehat{\mathcal M}$ and hence +is contained in $\Omega^0$ (or, more precisely, in the case of +strong $2\epsilon$-necks in the union of maximum backward flow lines +ending at points of $\Omega^0$). If the canonical neighborhood of +$(x,T)\in \Omega^0$ is a $2C$-component or is an $2\epsilon$-round +component, then of course $\Omega^0$ is that $2C$-component or +$2\epsilon$-round component. Otherwise, each point of $\Omega^0$ is +either the center of a strong $2\epsilon$-neck or is contained in +the core of a $(2C,2\epsilon)$-cap. We have chosen $2\epsilon$ +sufficiently small so that the result follows from +\entryref{fa8d9a43f34a}. + +\entryblock{1c2d9d005b0e} + +It follows immediately from this lemma that if $X$ is a component of $\Omega$ +not containing any point of $\Omega_\rho$, then every end of $X$ is contained +in a strong $2\epsilon$-tube. To complete the proof of \entryref{af00774dbaa9}, it +remains only to establish the same result for the components of $\Omega$ that +meet $\Omega_\rho$. That is part of the content of the next lemma. + +\entryblock{01f0d3baa28f} + +**Proof.** Since $R\colon \Omega\times\{T\}\to \Ar$ is a proper function +bounded below, $\Omega_\rho$ is compact. Hence, there are only +finitely many components of $\Omega$ containing points of +$\Omega_\rho$. Let $\Omega^0$ be a non-compact component of $\Omega$ +containing a point of $\Omega_\rho$, and let ${\mathcal E}$ be an +end of $\Omega^0$. Let + +$$ +X=\{x\in \Omega^0\bigl|\bigr. +R(x)\ge 2C^2\rho^{-2}\}. +$$ + + Then $X$ is a closed set and contains a +neighborhood of the end ${\mathcal E}$. Since $\Omega^0$ contains a +point of $\Omega_\rho$, $\Omega^0\setminus X$ is non-empty. Let +$X_0$ be the connected component of $X$ that contains a neighborhood +of ${\mathcal E}$. This is a closed, connected set every point of +which has a $(2C,2\epsilon)$-canonical neighborhood. Since $X_0$ +includes an end of $\Omega^0$, no point of $X_0$ can be contained in +an $\epsilon$-round component nor in a $C$-component. Hence, every +point of $X_0$ is either the center of a strong $2\epsilon$-neck or +is contained in the core of a $(2C,2\epsilon)$-cap. Since +$2\epsilon$ is sufficiently small to invoke +\entryref{0e9a1a167caa}, the latter implies that $X_0$ is +contained either in a $2\epsilon$-tube which is a union of strong +$2\epsilon$-necks centered at points of $X_0$ or $X_0$ is contained +in a $2C$-capped $2\epsilon$-tube where the core of the cap contains +a point of $X_0$. ($X_0$ cannot be contained in a double capped +$2\epsilon$-tube since the latter is compact.) In the second case, +since this capped tube contains an end of $\Omega^0$, it is in fact +equal to $\Omega^0$. Since a point of $X_0$ is contained in the core +of the $(2C,2\epsilon)$-cap, the curvature of this point is at most +$2C^2\rho^{-2}$ and hence the curvature at any point of the cap is +at least $2C\rho^{-2}>\rho^{-2}$. This implies that the cap is +disjoint from $\Omega_{\rho}$. Of course, any $2\epsilon$-neck +centered at a point of $X_0$ has curvature at least $C^2\rho^{-2}$ +and hence is also disjoint from $\Omega_\rho$. Hence, if $\Omega^0$ +is a $2C$-capped $2\epsilon$-tube and there is a point of $X_0$ in +the core of the cap, then this component is disjoint from +$\Omega_\rho$, which is a contradiction. Thus, $X_0$ is contained +in a $2\epsilon$-tube made up of strong $2\epsilon$-necks centered +at points of $X_0$. + +This proves that $X_0$ is contained in a strong $2\epsilon$-tube, +$Y$, every point of which has curvature $\ge C^2\rho^{-2}$. Since +$X_0$ is closed but not the entire component $\Omega^0$, it follows +that $X_0$ has a frontier point $y$. Of course, +$R(y)=2C^2\rho^{-2}$. Let $N$ be the strong $2\epsilon$-neck +centered at $y$ and let $S_N^2$ be its central $2$-sphere. Clearly, +every $y'\in S^2_N$ satisfies $R(y')\le 4C^2\rho^{-2}$, so that +$S^2_N$ is contained in $\Omega_{\rho/2C}$. Let $Y'\subset Y$ be the +complementary component of $S^2_N$ in $Y$ that contains a +neighborhood of the end ${\mathcal E}$. Then the closure of $Y'$ is +the required strong $2\epsilon$-horn containing a neighborhood of +${\mathcal E}$, disjoint from $\Omega_\rho$ and with boundary +contained in $\Omega_{\rho/2C}$. + + The last thing to see is +that there are only finitely many such ends in a given component +$\Omega^0$. First suppose that the boundary $2$-sphere of one of the +$2\epsilon$-horns is homotopically trivial in $\Omega^0$. Then this +$2$-sphere separates $\Omega^0$ into two components one of which is +compact and hence $\Omega^0$ has only one boundary component. Thus, +we can assume that all the boundary $2$-spheres of the +$2\epsilon$-horns are homotopically non-trivial. Suppose that two of +these $2\epsilon$-horns containing different ends of $\Omega^0$ have +non-empty intersection. Let $N$ be the $2\epsilon$-neck whose +central $2$-sphere is the boundary of one of the $2\epsilon$-horns. +Then the boundary of the other $2\epsilon$-horn is also contained in +$N$. This means that + the +union of the two $2\epsilon$-horns and $N$ is a component of +$\Omega$. Clearly, this component has exactly two +ends. Thus, we can assume that all the $2\epsilon$-horns with +boundary in $\Omega_{\rho/2C}$ are disjoint. If two of the +$2\epsilon$-horns have boundary components that are topologically +parallel in $\Omega^0\cap \Omega_{\rho/2C}$ (meaning that they are +the boundary components of a compact submanifold diffeomorphic to +$S^2\times I$), then $\Omega^0$ is diffeomorphic to $S^2\times +(0,1)$ and has only two ends. + By compactness of +$\Omega_{\rho/2C}$, there can only be finitely many disjoint +$2\epsilon$-horns with non-parallel, homotopically noon-trivial +boundaries in $\Omega^0\cap \Omega_{\rho/2C}$. This completes the +proof of the fact that each component of $\Omega_{\rho/2C}$ has only +finitely many ends. + +This completes the proof of \entryref{af00774dbaa9}. + +## Existence of strong $\delta$-necks sufficiently deep in a $2\epsilon$-horn + +We keep the notation and assumptions of the previous section. + +\entryblock{074e02b067b8} + +**Proof.** The proof of the first statement is by contradiction. Fix $\rho>0$ +and $\delta>0$ and suppose that there is no $0\epsilon^{-1}\rho/2C$, and thus +$d_{Q_nG_n}(x_n,z)>Q^{1/2}_n\epsilon^{-1}\rho/2C$. This implies +that, given $A<\infty$, for all $n$ sufficiently large, $z\not\in +B_{Q_nG_n}(x_n,0,A)$. Since this is true for all $z\in \partial +{\mathcal H}_n$, it follows that for all $n$ sufficiently large +$B_{Q_n,G_n}(x_n,0,A)\subset {\mathcal H}_n$. Next, we must show +that, for all $n$ sufficiently large, this ball has compact closure. +That is to say, we must show that for every $A$ for all $n$ +sufficiently large the distance from $x_n$ to the end of the horn +${\mathcal H}_n$ is greater than $AQ_n^{-1/2}$. If not, then since +the curvature at the end of ${\mathcal H}_n$ goes to infinity for +each $n$, this sequence would violate \entryref{ecd62fc2ea2b}. + +Because $B(x_n,0,AQ_n^{-1/2})$ is contained in a $2\epsilon$-horn, +it is $\kappa$-non-collapsed on scales $\le r$ for a universal +$\kappa>0$ and $r>0$. Also, because every point in the horn is the +center of a strong $2\epsilon$-neck, for every $n$ sufficiently +large and every $y\in B(x_n,0,AQ_n^{-1/2})$ the flow is defined on +an interval through $y$ defined for backward time $R(y)^{-1}$. + +This completes the proof that all the hypotheses of +\entryref{d51aab8a8065} hold and establishes \entryref{dff170186455}. + +We form a new sequence of generalized Ricci flows from the +$({\mathcal M}_n, G_n)$ by translating by $-T_n$, so that the final +time-slice is at $\mathbf{t}_n=0$, where $\mathbf{t}_n$ is the time +function for ${\mathcal M}_n$. + +\entryref{d51aab8a8065} implies that, after passing to a +subsequence, there is a limit flow +$(M_\infty,g_\infty(t),(x_\infty,0)),\ t\in[-t_0,0])$ defined for +some $t_0>0$ for the sequence $(Q_n\widehat{\mathcal +M}_n,Q_n\widehat G_n,x_n)$. Because of the curvature pinching toward +positive assumption, by \entryref{2ddfbc74b081}, the limit Ricci +flow has non-negative sectional curvature. Of course, +$R(x_\infty)=1$ so that the limit $(M_\infty,g_\infty(0))$ is +non-flat. + +\entryblock{ee7e22fb33f6} + +**Proof.** Because of the fact that the $({\mathcal M}_n,G_n)$ +have curvature pinched toward positive, and since $Q_n$ tend to +$\infty$ as $n$ tends to infinity, it follows that the geometric +limit $(M_\infty,g_\infty)$ has non-negative curvature. In +${\mathcal H}_n$ take a minimizing geodesic ray $\alpha_n$ from +$x_n$ to the end of ${\mathcal H}_n$ and a minimizing geodesic +$\beta_n$ from $x_n$ to $\partial {\mathcal H}_n$. As we have seen, +the lengths of both $\alpha_n$ and $\beta_n$ tend to $\infty$ as +$n\rightarrow \infty$. By passing to a subsequence, we can assume +that the $\alpha_n$ converge to a minimizing geodesic ray $\alpha$ +in $(M_\infty,g_\infty)$ and that the $\beta_n$ converge to a +minimizing geodesic ray $\beta$ in $(M_\infty,g_\infty)$. Since, for +all $n$, the union of $\alpha_n$ and $\beta_n$ forms a piecewise +smooth ray in ${\mathcal H_n}$ meeting the central $2$-sphere of a +$2\epsilon$-neck centered at $x_n$ in a single point and at this +point crossing from one side of this $2$-sphere to the other, the +union of $\alpha$ and $\beta$ forms a proper, piecewise smooth map +of $\Ar$ to $M_\infty$ that meets the central $2$-sphere of a +$2\epsilon$-neck centered at $x_\infty$ in a single point and +crosses from one side to the other at the point. This means that +$M_\infty$ has at least two ends. Since $(M_\infty,g_\infty)$ has +non-negative curvature, according to \entryref{e5b196773169}, this +implies that $M_\infty$ is a product of a surface with $\Ar$. Since +$M$ has non-negative curvature, the surface has non-negative +curvature. Since $M$ has positive curvature at at least one point, +the surface is diffeomorphic to the $2$-sphere. + +According to \entryref{d51aab8a8065}, after passing to a subsequence +there is a limit flow defined on some interval of the form +$[-t_0,0]$ for $t_0>0$. Suppose that, after passing to a subsequence +there is a limit flow defined on $[-T,0]$ for some $00$ and an isometry -$$\psi\colon (S^2\times -(-\infty,4],h_0\times ds^2)\to (\Ar^3\setminus B(p_0,A_0),g_0).$$ The -composition of $\psi^{-1}$ followed by projection onto the second -factor defines a map $s_1\colon \Ar^3\setminus B(p_0,A_0)\to -(-\infty,4]$. Lastly, there is $00,\end{cases}$$ -and then we define the metric $\tilde g$ on ${\mathcal S}$ by first -defining a metric: -$$\hat g=\begin{cases} exp(-2f(s))R(x_0)\rho^*g & \ \ on \ \ -s^{-1}(-\infty,1] \\ -exp(-2f(s))\left(\alpha(s)R(x_0)\rho^*g+(1-\alpha(s))\eta -g_0\right) & \ \ on\ \ s^{-1}([1,2]) \\ -exp(-2f(s))\eta g_0 & \ \ on\ \ s^{-1}([2,A_{r_0}]\\ -\left[\beta(s)exp(-2f(s))+(1-\beta(s))exp(-2f(4+A_0))\right]\eta -g_0 & \ \ on \ \ s^{-1}([A_{r_0},A']),\end{cases}$$ where -$A_{r_0}=4+A_0-r_0$ and $A'=A_0+4$. - Then we define -$$\widetilde g=R(x_0)^{-1}\hat g.$$ -See Fig.~. - -**Theorem.** -There are constants $C_0,q,R_0<\infty$ and $\delta'_0>0$ such that the -following hold for the result $({\mathcal S},\widetilde g)$ of surgery on -$(N,g)$ provided that $R(x_0)\ge R_0$, $0<\delta\le \delta'_0$. Define $f(s)$ -as above with the constants $C_0,\delta$ and then use $f$ to define surgery on -a $\delta$-neck $N$ to produce $({\mathcal S},\tilde g)$. Then the following -hold. -\begin{itemize} -\item Fix $t\ge 0$. For any $p\in N$, -let $X(p)=max(0,-\nu_{g}(p))$, where $\nu_ g(p)$ is the -smallest eigenvalue of $Rm_{ g}(p)$. Suppose that for all $p\in N$ -we have: -\begin{enumerate}\item[(1)] $R(p)\ge \frac{-6}{1+4t}$, and -\item[(2)] $R(p)\geq 2X(p)\left(logX(p)+log(1+t)-3\right)$, whenever $00$ there is $\delta'_1=\delta'_1(\delta'')>0$ such that if -$\delta\le min(\delta'_1,\delta'_0)$, then the restriction of -$\hat g$ to $B_{\hat g}(p_0,(\delta'')^{-1})$ in $({\mathcal S},\hat -g)$ is $\delta''$-close in the $C^{[1/\delta'']}$-topology to the -restriction of the standard initial metric $g_0$ to -$B_{g_0}(p_0,(\delta'')^{-1})$. -\end{itemize} - -The rest of this chapter is devoted to the proof of this theorem. - -Before starting the curvature computations let us make a remark about the surgery cap. - -**Definition.** -The image in ${\mathcal S}$ of $B_{g_0}(p_0,0,A_0+4)$ is called the *surgery cap*. - -The following is immediate from the definitions provided that -$\delta>0$ is sufficiently small. - -**Lemma.** -The surgery cap in $({\mathcal S},\tilde g)$ has a metric that -differs from the one coming from a rescaled version of the standard -solution. Thus, the image of this cap is not necessarily a metric -ball. Nevertheless for $\epsilon<1/200$ the image of this cap will -be contained in the metric ball in ${\mathcal S}$ centered at $p_0$ -of radius $R(x_0)^{-1/2}(A_0+5)$ and will contain the metric ball -centered at $p_0$ of radius $R(x_0)^{-1/2}(A_0+3)$. Notice also that -the complement of the closure of the surgery cap in ${\mathcal S}$ -is isometrically identified with $N^-$. - -## Preliminary computations - -We shall compute in a slightly more general setup. - Let $I$ be an open interval contained in $(-\delta^{-1},4+A_0)$ and let -$h$ be a metric on $S^2\times I$ within $\delta$ in the -$C^{[1/\delta]}$-topology of the restriction to this open -submanifold of the standard metric $h_0\times ds^2$. We let $\hat -h=e^{-2f}h$. Fix local coordinates near a point $y\in S^2\times I$. -We denote by $\nabla$ the covariant derivative for $h$ and by -$\widehat\nabla$ the covariant derivative for $\hat h$. - We also denote by -$(R_{ijkl})$ the matrix of the Riemann curvature operator of $h$ -in the associated basis of $\wedge^2T(S^2\times I)$ and by $(\hat -R_{ijkl})$ the matrix of the Riemann curvature operator of $\hat -h$ with respect to the same basis. Recall the formula for the -curvature of a conformal change of metric (see, (3.34) on p.51 of -): - -$$ - -$$ -\begin{aligned} -\hat R_{ijkl} & = & -e^{-2f}\left(R_{ijkl}-f_jf_kh_{il}+f_jf_lh_{ik}+f_if_kh_{jl}-f_if_lh_{jk}\right. -\\ & & \left.- -(\wedge^2h)_{ijkl}|\nabla -f|^2-f_{jk}h_{il}+f_{ik}h_{jl}+f_{jl}h_{ik}-f_{il}h_{jk}\right).\nonumber -\end{aligned} -$$ - -$$ - -Here, $f_i$ means $\partial_if$, -$$f_{ij}=Hess_{ij}(f)=\partial_if_j-f_l\Gamma_{ij}^l,$$ -and $\wedge^2h$ is the metric induced by $h$ on $\wedge^2TN$, so -that -$$\wedge^2h_{ijkl}=h_{ik}h_{jl}-h_{il}h_{jk}.$$ - -Now we introduce the notation $O(\delta)$. When we say that a -quantity is $O(\delta)$ we mean that there is some universal -constant $C$ such that, provided that $\delta>0$ is sufficiently -small, the absolute value of the quantity is $\le C\delta$. The -universal constant is allowed to change from inequality to -inequality. - -In our case we take local coordinates adapted to the $\delta$-neck: -$(x^0,x^1,x^2)$ where $x^0$ agrees with the $s$-coordinate and -$(x^1,x^2)$ are Gaussian local coordinates on the $S^2$ such that -$dx^1$ and $dx^2$ are orthonormal at the point in question in the -round metric $h_0$. The function $f$ is a function only of $x^0$. -Hence $f_i=0$ for $i=1,2$. Also, $f_0=\frac{q}{s^2}f$. It follows -that -$$|\nabla f|_h=\frac{q}{s^2}f\cdot(1+O(\delta)),$$ so that -$$|\nabla -f|_h^2=\frac{q^2}{s^4}f^2\cdot (1+O(\delta)).$$ - -Because the metric $h$ is $\delta$-close to the product $h_0\times -ds^2$, we see that $h_{ij}(y)=(h_0)_{ij}(y)+O(\delta)$ and the -Christoffel symbols $\Gamma_{ij}^k(y)$ of $h$ are within $\delta$ in -the $C^{([1/\delta]-1)}$-topology of those of the product metric -$h_0\times ds^2$. In particular, $\Gamma_{ij}^0=O(\delta)$ for all -$ij$. - The components $f_{ij}$ of the Hessian with -respect to $h$ are given by -$$f_{00}=\left(\frac{q^2}{s^4}-\frac{2q}{s^3}\right)f+\frac{q}{s^2}fO(\delta),$$ -$$f_{i0}=\frac{q}{s^2}fO(\delta)\ \ \ for\ \ 1\le i\le 2,$$ -$$f_{ij}=\frac{q}{s^2}fO(\delta)\ \ \ for\ \ 1\le i,j\le 2.$$ -In the following $a,b,c,d$ are indices taking values $1$ and $2$. -Substituting in Equation~() yields - -$$ - -$$ -\begin{aligned} -\hat -R_{0a0b} & = & -e^{-2f}\left(R_{0a0b}+\frac{q^2}{s^4}f^2h_{ab}-h_{ab}(\frac{q^2}{s^4})f^2(1+O(\delta)) -+\left(\frac{q^2}{s^4}-\frac{2q}{s^3}\right)fh_{ab} \right. \\ -& & \left. +\frac{q}{s^2}fO(\delta)\right)\\ -& = & -e^{-2f}\left(R_{0a0b}+\left(\frac{q^2}{s^4}-\frac{2q}{s^3}\right)fh_{ab}+ -h_{ab}(\frac{q^2}{s^4})f^2O(\delta) -+\frac{q}{s^2}fO(\delta)\right) -\end{aligned} -$$ - -$$ - Also, we have - -$$ - -$$ -\begin{aligned} -\hat -R_{ab0c} & = & -e^{-2f}\left(R_{ab0c}-(\wedge^2h)_{ab0c}(\frac{q^2}{s^4})f^2(1+O(\delta)) -+\frac{q}{s^2}fO(\delta)\right) \\ -& = & e^{-2f}\left(R_{ab0c}+(\frac{q^2}{s^4})f^2O(\delta) -+\frac{q}{s^2}fO(\delta)\right). -\end{aligned} -$$ - -$$ - - Lastly, - -$$ - -$$ -\begin{aligned} -\hat R_{1212} & = & -e^{-2f}\left(R_{1212}-(\wedge^2h)_{1212}\frac{q^2}{s^4}f^2(1+O(\delta))+\frac{q}{s^2}fO(\delta)\right) -\\ -& = & -e^{-2f}\left(R_{1212}-\frac{q^2}{s^4}f^2(1+O(\delta))+\frac{q}{s^2}fO(\delta)\right). -\end{aligned} -$$ - -$$ - -Now we are ready to fix the constant $q$. We fix it so that for all $s\in -[0,4+A_0]$ we have -$$ - - q \gg (4+A_0)^2\ \ \ and\ \ \ \frac{q^2}{s^4}e^{-q/s} \ll 1. -$$ - -It follows immediately that $q^2/s^4\gg q/s^3$ for all $s\in [0,4+A_0]$. - We are not yet ready -to fix the constant $C_0$, but once we do we shall always require $\delta$ to -satisfy -$\delta\ll C_0^{-1}$ so that for all $s\in [0,4+A_0]$ we have -$$\frac{q}{s^2}f^2\ll \frac{q^2}{s^4}f^2\ll \frac{q}{s^2}f\ll 1.$$ -(These requirements are not circular, since $C_0$ and $q$ are chosen -independent of $\delta$.) - - Using -these inequalities and putting our computations in matrix form show -the following. - -**Corollary.** -There is $\delta'_2>0$, depending on $C_0$ and $q$, -such that if $\delta\le \delta'_2$ then we have - -$$ - -\left(\hat -R_{ijkl}\right)=e^{-2f}\left[\left(R_{ijkl}\right)+\begin{pmatrix} --\frac{q^2}{s^4}f^2 & 0 \\ 0 & -\left(\frac{q^2}{s^4}-\frac{2q}{s^3}\right)f\begin{pmatrix} 1 & 0 -\\ 0 & 1\end{pmatrix} -\end{pmatrix}+\left(\frac{q}{s^2}fO(\delta)\right)\right]. -$$ - -Similarly, we have the equation relating scalar curvatures -$$\hat R=e^{2f}\left(R+4\triangle f-2|\nabla f|^2\right),$$ -and hence -$$\hat R=e^{2f}\left(R+4\left(\frac{q^2}{s^4}-\frac{2q}{s^3}\right)f-2\frac{q^2}{s^4}f^2+ -\frac{q}{s^2}fO(\delta)\right).$$ - -**Corollary.** -For any constant $C_0<\infty$ and any $\delta0$ such that the following hold if $\delta\le -\delta_3'$. Let $\{e_0,e_1,e_2\}$ be an orthonormal basis for the -tangent space at a point $y\in S^2\times I$ for the metric -$h_0\times ds^2$ with the property that $e_0$ points in the -$I$-direction. Then there is a basis $\{f_0,f_1,f_2\}$ for this -tangent space so that the following hold: -\begin{enumerate} -\item[(1)] The basis is orthonormal in the metric $h$. -\item[(2)] The change of basis matrix expressing the $\{f_0,f_1,f_2\}$ -in terms of $\{e_0,e_1,e_2\}$ is of the form $Id+O(\delta)$. -\item[(3)] The Riemann curvature of $h$ in the basis $\{f_0\wedge f_1,f_1\wedge f_2,f_2\wedge f_0\}$ -of $\wedge^2T_y(S^2\times I)$ is -$$\begin{pmatrix} 1/2 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 -\end{pmatrix}+O(\delta).$$ -\end{enumerate} - -**Proof.** Since $h$ is within $\delta$ of $h_0\times ds^2$ in the -$C^{[1/\delta]}$-topology, it follows that the matrix for $h(y)$ in -$\{e_0,e_1,e_2\}$ is within $O(\delta)$ of the identity matrix, and the matrix -for the curvature of $h$ in the associated basis of $\wedge^2T_y(S^2\times I)$ -is within $O(\delta)$ of the curvature matrix for $h_0\times ds^2$, the latter -being the diagonal matrix with diagonal entries $\{1/2,0,0\}$. Thus, the usual -Gram-Schmidt orthonormalization process constructs the basis $\{f_0,f_1,f_2\}$ -satisfying the first two items. Let $A=(A^{ab})$ be the change of basis matrix -expressing the $\{f_a\}$ in terms of the $\{e_b\}$, so that $A=Id+O(\delta)$. The curvature of $h$ in this basis is then given by -$B^tr(R_{ijkl})B$ where $B=\wedge^2A$ is the induced change of basis matrix -expressing the basis $\{f_0\wedge f_1,f_1\wedge f_2,f_2\wedge f_0\}$ in terms -of $\{e_0\wedge e_1,e_1\wedge e_2,e_2\wedge e_0\}$. Hence, in the basis -$\{f_0\wedge f_1,f_1\wedge f_2,,f_2\wedge f_0\}$ the curvature matrix for $h$ -is within $O(\delta)$ of the same diagonal matrix. For $\delta$ sufficiently -small then the eigenvalues of the curvature matrix for $h$ are within -$O(\delta)$ of $(1/2,0,0)$. - -**Corollary.** -The following holds provided that $\delta\le \delta_3'$. It is -possible to choose the basis $\{f_0,f_1,f_2\}$ satisfying the -conclusions of Lemma~ so that in addition the -curvature matrix for $(R_{ijkl}(y))$ is of the form -$$\begin{pmatrix} \lambda & 0 & 0 \\ 0 & \alpha & \beta \\ 0 & \beta & \gamma\end{pmatrix}$$ -with $|\lambda-\frac{1}{2}|\le O(\delta)$ and -$|\alpha|,|\beta|,|\gamma|\le O(\delta)$. - -**Proof.** We have an $h$-orthonormal basis $\{f_0\wedge f_1,f_1\wedge -f_2,f_2\wedge f_0\}$ for $\wedge^2T_y(S^2\times \Ar)$ in which the -quadratic form $(R_{ijkl}(y)$ is -$$\begin{pmatrix} 1/2 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{pmatrix} -+O(\delta).$$ It follows that the restriction to the $h$-unit sphere -in $\wedge^2T_y(S^2\times \Ar)$ of this quadratic form achieves its -maximum value at some vector $v$, which, when written out in this -basis, is given by $(x,y,z)$ with $|y|,|z|\le O(\delta)$ and -$|x-1|\le O(\delta)$. Of course, this maximum value is within -$O(\delta)$ of $1/2$. Clearly, on the $h$-orthogonal subspace to -$v$, the quadratic form is given by a matrix all of whose entries -are $O(\delta)$ in absolute value. This gives us a new basis of -$\wedge^2T_y(S^2\times I)$ within $O(\delta)$ of the given basis in -which $(R_{ijkl}(y))$ is diagonal. The corresponding basis for -$T_y(S^2\times\Ar)$ is as required. - -Now we consider the expression $(\hat R_{ijkl}(y))$ in this basis. - -**Lemma.** Set $\delta'_4=min(\delta_2',\delta_3')$. Suppose -that $\delta\le min(\delta'_4,C_0^{-1})$. Then in the basis -$\{f_0,f_1,f_2\}$ for $T_y(S^2\times I)$ as in Corollary~ we -have -$$(\hat R_{ijkl}(y)) =e^{-2f}\left[\begin{pmatrix} \lambda & 0 & 0 -\\ 0 & \alpha & \beta \\ 0 & \beta & \gamma\end{pmatrix} -+\begin{pmatrix} -\frac{q^2}{s^4}f^2 & 0 \\ 0 & -\left(\frac{q^2}{s^4}-\frac{2q}{s^3}\right)f\begin{pmatrix} 1 & 0 -\\ 0 & 1\end{pmatrix} -\end{pmatrix}+\left(\frac{q^2}{s^4}fO(\delta)\right)\right]$$ -where $\lambda,\alpha,\beta,\gamma$ are the constants in -Lemma~ and the first matrix is the expression for -$(R_{ijkl}(y))$ in this basis. - -**Proof.** We simply conjugate the expression in Equation~() by the change of -basis matrix and use the fact that by our choice of $q$ and the fact that -$C_0\delta<1$, we have $f\gg f^2$ and $q/s^3\ll q^2/s^4$. - -**Corollary.** -Assuming that $\delta\le min(\delta'_4,C_0^{-1})$, there is an -$h$-orthonormal basis $\{f_0,f_1,f_2\}$ so that in the associated basis for -$\wedge^2T_y(S^2\times I)$ the matrix $(R_{ijkl}(y))$ is diagonal and given by -$$\begin{pmatrix} \lambda & 0 & 0 \\ 0 & \mu & 0 \\ 0 & 0 & -\nu\end{pmatrix}$$ with $|\lambda-1/2|\le O(\delta)$ and -$|\mu|,|\nu|\le O(\delta)$. Furthermore, in this same basis the -matrix $(\hat R_{ijkl}(y))$ is -$$e^{-2f}\left[\begin{pmatrix}\lambda & 0 & 0 \\ 0 & \mu & 0 \\ 0 & 0 & -\nu\end{pmatrix}+\begin{pmatrix} -\frac{q^2}{s^4}f^2 & 0 \\ 0 & -\left(\frac{q^2}{s^4}-\frac{2q}{s^3}\right)f\begin{pmatrix} 1 & 0 \\ -0 & 1\end{pmatrix}\end{pmatrix}+\frac{q^2}{s^4}fO(\delta)\right].$$ - -**Proof.** To diagonalize $(R_{ijkl}(y))$ we need only rotate in the -$\{f_1\wedge f_2,f_2\wedge f_3\}$-plane. Applying this rotation to -the expression in Lemma~ gives the result. - -**Corollary.** -There is a constant $A<\infty$ such that the following holds for the given value of $q$ -and any $C_0$ provided that $\delta$ is sufficiently small. -Suppose that the eigenvalues for the curvature matrix of $h$ at $y$ -are $\lambda\ge \mu\ge \nu$. -Then the eigenvalues for the curvature of $\hat h$ at the point $y$ -are given by $\lambda',\mu',\nu'$, where -$$\left|\lambda'-e^{2f}\left(\lambda-\frac{q^2}{s^4}f^2\right)\right| -\le \frac{q^2}{s^4}fA\delta$$ -$$\left|\mu'-e^{2f}\left(\mu+\left(\frac{q^2}{s^4}-\frac{2q}{s^3}\right)f\right)\right|\le -\frac{q^2}{s^4}fA\delta$$ -$$\left|\nu'-e^{2f}\left(\nu+\left(\frac{q^2}{s^4}-\frac{2q}{s^3}\right)f\right)\right|\le -\frac{q^2}{s^4}fA\delta.$$ - -In particular, we have -$$\nu'\ge e^{2f}\left(\nu+\frac{q^2}{2s^4}f\right)$$ -$$\mu'\ge e^{2f}\left(\mu+\frac{q^2}{2s^4}f\right).$$ - -**Proof.** Let $\{f_0,f_1,f_2\}$ be the $h$-orthonormal basis given in -Corollary~. Then $\{e^ff_0,e^ff_1,e^ff_2\}$ is orthonormal for $\hat -h=e^{-2f}h$. This change multiplies the curvature matrix by $e^{4f}$. Since -$f\ll 1$, $e^{4f}<2$ so that the expression for $(\hat R_{ijkl}(y))$ in this -basis is exactly the same as in Lemma~ except that the factor -in front is $e^{2f}$ instead of $e^{-2f}$. Now, it is easy to see that since -$((q^2/s^4)fA\delta)^2\ll (q^2/s^4)fA\delta$ the eigenvalues will differ from -the diagonal entries by at most a constant multiple of $(q^2/s^4)fA\delta$. - - The first three inequalities are immediate -from the previous corollary. The last two follow since $q^2/s^4\gg q/s^3$ and -$\delta\ll 1$. - -One important consequence of this computation is the following: - -**Corollary.** -For the given value of $q$ and for any $C_0$, -assuming that $\delta>0$ is sufficiently small, the smallest -eigenvalue of $Rm_{\hat h}$ is greater than the smallest eigenvalue -of $Rm_h$ at the same point. Consequently, at any point where $h$ -has non-negative curvature so does $\hat h$. - -**Proof.** Since $|\lambda-1/2|, |\mu|, |\nu|$ are all $O(\delta)$ and since -$\frac{q^2}{s^4}f\ll 1$, it follows that the smallest eigenvalue of $(\hat -R_{ijkl}(y))$ is either $\mu'$ or $\nu'$. But it is immediate from the above -expressions that $\mu'>\mu$ and $\nu'>\nu$. This completes the proof. - -Now we are ready to fix $C_0$. -There is a universal constant $K$ such that for all $\delta>0$ sufficiently small -and for any $\delta$-neck $(N,h)$ of -scale one, every eigenvalue of $Rm_h$ is at least $-K\delta$. -We set -$$C_0=2Ke^{q}.$$ - -**Lemma.** -With these choices of $q$ and $C_0$ for any $\delta>0$ sufficiently small we have -$\nu'>0$ and $\mu'>0$ for $s\in [1,4+A_0]$ and $\lambda'>1/4$. - -**Proof.** Then by the previous result we have -$$\nu'\ge e^{2f}\left(\nu+\frac{q^2}{2s^4}f\right).$$ -It is easy to see that since $q\gg (4+A_0)$ the function $(q^2/2s^4)f$ is an -increasing function on $[1,4+A_0]$. Its value at $s=1$ is -$(q^2/2)e^{-q}C_0\delta>K\delta$. Hence $\nu+\frac{q^2}{2s^4}f>0$ for all $s\in -[1,4+A_0]$ and consequently $\nu'>0$ on this submanifold. The same argument -shows $\mu'>0$. Since $q^2/s^4 f^2\ll 1$ and $00$ sufficiently small. Suppose that for some $t\ge 0$ -and every point $p\in N$ the curvature of $h$ satisfies: -\begin{enumerate} -\item[(1)] $R(p)\ge \frac{-6}{1+4t}$, and -\item[(2)] $R(p)\geq 2X(p)\left(logX(p)+log(1+t)-3\right)$ whenever $02X_{\hat p}(p)(log(X_{\hat h}(p)+log(1+t)-3).$$ - -This completes the proof in both cases. - -This establishes the first item in the conclusion of -Theorem~ for $\delta>0$ sufficiently small on -$s^{-1}(-\delta^{-1},4)$. As we have seen in Lemma~, -the curvature is positive on $s^{-1}[1,4)$. - -### Proof of the first two items for $s\ge 4$ - -Now let us show that the curvature on $\tilde g$ is positive in the region -$s^{-1}([4,4+A_0])$. First of all in the preimage of the -interval $[4,4+A_0,-r_0]$ this follows -from Corollary~ and the fact that $\eta g_0$ has non-negative -curvature. As for the region $s^{-1}([4+A_0-r_0,4+A_0])$, as $\delta$ tends to -zero, the metric here tends smoothly to the restriction of the metric $g_0$ to -that subset. The metric $g_0$ has positive curvature on -$s^{-1}([4+A_0-r_0,4+A_0])$. Thus, for all $\delta>0$ sufficiently small the -metric $\tilde g$ has positive curvature on all of $s^{-1}([4+A_0-r_0,4+A_0])$. -This completes the proof of the first two items. - -### Proof of the third item - - By -construction the restriction of the metric $\widetilde g$ to -$s^{-1}((-\delta^{-1},0])$ is equal to the metric $\rho^*g$. Hence, -in this region the mapping is an isometry. In the region -$s^{-1}([0,4])$ we have $R(x_0)\rho^*g\ge \eta g_0$ so that by -construction in this region $\rho^*g\ge \tilde g$. Lastly, in the -region $s^{-1}([4,A_0+4])$ we have $R(x_0)^{-1}\eta g_0\ge \tilde -g$. On the other hand, it follows from Lemma~ that -the map from $([0,\delta^{-1}]\times S^2,R(x_0)\rho^* g)$ to -$(B(p_0,4+A_0),\eta g)$ is distance decreasing. This completes the -proof of the third item. - -### Completion of the proof - -As $\delta$ goes to zero, $f$ tends to zero in the -$C^\infty$-topology and $\eta$ limits to $1$. From this the fourth -item is clear. - -This completes the proof of Theorem~. - -## Other properties of the result of surgery - -**Lemma.** -Provided that $\delta>0$ is sufficiently small the following holds. Let $(N,g)$ -be a $\delta$-neck and let $({\mathcal S},\tilde g)$ be the result of surgery -along the cental $2$-sphere of this neck. Then for any $00$ and an embedding $i_U\colon -U\times[t,t+\epsilon)\subset {\mathcal M}$ compatible with time and -the vector field. Of course, two such embeddings agree on their -common domain of definition. Notice also that for each $t'\in -[t,t+\epsilon)$ the restriction of the map $i_U$ to $U\times \{t'\}$ -induces an diffeomorphism from $U$ to an open subset $U_{t'}$ of -$M_{t'}$. It follows that the local flow generated by the vector -field $\chi$ preserves the horizontal subbundle. Hence, the vector -field $\chi$ acts by Lie derivative on the sections of ${\mathcal -HT}({\mathcal M})$ and on all associated bundles (for example the -symmetric square of the dual bundle). - -### The equation - -**Definition.** A *Ricci flow with surgery* is a pair $({\mathcal M},G)$ -consisting of a surgery space-time ${\mathcal M}$ and a horizontal -metric $G$ on ${\mathcal M}$ such that for every $x\in {\mathcal M}$ - we have - -$$ - -{\mathcal L}_\chi(G)(x)=-2Ric_G(x)) -$$ - as sections -of the symmetric square of the dual to ${\mathcal HT}({\mathcal -M})$. If space-time is $(n+1)$-dimensional, then we say that the -Ricci flow with surgery is $n$-dimensional (meaning of course that -each time-slice is an $n$-dimensional manifold). - -**Remark.** Notice that at an exposed point and at points at the initial and the final time - the Lie derivative is a one-sided derivative. - -### Examples of Ricci flows with surgery - -\begin{exam} -One example of a Ricci flow with surgery is ${\mathcal M}=M_0\times -[0,T)$ with time function $**t**$ and the vector field $\chi$ -coming from the second factor. In this case the Lie derivative -${\mathcal L}_\chi$ agrees with the usual partial derivative in the -time direction, and hence our generalized Ricci flow equation is the -usual Ricci flow equation. This shows that an ordinary Ricci flow is indeed -a Ricci flow with surgery. -\end{exam} - -The next lemma gives an example of a Ricci flow with surgery where -the topology of the time-slices changes. - -**Lemma.** -Suppose that we have manifolds $M_1\times (a,b]$ and $M_2\times -[b,c)$ and compact, smooth codimension-$0$ submanifolds -$\Omega_1\subset M_1$ and $\Omega_2\subset M_2$ with open -neighborhoods $U_1\subset M_1$ and $U_2\subset M_2$ respectively. -Suppose we have a diffeomorphism $\psi\colon U_1\to U_2$ carrying -$\Omega_1$ onto $\Omega_2$. Let $(M_1\times (a,b])_0$ be the subset -obtained by removing $(M_1\setminus \Omega_1)\times \{b\}$ from -$M_1\times (a,b]$. Form the topological space -$${\mathcal M}=(M_1\times (a,b])_0\cup M_2\times [b,c)$$ -where $\Omega_1\times\{b\}$ in $(M_1\times (a,b])_0$ is identified -with $\Omega_2\times \{b\}$ using the restriction of $\psi$ to -$\Omega_1$. Then ${\mathcal M}$ naturally inherits the structure of -a surgery space-time where the time function restricts to -$(M_1\times (a,b])_0$ and to $M_2\times [b,c)$ to be the projection -onto the second factor and the vector field $\chi$ agrees with the -vector fields coming from the second factor on each of $(M_1\times -(a,b])_0$ and $M_2\times [b,c)$. - - Lastly, given Ricci -flows $(M_1,g_1(t)),\ ab$). At any point of -$(M_2\setminus\Omega_2)\times \{b\}$ there is the smooth manifold -with boundary coordinate patch coming from $M_2\times[b,c)$. For any -point in $int(\Omega_1)\times\{b\}$ we have the smooth -manifold structure obtained from gluing $(int(\Omega_1))\times -(a,b]$ to $int(\Omega_2)\times [b,c)$ along the $b$ time-slice -by $\psi$. Thus, at all these points we have neighborhoods on which -our data determine a smooth manifold structure. Lastly, let us -consider a point $x\in -\partial \Omega_1\times \{b\}$. -Choose local coordinates $(x^1,\ldots,x^n)$ for a neighborhood $V_1$ -of $x$ such that $\Omega_1\cap V_1=\{x^n\le 0\}$. We can assume that -$\psi$ is defined on all of $V_1$. Let $V_2=\psi(V_1)$ and take the - local coordinates on $V_2$ induced from the $x^i$ on $V_1$. - Were we to identify $V_1\times (a,b]$ -with $V_2\times [b,c)$ along the $b$ time-slice using this map, then -this union would be a smooth manifold. There is a -neighborhood of the point $(x,b)\in {\mathcal M}$ which is obtained -from the smooth manifold $V_1\times (a,b]\cup_\psi V_2\times [b,c)$ -by inducing a new topology where the open subsets are, in addition -to the usual ones, any open subset of the form $\{x^n>0\}\times -[b,b')$ where $b0$ we define -$B(x,t,r)\subset M_t$ to be the metric ball of radius $r$ centered -at $x$ in the Riemannian manifold $(M_t,G(t))$. - -**Definition.** Let $({\mathcal M},G)$ be a Ricci flow with surgery, and let $x$ be -a point of space-time. Set $t=**t**(x)$. For any $r>0$ and $\Delta -t>0$ we say that the *backward parabolic neighborhood* -$P(x,t,r,-\Delta t)$ *exists* in ${\mathcal M}$ if there is an -embedding $B(x,t,r)\times (t-\Delta t,t]\to {\mathcal M}$ compatible -with time and the vector field. Similarly, we say -that the *forward parabolic neighborhood* $P(x,t,r,\Delta t)$ -*exists* in ${\mathcal M}$ if there is an embedding -$B(x,t,r)\times [t,t+\Delta t)\to {\mathcal M}$ compatible with time -and the vector field. A *parabolic neighborhood* is either a -forward or backward parabolic neighborhood. - -**Definition.** -Fix $\kappa>0$ and $r_0>0$. We say that a Ricci flow with surgery -$({\mathcal M},G)$ is *$\kappa$-noncollapsed on scales $\le -r_0$* if the following holds for every point $x\in {\mathcal M}$ and -for every $r\le r_0$. Denote $**t**(x)$ by $t$. If the parabolic -neighborhood $P(x,t,r,-r^2)$ exists in ${\mathcal M}$ and if -$|Rm_G|\le r^{-2}$ on $P(x,t,r,-r^2)$, then $Vol\,B(x,t,r)\ge \kappa r^3$. - -**Remark.** For $\epsilon>0$ sufficiently small, -an $\epsilon$-round component satisfies the first condition in the above definition - for some $\kappa>0$ depending only on -the order of the fundamental group of the underlying manifold, but there is no universal -$\kappa>0$ that works for all $\epsilon$-round manifolds. Fixing an integer $N$ let ${\mathcal C}_N$ -be the class of closed $3$-manifolds - with the property that any finite free factor of $\pi_1(M)$ -has order at most $N$. Then any $\epsilon$-round component of any -time-slice of any Ricci flow $({\mathcal M},G)$ whose initial -conditions consist of a manifold in ${\mathcal C}_N$ will have -fundamental group of order at most $N$ and hence will satisfy the first condition -in the above definition for some $\kappa>0$ depending only on $N$. - -We also have the notion of the curvature being pinched toward -positive, analogous to the notions for Ricci flows and generalized Ricci flows. - -**Definition.** Let $({\mathcal M},G)$ be a $3$-dimensional Ricci flow with surgery, -whose time domain of definition is contained in $[0,\infty)$. For -any $x\in {\mathcal M}$ we denote the eigenvalues of $Rm(x)$ by -$\lambda(x)\ge \mu(x)\ge \nu(x)$ and we set $X(x)=max(0,-\nu(x))$. - We say -that its *curvature is pinched toward positive* if the following -hold for every $x\in {\mathcal M}$: -\begin{enumerate} -\item[(1)] $R(x)\ge \frac{-6}{1+4**t**(x)}$. -\item[(2)] $R(x)\geq 2X(x)\left(logX(x)+log(1+**t**(x))-3\right)$, whenever $01$ so that the limit is -defined for at least time $t\in [0, 1+t_{0,\infty})$. If -$t_{0,\infty}>0$, then, restricting to the appropriate subset of -this limit, a subset with compact closure in space-time, it follows -immediately that for all $n$ sufficiently large there is a strong -$\epsilon$-neck centered at $(x,0)$. This contradicts the assumption -that we began with a sequence of counterexamples to the -proposition. - -Let us consider the case when $t_{0,\infty}=0$. In this case the -smooth limit is an evolving round cylinder defined for time -$(-1,0]$. - Since $t_{1,n}>1$ we -see that for any $A<\infty$ for all $n$ sufficiently large the ball -$B(x_n,0,A)$ has compact closure in every time-slice and there are -uniform bounds to the curvature on $B(x_n,0,A)\times (-1,0]$. This -means that the limit is uniform for time $(-1,0]$ on all these -balls. Thus, once again for all $n$ sufficiently large we see that -$(x,0)$ is the center of a strong $\epsilon$-neck in the union. In -either case we have obtained a contradiction, and hence we have -proved the result. See Fig.~. - -### First assumptions - -**Choice of $C$ and $\epsilon$:** The first thing we need -to do is fix for the rest of the argument $C<\infty$ and -$\epsilon>0$. We do this in the following way. We fix $0<\epsilon\le -min(1/200,\left(\sqrt{D}(A_0+5)\right)^{-1},\bar\epsilon_1/2,\bar -\epsilon'/2,\epsilon_0)$ where $\bar\epsilon_1$ is the constant from -Proposition~, $\bar\epsilon'$ is the constant from -Theorem~, $\epsilon_0$ is the constant from Section~, -and $A_0$ and $D$ are the constants from -Lemma~. - We fix $\beta<1/2$, the constant from -Proposition~. Then we let $C$ be the maximum of the -constant $C(\epsilon)$ as in Corollary~ and -$C'(\beta\epsilon/3)+1$ as in Theorem~. - -For all such $\epsilon$, Theorem~ holds for $\epsilon$ and -Proposition~, Proposition~ and -Corollaries~ and~ and Theorems~ and~ - hold for $2\epsilon$. Also, all the topological results of the Appendix hold -for $2\epsilon$ and $\alpha=10^{-2}$. - -Now let us turn to the assumptions we shall make on the Ricci flows -with surgery that we shall consider. Let ${\mathcal M}$ be a -space-time. Our first set of assumptions are basically topological -in nature. They are: - - **Assumption (1). Compactness and dimension:** *Each -time-slice $M_t$ of space-time is a compact $3$-manifold -containing no embedded $\Ar P^2$ with trivial normal bundle.* - -**Assumption (2). Discrete singularities:** *The set of -singular times is a discrete subset of $\Ar$.* - - **Assumption (3). Normalized initial conditions:** *$0$ is the initial time of the Ricci flow with surgery and the -initial metric $(M_0,G(0))$ is normalized.* - - It follows -from Assumption (2) that for any time $t$ in the time-interval of -definition of a Ricci flow with surgery, with $t$ being distinct -from the initial and final times (if these exist), for all -$\delta>0$ sufficiently small, the only possible singular time in -$[t-\delta,t+\delta]$ is $t$. Suppose that $t$ is a singular time. -The singular locus at time $t$ is a closed, smooth subsurface -$\Sigma_{t}\subset M_{t}$. From the local -model, near every point of $x\in \Sigma_{t}$ we see that this -surface separates $M_{t}$ into two pieces: -$$M_{t}=C_{t}\cup_{\Sigma_{t}}E_{t},$$ -where $E_{t}$ is the exposed region at time $t$ and $C_{t}$ is the -complement of the interior of $E_{t}$ in $M_{t}$. We call $C_t$ the -*continuing region*. $C_t\subset M_t$ is the maximal subset of -$M_t$ for which there is $\delta>0$ and an embedding $C_t\times -(t-\delta,t]\to {\mathcal M}$ compatible with time and the vector -field. - - **Assumption (4). Topology of the exposed regions:** {\em -At all singular times $t$ we require that $E_{t}$ be a finite -disjoint union of $3$-balls. In particular, $\Sigma_{t}$ is a -finite disjoint union of $2$-spheres.} - -The next assumptions are geometric in nature. -Suppose that $t$ is a surgery time. Let ${\mathcal M}_{(-\infty,t)}$ be $**t**^{-1}((-\infty,t))$ -and let $(\widehat {\mathcal M}_{(-\infty,t)},\widehat G)$ be the maximal extension of -$({\mathcal M}_{(-\infty,t)},G)$ to time $t$, as given in Definition~. - -**Assumption (5). Boundary components of the exposed -regions:** {\em There is a surgery control parameter -function, -$\delta(t)>0$, a non-increasing function of -$t$, such that each component of $\Sigma_{t}\subset M_{t}$ is the -central $2$-sphere of a strong $\delta(t)$-neck in $(\widehat{\mathcal -M}_{(-\infty,t)},\widehat G)$.} - -Suppose that $t$ is a singular time. Then for all $t^-0$ the manifold $M_T$ is empty, then $M$ is a -connected sum of manifolds diffeomorphic to $2$-sphere bundles over -$S^1$ and $3$-dimensional space-forms, i.e., compact -$3$-manifolds that admit a metric of constant positive curvature. -\item[(2)] -If for some $T>0$ the manifold $M_T$ is empty and if $M$ is connected and simply connected, -then $M$ is diffeomorphic to $S^3$. -\item[(3)] -If for some $T>0$ the manifold $M_T$ is empty and if $M$ has finite fundamental group, then $M$ -is a $3$-dimensional space-form. -\end{enumerate} - -**Proof.** Suppose that $M_T$ satisfies the Thurston Geometrization Conjecture and that $t_0$ is -the largest surgery time $\le T$. (If there is no such surgery time -then $M_T$ is diffeomorphic to $M$ and the result is established.) -Let $T'0$ also decays to zero rapidly as time goes to infinity. -Here then are the functions that will play the crucial role in defining -the surgery process. - -**Definition.** -We have: (i) *a canonical neighborhood parameter*, - $r(t)>0$, and -(ii) *a surgery control parameter* $\overline\delta(t)>0$. We use these -to define the surgery scale function $h(t)$. Set -$\rho(t)=\overline\delta(t)r(t)$. Let $h(t)= -h(\rho(t),\overline\delta(t))\le -\rho(t)\cdot\overline\delta(t)=\overline\delta^2(t)r(t)$ be the -function given by Theorem~. We require that $h(0)\le -R_0^{-1/2}$ where $R_0$ is the constant from -Theorem~. - - In addition, there is a -function $\kappa(t)>0$ called the *non-collapsing parameter*. -All three functions $r(t)$, $\overline\delta(t)$ and $\kappa(t)$ are -required to be positive, non-increasing functions of $t$. - -We shall consider Ricci flows with surgery $({\mathcal M},G)$ that -satisfy Assumptions (1) -- (7) and also satisfy: - - {\bf For any singular time $t$ the surgery at time $t$ is -performed with control $\overline\delta(t)$ and at scale -$h(t)=h(\rho(t),\overline\delta(t))$, where -$\rho(t)=\overline\delta(t)r(t)$, in the sense that each boundary -component of $C_t$ is the central $2$-sphere of a strong -$\overline\delta(t)$-neck centered at a point $y$ with -$R(y)=h(t)^{-2}$.} - -There is quite a bit of freedom in the choice of these parameters. -But it is not complete freedom. They must decay rapidly enough as -functions of $t$. We choose to make $r(t)$ and $\kappa(t)$ step -functions, and we require $\overline\delta(t)$ to be bounded above -by a step function of $t$. Let us fix the step sizes. - -**Definition.** We set $t_0=2^{-5}$, and for any $i\ge 0$ we define $T_i=2^it_0$. - -The steps we consider are $[0,T_0]$ and then $[T_i,T_{i+1}]$ for -every $i\ge 0$. The first step is somewhat special. Suppose that -$({\mathcal M},G)$ is a Ricci flow with surgery with normalized -initial conditions. Then - according to -Claim~ the flow exists on $[0,T_1]$ and the norm -of the Riemann curvature is bounded by $2$ on $[0,T_1]$, so that -by Assumption (7) there are no surgeries in this time interval. Also, -by Claim~ there is a $\kappa_0>0$ so -that $Vol\,B(x,t,r)\le \kappa_0r^3$ for every $t\le T_1$ and -$x\in M_t$ and every $r\le \epsilon$. - -**Definition.** -*Surgery parameter sequences* are sequences -\begin{enumerate} -\item[(i)] $**r**=r_0\ge r_1\ge r_2\ge \cdots >0$, with $r_0=\epsilon$, -\item[(ii)] $**K**=\kappa_0\ge \kappa_1\ge \kappa_2\ge \cdots >0$ -with $\kappa_0$ as in Claim~, and \item[(iii)] $ -\Delta=\delta_0\ge \delta_1\ge \delta_2\ge \cdots ->0$ with $\delta_0=min(\beta\epsilon/3,\delta_0',K^{-1},D^{-1})$ - where $\delta_0'$ is the constant -from Theorem~ and $\beta<1/2$ is the constant from -Proposition~, $\epsilon$ is the constant that we have -already fixed, and $K$ and $D$ are the constants from -Lemma~. -\end{enumerate} -We shall also refer to partial sequences defined for indices $0,\ldots, i$ for some $i>0$ -as surgery parameter sequences if they are positive, non-increasing and if their initial -terms satisfy the conditions given above. - -We let $r(t)$ be the step function whose value on $[T_i,T_{i+1})$ is $r_{i+1}$ -and whose value on $[0,T_0)$ is $r_0$. We say that a Ricci flow with surgery -satisfies the strong $(C,\epsilon)$-canonical neighborhood assumption with parameter $**r**$ if it satisfies this condition with respect to the step function $r(t)$ -associated with $**r**$. This means that any $x\in {\mathcal M}$ with -$R(x)\ge r^{-2}(**t**(x))$ has a strong $(C,\epsilon)$-canonical neighborhood in -${\mathcal M}$. Let $\kappa(t)$ be the step function whose value on -$[T_i,T_{i+1})$ is $\kappa_{i+1}$ and whose value on $[0,T_0)$ is $\kappa_0$. -Given $\kappa>0$, we say that a Ricci flow defined on $[0,t]$ is $\kappa$-non-collapsed -on scales $\le \epsilon$ provided that for every point $x$ not contained in a -component of its time-slice with positive sectional curvature, -if for some $r\le \epsilon$, the parabolic neighborhood $P(x,**t**(x),r,-r^2)$ exists in ${\mathcal M}$ -and the norm of the Riemann curvature is bounded on this backward -parabolic neighborhood by $r^{-2}$, then $Vol\,B(x,**t**(x),r)\ge -\kappa r^3$. -We say that a Ricci flow with surgery is $**K**$-non-collapsed on scales -$\epsilon$ if for every $t\in [0,\infty)$ the restriction of the flow to -$[0,t]$ is $\kappa(t)$-non-collapsed on scales $\le \epsilon$. - Lastly, we fix a non-increasing function $\overline\delta(t)>0$ -with $\overline\delta(t)\le \delta_{i+1}$ if $t\in [T_i,T_{i+1})$ for all $i\ge -0$ and $\overline\delta(t)\le \delta_0$ for $t\in [0,T_0)$. We denote the fact -that such inequalities hold for all $t$ by saying $\overline\delta(t)\le -\Delta$. - -Having fixed surgery parameter sequences $**K**$, $**r**$ and -$\Delta$, defined step functions $r(t)$ and $\kappa(t)$, and fixed -$\overline\delta(t)\le \Delta$ as above, we shall consider only -Ricci flows with surgery where the surgery at time $t$ is defined -using the surgery parameter functions $r(t)$ and -$\overline\delta(t)$. In addition, we require that these Ricci -flows with surgery satisfy Assumptions (1) -- (7). - -What we shall show is that there are surgery parameter -sequences $**r**$, $**K**$ and $\Delta$ with the property that -for any normalized initial metric and any positive, -non-increasing function $\overline\delta(t)\le \Delta$, it is -possible to construct a Ricci flow with surgery using the surgery -parameters $r(t)$ and $\overline\delta(t)$ with the given initial -conditions and furthermore that this Ricci flow with surgery -satisfies the Assumptions (1) -- (7), has curvature pinched toward -positive, satisfies the canonical neighborhood assumption, and satisfies the -non-collapsing assumption using these parameters. - -In fact we shall prove this inductively, constructing the step -functions inductively one step at a time. Thus, given surgery -parameter sequences indexed by $0,\ldots,i$ we show that there are appropriate choices of -$r_{i+1}, \kappa_{i+1}$ and $\delta_{i+1}$ such that the following is true. -Given a Ricci flow with -surgery defined on time $[0,T_i)$ satisfying all the properties with -respect to the first set of data, that Ricci -flow with surgery extends to one defined for time $[0,T_{i+1})$ and satisfies -Assumptions (1) -- (7), the canonical neighborhood assumption and the -non-collapsing assumption with respect to the extended surgery -parameter sequences, and has curvature pinched toward positive. -As stated this is not quite true; there is a slight twist: we must also assume -that $\overline\delta(t)\le \delta_{i+1}$ for all $t\in [T_{i-1},T_{i+1})$. -It is for this reason that we consider -pairs consisting of sequences $\Delta$ and a surgery control -parameter $\overline\delta(t)$ bounded above by $\Delta$. - -## The process of surgery - -Suppose given surgery parameter sequences $\{r_0,\ldots,r_{i+1}\}$, -$\{\kappa_0,\ldots,\kappa_{i+1}\}$ and -$\Delta_i=\{\delta_0,\ldots,\delta_i\}$ and also given a positive, -decreasing function $\overline\delta(t)\le \Delta_i$, defined for -$t\le T_{i+1}$ with $\delta_0=min(\alpha\epsilon/3,\delta_0',K^{-1},D^{-1})$ as above. Suppose -that $({\mathcal M},G)$ is a Ricci flow with surgery defined for -$t\in [0,T)$ that goes singular at time $T\in (T_i,T_{i+1}]$. We -suppose that it satisfies Assumptions (1) -- (7). Since the flow has -normalized initial conditions and goes singular at time $T$, it -follows that $i\ge 1$. We suppose that $({\mathcal M},G)$ satisfies -the $(C,\epsilon)$-canonical neighborhood assumption with parameter -$r_{i+1}$ and that its curvature is pinched toward positive. By -Theorem~ we know that there is a maximal extension -$(\widehat{\mathcal M},\widehat G)$ of this generalized flow to time -$T$ with the $T$ time-slice being $(\Omega(T),G(T))$. Set -$\rho=\overline\delta(T)r_{i+1}$, and set -$h(T)=h(\rho(T),\overline\delta(T))$ as in Theorem~. -Since $\overline\delta(T)\le \delta_0<1$, we see that -$\rhoT$. - The resulting Ricci flow with -surgery satisfies Assumptions (1) -- (7). It also has curvature pinched -toward positive. - -**Proof.** It is immediate from the construction and Lemma~ that -the result of performing the surgery operation at time $T$ on a Ricci flow with -surgery produces a new Ricci flow with surgery. Assumptions (1) -- (3) -clearly hold for the result. and Assumptions (4) and (5) hold because of -the way that we do surgery. Let us consider Assumption (6). Fix $t'\rho^{-2}>r_i^{-2}.$$ -Since $M_{t'}\setminus \psi(int\,C_T\times \{t'\})$ is compact -for every $t'$, there is $T_1r_i^{-2}$ for -all $p\in M_{t'}\setminus \psi(intC_T\times \{t'\})$ and all -$t\in [T_1,T)$. Hence, by our assumptions all these points have -strong $(C,\epsilon)$-canonical neighborhoods. This establishes that -Assumption (6) holds at the singular time $T$. By hypothesis -Assumption (6) holds at all earlier singular times. Clearly, from the -construction the Ricci flow on $[T,T')$ is maximal. Hence, -Assumption (7) holds for the new Ricci flow with surgery. - - From Theorem~ the -fact that $\delta(T)\le \delta_i\le \delta_0\le\delta'_0$ and -$h(T)\le R_0^{-1/2}$ imply that the Riemannian manifold $(M_T,G(T))$ -has curvature pinched toward positive for time $T$. It then follows -from Corollary~ that the Ricci flow defined on $[T,T')$ -with $(M_T,G(T))$ as initial conditions has curvature pinched toward -positive. The inductive hypothesis is that on the time-interval -$[0,T)$ the Ricci flow with surgery has curvature pinched toward -positive. This completes the proof of the lemma. - -**Proposition.** -Suppose that $({\mathcal M},G)$ is a Ricci flow with surgery -satisfying Assumptions (1) -- (7) in Section~. -Suppose that $T$ is a surgery time, suppose that the surgery -control parameter $\delta(T)$ is less than $\delta_0$ in -Definition~, and suppose that the scale of the -surgery $h(T)$ is less than $R_0^{-1/2}$ where $R_0$ is the constant -from Theorem~. Fix $t'0$. -Then there is $\delta(r_{i+1})>0$ and $\kappa_{i+1}>0$ such that, -provided that $\overline\delta(t)\le \delta(r_{i+1})$ for all $t\in -[T_{i-1},T)$, the Ricci flow with surgery $({\mathcal M},G)$ is -$\kappa_{i+1}$ non-collapsed on scales $\le \epsilon$. - -In Section~ we show that the strong $(C,\epsilon)$-canonical neighborhood -assumption extends for some parameter $r_{i+1}$, assuming again that -$\overline\delta(t)\le \delta(r_{i+1})$ for all $t\in [T_{i-1},T)$. - -Lastly, in Section~ we complete the proof by -showing that the number of surgeries possible in $[0,T_{i+1})$ is -bounded in terms of the initial conditions and $\overline\delta(T)$. -The argument for this is a simple volume comparison argument -- -under Ricci flow with normalized initial conditions, the volume -grows at most at a fixed exponential rate and under each surgery an -amount of volume, bounded below by a positive constant depending -only on $\overline \delta(T_{i+1})$, is removed. - -# Proof of the non-collapsing - -The precise statement of the non-collapsing result is given in the -next section. Essentially, the proof of non-collapsing in the -context of Ricci flow with surgery is the same as the proof in the -case of ordinary Ricci flows. Given a point $x\in {\mathcal M}$, one -finds a parabolic neighborhood whose size, $r'$, is determined by -the constants $r_i$, $C$ and $\epsilon$, contained in $**t**^{-1}([T_{i-1},T_i))$ and on which the curvature is bounded by -$(r')^{-2}$. Hence, by the inductive hypothesis, the final -time-slice of this neighborhood is $\kappa_i$-non-collapsed. -Furthermore, we can choose this neighborhood so that the reduced -${\mathcal L}$-length of its central point from $x$ is bounded by -$3/2$. This allows us to produce an open subset at an earlier time -whose reduced volume is bounded away from zero. Then using -Theorem~ we transfer this conclusion to a non-collapsing -conclusion for $x$. The main issue in this argument is to show that -there is a point in each earlier time-slice whose reduced length -from $x$ is at most $3/2$. We can argue as in the case of a Ricci -flow if we can show that any curve parameterized by backward time -starting at $x$ (a point where the hypothesis of -$\kappa$-non-collapsing holds) that comes close to a surgery cap -either from above or below must have large ${\mathcal L}$-length. In -establishing the relevant estimates we are forced to require that -$\delta_i$ be sufficiently small. - -## The statement of the non-collapsing result - -Here, we shall assume that after surgery the strong canonical -neighborhood assumption holds, and we shall establish the -non-collapsing result. - -**Proposition.** -Suppose that for some $i\ge 0$ we have surgery parameter sequences $\delta_0\ge -\delta_1\ge\cdots\ge \delta_i>0$, $\epsilon=r_0\ge r_1\ge \cdots\ge -r_i>0$ and $\kappa_0\ge \kappa_1\ge \cdots\ge \kappa_i>0$. - Then there is $0<\kappa\le \kappa_i$ and for any $00$ depending only -on $C$ such that ${\mathcal M}$ is $\kappa$-non-collapsed at $x$; that is to say, -if $R(x)=r^{-2}$ with $r\le r_{i+1}$, then $Vol\,B(x,t,r)\ge \kappa r^3$, or $x$ is contained in a component of $M_t$ with positive sectional curvature. - -**Proof.** Since -$R(x)\ge r_{i+1}^{-2}$, by assumption any such $x$ has a strong -$(C,\epsilon)$-canonical neighborhood. If this neighborhood is a -strong $\epsilon$-neck centered at $x$ then the result is clear for -a non-collapsing constant $\kappa$ which is universal. If the -neighborhood is an $\epsilon$-round component containing $x$, then -$x$ is contained in a component of positive sectional curvature. Likewise, -if $x$ is contained in a $C$-component then by definition it is contained in a component of its -time-slice with positive sectional curvature. - -Lastly, we suppose that $x$ is contained in the core $Y$ of a -$(C,\epsilon)$-cap ${\mathcal C}$. Let $r'>0$ be such that the -supremum of $|Rm|$ on $B(x,t,r')$ is $(r')^{-2}$. -Then, by the -definition - of a $(C,\epsilon)$-cap, -$vol\,B(x,t,r')\ge C^{-1}(r')^3$. -Clearly, $r'\le r$ and there is a point $y\in \overline{B(x,t,r')}$ -with $R(y)=(r')^{-2}$. On the other hand, by the -definition of a $(C,\epsilon)$-cap, we have $R(y)/R(x)\le C$, so that $r'/r\ge C^{-1/2}$. -Thus, the -volume of $B(x,t,r)$ is at least $C^{-5/2}r^3$. - -This completes an examination of all cases and establishes the -proposition. - -## Minimizing ${\mathcal L -$-geodesics exist when -$R(x)\le r^{-2}_{i+1}$: the statement} - -The proof of the non-collapsing result when $R(x)=r^{-2}$ with -$r_{i+1}0$ (depending implicitly on $t_0$, $C$, -$\epsilon$, and $i$) such that if $\overline\delta(t)\le \delta$ for -all $t\in [T_{i-1},T_{i+1}]$ then the following holds. Let -$({\mathcal M},G)$ be a Ricci flow with surgery satisfying the -hypothesis of Proposition~ with respect to the given -sequences and $r_{i+1}$, and let $x\in {\mathcal M}$ have $**t**(x)=T$ with $T\in [T_i,T_{i+1})$. Suppose that for some $r\ge -r_{i+1}$ the parabolic neighborhood $P(x,r,T,-r^2)$ exists in -${\mathcal M}$ and $|Rm|\le r^{-2}$ on this neighborhood. Then -there is an open subset $U$ of $**t**^{-1}[T_{i-1},T)$ contained -in the open subset of smooth manifold points of ${\mathcal M}$ with -the following properties: -\begin{enumerate} -\item[(1)] For every $y$ in $U$ there is a minimizing ${\mathcal L}$-geodesic -connecting $x$ to $y$. -\item[(2)] $U_t=U\cap **t**^{-1}(t)$ is non-empty for every $t\in [T_{i-1},T)$ -\item[(3)] For each $t\in [T_{i-1},T)$ the restriction of ${\mathcal -L}$ to $U_t$ achieves its minimum and that minimum is at most -$3\sqrt{(T-t)}$. -\item[(4)] The subset of $U$ consisting of all $y$ with the property that -${\mathcal L}(y)\le {\mathcal L}(y')$ for all $y'\in **t**^{-1}(**t**(y))$ has the property that its intersection with -$**t**^{-1}(I)$ is compact for every compact interval $I\subset -[T_{i-1},T)$. -\end{enumerate} - -The basic idea in proving this result is to show that all paths -beginning at $x$ and parameterized by backward time that come close -to the exposed regions have large ${\mathcal L}$-length. If we can -establish this, then the existence of such paths will not be an -impediment to using the analytic estimates from -Chapter~ to show that for each $t\in [T_{i-1},T)$ -there is a point whose ${\mathcal L}$-length from $x$ is at most -$3\sqrt{T-t}$, and that the set of points that minimize the -${\mathcal L}$-length from $x$ in a given time-slice form a compact -set. - -Given Proposition~, arguments from -Chapter~ will be applied to complete the proof of -Proposition~. - -## Evolution of neighborhoods of surgery caps - - We begin this analysis required to prove Proposition~ - by studying the -evolution of surgery caps. -Proposition~ below is the main result along these lines. -Qualitatively, it says that if the surgery control parameter -$\delta$ is sufficiently small, then as a surgery cap evolves in a -Ricci flow with surgery it stays near the rescaled version of the -standard flow for any rescaled time less than one unless the entire -cap is removed (all at once) by some later surgery. In that case, -the evolution of the cap is close to the rescaled version of the -standard flow until it is removed. Using this result we will show -that if a path parameterized by backward time has final point near -a surgery cap and has initial point with scalar curvature not too -large, then this path must enter this evolving neighborhood either -from the `top' or `size' and because of the estimates that we derive -in this chapter such a path must have large ${\mathcal L}$-length. - -**Proposition.** -Given $A<\infty$, $\delta''>0$ and $0<\theta<1$, there is -$\delta''_0=\delta''_0(A,\theta,\delta'')$ ($\delta_0''$ also -depends on $r_{i+1}$, $C$, and $\epsilon$, which are all now fixed) -such that the following holds. Suppose that $({\mathcal M},G)$ is a -Ricci flow with surgery defined for $0\le t0$ sufficiently small and also -independent of the strong $\delta'$-neck $N$. Gluing in the cap with -a $C^\infty$-metric that converges smoothly to the standard initial -metric $g_0$ as $\delta'$ tends to zero using a fixed -$C^\infty$-partition of unity produces a family of manifolds -uniformly bounded in the $C^\infty$-topology. - -This leads immediately to: - -**Corollary.** Given a sequence of $\delta'_n\rightarrow 0$ and strong -$\delta'_n$-necks $(N(n),g_{N(n)})$ of scales $h_n$ and results of -surgery $(S_0(N(n)),g(n))$ with tips $p_n$ as in the previous claim, -then after passing to a subsequence there is a smooth limit -$(S_\infty,g_\infty,p_\infty)$ of a subsequence of the -$(S_0(N(n)),h_n^{-2}g_0(n)),p_n)$. This limit is the metric from Section~ -that gives the standard initial conditions for a surgery cap. - -**Proof.** That there is a smooth limit of a subsequence is immediate from the -previous claim. Since the $\delta_n$ tend to zero, it is clear that the -limiting metric is the standard initial metric. - -**Lemma.** Suppose that we have a sequence of $3$-dimensional Ricci flows -with surgeries $({\mathcal M}_n,G_n)$ that satisfy the strong -$(C,\epsilon)$-canonical neighborhood assumption with parameter -$r_{i+1}$, and have curvature pinched toward positive. Suppose that -there are surgeries in ${\mathcal M}_n$ at times $t_n$ with surgery -control parameters $\delta'_n$ and scales $h_n$. Let $p_n$ be the -tip of a surgery cap for the surgery at time $ t_n$. Also suppose -that there is $0\le\theta_n<1$ such that for every $A<\infty$, -for all $n$ sufficiently large -there are embeddings -$B(p_a, t_n,A h_n)\times [t_n, t_n+ h^2_n\theta_n)\to {\mathcal -M}_n$ compatible with time and the vector field. Suppose that -$\delta'_n\rightarrow 0$ and $\theta_n\rightarrow \theta<1$ as $n\rightarrow -\infty$. Let $({\mathcal M}'_n,G'_n,p_n)$ be the Ricci flow with -surgery obtained by shifting time by $-t_n$ so that surgery occurs -at $**t**=0$ and rescaling by $h_n^{-2}$ so that the scale of the -surgery becomes one. Then, after passing to a subsequence, the -sequence converges smoothly to a limiting flow -$(M_\infty,g_\infty(t),(p_\infty,0)),\ 0\le t<\theta$. This limiting flow is isomorphic to -the restriction of the standard flow to time $0\le t<\theta$. - -**Proof.** Let $Q<\infty$ be an upper bound for the scalar curvature of the -standard flow on the time interval $[0,\theta)$. Since -$\delta'_n\rightarrow 0$, according to the previous corollary, there is -a smooth limit at time $0$ for a subsequence, and this limit is the -standard initial metric. Suppose that, for some $0\le -\theta'<\theta$, we have established that there is a smooth limiting -flow on $[0,\theta']$. Since the initial conditions are the standard -solution, it follows from the uniqueness statement in -Theorem~ that in fact the limiting flow is -isomorphic to the restriction of the standard flow to this time -interval. Then the scalar curvature of the limiting flow is bounded -by $Q$. Hence, for any $A<\infty$, for all $n$ sufficiently large, -the scalar curvature of the restriction of $G'_n$ to the image of -$B_{G'_n}(p_n,0,2A)\times[0,\theta']$ is bounded by $2Q$. According -to Lemma~ there is an $\eta>0$ and a constant -$Q'<\infty$, each depending only on $Q$, $r_{i+1}$, $C$ and -$\epsilon$, such that for all $n$ sufficiently large, the scalar -curvature of the restriction of $G_n'$ to $B_{G_n'}(p_n,0,A)\times -[0,min(\theta'+\eta,\theta_n))$ is bounded by $Q'$. Because of -the fact that the curvature is pinched toward positive, this implies -that on the same set the sectional curvatures are uniformly bounded. -Hence, by Shi's theorem with derivatives -(Theorem~), it follows that there are uniform bounds -for the curvature in the $C^\infty$-topology. Thus, passing to a -subsequence we can extend the smooth limit to the time interval -$[0,\theta'+\eta/2]$ unless $\theta'+\eta/2\ge \theta$. Since $\eta$ -depends on $\theta$ (through $Q$), but is independent of $\theta'$, -we can repeat this process extending the time-interval of definition -of the limiting flow by $\eta/2$ until $\theta'+\eta/2\ge \theta$. -Now suppose that $\theta'+\eta/2\ge \theta$. Then the argument shows -that by passing to a subsequence we can extend the limit to any -compact subinterval of $[0,\theta)$. Taking a diagonal sequence -allows us to extend it to all of $[0,\theta)$. By the uniqueness of -the standard flow, this limit is the standard flow. - -**Corollary.** -With the notation and assumptions of the previous lemma, for all -$A<\infty$, and any $\delta''>0$, then for all $n$ sufficiently -large, the restriction of $G_n'$ to the image -$B_{G_n'}(p_n,0,A)\times [0,\theta_n)$ is within $\delta''$ in the -$C^{[1/\delta'']}$-topology of the restriction of the standard -solution to the ball of radius $A$ about the tip for time $0\le -t<\theta_n$. - -**Proof.** Let $\eta>0$ depending on $\theta$ (though $Q$) as well as -$r_{i+1}$, $C$ and $\epsilon$ be as in the proof of the previous -lemma, and take $0<\eta'<\eta$. For all $n$ sufficiently large -$\theta_n>\theta-\eta'$, and consequently for all $n$ sufficiently -large there is an embedding $B_{G_n}(p_n,t_n,Ah_n)\times -[t_n,t_n+h_n^2(\theta-\eta')]$ into ${\mathcal M}_n$ compatible with -time and with the vector field. For all $n$ sufficiently large, we -consider the restriction of $G_n'$ to $B_{G_n'}(p_n,0,A)\times -[0,\theta-\eta']$. These converge smoothly to the restriction of the -standard flow to the ball of radius $A$ on the time interval -$[0,\theta-\eta']$. In particular, for all $n$ sufficiently large, -the restrictions to these time intervals are within $\delta''$ in -the $C^{[1/\delta'']}$-topology of the standard flow. Also, for all $n$ sufficiently -large, $\theta_n-(\theta-\eta')<\eta$. Thus, by -Lemma~, we see that the scalar curvature of $G_n'$ -is uniformly bounded (independent of $n$) on -$B_{G_n'}(p_n,0,A)\times [0,\theta_n)$. By the assumption that the -curvature is pinched toward positive, this means that the sectional -curvatures of the $G_n'$ are also uniformly bounded on these sets, -and hence so are the Ricci curvatures. (Notice that these bounds are -independent of $\eta'>0$.) By Shi's theorem with derivatives -(Theorem~), we see that there are uniform bounds on -the curvatures in the $C^\infty$-topology on these subsets, and -hence bounds in the $C^\infty$-topology on the Ricci curvature. -These bounds are independent of both $n$ and $\eta'$. Thus, choosing -$\eta'$ sufficiently close to zero, so that $\theta_n-\eta'$ is also -close to $\theta$ for all $n$ sufficiently large, we see that for all such large $n$ -and all -$t\in [\theta-\eta',\theta)$, the -restriction of $G_n'$ to $B_{G_n'}(p_n,0,A)\times \{t\}$ is -arbitrarily close in the $C^{[1/\delta'']}$-topology to -$G_n'(\theta-\eta')$. The same is of course true of the standard -flow. This completes the proof of the corollary. - -Now we turn to the proof proper of Proposition~. We fix -$A<\infty$, $\delta''>0$ and $\theta<1$. We are free to make $A$ -larger so we can assume by Proposition~ that for the -standard flow the restriction of the flow to $B(p_0,0,A)\setminus -B(p_0,0,A/2)$ remains close to a standard evolving $S^2\times -[A/2,A]$ for time $[0,\theta]$. Let $K<\infty$ be a constant with -the property that $R(x,t)\le K$ for all $x\in B(p_0,0,A)$ in the -standard flow and all $t\in [0,\theta]$. If there is no -$\delta''_0>0$ as required, then we can find a sequence -$\delta'_n\rightarrow 0$ as $n\rightarrow\infty$ and Ricci flows with -surgery $({\mathcal M}_n,G_n)$ with surgeries at time $t_n$ with -surgery control parameter $\delta_n(t_n)\le \delta_n'$ and surgery -scale parameter $ h_n=h(r_{i+1}\delta_n( t_n),\delta_n(t_n))$ -satisfying the hypothesis of the lemma but not the conclusion. Let -$T'_n$ be the final time of $({\mathcal M}_n,G_n)$. Let $\theta_n\le -\theta$ be maximal subject to the condition that there is an -embedding $\rho_n\colon B_{G_n}(x, t_n,Ah_n)\times [ t_n, -t_n+h_n^2\theta_n)\to {\mathcal M}_n$ compatible with time and the -vector field. Let $G'_n$ be the result of shifting the time by -$-t_n$ and scaling the result by $h_n^{-2}$. According to -Lemma~, for all $n$ sufficiently large, the -restriction of $G_n'$ to the image of $\rho_n$ is within $\delta''$ -in the $C^{[1/\delta'']}$-topology of the standard flow restricted -to the ball of radius $A$ about the tip of the standard solution on -the time interval $[0,\theta_n)$. If $\theta_n=min(\theta,(T_n'-t_n)/h_n^2)$, then the first conclusion of -Proposition~ holds for $({\mathcal M}_n,G_n)$ for all -$n$ sufficiently large which contradicts our assumption that the -conclusion of this proposition holds for none of the $({\mathcal -M}_n,G_n)$. If on the other hand $\theta_n0$ be the constant from Proposition~. Fix -$0<\overline\delta_0<1/4$ such that if $g$ is within -$\overline\delta_0$ of $g_0$ in the $C^{[1/\overline -\delta]}$-topology then $|R_{g'}(x)-R_{g_0}(x)|0$ -from Proposition~ the following holds. Suppose that -$({\mathcal M},G)$ is a Ricci flow with surgery defined for $0\le -t< T<\infty$. Suppose that it satisfies the strong -$(C,\epsilon)$-canonical neighborhood assumption at all points $x$ -with $R(x)\ge r_{i+1}^{-2}$. Suppose also that the solution has -curvature pinched toward positive. Suppose that there is a surgery -at some time $\bar t$ with $T_{i-1}\le \bar t\ell.$$ -See Fig.~. - -**Proof.** The logic of the proof is as follows. We -fix $\ell<\infty$. We shall determine the relevant value of -$\theta_0$ and then of $A_0$ in the course of the argument. Then for any $A\ge A_0$ -we define - $\delta''(A)= \delta''_0(A,\theta_0,\overline\delta_0)$, as in -Proposition~. - -The integral expression is invariant under time translation and also -under rescaling. Thus, we can (and do) assume that $\bar t=0$ and -that the scale $h$ of the surgery is $1$. We use the embedding of -$P(p,0,A,T')\to {\mathcal M}$ and write the restriction of the flow -to this subset as a one-parameter family of metrics $g(t),\ 0\le -t\le T'$, on $B(p,0,A)$. With this renormalization, $0\le t'\le -1/2$, also $T'\le \theta_0$, and $\tau=T'-t$. - -Let us first consider the case when $T'=\theta_0\le T$. Consider the -standard flow $(\Ar^3,g_0(t))$, and let $p_0$ be its tip. According -to Proposition~, for all $x\in \Ar^3$ and all $t\in -[0,1)$ we have $R_{g_0}(x,t)\ge c/(1-t)$. By Lemma~ and -since we are assuming that $\overline\delta(\bar t)\le \delta''= -\delta''_0(A,\theta_0,\overline\delta_0)$, we have that $R(a,t)\ge -c/2(1-t)$ for all $a\in B(p,0,A)$ and all $t\in [0,\theta]$. Thus, - we have - -$$ - -$$ -\begin{aligned} -\int_0^{\theta_0-t'}\left(R(\gamma(\tau))+|X_\gamma(\tau)|^2\right)d\tau -& \ge & \int_{t'}^{\theta_0} -\frac{c}{2(1-t)}dt \\ -& = & \frac{-c}{2}\left(log(1-\theta_0)-log(1-t')\right)dt \\ -& \ge & \frac{-c}{2}\left(log(1-\theta_0)+log(2)\right). -\end{aligned} -$$ - -$$ - - Hence, if -$\theta_0<1$ sufficiently close to $1$, the integral will be $> \ell$. -This fixes the value of $\theta_0$. - -**Claim.** There is $A'_0<\infty$ with the property that for any $A\ge A'_0$ -the restriction of the standard solution $g_0(t)$ to -$\left(B(p_0,0,A)\setminus B(p_0,0,A/2)\right)\times [0,\theta_0]$ -is close to an evolving family $(S^2\times [A/2,A],h_0(t)\times -ds^2)$. In particular, for any $t\in [0,\theta_0]$, the -$g_0$-distance at time $t$ from $B(p_0,0,A/2)$ to the complement of -$B(p_0,0,A)$ in the standard solution is more than $A/4$. - -\begin{proof} -This is immediate from Proposition~ and the fact that -$\theta_0<1$. - -Now fix $A_0=max(A_0', 10\sqrt{\ell})$ and let $A\ge A_0$. - -Since $\overline\delta_0<1/4$ and since $T'\le \theta_0$, for $\overline\delta(\bar t)\le -\delta''_0(A,\theta_0,\overline\delta_0)$ -by Proposition~ -the $g(T')$-distance between $B(p,0,A/2)$ and $\partial -B(p,0,A)$ is at least $A/5$. - - Since the -flow on $B(p,0,A)\times[0,T']$ is within $\overline\delta_0$ of -the standard solution, and since the curvature of the standard -solution is non-negative, for any horizontal tangent vector $X$ at -any point of $B(p,0,A)\times [0,T']$ we have that -$$Ric_g(X,X)\ge -\frac{1}{4}|X|_{g_0}^2\ge -\frac{1}{2}|X|^2_g,$$ and hence -$$\frac{d}{dt}|X|_g^2\le |X|_g^2.$$ -Because $T'\le 1$, we see that -$$|X|^2_{g(T')}\le e\cdot|X|^2_{g(t)}<3|X|^2_{g(t)}$$ -for any $t\in [0,T']$. - -Now suppose that $\gamma(0)\in \partial B(p,0,A)\times\{T'\}$. Since -the image of $\gamma$ is contained in the closure of $P(p,0,A,T')$ -for every $\tau\in [0,T']$ we have -$\sqrt{3}|X_\gamma(\tau)|_{g(T'-\tau)}\ge |X_\gamma(\tau)|_{g(T')}$. -Since the flow $g(t)$ on $P(p,0,A, T')$ is within -$\overline\delta_0$ in the $C^{[1/\overline\delta_0]}$-topology of -the standard flow on the corresponding parabolic neighborhood, -$R(\gamma(t))\ge 0$ for all $t\in [0,T']$. Thus, because of these -two estimates we have - -$$ - -\int_0^{T'-t'}\left(R(\gamma(\tau))+|X_\gamma(\tau)|^2\right)d\tau\ge -\int_0^{T'-t'} -\frac{1}{3}|X_\gamma(\tau)|_{g(T')}^2d\tau. -$$ - - Since $\gamma(0)\in -\partial B(p,0,A)\times \{T'\}$ and $\gamma(T')\in B(p,0,A/2)$, it follows from -Cauchy-Schwarz that - -$$ - -$$ -\begin{aligned} -(T'-t')^2\int_0^{T'}|X_\gamma(\tau)|_{g(T')}^2d\tau & -\ge & \left(\int_0^{T'-t'}|X_\gamma(\tau)|_{g(T')}d\tau\right)^2 -\\ -& \ge & \left(d_{g(T')}(B(p,0,A/2),\partial -B(p,0,A))\right)^2\ge \frac{A^2}{25}. -\end{aligned} -$$ - -$$ - - Since $T'-t'<1$, it -immediately follows from this and Equation~() that -$$\int_0^{T'-t'}\left(R(\gamma(\tau))+|X_\gamma(\tau)|^2\right)d\tau\ge -\frac{A^2}{75}.$$ Since $A\ge A_0\ge 10\sqrt{ \ell}$, this -expression is $>\ell$. This completes the proof of -Proposition~ -\end{proof} - -### Paths with short ${\mathcal L -_+$-length avoid the surgery caps} - -Here we show that a path parameterized by backward time that ends in -a surgery cap (or comes close to it) must have -long ${\mathcal L}$-length. Let $({\mathcal M},G)$ be a Ricci flow -with surgery, and let $x\in {\mathcal M}$ be a point with $**t**(x)=T\in (T_i,T_{i+1}]$. We suppose that these data satisfy the -hypothesis of Proposition~ with respect to the given -sequences and $r\ge r_{i+1}>0$. In particular, the parabolic -neighborhood $P(x,T,r,-r^2)$ exists in ${\mathcal M}$ and $|Rm|$ is -bounded on this parabolic neighborhood by $r^{-2}$. - -Actually, here we do not work directly with the length function -${\mathcal L}$ defined from $x$, but rather with a closely related -function. We set $R_+(y)=max(R(y),0)$. - -**Lemma.** -Given $L_0<\infty$, there is -$\overline\delta_1=\overline\delta_1(L_0,r_{i+1})>0$, independent of -$({\mathcal M},G)$ and $x$, such that if $\overline\delta(t)\le -\overline\delta_1$ for all $t\in [T_{i-1},T)$, then for any curve -$\gamma(\tau),\ 0\le \tau\le \tau_0$, with $\tau_0\le T-T_{i-1}$, -parameterized by backward time with $\gamma(0)=x$ and with -$${\mathcal L}_+(\gamma)=\int_0^{\tau_0}\sqrt{\tau} -\left(R_+(\gamma(\tau))+|X_\gamma|^2\right)d\tau< L_0$$_+$|ii} the -following two statements hold: \begin{enumerate} \item[(2)] Set -$$\tau'=min\left(\frac{r_{i+1}^4}{(256)L_0^2},ln(\root 3\of{2})r_{i+1}^2\right).$$ -Then for all $\tau\le min(\tau',\tau_0)$ we have $\gamma(\tau)\in P(x,T,r/2,-r^2)$. -\item[(2)] Suppose that $\bar t\in [T-\tau_0,T)$ is a surgery time with $p$ -being the tip of the surgery cap at time $\bar t$ and with the scale -of the surgery being $\bar h$. Suppose $t'\in [\bar t,\bar t+\bar -h^2/2]$ is such that there is an embedding -$$\rho\colon B(p,\bar t,(50+A_0)\bar h)\times [\bar t,t']\to {\mathcal M}$$ -compatible with time and the vector field. Then the image of $\rho$ -is disjoint from the image of $\gamma$. See Fig.~. -\end{enumerate} - -**Remark.** Recall that $(A_0+4)\bar h$ is the radius of the surgery cap (measured in the -rescaled version of the standard initial metric) that is glued in when -performing surgery with scale $\bar h$. - -**Proof.** We define $\ell=L_0/\sqrt{\tau'}$, then define $A=max(A_0(\ell),2(50+A_0))$ and $\theta=\theta_0(\ell)$. Here, -$A_0(\ell)$ and $\theta_0(\ell)$ are the constants in -Proposition~. - Lastly, we require $\overline\delta_1\le \delta''(A)$ from -Proposition~. Notice that, by construction, - $\delta''(A)=\delta''_0(A,\theta,\overline\delta_0)$ from -Proposition~. Thus, if $p$ is the tip of a surgery cap at time $\bar t$ -with the scale of the surgery being $\bar h$, then it follows that -for any $\Delta t\le \theta$, if there is an embedding -$$\rho\colon B(p,\bar t,A\bar h)\times [\bar t,\bar t+\bar h^2\Delta t)\to {\mathcal M}$$ - compatible with time and the vector field, then the induced flow (after time shifting by $-\bar t$ - and scaling by $(\bar h)^{-2}$ is within $\overline\delta_0$ in the - $C^{[1/\overline\delta_0]}$-topology of the standard solution. - In particular, the scalar curvature -at any point of the image of $\rho$ is positive and is within a -multiplicative factor of two of the scalar curvature at the -corresponding point of the standard flow. - - Recall that we have $r\ge r_{i+1}$ and -that $P(x,T,r,-r^2)$ exists in ${\mathcal M}$ and that $|Rm|\le -r^{-2}$ on this parabolic neighborhood. We begin by proving by -contradiction that there is no $\tau\le \tau'$ with the property -that $\gamma(\tau)\not\in P(x,T,r/2,-r^2)$. Suppose there is such a -$\tau\le \tau'$. Notice that by construction -$\tau' r/2\sqrt{2}$. - -\begin{proof} -Since $|Rm|\le r^{-2}$ on $P(x,T,r,-r^2)$, we have $|Ric|\le -2r^{-2}$ on $P(x,T,r,-\tau'')$. Thus, for any tangent vector $v$ at -a point of $B(x,T,r)$ we have -$$\left|\frac{d(\langle v,v\rangle_{G(T-\tau)})}{d\tau}\right|\le 2r^{-2}\langle v,v\rangle_{G(T-\tau)}$$ -for all -$\tau\in [0,\tau'']$. Integrating gives that for any $\tau\le -\tau''$ we have -$$exp(-2r^{-2}\tau'')\langle v,v\rangle_{G(T)}\le \langle -v,v\rangle_{G(T-\tau)} \le exp(2r^{-2}\tau'')\langle -v,v\rangle_{G(T)}.$$ Since $\tau''\le \tau'$ and $r\ge r_{i+1}$ by -the assumption on $\tau'$ we have -$$exp(2r^{-2}\tau'')\le exp(2\root 3\of{2})< 2.$$ This -implies that for all $\tau\le \tau''$ we have -$$\frac{1}{\sqrt{2}}| X_\gamma(\tau)|_{G(T)}< -|X_\gamma(\tau)|_{G(T-\tau)}<\sqrt{2}|X_\gamma(\tau)|_{G(T)},$$ -and hence -$$\int_0^{\tau''}|X_\gamma(\tau)|d\tau> -\frac{1}{\sqrt{2}}\int_0^{\tau''}|X_\gamma(\tau)|_{G(T)}\ge -\frac{r}{2\sqrt{2}},$$ where we use the fact that -$d_T(\gamma(0),\gamma(\tau''))=r/2$. - -Applying Cauchy-Schwarz to $\tau^{1/4}|X_\gamma|$ and $\tau^{-1/4}$ on the -interval $[0,\tau'']$ yields - -$$ - -$$ -\begin{aligned} -\int_0^{\tau''}\sqrt{\tau}\left(R_+(\gamma(\tau))+|X_\gamma(\tau)|^2\right)d\tau & -\ge & \int_0^{\tau''}\sqrt{\tau}|X_\gamma(\tau)|^2d\tau -\\ -& \ge & -\frac{\left(\int_0^{\tau''}|X_\gamma(\tau)|d\tau\right)^2}{\int_0^{\tau''}\tau^{-1/2}d\tau} -\\ & > & -\frac{r^2}{16\sqrt{\tau''}}\ge L_0. -\end{aligned} -$$ - -$$ - -Of course, the integral from $0$ to $\tau''$ is less than or equal -the entire integral from $0$ to $\tau_0$ since the integrand is -non-negative, contradicting the assumption that ${\mathcal -L}_+(\gamma)\le L_0$. This completes the proof of the first -numbered statement. - -We turn now to the second statement. We impose a further condition -on $\overline\delta_1$. Namely, require that -$\overline\delta_1^22r_{i+1}^{-2}$). Thus, if -$\tau'\ge \tau_0$ we have completed the proof. Suppose that -$\tau'<\tau_0$. It suffices to establish that for every $\tau_1\in -[\tau',\tau_0]$ the point $\gamma(\tau_1)$ is not contained in the -image of $\rho$ for any surgery cap and any $t'$ as in the -statement. Suppose that in fact there is $\tau_1\in [\tau', \tau_0]$ -with $\gamma(\tau_1)$ contained in the image of $\rho(B(p,\bar -t,(A_0+50)\bar h)\times [\bar t,t'])$ where $\bar t\le t'\le \bar -t+(\bar h)^2/2$ and where $p$ is the tip - of some surgery cap at time $\bar t$. We estimate - -$$ - -$$ -\begin{aligned} -\nonumber -\lefteqn{\int_0^{\tau_0}\sqrt{\tau}\left(R_+(\gamma(\tau))+|X_\gamma(\tau)|^2\right)d\tau} & & \\ -& \ge & -\int_{\tau'}^{\tau_0}\sqrt{\tau}\left(R_+(\gamma(\tau))+|X_\gamma(\tau)|^2\right)d\tau -\nonumber -\\ & \ge & -\sqrt{\tau'}\int_{\tau'}^{\tau_1}\left(R_+(\gamma(\tau))+|X_\gamma(\tau)|^2\right)d\tau. - -\end{aligned} -$$ - -$$ - - Let $\Delta t\le -T-\bar t$ be the supremum of the set of $s$ for which there is a -parabolic neighborhood $P(p,\bar t,A\bar h,s)$ embedded in $**t**^{-1}((-\infty,T])\subset {\mathcal M}$. Let $\Delta t_1=min(\theta \bar h^2,\Delta t)$. We consider $P(p,\bar t,A\bar -h,\Delta t_1)$. First, notice that since $\bar h\le -\overline\delta_1^2r_{i}r_{i+1}^{-2}\ge r^{-2}$. In particular, the parabolic -neighborhood $P(x,T,r,-r^2)$ is disjoint from $P(p,\bar t,A\bar -h,\Delta t_1)$. This means that there is some $\tau''\ge \tau'$ such -that $\gamma(\tau'')\in -\partial P(p,\bar t,A\bar h,\Delta t_1)$ and $\gamma|_{[\tau'',\tau_1]}\subset -P(p,\bar t,A\bar h,\Delta t_1)$. There are two cases to consider. -The first is when $\Delta t_1=\theta \bar h^2$, $\tau''=T-(\bar -t+\Delta t_1)$ and $\gamma(\tau'')\in B(p,\bar t,A\bar -h)\times\{\bar t+\Delta t_1\}$. Then, according to -Proposition~, - -$$ - -\int_{\tau''}^{\tau_1}R_+(\gamma(\tau))d\tau> \ell. -$$ - -Now let us consider the other case. If $\Delta t_1<\theta \bar h^2$, -this means that either $\bar t+\Delta t_1=T$ or, according to -Proposition~, at the time $\bar t+\Delta t_1$ there is a -surgery that removes all of $B(p,\bar t,A\bar h)$. Hence, under -either possibility it must be the case that $\gamma(\tau'')\in -\partial B(p,\bar t,A\bar h)\times \{T-\tau''\}$. Thus, the remaining -case to consider is when, whatever $\Delta t_1$ is, -$\gamma(\tau'')\subset -\partial B(p,\bar t,A\bar h)\times \{T-\tau''\}$. -Lemma~ and the fact that $R\ge 0$ on $P(p,\bar t,A\bar -h,\Delta t_1)$ imply that -$$\ell<\int_{\tau''}^{\tau_1}\left(R(\gamma(\tau))+|X_\gamma(\tau)|^2\right)d\tau = -\int_{\tau''}^{\tau_1}\left(R_+(\gamma(\tau))+|X_\gamma(\tau)|^2\right)d\tau.$$ - -Since $\ell=L_0/\sqrt{\tau'}$ and $\tau''\ge \tau'$, it follows from -Equation~() that in both -cases -$${\mathcal L}_+(\gamma)\ge -\int_{\tau''}^{\tau_1}\sqrt{\tau}\left(R_+(\gamma(\tau))+|X_\gamma(\tau)|^2\right)d\tau> -\ell\sqrt{\tau'}=L_0,$$ which contradicts our hypothesis. This -completes the proof of Lemma~. -\end{proof} - -### Paths with small energy avoid - the disappearing regions - -At this point we have shown that paths of small energy do not approach -the surgery caps from above. We also need to rule out that they can be arbitrarily -close from below. That is to say, - we need to see that paths whose ${\mathcal L}$-length is not -too large avoid neighborhoods of the disappearing regions at all -times just before the surgery time at which they disappear. Unlike -the previous estimates which were universal for all $({\mathcal -M},G)$ satisfying the hypothesis of Proposition~, in -this case the estimates will depend on the Ricci flow with surgery. -First, let us fix some notation. - -**Definition.** - Suppose that $\bar t$ is a surgery -time, that $\tau_1>0$, and that there are no other surgery times in -the interval $(\bar t-\tau_1,\bar t]$. Let $\{\Sigma_i(\bar t)\}_i$ -be the $2$-spheres on which we do surgery at time $\bar t$. Each -$\Sigma_i$ is the central $2$-sphere of a strong $\delta$-neck -$N_i$. We can flow the cylinders $J_0(\bar -t)=\cup_is_{N_i}^{-1}(-25,0])$ backward to any time - $t\in (\bar t-\tau_1,\bar t]$. Let $J_0(t)$ be -the result. There is an induced function, denoted -$\coprod_is_{N_i}(t)$, on $J_0(t)$. It takes values in $ (-25,0]$. -We denote the boundary of $J_0(t)$ by $\coprod_i\Sigma_i(t)$. Of -course, this boundary is the result of flowing -$\coprod_i\Sigma_i(\bar t)$ backward to time $t$. (These backward -flows are possible since there are no surgery times in $(\bar -t-\tau_1,\bar t)$.) For each $t\in [\bar t-\tau_1,\bar t)$ we also -have the disappearing region $D_t$: -- the region that disappears at -time $\bar t$. It is an open submanifold whose boundary is -$\coprod_i\Sigma_i(t)$. Thus, for every $t\in (\bar t-\tau_1,\bar -t)$ the subset $J(t)=J_0(t)\cup D_t$ is an open subset of $M_t$. We -define -$$J(\bar t-\tau_1,\bar t)=\cup_{t\in (\bar t-\tau_1,\bar t)}J(t).$$ -Then $J(\bar t-\tau_1,\bar t)$ is an open subset of ${\mathcal M}$. -See Fig.~. - -**Lemma.** -Fix a Ricci flow with surgery $({\mathcal M},G)$, a point $x\in -{\mathcal M}$ and constants $r\ge r_{i+1}>0$ as in the statement of -Proposition~. For any $1<\ell<\infty$ the following -holds. Suppose that $\bar t\in [T_{i-1},T)$ is a surgery time and -that $\gamma(\tau)$ is a path with $\gamma(\tau)\in M_{\bar -t-\tau}$. Let $\{p_1,\ldots,p_k\}$ be the tips of all the surgery -caps at time $\bar t$ and let $\bar h$ be the scale of surgery at -time $\bar t$. Suppose that for some $0<\tau_1\le \ell^{-1} \bar -h^2$ there are no surgery times in the interval $(\bar t-\tau_1,\bar -t)$. We identify all $M_t$ for $t\in [\bar t-\tau_0,\bar t)$ with -$M_{\bar t-\tau_1}$ using the flow. Suppose that $\gamma(0)\in -M_{\bar t}\setminus \cup_{i=1}^kB(p_i,\bar t,(50+A_0)\bar h)$, and -lastly, suppose that -$$\int_{0}^{\tau_1}|X_\gamma(\tau)|^2d\tau\le \ell.$$ -Then $\gamma$ is disjoint from the open subset $J(\bar t-\tau_1,\bar -t))$ of ${\mathcal M}$. - -**Proof.** Suppose that the lemma is false and let $\gamma\colon [0,\bar -\tau]\to {\mathcal M}$ be a path satisfying the hypothesis of the -lemma with $\gamma(\bar \tau)\in J(\bar t-\tau_1,\bar t)$. - Since $$\gamma(0)\in -M_{\bar t}\setminus \cup_iB(p_i,\bar t,(50+A_0)\bar h),$$ if follows -that $\gamma(0)$ is separated from the boundary of -$s_{N_i}^{-1}(-25,0]$ by distance at least $20\bar h$. Since the $ -J_0(t)$ are contained in the disjoint union of strong $\delta$-necks -$N_i$ centered at the $2$-spheres along which we do surgery, and -since $\tau_1\le \bar h^2/\ell<\bar h^2$, it follows that, provided -that $\delta$ is sufficiently small, for every $t\in [\bar -t-\tau_1,\bar t)$, the metric on $J_0(t)$ is at least -$1/2$ the metric on $ J_0(\bar t)$. - It follows that, for $\delta$ sufficiently small, if there is a $\tau\in [0,\tau_1]$ -with $\gamma(\tau)\in J(t)$ then $\int_0^{\tau_1}|X_\gamma|d\tau> 10\bar h$. -Applying Cauchy-Schwarz we see that -$$\int_0^{\tau_1}|X_\gamma(\tau)|^2d\tau\ge (10\bar h)^2/\tau_1.$$ -Since $\tau_1\le \ell^{-1}(\bar h)^2$, we see that -$$\int_0^{\tau'}|X_\gamma(\tau)|^2d\tau>\ell,$$ -contradicting our hypothesis. - -### Limits of a sequence of paths with short ${\mathcal L -$-length} - -Now using Lemmas~ and~ we show that it is possible to take limits -of certain types of sequences of paths parameterized by backward time to create -minimizing ${\mathcal L}$-geodesics. - -We shall work with a compact subset of $**t**^{-1}([T_{i-1},T])$ that is -obtained by removing appropriate open neighborhoods of the exposed regions. - -**Definition.** Fix $\ell<\infty$. Let $\theta_0=\theta_0(\ell)$ be as in -Proposition~. For each surgery time $\bar t\in -[T_{i-1},T]$, let $h(\bar t)$ be the scale of the surgery. - Let $p_1,\ldots,p_k$ be -the tips of the surgery caps at time $\bar t$. For each $1\le j\le k$, -we consider $B_j(\bar -t)=B(p_j,\bar t,(A_0+10)h(\bar t))$, and we let $\Delta t_j\le min(\theta_0,(T-\bar t)/h^2(\bar t))$ be maximal subject to the condition that -there is an embedding $\rho_j\colon B_j(\bar t)\times [\bar t,\bar t+h^2(\bar -t)\Delta t_j)$ into ${\mathcal M}$ compatible with time and the vector field. -Clearly, $B'_j=B(p_j,\bar t,(10+A_0)h)\cap C_{\bar t}$ is contained in $J(\bar -t)$. Let $\bar t'$ be the previous surgery time if there is one, otherwise set -$\bar t'=0$. Also for each $\bar t$ we set $\tau_1(\ell,\bar t)=min\left(h(\bar t)^2/\ell,\bar t-\bar t'\right)$. For each $t\in (\bar -t-\tau_1(\ell,\bar t),\bar t)$ let $\widetilde J(t)\subset J(t)$ be the union -of $D_t$, the disappearing region at time $t$, and $\coprod_iB_i'(t)$, the -result of flowing $\coprod_iB'_i$ backward to time $t$. Then we set -$\widetilde J(\bar t-\tau_1(\ell,\bar t),\bar t)\subset J(\bar -t-\tau_1(\ell,\bar t),\bar t)$ equal to the union over $t\in (\bar -t-\tau_1(\ell,\bar t),\bar t)$ of $\widetilde J(t)$. - - By construction, -for each surgery time $\bar t$, the union $$\nu_sing(\ell,\bar -t)=\widetilde J(\bar t-\tau_1(\ell,\bar t),\bar t)\cup\bigcup -_iB_i\times [\bar t,\bar t+h^2(\bar t)\Delta t_i)$$ is an open -subset of ${\mathcal M}$ containing all the exposed regions and -singular points at time $\bar t$. - -We define $Y(\ell )\subset **t**^{-1}([T_{i-1},T])$ to be the -complement of the $\cup_{\bar t}\nu_sing(\ell,\bar t)$ where -the union is over all surgery times $\bar t\in [T_{i-1},T]$. -Clearly, $Y(\ell )$ is a closed subset of $**t**^{-1}([T_{i-1},T])$ and hence $Y(\ell )$ is a compact subset -contained in the open subset of smooth points of ${\mathcal M}$. -(Notice that $Y(\ell )$ depends on $\ell $ because $\tau_1(\ell,\bar -t)$ and $\theta_0$ depend on $\ell$.) - -**Proposition.** -Fix $0\tau'$. We turn our -attention to the paths $\gamma_n|_{[\tau',\bar\tau]}$. Let -$T_{i-1}<\bar t\le T-\tau'$ be either a surgery time or $T-\tau'$, and let $t'$ be the -maximum of the last surgery time before $\bar t$ and $T_{i-1}$. We consider the -restriction of the $\gamma_n$ to the interval $[T-\bar t,T-t']$. As -we have seen, these restrictions are disjoint from $J(\bar -t-\tau_1(\bar t),\bar t)$ and also from the exposed region at time -$\bar t$, which is denoted $E(\bar t)$, and from $J_0(\bar t)$. Let -$$Y=**t**^{-1}([T-t',T-\bar t])\setminus -\left(J(\bar t-\tau_1(\bar t),\bar t)\cup (E(\bar t)\cup J_0(\bar -t))\right).$$ This is a compact subset with the property that any -point $y\in Y$ is connected by a backward flow line lying entirely -in $Y$ to a point $y(t')$ contained in $M_{t'}$. - - Since $Y$ is compact there is a finite upper bound on the -Ricci curvature on $Y$, and hence to ${\mathcal L}_\chi(G)$ at any -point of $Y$. Since all backward flow lines from points of $Y$ -extend all the way to $M_{t'}$, it follows that there is a constant -$C'$ such that -$$|X_{\gamma_n}(\tau)|_{G(t')}\le C'|X_{\gamma_n}(\tau)|_{G(t)}$$ -for all $t\in [t',\bar t]$. Our hypothesis that the ${\mathcal -L}(\gamma_n)$ are uniformly bounded, the fact that the curvature is -pinched toward positive and the fact that there is a uniform bound -on the lengths of the $\tau$-intervals implies that the -$$\int_{T-\bar t}^{T-t'} \sqrt{\tau}|X_{\gamma_n}(\tau)|^2d\tau$$ -are uniformly bounded. Because $T-\bar t$ is at least $\tau'>0$, it -follows that the $\int_{T-\bar t}^{T-t'}|X_{\gamma_n}|^2d\tau$ have -a uniform upper bound. This then implies that there is a constant -$C_1$ such that for all $n$ we have -$$\int_{T-\bar t}^{T-t'}|X_{\gamma_n}(\tau)|_{G(t')}^2d\tau\le -C_1.$$ Thus, after passing to a subsequence, the $\gamma_n$ converge -uniformly to a continuous $\gamma$ defined on $[T-\bar t,T-t']$. -Furthermore, we can arrange that the convergence is a weak -convergence in $W^{1,2}$. This means that $\gamma$ has a derivative -in $L^2$ and -$$\int_{T-\bar t}^{T-t'}|X_{\gamma}(\tau)|^2d\tau\le liminf_{n\rightarrow -\infty}\int_{T-\bar t}^{T-t'}|X_{\gamma_n}(\tau)|^2d\tau.$$ - -Now we do this simultaneously for all $\bar t=T-\tau'$ and for all the finite number of surgery times -in $[T_{i-1},T-\tau']$. This -gives a limiting path $\gamma\colon [\tau',\bar\tau]\to {\mathcal M}$. -Putting together the above inequalities we see that the limit -satisfies - -$$ - -\int_{\tau'}^{\bar\tau}\sqrt{\tau}\left(R(\gamma(\tau))+|X_\gamma(\tau)|^2\right)d\tau\le -liminf_{n\rightarrow\infty}\int_{\tau'}^{\bar\tau}\sqrt{\tau}\left(R(\gamma_n(\tau))+ -|X_{\gamma_n}(\tau)|^2\right)d\tau. -$$ - - Since we have already -arranged that there is a limit on $[0,\tau']$, this produces a -limiting path $\gamma\colon [0,\tau_0]\to {\mathcal M}$. By -Inequalities~ and~ we see that -$${\mathcal L}(\gamma)\le liminf_{i\rightarrow \infty}{\mathcal -L}(\gamma_n).$$ -The limit lies in the compact subset $Y(\ell)$ and hence is contained -in the open subset of smooth points of ${\mathcal M}$. - This completes the proof of the first statement of the proposition. - -Now suppose, in addition to the above, that all the $\gamma_n$ -have the same endpoint $y\in M_{T-\tau_0}$ and that $lim_{n\rightarrow\infty}{\mathcal L}(\gamma_n)$ is less than or equal -to the ${\mathcal L}$-length of any path parameterized by backward -time connecting $x$ to $y$. Let $\gamma$ be the limit of a -subsequence as constructed in the proof of the first part of this -result. Clearly, by what we have just established, $\gamma$ is a path -parameterized by backward time from $x$ to $y$ and ${\mathcal -L}(\gamma)\le lim_{n\rightarrow\infty}{\mathcal L}(\gamma_n)$. -This means that $\gamma$ is a minimizing ${\mathcal L}$-geodesic -connecting $x$ to $y$, an ${\mathcal L}$-geodesic contained in the open subset of smooth -points of ${\mathcal M}$. - -This completes the proof of the proposition. -\end{proof} - -**Corollary.** -Given $L<\infty$, let -$\overline{\delta}_1=\overline\delta_1(L+4(T_{i+1}^{3/2},r_{i+1})$ be as given in Lemma~. -If -$\overline\delta(t)\le \overline\delta_1$ for all $t\in [T_{i-1},T_{i+1}]$, -then for any $x\in **t**^{-1}([T_i,T_{i+1}))$ and for any $y\in -M_{T_{i-1}}$, if there is a path $\gamma$ parameterized by -backward time connecting $x$ to $y$ with ${\mathcal L}(\gamma)\le L$, -then there is a minimizing ${\mathcal L}$-geodesic contained in the -open subset of smooth points of ${\mathcal M}$ connecting $x$ to -$y$. - -**Proof.** Choose an ${\mathcal L}$-minimizing sequence of paths from $x$ to -$y$ and apply the previous proposition. - -### Completion of the proof of Proposition~\protect{} - -Having found a compact subset of the open subset of smooth points of ${\mathcal M}$ -that contains all paths parameterized by backward time whose ${\mathcal L}$-length is not -too large, we are in a position to prove Proposition~, which states -the existence of a minimizing ${\mathcal L}$-geodesics in ${\mathcal M}$ from $x$ -and gives estimates on their ${\mathcal L}$-lengths. - -**Proof.** (of Proposition~). -Fix $r\ge r_{i+1}>0$. Let $({\mathcal M},G)$ and $x\in {\mathcal M}$ be -as in the statement of Proposition~. We set -$L=8\sqrt{T_{i+1}}(1+T_{i+1})$, and we set -$$\delta=min\left(\delta_i,\overline \delta_1(L+4(T_{i+1})^{3/2},r_{i+1})\right),$$ -where $\overline\delta_1$ is as given in Lemma~. -Suppose that $\overline\delta(t)\le -\delta$ for all $t\in [T_{i-1},T_{i+1})$. We set $U$ equal to the -subset of $**t**^{-1}([T_{i-1},T))$ consisting of all points $y$ -for which there is a path $\gamma$ from $x$ to $y$, parameterized by -backward time, with ${\mathcal L}(\gamma)0$ sufficiently small. Also, -extending $\gamma$ by adding a backward vertical flow line from $z$ -shows that the minimum value of ${\mathcal L}_x$ on $U_{t-\mu}$ is -at most $(D'+D)/2$ for all $\mu>0$ sufficiently small. (Such a -vertical flow line backward in time exists since $z\in U$ and hence -$z$ is contained in the smooth part of ${\mathcal M}$.) This -contradicts the fact that limit of the minimum values of ${\mathcal -L}_x$ on $U_{t_n}$ converge to $D$ as $t_n$ converges to $t$. This -contradiction proves that the minimum value of ${\mathcal L}_x$ on -$U_t$ is $D$, and consequently the point $y\in Z'$. This proves that -$Z'\cap **t**^{-1}(I)$ is compact, establishing the claim. - -At this point we have established that Properties (1),(2), and (4); -So it remains only to prove Property (3) of Proposition~. - To do this we define the reduced length function -$l_x\colon U\to \Ar$ by -$$l_x(q)=\frac{{\mathcal L}_x(q)}{2\sqrt{T-**t**(q)}} \ \ and \ \ -l_x^min(\tau)=min_{q\in M_t}l_x(q).$$ We consider the -subset ${\mathcal S}$ of $\tau'\in (0,T-T_{i-1}]$ with $l_x^min(\tau)\le L/2$ for all $\tau\le \tau'$. Recall that by the -choice of $L$, we have $3\sqrt{T-T_{i-1}}< L/2$. Clearly, the -minimum value of $l_x$ on $U_{T-\tau}$ converges to $0$ -as $\tau\rightarrow 0$, implying that this set is non-empty. Also, from -its definition, ${\mathcal S}$ is an interval with $0$ being one -endpoint. - -**Lemma.** -Let $l_x^min(\tau')$ be the minimum value of $l_x$ on -$U_{T-\tau'}$. For any $\tau\in {\mathcal S}$ we have $l_x^min(\tau)\le 3/2$. - -**Proof.** Given that we have already established Properties 1,2 and 4 of -Proposition~, this is immediate from -Corollary~. - -Now let us establish that ${\mathcal S}=(0,T-T_{i-1}]$. As we remarked above, -${\mathcal S}$ is a non-empty interval with $0$ as one endpoint. Suppose that -it is of the form $(0,\tau]$ for some $\tau\tau$ but sufficiently close to -$\tau$, there is a point $y(\tau')\in U_{T-\tau'}$ with ${\mathcal -L}_x(y(\tau'))0.85$ and -$(2/3)^{3/2}\le 0.6$, the first inequality follows. - -The second inequality is clear since $T'<(T-T_{i-1})$. - -The last inequality is clear from the fact that -$T''=T-T_{i-1}-\epsilon^2$ and $\epsilon\le \sqrt{t_0/50}$. - - Putting these together yields - $${\mathcal L}(\gamma)\ge - \left[\left(\frac{1}{6}r_i^{-2}-4\right)\left(T-T_{i-1}\right)-\frac{2t_0}{25}\right] - \sqrt{T-T_{i-1}}.$$ - -Since -$$r_i^{-2}\ge r_0^{-2}\ge \epsilon^{-2}\ge 50/t_0,$$ -and $T-T_{i-1}\ge t_0$ we see that -$$ - -$$ -\begin{aligned} -{\mathcal -L}(\gamma) & \ge & -\left[\left(\frac{50}{6t_0}-4\right)t_0-\frac{2t_0}{25}\right]\sqrt{T-T_{i-1}} -\\ -& \ge & (8-5t_0)\sqrt{T-T_{i-1}} \\ -& \ge & 4\sqrt{T-T_{i-1}}. -\end{aligned} -$$ - -$$ - -(The last inequality uses the fact that $t_0=2^{-5}$.) But this -contradicts the fact that ${\mathcal L}(\gamma)\le -3\sqrt{T-T_{i-1}}$. -\end{proof} - -Now fix $\tau_0$ satisfying Claim~. - Let $\gamma_1$ be the -restriction of $\gamma$ to the subinterval $[0,\tau_0]$, and let -$y=\gamma_1(\tau_0)$. Again using the fact that $R(\gamma(\tau))\ge --6$ for all $\tau$, we see that - -$$ - -{\mathcal L}(\gamma_1)\le {\mathcal L}(\gamma)+4(T-T_{i-1})^{3/2}\le -3(T_{i+1})^{1/2}+4(T_{i+1})^{3/2}. -$$ - -Set $t'=T-\tau_0$. Notice that from the definition we -have $t'\le T_i$. Consider $B=B(y,t',\frac{r_i}{2C})$, and -define $\Delta=min(r_i^2/16C,\epsilon^2)$. According to -Lemma~ every point $z$ on a backward flow line -starting in $B$ and defined for time at most $ \Delta$ has the -property that $R(z)\le 2r_i^{-2}$. For any surgery time $\bar t$ in -$[t'-\Delta,t')\subset [T_{i-1},T)$ the scale $\bar h$ of the -surgery at time $\bar t$ is $\le \delta(\bar t)^2 r_i$, and hence -every point of the surgery cap has scalar curvature at least -$D^{-1}\delta(\bar t)^{-4}r_i^{-2}$. Since $\overline\delta(\bar t)\le -\overline \delta\le \delta_0\le min(D^{-1},1/10)$, it follows -that every point of the surgery cap has curvature at least -$\delta_0^{-3}r_i^{-2}\ge 1000r_i^{-2}$. Thus, - no point -$z$ as above can lie in a surgery cap. This means that the entire -backward parabolic neighborhood $P(y,t',\frac{r_i}{2C},-\Delta)$ -exists in ${\mathcal M}$, and the scalar curvature is bounded by $2 -r_i^{-2}$ on this backward parabolic neighborhood. Because of the -curvature pinching toward positive assumption, there is $C'<\infty$ -depending only on $r_i$ and such that the Riemann curvature is -bounded by $C'$ on $P(y,t',\frac{r_i}{2C},-\Delta)$. - -Consider the one-parameter family of metrics $g(\tau),\ 0\le \tau\le -\Delta$, on $B(y,t',\frac{r_i}{2C})$ obtained by restricting the -horizontal metric $G$ to the backward parabolic neighborhood. -There is $0<\Delta_1\le \Delta /2$ -depending only on $C'$ such that for every $\tau\in [0,\Delta_1]$ and -every non-zero tangent vector $v$ at a point of -$B(y,t',\frac{r_i}{2C})$ we have -$$\frac{1}{2}\le \frac{|v|^2_{g(\tau)}}{|v|^2_{g(0)}}\le 2.$$ -Set $\hat r=min(\frac{r_i}{32C},\Delta_1/2)$, so that $\hat r$ depends -only on $r_i$, $C$, and $\epsilon$. Set $t''=t'-\Delta_1$. Clearly, -$B(y, t'',\hat r)\subset B(y,t',\frac{r_i}{2C})$ so that $B(y,t'',\hat r)\subset -P(y,t',\frac{r_i}{2C},-\Delta)$. Of course, it then follows that the parabolic neighborhood -$P(y,t'',\hat r,-\Delta_1)$ exists in ${\mathcal M}$ and -$$P(y,t'',\hat r,-\Delta_1)\subset P(y,t',\frac{r_i}{2 C},-\Delta),$$ so that the -Riemann curvature is bounded above by $C'$ on the parabolic -neighborhood $P(y,t'',\hat r,-\Delta_1)$. We set $r'=min(\hat r,(C')^{-1/2},\sqrt{\Delta_1}/2)$, so that $r'$ depends only on -$r_i$, $C$, and $\epsilon$. Then the parabolic neighborhood -$P(y,t'',r',-(r')^2)$ exists in ${\mathcal M}$ and $|Rm|\le -(r')^{-2}$ on $P(y,t'',r',-(r')^2)$. Hence, by the inductive -non-collapsing assumption either $y$ is contained in a component of $M_{t''}$ of positive sectional curvature or -$$Vol\,B(y,t'',r')\ge -\kappa_i(r')^3.$$ - -If $y$ is contained in a component of $M_{t''}$ of positive sectional curvature, then by - Hamilton's result, Theorem~, -under Ricci flow the component of $M_{t''}$ containing $y$ flows forward as -a family of components of positive sectional curvature until it disappears. Since there is path moving backwards in time -from $x$ to $y$, this means that the original point $x$ is contained -in a component of its time-slice with positive sectional curvature. - -Let us consider the other possibility when $Vol\,B(y,t'',r')\ge -\kappa_i(r')^3$. - For each $z\in B(y,t'',r')$ let - $$\mu_z\colon [T-t',T-t'']\to B(y,t',\frac{r_i}{2C})$$ be - the $G(t')$-geodesic connecting $y$ to $z$. Of course - $$|X_{\mu_z}(\tau)|_{G(t')}\le \frac{r'}{\Delta_1}$$ - for every $\tau\in [0,\Delta_1]$. Thus, - $$|X_{\mu_z}(\tau)|_{G(T-\tau)}\le - \frac{\sqrt{2}r'}{\Delta_1}$$ for all $\tau\in [T-t',T-t'']$. - Now we let $\tilde \mu_z$ be the resulting path parameterized by backward time - on the time-interval $[T-t',T-t'']$. - We estimate - -$$ - -$$ -\begin{aligned} -{\mathcal - L}(\tilde \mu_z) & = & - \int_{T-t'}^{T-t''}\sqrt{\tau}\left(R(\tilde\mu_z(\tau))+|X_{\tilde\mu_z}(\tau)|^2\right)d\tau - \\ -& \le & \sqrt{T-t''}\int_{T-t'}^{T-t''}\left(2 r_i^{-2}+\frac{2(r')^2}{\Delta_1^2}\right)d\tau -\\ -& \le & \sqrt{T-t''}(2r_i^{-2}\Delta_1+\frac{1}{2}) \le -\sqrt{T}\left(\frac{1}{16C}+\frac{1}{2}\right). -\end{aligned} -$$ - -$$ - -In passing to the last inequality we use the fact, from the -definitions that $r'\le \sqrt{\Delta_1}/2$ and $\Delta\le -r_i^2/16C$, whereas $\Delta_1\le \Delta/2$. - -Since $C>1$, we see that -$${\mathcal L}(\tilde \mu_z)\le \sqrt{T}.$$ -Putting this together with the estimate, Equation~(), -for ${\mathcal L}(\gamma_1)$ tells us that for each $z\in -B(y,t'',r')$ we have -$${\mathcal L}(\gamma_1*\tilde\mu_z)\le -4(T_{i+1})^{1/2}+4(T_{i+1})^{3/2}\le L/2.$$ Hence, by -Proposition~ and the choice of $L$, there is a -minimizing ${\mathcal L}$-geodesic from $x$ to each point of -$B(y,t'',r')$ of length $\le L/2$, and these geodesics are contained -in the smooth part of ${\mathcal M}$. In fact, by -Proposition~ there is a compact subset $Y$ of -the open subset of smooth points of ${\mathcal M}$ that contains all -the minimizing ${\mathcal L}$-geodesics from $x$ to points of -$B(y,t'',r')$. - -Then, by Corollary~ (see also, -Proposition~), the intersection, $B'$, of ${\mathcal U}_x$ -with $B(y,t'',r')$ - is an open subset of full measure in $B(y,t'',r')$. -Of course, $Vol\,B'=Vol\,B(y,t'',r')\ge \kappa_i(r')^3$ -and the function $l_x$ is bounded by $L/2$ on $B'$. It now follows -from Theorem~ that there is $\kappa>0$ depending only on -$\kappa_i$, $r'$, $\epsilon$ and $L$ such that $x$ is $\kappa$ -non-collapsed on scales $\le \epsilon$. Recall that $L$ depends only -on $T_{i+1}$, and $r'$ depends only on $r_i,C,C'$ and $\epsilon$, -whereas $C'$ depends only on $r_i$. Thus, in the final analysis, -$\kappa$ depends only on $\kappa_i$ and $r_i$ (and $C$ and -$\epsilon$ which are fixed). This entire analysis assumed that for -all $t\in [T_{i-1},T_{i+1})$ we have the inequality -$\overline\delta(t)\le -\overline\delta_1(L+4(t_{i+1})^{3/2},r_{i+1})$ as in -Lemma~. Since $L$ depends only on $i$ and $t_0$, this shows -that the upper bound for $\delta$ depends only on $r_{i+1}$ (and on -$i$, $t_0$, $C$, and $\epsilon$). This completes the proof of -Proposition~. - -# Completion of the proof of Theorem~\protect{} - -We have established the requisite non-collapsing result assuming the -existence of strong canonical neighborhoods. In order to complete -the proof of Theorem~ it remains for us to show the -existence of strong canonical neighborhoods. This is the result of -the next section. - -## Proof of the strong canonical neighborhood assumption - -**Proposition.** -Suppose that for some $i\ge 0$ we have surgery parameter sequences -$\delta_0\ge \delta_1\ge\cdots\ge \delta_i>0$, $\epsilon=r_0\ge -r_1\ge \cdots\ge r_i>0$ and $\kappa_0\ge \kappa_1\ge \cdots\ge -\kappa_i>0$. For any $r_{i+1}\le r_i$, let $\delta(r_{i+1})>0$ be -the constant in Proposition~ associated to these -three sequences and to $r_{i+1}$. - Then there are positive -constants $r_{i+1}\le r_i$ and $\delta_{i+1}\le\delta(r_{i+1})$ such that the -following holds. Suppose that $({\mathcal M},G)$ is a Ricci flow with surgery -defined for $0\le t0$ and for any counterexample point -$x\in {\mathcal M}_{(a,b)}$ eventually $**t**(x_{n})< **t**(x)+\xi$. Let $t'=t_{(a,b)}'=lim_{n\rightarrow\infty}**t**(x_{n})$. Clearly, $t'\in [T_i,T_{(a,b)})$, and by construction -the restriction of $({\mathcal M}_{(a,b)},G_{(a,b)})$ to $**t**^{-1}([0,t'))$ satisfies the $(C,\epsilon)$-canonical neighborhood assumption -with parameter $r_a$. Since the surgery times are discrete, there is -$t''=t_{(a,b)}''$ with $t'0$ for all $b$ sufficiently large, we -have $\delta_{(a,b)}\le \delta''(A,\theta_1,\overline\delta)$, where -$\delta''(A,\theta_1,\overline\delta)$ -is the constant given in Proposition~. - -**Claim.** Suppose that $b$ is sufficiently large so that -$\delta_{(a,b)}\le \delta''(A,\theta_1,\overline\delta_0)$, where $\overline\delta_0$ is the constant -given in Definition~. Then -$\tilde t_{(a,b)}'\le \theta_1$. - -\begin{proof} -In this proof we shall fix $(a,b)$, so we drop these indices from -the notation. - Consider -$s\le \theta_1$ maximal so that there is an embedding -$$\psi=\psi_{(a,b)}\colon -B(p_0,0,A)\times [0,s)\to \widetilde{\mathcal -M}_{(a,b)}$$ compatible with time and the vector field. First -suppose that $s<\theta_1$. Then according to -Proposition~ either the entire ball $B(p,0,A)$ -disappears at time $s$ or $s$ is the final time of the -time interval of definition for the flow $(\widetilde{\mathcal -M}_{(a,b)},\widetilde G_{(a,b)})$. Since we have the flow line from $z\in -B(p_0,0,A)$ extending to time $\tilde t'=\tilde t'_{(a,b)}$, in either case this -implies that $\tilde t'0$, for -all $b$ sufficiently large the pullback under this map of $\widetilde G_{(a,b)}$ is within $\delta$ -in the $C^{[1/\delta]}$-topology of the restriction of the standard solution. -Let $w_{(a,b)}$ be the result of flowing $x_{(a,b)}$ backward to time $0$. - -**Claim.** For all $b$ sufficiently large, $w_{(a,b)}\in \psi_{(a,b)}(B(p_0,0,A)\times\{0\})$. - -**Proof.** First notice that, by our choice of $\epsilon$, - every point in the $0$ time-slice of the closure of the strong $\epsilon$-neck -centered at $x_{(a,b)}$ is within distance $(1.1)\tilde Q_{(a,b)}^{-1}\epsilon^{-1}$ -of $w_{(a,b)}$. In particular, -$$d_{\widetilde G_{(a,b)}}(w_{(a,b)},y_{(a,b)})<(1.1)\tilde Q_{(a,b)}^{-1/2}\epsilon^{-1}.$$ -Since $y_{(a,b)}$ is contained in the surgery cap and the scale of the surgery -at this time is $1$, $y_{(a,b)}$ is within distance $A_0+5$ of $p_{(a,b)}$. -Hence, by the triangle inequality and Inequality~(), we have - -$$ - -$$ -\begin{aligned} -d_{\widetilde G_{(a,b)}}(w_{(a,b)},p_{(a,b)}) & < & (1.1)\tilde Q_{(a,b)}^{-1/2}\epsilon^{-1}+(A_0+5) \\ -& < & (1.1)\sqrt{5D}\epsilon^{-1}+(A_0+5). -\end{aligned} -$$ - -$$ - -For $b$ sufficiently large, the image $\psi_{(a,b)}(B(p_0,0,A))$ -contains the ball of radius $(0.95)A$ -centered at $p_{(a,b)}$. Since by our choice of $A$ we have $(0.95)A>(1.1)\sqrt{5D}\epsilon^{-1}+(A_0+5)$, -the claim follows. - -We define $q_{(a,b)}\in B(p_0,0,A)$ so that -$\psi_{(a,b)}(q_{(a,b)},0)=w_{(a,b)}$. Of course, -$$\psi_{(a,b)}(q_{(a,b)},\tilde t'_{(a,b)})=x_{(a,b)}.$$ If follows from the -above computation that for all $b$ sufficiently large we have -$$d_0(q_{(a,b)},p_0)<(1.15)\tilde Q_{(a,b)}^{-1/2}\epsilon^{-1}+(1.05)(A_0+5).$$ -Since the standard flow has non-negative curvature, it is a distance non-increasing flow. Therefore, -$$d_{\tilde t'_{(a,b)}}(q_{(a,b)},p_0)<(1.15)\tilde Q_{(a,b)}^{-1/2}\epsilon^{-1}+(1.05)(A_0+5).$$ - -Suppose that a point $(q,\tilde t'_{(a,b)})$ in the standard solution were the center of a -$\beta\epsilon/3$-neck, where $\beta$ is the constant from Proposition~. - Of course, for all $b$ sufficiently large, $R(q,\tilde -t'_{(a,b)})>(0.99)\tilde Q_{(a,b)}$. Since $\beta<1/2$ and -$\epsilon<\sqrt{5D}(A_0+5)/2$ and $\tilde Q_{(a,b)}\le 5D$, it follows from the -above distance estimate that this neck would contain $(p_0,\tilde t'_{(a,b)})$. -But this is impossible: since $(p_0,\tilde t'_{(a,b)})$ is an isolated fixed -point of an isometric $SO(3)$-action on the standard flow, all the sectional -curvatures at $(p_0,\tilde t'_{(a,b)})$ are equal, and this is in contradiction -with estimates on the sectional curvatures at any point of an $\epsilon$-neck -given in Lemma~. We can then conclude from Theorem~ -that for all $b$ sufficiently large, the point $(p_0,\tilde t'_{(a,b)})$ is -contained in the core of a $(C(\beta\epsilon/3),\beta\epsilon/3)$-cap -$Y_{(a,b)}$ in the $\tilde t'_{(a,b)}$ time-slice of the standard solution. -Now note that for -all $b$ sufficiently large, the scalar curvature of $(q_{(a,b)},\tilde -t'_{(a,b)})$ is at least $(0.99)\tilde Q_{(a,b)}$, since the scalar curvature of $x_{(a,b)}$ -is equal to $Q_{(a,b)}$. This implies that the -diameter of $Y_{(a,b)}$ is at most -$$(1.01)\tilde Q^{-1/2}_{(a,b)}C(\beta\epsilon/3)<(1.1)\sqrt{5D}C(\beta\epsilon/3).$$ -Since $B(p_0,0,A)$ contains $B(p_0,\tilde t'_{(a,b)},A/2)$, and since -$C>C(\beta\epsilon/3)$, it follows from the definition of $A$, the above -distance estimate, and the triangle inequality that for all $b$ sufficiently -large $B(p_0,0,A)\times\{\tilde t'_{(a,b)}\}$ contains $Y_{(a,b)}$. - - Since $C> -C(\beta\epsilon/3)+1$ and since for $b$ sufficiently large -$\psi_{(a,b)}^*\widetilde G_{(a,b)}$ is arbitrarily close to the -restriction of the standard solution metric, it follows from -Lemma~ that for all $b$ sufficiently large, the image $\psi_{(a,b)}(Y_{(a,b)})$ - is a $(C,\epsilon)$-cap whose core contains $x_{(a,b)}$. -As we have already remarked, this contradicts the assumption that no $x_n$ -has a strong $(C,\epsilon)$-canonical neighborhood. - - This completes the proof in the last case and establishes -Lemma~. -\end{proof} - -**Remark.** Notice that even though $x_{(a,b)}$ is the center of a strong -$\epsilon$-neck, the canonical neighborhoods of the $x_n$ constructed -in the second case are not a strong $\epsilon$-necks but rather are -$(C,\epsilon)$-caps coming from applying the flow to a -neighborhood of the surgery cap ${\mathcal C}$. - -Now we return to the proof of Proposition~. For each $a$, we pass -to a subsequence (in $b$) so that Lemma~ holds for all $(a,b)$. For -each $(a,b)$, let $t_{(a,b)}$ be as in that lemma. We fix a point -$x_{(a,b)}\in **t**^{-1}(t_{(a,b)})\subset {\mathcal M}_{(a,b)}$ at which the -canonical neighborhood assumption with parameter $r_a$ fails. For each $a$ -choose $b(a)$ such that $\delta_{b(a)}\rightarrow 0$ as $a\rightarrow \infty$. For each -$a$ we set $({\mathcal M}_a,G_a)=({\mathcal M}_{(a,b(a))},G_{(a,b(a))})$, we -set $t_a=t_{(a,b(a))}$, and we let $x_a=x_{(a,b(a))}\in {\mathcal M}_a$. Let -$(\widetilde {\mathcal M}_a,\widetilde G_a)$ be the Ricci flow with surgery -obtained from $({\mathcal M}_a,G_a)$ by shifting $t_a$ to $0$ and rescaling the -metric and time by $R(x_a)$. We have the points $\tilde x_a$ in the $0$ -time-slice of $\widetilde {\mathcal M}_a$ corresponding to $x_a\in {\mathcal -M}_a$. Of course, by construction $R_{\widetilde G_a}(\tilde x_a)=1$ for all -$a$. - -We shall take limits of a subsequence of this sequence of based -Ricci flows with surgery. Since $r_a\rightarrow 0$ and $R(x_a)\ge -r_a^{-2}$, it follows that $R(x_a)\rightarrow \infty$. By -Proposition~, since $\delta_{b(a)}\le \delta(r_a)$ -it follows that the restriction of $(\widetilde {\mathcal -M}_a,\widetilde G_a)$ to $**t**^{-1}(-\infty,0)$ is -$\kappa$-non-collapsed on scales $\le \epsilon R_{G_a}^{1/2}(x_a)$. -By passing to a subsequence we arrange that one of the following two -possibilities holds: -\begin{enumerate} -\item[(i)] There is $A<\infty$ and $t'<\infty$ such that, -for each $a$ there is a flow line through a point $y_a$ of $B_{\widetilde -G_a}(\tilde x_a,0,A)$ that is not defined on all of $[-t',0]$. (See -Fig.~.) -\item[(ii)] For every $A<\infty$ and every $t'<\infty$, for all $a$ -sufficiently large all flow lines through points of $B_{\widetilde -G_a}(\tilde x_a,0,A)$ are defined on the interval $[-t', 0]$. -\end{enumerate} - -Let us consider the second case. By Proposition~ -these rescaled solutions are $\kappa$-non collapsed on scales $\le -\epsilon R_{G_a}(x_a)^{1/2}$ for all $t<0$. Since this condition is -a closed constraint, the same is true if $t=0$. Since $R(x_a)\ge -r_a^{-2}$, by construction every point $\tilde -x\in(\widetilde{\mathcal M}_a,\widetilde G_a)$ with $R(\tilde x)\ge -1$ and $**t**(\tilde x)<0$ has a strong $(C,\epsilon)$-canonical -neighborhood. - -**Claim.** For all $a$ sufficiently large, -every point $\tilde x\in(\widetilde{\mathcal M}_a,\widetilde G_a)$ with $R(\tilde x)>1$ and -$**t**(\tilde x)=0$ has a $(2C,2\epsilon)$-canonical neighborhood. - -**Proof.** Assume that $\tilde x\in \widetilde{\mathcal M}$ has $R(\tilde -x)>1$. Suppose that $\tilde x$ is an exposed point. If $a$ is -sufficiently large, then $\delta_{b(a)}$ is arbitrarily close to -zero and hence by the last item in Theorem~ and -the structure of the standard initial condition, we see that -$\tilde x$ is contained in the core of a $(2C,2\epsilon)$-cap. - -Suppose now that $\tilde x$ is not an exposed point. - Then we can take a sequence of points -$\tilde y_n\in \widetilde{\mathcal M}_a$ all lying on the flow line for the vector field through $\tilde x$ -converging to $\tilde x$ with $**t**(\tilde y_n)<0$. Of course, for all $n$ sufficiently large -$R(\tilde y_n)>1$, -which implies that for all $n$ sufficiently large $\tilde y_n$ has a strong $(C,\epsilon)$-canonical -neighborhood. Passing to a subsequence, we can arrange that all of these canonical neighborhoods -are of the same type. If they are all $\epsilon$-round components, all $C$-components, or all -$(C,\epsilon)$-caps whose cores contain $y_n$, then by taking limits and arguing as in the proof of -Lemma~ we see that $\tilde x$ has a strong -$(2C,2\epsilon)$-canonical neighborhood of the same type. On the other hand, if $\tilde y_n$ is the center -of a strong $\epsilon$-neck for all $n$, then according to Claim~, -the limit point $\tilde x$ is the center -of a strong $2\epsilon$-neck. - -Since we have chosen $\epsilon>0$ sufficiently small so that Theorem~ -applies with $\epsilon$ replaced by $2\epsilon$, applying this theorem shows -that we can pass to a subsequence and take a smooth limiting -flow of a subsequence of the rescaled flows $(\widetilde {\mathcal -M}_a,\widetilde G_a)$ based at $\tilde x_a$ and defined for all -$t\in (-\infty,0]$. Because the $({\mathcal M}_a,G_a)$ all have -curvature pinched toward positive and since $R(x_a)\rightarrow \infty$ -as $a$ tends to infinity, this result says that the limiting flow -has non-negative, bounded curvature and is $\kappa$-non-collapsed on -all scales. That is to say, the limiting flow is a -$\kappa$-solution. By Corollary~ this contradicts -the fact that the strong $(C,\epsilon)$-canonical neighborhood -assumption fails at $x_a$ for every $a$. This contradiction shows -that in the second case there is a subsequence of the $a$ such that -$x_a$ has a strong canonical neighborhood and completes the proof of the second case. - -Let us consider the first case. In this case we will arrive at a -contradiction by showing that for all $a$ sufficiently large, the -point $x_a$ lies in a strong $(C,\epsilon)$-canonical neighborhood -coming from a surgery cap. Here is the basic result we use to find -that canonical neighborhood. - -**Lemma.** -Suppose that there are $A',D',t'<\infty$ such that the following -holds for all $a$ sufficiently large. There is a point $y_a\in -B_{\widetilde G_a}(\tilde x_a,0,A')$ and a flow line of $\chi$ - beginning at $y_a$, defined for backward time and ending at a - point $z_a$ in a -surgery cap ${\mathcal C}_a$ at time $-t_a$ for some $t_a\le t'$. We -denote this flow line by $y_a(t), -t_a\le t\le 0$. Furthermore, -suppose that the scalar curvature on the flow line from $y_a$ to -$z_a$ is bounded by $D'$. Then for all $a$ sufficiently large, $x_a$ -has a strong $(C,\epsilon)$-canonical neighborhood. - -**Proof.** The proof is by contradiction. Suppose -the result does not hold. Then there are $A',D',t'<\infty $ and we -can pass to a subsequence (in $a$) such that the hypotheses of the -lemma hold for every $a$ but no $x_a$ has a strong -$(C,\epsilon)$-canonical neighborhood. The essential point of the -argument is to show that **in the units of the surgery scale** the -elapsed time between the surgery time and $0$ is less than $1$ and -the distance from the point $z_a$ to the tip of the surgery cap is -bounded independent of $a$. - - By Lemma~, the fact -that the scalar curvature at $z_a$ is bounded by $D'$ implies that -for all $a$ sufficiently large the scale $\bar h_a$ of the surgery -at time $-t_a$ satisfies - -$$ - -\bar h_a^2\ge (2D'D)^{-1}. -$$ - -(Recall that we are working in the rescaled flow $(\widetilde {\mathcal M}_a,\widetilde G_a)$.) - -Now we are ready to show that the elapsed time is bounded less than -one in the surgery scale. - -**Claim.** There is $\theta_1<1$, depending on $D'$ and $t'$, such that for all -$a$ sufficiently large we have $t_a< \theta_1\bar h_a^2$. - -\begin{proof} -We consider two cases: either $t_a\le \bar h_a^2/2$ or $\bar -h_a^2/2D'$ as - $t_a\ge \bar h_a^2/2$. Hence, -$R(y_a,(-t_a+(2\theta_1-1)\bar h_a^2))\le 2t_aD'\bar -h_a^{-2}\le 2t'D'\bar h_a^{-2}$. This contradiction -shows that if $a$ is sufficiently large then $t_a<\theta_1\bar -h_a^2$. - -We pass to a subsequence so that $t_a\bar h_a^{-2}$ converges to -some $\theta\le \theta_1$. We define $\widetilde C$ to be the -maximum of $C$ and $3\epsilon^{-1}\beta^{-1}$. Now, using Part 5 of -Theorem~ we set $A''\ge (9\widetilde -C+3A')\sqrt{2DD'}+6(A_0+5)$ sufficiently large so that in the - standard flow $B(p_0,0,A'')$ contains -$B(p_0,t,A''/2)$ for any $t\le (\theta_1+1)/2$. This constant is chosen only to -depend on $\theta_1$, $A'$, and $C$. As $a$ tends to infinity, $\delta_a$ tends -to zero which means, by Proposition~, that for all $a$ sufficiently -large there is an embedding $\rho_a\colon B(p_0,-t_a,A''\bar h_a)\times -[-t_a,0]\to \widetilde {\mathcal M}_a$ compatible with time and the vector -field such that (after translating by $t_a$ to make the flow start at time $0$ -and scaling by $\bar h_a^{-2}$) the restriction of $\widetilde G_a$ to this -image is close in the $C^\infty$-topology to the restriction of the standard -flow to $B(p_0,0,A'')\times [0,\bar h_a^{-2}t_a]$. The image $\rho_a(p_0,-t_a)$ -is the tip $p_a$ of the surgery cap ${\mathcal C}_a$ in $\widetilde {\mathcal -M}_a$. In particular, for all $a$ sufficiently large the image -$\rho_a\left(B(p_0,-t_a,A''\bar h_a)\times\{0\}\right)$ contains the $A''\bar -h_a/3$-neighborhood of the image $\rho_a(p_0,0)$ of the tip of the surgery cap -under the flow forward to time $0$. By our choice of $A''$, and -Equation~(), this means that for all $a$ sufficiently large -$\rho_a\left(B(p_0,-t_a,A''\bar h_a)\times\{-t_a\}\right)$ contains the -$(3\widetilde C+A') +2(A_0+5)\bar h_a$ neighborhood of $p_a=\rho_a(p_0,-t_a)$. -Notice also that, since the standard solution has positive curvature and hence -the distance between points is non-increasing in time by -Lemma~, the distance at time $0$ between $\rho_a(p_0,0)$ and -$y_a$ is less than $2(A_0+5)\bar h_a$. By the triangle inequality, we conclude -that for all $a$ sufficiently large, $\rho_a\left(B(p_0,-t_a,A''\bar h_a)\times -\{0\}\right)$ contains the $3\widetilde C$-neighborhood of $x_a$. Since the -family of metrics on $\rho_a\left(B(p_0,-t_a,A''\bar h_a)\times -[-t_a,0]\}\right)$ (after time-shifting by $t_a$ and rescaling by $\bar -h_a^{-2}$) are converging smoothly to the ball $B(p_0,0,A'')\times [0,\theta]$ -in the standard flow, - for all $a$ sufficiently large then the flow from time $-t_a$ to $0$ on the -$3\widetilde C$-neighborhood of $x_a$ is, after rescaling by $\bar -h_a^{-2}$, - very nearly isometric to the restriction of the standard flow from time $0$ to $\bar h^{-2}_at_a$ on - the $3\widetilde C\bar h_a^{-1}$-neighborhood of some -point $q_a$ in the standard flow. Of course, since the scalar -curvature of $x_a$ is one, $R(q_a,\bar h_a^{-2}t_a)$ in the standard flow is -close to $\bar h_a^{-2}$. Hence, by Theorem~ -there is a neighborhood $X$ of $(q_a,\bar h_a^{-2}t_a)$ in the standard solution -that either is a $(C,\epsilon)$-cap, or is an evolving -$\beta\epsilon/3$-neck centered at $(q_a,\bar h_a^{-2}t_a)$. In the latter case -either the evolving neck is defined for backward time -$(1+\beta\epsilon/3)$ or its initial time-slice is the zero -time-slice and this initial time-slice lies at distance at least $1$ -from the surgery cap. Of course, $X$ is contained in the -$CR(q_a,\bar h_a^{-2}t_a)^{-1/2}$ neighborhood of $(q_a,\bar h_a^{-2}t_a)$ in the standard -solution. Since $\widetilde C\ge C$ and $R(q_a,\bar h_a^{-2}t_a)$ is close to -$\bar h_a^{-2}$, the neighborhood $X$ is contained in the -$2\widetilde C\bar h_a^{-1}$-neighborhood of $(q_a,\bar h_a^{-2}t_a)$ in the -standard solution. Hence, after rescaling, the corresponding -neighborhood of $x_a$ is contained in the $3\widetilde -C$-neighborhood of $x_a$. If either of the first two cases in Theorem~ occurs -for a subsequence of $a$ tending to infinity, then by -Lemma~ and the fact that $\widetilde C>max(C,\epsilon^{-1})$, we see that there is a subsequence of $a$ -for which $x_a$ either is contained in the core of a -$(C,\epsilon)$-cap or is the center of a strong $\epsilon$-neck. - -We must examine further the last case. We suppose that for every $a$ -this last case holds. Then for all $a$ sufficiently large we have an -$\beta\epsilon/3$-neck $N_a$ in the zero time-slice of $\widetilde{\mathcal -M}_a$ centered at $x_a$. It is an evolving neck and there is an -embedding $\psi\colon N_a\times [-t_a,0]\to \widetilde{\mathcal -M}_a$ compatible with time and the vector field so that the initial -time-slice $\psi(N_a\times \{-t_a\})$ is in the surgery time-slice -$M_{-t_a}$ and is disjoint from the surgery cap, so in fact it is -contained in the continuing region at time $-t_a$. As we saw above, -the image of the central $2$-sphere $\psi(S^2_a\times\{-t_a\})$ lies -at distance at most $ A''\bar h_a$ from the tip of the surgery cap -$p_a$ (where, recall, $A''$ is a constant independent of $a$). The -$2$-sphere, $\Sigma_a$, along which we do surgery, creating the surgery -cap with $p_a$ as its tip, is the central $2$-sphere of a strong -$\delta_{b(a)}$-neck. As $a$ tends to infinity the surgery control -parameter $\delta_{b(a)}$ tends to zero. Thus, for $a$ sufficiently -large this strong $\delta_{b(a)}$-neck will contain a strong -$\beta\epsilon/2$- neck $N'$ centered at $\psi(x_a,-t_a)$. Since we -know that the continuing region at time $-t_a$ contains a -$\beta\epsilon/3$-neck centered at $(x_a,-t_a)$, it follows that -$N'$ is also contained in $C_{-t_a}$. That is to say, $N'$ is -contained in the negative half of the $\delta_{b(a)}$-neck centered -at $\Sigma_a$. Now we are in the situation of -Proposition~. Applying this result tells us that $x_a$ -is the center of a strong $\epsilon$-neck. - -This completes the proof that for all $a$ sufficiently large, $x_a$ -has a $(C,\epsilon)$-canonical neighborhood in contradition to our -assumption. This contradiction completes the proof of -Lemma~. -\end{proof} - -There are several steps required to complete the proof of -Proposition~. The first step helps us apply the previous claim to -find strong $(C,\epsilon)$-canonical neighborhoods. - -**Claim.** -Given any $A<\infty$ there is $D(A)<\infty$ and $\delta(A)>0$ such that for all $a$ -sufficiently large, $|Rm|$ is bounded by $D(A)$ along all backward flow lines -beginning at a point of $B_{\widetilde -G_a}(\tilde x_a,0,A)$ and defined for backward time at most $\delta(A)$. - -**Proof.** Since all points $y\in ({\mathcal M}_a,G_a)$ with $R_{G_a}(y)\ge -r^{-2}_a$ and $**t**(y)<**t**(x_a)$ have strong -$(C,\epsilon)$-canonical neighborhoods, and since $R(x_a)=r^{-2}_a$, -we see that all points $y\in (\widetilde{\mathcal M}_a,\widetilde -G_a)$ with $**t**(y_a)<0$ and with $R_{\tilde G_a}(y_a)\ge 1$ have -strong $(C,\epsilon)$-canonical neighborhoods. It follows that all -points in $(\widetilde {\mathcal M}_a,\widetilde G_a)$ with $**t**(y)\le 0$ and $R(y)>1$ have strong $(2C,2\epsilon)$-canonical -neighborhoods. Also, since $\delta_a\le \delta(r_a)$, where -$\delta(r_a)$ is the constant given in Proposition~, -and since the condition of being $\kappa$-non-collapsed is a closed -constraint, it follows from Proposition~ that these -Ricci flows with surgery are $\kappa$-non-collapsed for a fixed -$\kappa>0$. It is now immediate from Theorem~ that there -is a constant $D_0(A)$ such that $R$ is bounded above on -$B_{\widetilde G_a}(\tilde x_a,0,A)$ by $D_0(A)$. Since every point -$y\in ({\mathcal M}_a,G_a)$ with $R(y)>1$ of the sequence of with -scalar curvature at least $1$ has a $(C,\epsilon)$ canonical -neighborhood, it follows from the definition that for every such -point $y$ we have $\left|\partial R(y)/\partial t\right|0$ and a bound $D'(A)$, both depending only in -$D_0(A)$, - for the scalar curvature at all points -of backward flow lines beginning in $B_{\widetilde -G_a}(\tilde x_a,0,A)$ and defined for backward time at most $\delta(A)$. Since the curvature is pinched toward positive, -it follows that there is a bound $D(A)$ depending only on $D'(A)$ to $|Rm|$ on the same flow lines. - -**Claim.** After passing to a subsequence (in $a$), either: -\begin{enumerate} -\item[(1)] -for each $A<\infty$ there are $D(A)<\infty$ and $t(A)>0$ such that for all $a$ -sufficiently large $P_{\widetilde G_a}(\tilde x_a,0,A,-t(A))$ exists in -$\widetilde {\mathcal M}_a$ and $|Rm|$ is bounded by $D(A)$ on this -backward parabolic neighborhood, or -\item[(2)] each $x_a$ has a strong -$(C,\epsilon)$-canonical neighborhood. -\end{enumerate} - -**Proof.** First notice that if there is $t(A)>0$ for which the backwards parabolic neighborhood -$P=P_{\widetilde G_a}(\tilde x_a,0,A,-t(A))$ exists, -then, by Claim~, -there are constants $D(A)<\infty$ and $\delta(A)>0$ such that, replacing $t(A)$ by $min(t(A),\delta(A))$, - $|Rm|$ is bounded by $D(A)$ on $P$. -Thus, either Item (1) holds or -passing to a subsequence, we can suppose that there is some $A<\infty$ for which no $t(A)>0$ as -required by Item (1) exists. -Then, for each $a$ we find a point -$y_a\in B_{\widetilde G_a}(\tilde x_a,0,A)$ such that the backwards -flow line from $y_a$ meets a surgery cap at a time $-t_a$ where -$lim_{a\rightarrow \infty}(t_a)=0$. Then, by the previous claim, for all $a$ sufficiently large, the -sectional curvature along any backward flow line beginning in $B_{\widetilde G_a}(\widetilde x_a,0,A)$ -and defined for backward time $t_a$ is bounded by -a constant $D(A)$ independent of $a$. Under our assumption this -means that for all $a$ sufficiently large, there is a point $y_a\in -B_{\widetilde G_a}(\tilde x_a,0,A)$ and a backwards flow line -starting at $y_a$ ending at a point $z_a$ of a surgery cap, and the -sectional curvature along this entire flow line is bounded by $D(A)<\infty$. Thus, -applying Lemma~ produces the strong -$(C,\epsilon)$-canonical neighborhood around $x_a$, proving the claim. - -But we are assuming that no $x_a$ has a strong $(C,\epsilon)$-canonical -neighborhood. Thus, the consequence of the previous claim is that for each -$A<\infty$ there is a $t(A)>0$ such that for all $a$ sufficiently large -$P_{\widetilde G_a}(\tilde x_a,0,A,-t(A))$ exists in $\widetilde {\mathcal -M}_a$ and there is a bound, depending only on $A$ for $|Rm|$ on -this backward parabolic neighborhood. Applying Theorem~ we see that, after passing to a -subsequence, there is a smooth limit $(M_\infty,g_\infty,x_\infty)$ to the zero -time-slices $(\widetilde{\mathcal M}_a,\widetilde G_a,\tilde x_a)$. -Clearly, since the curvatures of the sequence are pinched toward positive, this -limit has non-negative curvature. - -Lastly, we show that $(M_\infty,g_\infty)$ has bounded curvature. By Part 3 of Proposition~ -each point of $(M_\infty,g_\infty)$ with scalar curvature -greater than one has a $(2C,2\epsilon)$-canonical neighborhood. If a point lies in an $2\epsilon$-component -or in a $2C$-component, then $M_\infty$ is compact, and hence clearly has bounded curvature. -Thus, we can assume that each $y\in M_\infty$ with $R(y)>1$ is either the center of a $2\epsilon$-neck -or is contained in the core of a $(2C,2\epsilon)$-cap. According to Proposition~ - $(M_\infty,g_\infty)$ does not contain $2\epsilon$-necks of arbitrarily high curvature. - It now follows then that $(M_\infty,g_\infty)$ there is a bound to the scalar curvature of any - $2\epsilon$-neck and of any $(2C,2\epsilon)$-cap, and hence it follows that $(M_\infty,g_\infty)$ - has bounded curvature. - -**Claim.** If the constant $t(A)>0$ cannot be chosen independent of $A$, then -after passing to a subsequence, the $x_a$ have strong -$(C,\epsilon)$-canonical neighborhoods. - -**Proof.** Let $Q$ be the bound of the scalar curvature of -$(M_\infty,g_\infty,x_\infty)$. Then by Lemma~ - there is a constant -$\Delta t>0$ such that if $R_{\widetilde G_a}(y,0)\le 2Q$, then the -scalar curvature is bounded by $16Q$ on the backward flow line from -$y$ defined for any time $\le \Delta t$. Suppose that there is -$A<\infty$ and a subsequence of $a$ for which there is a flow line -beginning at a point $y_a \in B_{\widetilde G_a}(\tilde x_a,0,A)$ -defined for backward time at most $\Delta t$ and ending at a point -$z_a$ of a surgery cap. Of course, the fact that the scalar -curvature of $(M_\infty,g_\infty)$ is at most $Q$ implies that for -all $a$ sufficiently large, the scalar curvature of $B_{\widetilde -G_a}(\tilde x_a,0,A)$ is less than $2Q$. This implies that for all -$a$ sufficiently large the scalar curvature along the flow line -from $y_a$ to $z_a$ in a surgery cap is $\le 16Q$. Now invoking -Lemma~ we see that for all $a$ sufficiently large the -point $\widetilde x_a$ has a strong $(C,\epsilon)$-canonical -neighborhood. This is a contradiction, and this contradiction proves -that we can choose $t(A)>0$ independent of $A$. - -Since we are assuming that no $x_a$ has a strong -$(C,\epsilon)$-canonical neighborhood, this means that it is -possible to find a constant $t'>0$ such that $t(A)\ge t'$ for all -$A<\infty$. Now let $0T'$, after passing to a subsequence, for all -$a$ sufficiently large there is $A<\infty$ and a backwards flow -line, defined for a time less than $t$, starting at a point $y_a$ of -$B_{\widetilde G_a}(\tilde x_a,0,A)$ and ending at a point $z_a$ of -a surgery cap. Invoking Lemma~ again, we see that -for all $a$ sufficiently large, the scalar curvature is bounded on -the flow line from $y_a$ to $z_a$ by a constant independent of $a$. -Hence, as before, we see that for all $a$ sufficiently large $x_a$ -has a strong $(C,\epsilon)$-canonical neighborhood; again this is a contradiction. - -Hence, we have now shown that our assumption that the strong -$(C,\epsilon)$-canonical neighborhood assumption fails for all $r_a$ -and all $\delta_{a,b}$ leads to a contradiction and hence is false. - -This completes the proof of Proposition~. -\end{proof} - -## Surgery times don't accumulate - -Now we turn to the proof of Theorem~. - Given surgery parameter -sequences $$\Delta_i=\{\delta_0,\ldots,\delta_i\}$$ -$$**r_i**=\{r_0,\ldots,r_i\}$$ -$$**K_i**=\{\kappa_0,\ldots,\kappa_i\},$$ - we let -$r_{i+1}$ and $\delta_{i+1}$ be as in Proposition~ and -then set $\kappa_{i+1}=\kappa(r_{i+1})$ as in -Proposition~. - Set $${\bf -r_{i+1}}=\{**r_i**,r_{i+1}\}$$ -$${\bf K_{i+1}}=\{**K_i**,\kappa_{i+1}\}$$ -$$\Delta_{i+1}=\{\delta_0,\ldots,\delta_{i-1},\delta_{i+1},\delta_{i+1}\}.$$ -Of course, these are also surgery parameter sequences. - -Let $\overline \delta\colon [0,T]\to \Ar^+$ be any non-increasing positive -function and let $({\mathcal M},G)$ be a Ricci flow with surgery defined on -$[0,T)$ for some $T\in [T_i,T_{i+1})$ with surgery control parameter -$\overline\delta$. Suppose $\overline \delta\le \Delta_{i+1}$ and that this -Ricci flow with surgery satisfies the conclusion of Theorem~ with -respect to these sequences on its entire interval of definition. We wish to -extend this Ricci flow with surgery to one defined on $[0,T')$ for some $T'$ -with $T0$ and the canonical -neighborhood parameter $r$ at time $t_0$ is at least $r_{i+1}>0$, it follows -that $h(t_0)\ge h(T_{i+1})>0$. Thus, each surgery at time $ t_0\le T_{i+1}$ -along a $2$-sphere removes at least a fixed amount of volume depending on -$\overline \delta(T_{i+1})$ and $r_{i+1}$. Since under Ricci flow the volume -grows at most exponentially, we see that there is a bound depending only on -$\overline\delta(T_{i+1})$, $T_{i+1}$, $r_{i+1}$ and $Vol(M_0,g(0))$ to -the number of $2$-sphere surgeries that we can do in this time interval. On the -other hand, the number of components at any time $t$ is at most $N_0+S(t)-D(t)$ -where $N_0$ is the number of connected components of $M_0$, $S(t)$ is the -number of $2$-sphere surgeries performed in the time interval $[0,t)$ and -$D(t)$ is the number of connected components removed by surgeries at times in -the interval $[0,t)$. Hence, there is a bound on the number of components in -terms of $N_0$ and $S(T)$ that can be removed by surgery in the interval -$[0,T)$. Since the initial conditions are normalized, $N_0$ is bounded by the -volume of $(M_0,g(0))$. This completes the proof of the result. - -This lemma completes the proof of the fact that for any $T\le T_{i+1}$, we -encounter only a fixed bounded number surgeries in the Ricci flow with -surgery from $0$ to $T$. The bound depends on the volume of the initial -manifold as well as the surgery constants up to time $T_{i+1}$. In particular, -for a given initial metric $(M_0,g(0))$ there is a uniform bound, depending -only on the surgery constants up to time $T_{i+1}$, on the number of surgeries -in any Ricci flow with surgery defined on a subinterval of $[0,T_{i+1})$. It -follows that the surgery times cannot accumulate in any finite interval. This -completes the proof of Theorem~. - -To sum up, we have sequences $\Delta$, $**K**$ and $**r**$ as given in -Theorem~. Let $\overline \delta\colon [0,\infty)\to \Ar$ be a -positive, non-increasing function with $\overline \delta\le \Delta$. Let $M$ be -a compact $3$-manifold that contains no embedded $\Ar P^2$ with trivial normal -bundle. We have proved that for any normalized initial Riemannian metric -$(M_0,g_0)$ there is a Ricci flow with surgery with time-interval of definition -$[0,\infty)$ and with $(M_0,g_0)$ as initial conditions. This Ricci flow with -surgery is $**K**$-non-collapsed and satisfies the strong -$(C,\epsilon)$-canonical neighborhood theorem with respect to the parameter -$**r**$. It also has curvature pinched toward positive. Lastly, for any $T\in -[0,\infty)$ if there is a surgery at time $T$ then this surgery is performed -using the surgery parameters $\overline\delta(T)$ and $r(T)$, where if $T\in -[T_i,T_{i+1})$ then $r(T)=r_{i+1}$. In this Ricci flow with surgery, there are -only finitely many surgeries on each finite time interval. As far as we know -there may be infinitely many surgeries in all. diff --git a/projects/poincare-conjecture/.astrolabe/docs/13-the-standard-solution.mdx b/projects/poincare-conjecture/.astrolabe/docs/13-the-standard-solution.mdx new file mode 100644 index 00000000..9ab22bde --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/docs/13-the-standard-solution.mdx @@ -0,0 +1,2160 @@ + + +# Chapter 12 -- The standard solution + +The process of surgery involves making a choice of the metric on a +three-ball to 'glue in'. In order to match approximatively with the +metric coming from the flow, the metric we glue in must be +asymptotic to the product of a round two-sphere and an interval near +the boundary. There is no natural choice for this metric; yet it is +crucial to the argument that we choose an initial metric so that the +Ricci flow with these initial conditions has several properties. +Conditions on the initial metric that ensure the required +properties for the subsequence flow are contained in the following +definition. + +\entryblock{c5e007547ff3} + +Actually, one can work with an alternative weaker version of the +fourth condition, namely: + + (iv) $g_0$ is asymptotic at infinity in the +$C^\infty$-topology to the product of the round metric $h_0$ on +$S^2$ of scalar curvature $1$ with the usual metric $ds^2$ on the +real line. By this we mean that if $x_n\in \Ar^3$ is any sequence +converging to infinity, then the based Riemannian manifolds +$(\Ar^3,g_0,x_n)$ converge smoothly to $(S^2,h_0)\times (\Ar,ds^2)$. +But we shall only use standard initial metrics as given in +\entryref{c5e007547ff3}. + +\entryblock{f1d3021d32b8} + +**Proof.** We construct our Riemannian manifold as follows. Let +$(x_0,x_1,x_2,x_3)$ be Euclidean coordinates on $\Ar^4$. Let +$y=f(s)$ be a function defined for $s\ge 0$ and satisfying: + +- **(1)** $f$ is $C^\infty$ on $(0,\infty)$ +- **(2)** $f(s)>0$ for all $s>0$. +- **(3)** $f''(s)\le 0$ for all $s>0$. +- **(4)** There is $s_1>0$ such that $f(s)=\sqrt{2}$ for all $s\ge +s_1$. +- **(5)** There is $s_0>0$ such that +$f(s)=\sqrt{4s-s^2}$ for all $s\in [0,s_0]$. + +Given such a function $f$, consider the graph + +$$ +\Gamma=\{(x_0,x_1)\bigl|\bigr. x_0\ge 0 \text{ and } +x_1=f(x_0)\} +$$ + + in the $(x_0,x_1)$-plane. We define $\Sigma(f)$ by +rotating $\Gamma$ about the $x_0$-axis in four-space: + +$$ +\Sigma(f)=\{(x_0,x_1,x_2,x_3)\bigl|\bigr. x_0\ge 0 \text{ and } +x_1^2+x_2^2+x_3^2=f(x_0)^2\}. +$$ + + Because of the last condition on +$f$, there is a neighborhood of $0\in \Sigma(f)$ that is isometric +to a neighborhood of the north pole in the three-sphere of radius +$2$. Because of this and the first item, we see that $\Sigma(f)$ is +a smooth submanifold of $\Ar^4$. Hence, it inherits a Riemannian +metric $g_0$. Because of the fourth item, a neighborhood of infinity +of $(\Sigma(f),g_0)$ is isometric to $(S^2,h)\times (0,\infty)$, and +in particular, $(\Sigma(f),g_0)$ is complete. Clearly, the rotation +action of $S0(3)$ on $\Sigma(f)$, induced by the orthogonal action +on the last three coordinates in $\Ar^4$, is an isometric action +with the origin as the only fixed point. It is also clear that +$\Sigma(f)$ is diffeomorphic to $\Ar^3$ by a diffeomorphism that +send the $SO(3)$ action to the standard one on $\Ar ^3$. + +It remains to compute the sectional curvatures of $g_0$. Let $q\in +\Sigma(f)$ be a point distinct from the fixed point of the +$SO(3)$-action. Direct computation shows that the tangent plane to +the two-dimensional $SO(3)$-orbit through $q$ is a principal +direction for the curvature, and the sectional curvature on this + tangent two-plane is given by + +$$ +\frac{1}{f(q)^2(1+f'(q)^2)}. +$$ + +On the subspace in $\wedge^2T_q\Sigma(f)$ perpendicular to the line +given by this two-plane, the curvature is constant with eigenvalue + +$$ +\frac{-f''(q)}{f(q)(1+f'(q)^2)^2}. +$$ + +Under our assumptions about $f$, it is clear that $\Sigma(f)$ has +non-negative curvature and has constant sectional curvature $1/4$ +near the origin. It remains to choose the function $f$ satisfying +Items (1) -- (5) above. + +Consider the function $h(s)=(2-s)/\sqrt{4s-s^2}$. This function is +integrable from $0$ and the definite integral from zero to $s$ is +equal to $\sqrt{4s-s^2}$. Let $\lambda(s)$ be a non-increasing +$C^\infty$-function defined on $[0,1/2]$, with $\lambda$ identically +one near $0$ and identically equal to $0$ near $1/2$. We extend +$\lambda$ to be identically $1$ for $s<0$ and identically $0$ for +$s>1/2$. Clearly, + +$$ +\int_0^2h(s)\lambda(s-3/2)ds>\int_0^{3/2}h(s)ds>\sqrt{2} +$$ + +and + +$$ +\int_0^2h(s)\lambda(s)ds<\int_0^{1/2}h(s)<\sqrt{2}. +$$ + + Hence, +for some $s_0\in (1/2,3/2)$ we have + +$$ +\int_0^2h(s)\lambda(s-s_0)ds=\sqrt{2}. +$$ + +We define + +$$ +f(s)=\int_0^sh(\sigma)\lambda(\sigma-s_0)d\sigma. +$$ + + It +is easy to see that $f$ satisfies all the above conditions. + +The following lemma is clear from the construction. + +\entryblock{0b07beec84c0} + +### Uniqueness and properties: The statement + +**Fix once and for all a standard initial metric $g_0$ on +$\Ar^3$.** + +\entryblock{acada7b5d698} + +Here is the main result of this chapter. + +\entryblock{f44d182ef27e} + +The proof of this result occupies the next few subsections. All the +properties except the uniqueness are fairly straightforward to +prove. We establish uniqueness by reducing the Ricci flow to the +Ricci-DeTurck by establishing the existence of a solution to the +harmonic map flow in this case. This technique can be made to work +more generally in the case of complete manifolds of bounded +curvature, see [ChenZhu], but we preferred to give the more +elementary argument that covers this case, where the symmetries +allow us to reduce the existence problem for the harmonic map flow +to a problem that is the essentially one-dimensional. Also, in the +rest of the argument one does not need uniqueness, only a +compactness result for the space of all Ricci flows of bounded +curvature on each time-slice with the given initial conditions. +Kleiner and Lott pointed out to us that this uniqueness can be +easily derived from the other properties by arguments similar to +those used to establish the compactness of the space of +$\kappa$-solutions. + +## Existence of a standard flow + +For any $R<\infty$, denote by $B_R\subset \Ar^3$, the ball of radius +$R$ about the origin in the metric $g_0$. For $R\ge A_0+1$, a +neighborhood of the boundary of this ball is isometric to +$(S^2,h)\times ([0,1],ds^2)$. Thus, in this case, we can double the +ball, gluing the boundary to itself by the identity, forming a +manifold we denote by $S^3_R$. The doubled metric will be a smooth +Riemannian metric $g_R$ on $S^3_R$. Let $p\in S^3_R$ be the image of +the origin in the first copy of $B_R$. Now take a sequence, $R_n$, +tending to infinity to construct based Riemannian manifolds +$(S^3_{R_n},g_{R_n},p)$ that converge geometrically to +$(\Ar^3,g_0,p)$. For each $n$, let $(S^3_{R_n},g_{R_n}(t)),\ 0\le +tt_0$ for all $n$. + Since the Riemann +curvatures of the $(S^3_{R_n},g_{R_n}(t)),\ 0\le t0$ and +any compact subset $K\subset M\times [0,t_0]$ there is $\epsilon>0$ +and a function $\varphi\colon M\times [0,t_0]\to \Ar$ with the +following properties: + +- **(1)** $\varphi|_K\le \eta$. +- **(2)** $\varphi\ge \epsilon$ everywhere. +- **(3)** For each $t\in [0,t_0]$ the restriction of $\varphi$ to $M\times \{t\}$ +goes to infinity at infinity in the sense that for any $A<\infty$ +the pre-image $\varphi^{-1}([0,A]\cap (M\times\{t\})$ is compact. +- **(4)** On all of $M\times [0,t_0]$ we have +$ + \left(\frac{\partial}{\partial t}-\triangle\right)\varphi\ge C\varphi. +$ + +Recall from Section 4.2.1 that ${\mathcal T}$ is the curvature +tensor written with respect to an evolving orthonormal frame +$\{F_\alpha\}$ for the tangent bundle. Consider the symmetric, +horizontal two-tensor $\widehat{\mathcal T}={\mathcal T}+\varphi g$. +Let $\hat\mu(x,t)$ denote the smallest eigenvalue of this symmetric +two-tensor at $(x,t)$. Clearly, since the curvature is bounded, it +follows from the third property of $\varphi$ that for each $t\in +[0,t_0]$ the restriction of $\hat\mu$ to $M\times\{t\}$ goes to +infinity at infinity in $M$. In particular, the subset of $(x,t)\in +M\times [0,t_0]$ with the property that $\hat\mu(x,t)\le +\hat\mu(y,t)$ for all $y\in M$ is a compact subset of $M\times +[0,t_0]$. It follows from \entryref{37d7d059a526} that +$f(t)=\mathit{min}_{x\in M}\hat\mu(x,t)$ is a continuous function of +$t$. Choosing $\eta>0$ sufficiently small and $K$ to include +$(x,t_0)$, then $\widehat{\mathcal T}$ will have a negative +eigenvalue at $(x,t_0)$. Clearly, it has only positive eigenvalues +on $M\times \{0\}$. Thus, there is $00$ +and its value at $t_1$ is zero, this is a contradiction. This +establishes that the solution has non-negative curvature everywhere. +Indeed, by \entryref{d21a675e4ea8} it has strictly positive +curvature for every $t>0$. + +Now let us turn to the asymptotic behaviour of the flow. + +Fix $T'0$, for all $k$ sufficiently large, the restriction +of the flow to the cylinder of length $2R$ centered at $y_k$ is +within $\epsilon$ in the $C^{[1/\epsilon]}$-topology of the +shrinking cylindrical flow on time $[0,T']$. Given $\epsilon>0$ and +$R<\infty$ this statement is true for all $y$ outside a compact ball +$B$ centered at the origin. + +We have now established the following + +\entryblock{77ca4669ffb3} + +\entryblock{2db418fba1c1} + +**Proof.** If $T>1$, then we can apply the above result to $T'$ with +$10$ such that + +$$ +\begin{aligned} +\left( \frac{\partial}{\partial t}-\Delta\right) \left\vert h_{ij} +\right\vert ^{2} & =-2\left\vert \nabla_{k}h_{ij}\right\vert ^{2} ++4R^{ijkl}h_{jk}h_{il} \\ +\frac{\partial}{\partial t}\left\vert h_{ij}\right\vert ^{2} & \leq +\Delta\left\vert h_{ij}\right\vert ^{2}-2\left\vert \nabla_{k}h_{ij} +\right\vert ^{2}+C\left\vert h_{ij}\right\vert ^{2}. +\end{aligned} +$$ + +Note that $X(t)$ is a Killing vector field for $g(t)$ if and only if +$h_{ij}(t)=0$. Since Equation (12.1) is linear and since the +curvature is bounded on each time-slice, for any given bounded +Killing vector field $X(0)$ for metric $g(0)$, there is a bounded +solution $X^{i}\left( t\right) $ of Equation (12.1) for +$t\in[0,T]$. Then $\left \vert h_{ij}\left( t\right) \right +\vert^2 $ is a bounded function satisfying (12.6) and $\left +\vert h_{ij} \right \vert^2 (0)=0$. One can apply the maximum +principle to (12.6) to conclude that $h_{ij}(t)=0$ for all +$t\geq 0$. This is done as follows: Let $h(t)$ denote the maximum of +$|h_{ij}(x,t)|^2$ on the $t$ time-slice. Note that, for any fixed +$t$ the function $|h_{ij}(x,t)|^2$ approaches $0$ as $x$ tends to +infinity since the metric is asymptotic at infinity to the product +of a round metric on $S^2$ and the standard metric on the line. By +virtue of (12.6) and \entryref{37d7d059a526}, the +function $h(t)$ satisfies $dh/dt\le Ch$ in the sense of forward +difference quotients, so that $d(e^{-Ct}h)/dt\le 0$, also in the +sense of forward difference quotients. Thus, by +\entryref{67c378f87457}, since $h(0)=0$ and $h\ge 0$, it follows +that $e^{-Ct}h(t)=0$ for all $t\ge 0$, and consequently, $h(t)=0$ +for all $t\ge 0$. + +Thus, the evolving vector field $X( t) $ is a Killing vector field +for $g(t)$ for all $t\in [0,T)$. The following is a very nice +observation of Bennett Chow; we thank him for allowing us to use it +here. From $h_{ij}=0$ we have $\nabla_{j} X^i +\nabla_{i} X^{j}=0$. +Taking the $\nabla_{j}$ derivative and summing over $j$ we get +$\Delta X^i +R^i_kX^k =0$ for all $t$. Hence (12.2) gives +$\frac{\partial}{\partial t}X^{i}=0$ and $X(t)=X(0)$, i.e., the +Killing vector fields are stationary and remain Killing vector +fields for the entire flow $g(t)$. + Since at $t=0$ the Lie algebra $so(3)$ of the standard rotation action + consists of Killing vector fields, the same is true for all the metrics $g(t)$ + in the standard solution. + Thus, the +rotation group $SO(3)$ of $\mathbb{R}^3$ is contained in the +isometry group of $g(t)$ for every $t\in [0,T)$. We have shown: + +\entryblock{0d0c8f66ca5d} + +### Non-collapsing + +\entryblock{ed069175cdf9} + +**Proof.** Since the curvature of the standard solution is non-negative, it +follows directly that $2|\mathit{Ric}|^2\le R^2$. By +Equation (3.5) this gives + +$$ +\frac{\partial R}{\partial t}=\triangle R+2|\mathit{Ric}|^2\le \triangle +R+R^2. +$$ + + Let $C=\mathit{max}(2,\mathit{max}_{x\in \Ar^3}R(x,0))$. Suppose +that $t_0< T$ and $t_0<1/C$. + +\entryblock{f26ee81ded19} + +**Proof.** By the asymptotic condition, there is a compact subset $X\subset +\Ar^3$ such that for any point $p\in \Ar^3\setminus X$ and for any +$t\le t_0$ we have $R(p,t)<2/(1-t)$. Since $C\ge 2$, for all $t$ for +which $\mathit{sup}_{x\in \Ar^3}R(x,t)\le 2/(1-t)$, we also have + +$$ +R(x,t)\le \frac{C}{1-Ct}. +$$ + +Consider the complementary subset of $t$, that is to say the subset +of $[0,t_0]$ for which there is $x\in \Ar ^3$ with +$R(x,t)>C/(1-Ct)$. This is an open subset of $[0,t_0]$, and hence is +a disjoint union of relatively open intervals. Let $\{t_10$ such that for any metric ball +$B(x,0,r)$ on which $|\mathit{Rm}|\le r^{-2}$ we have $\mathit{Vol} B(x,r)\ge V r^3$. Since there is a uniform bound on the +curvature on $[0,1/2C]$, it follows that there is $V'>0$ so that any +ball $B(q,t,r)$ with $t\le 1/2C$ on which $|\mathit{Rm}|\le r^{-2}$ +satisfies $\mathit{Vol} B(q,t,r)\ge V'r^3$. Set $t_0=1/4C$. For any +point $x=(p,t)$ with $t\ge 1/2C$ there is a point $(q,t_0)$ such +that $l_x(q,t_0)\le 3/2$; this by \entryref{aebc3ff2bd3b}. Since +$B(q,0,1/\sqrt{R_\mathit{max}(0)})\subset \Ar^3$ has volume at least +$V/R_\mathit{max}(0)^{3/2}$, and clearly $l_x$ is bounded above on +$B(q,0,1/\sqrt{R_\mathit{max}(0)})$ by a uniform constant, we see that +the reduced volume of $B(q,0,1/\sqrt{R_\mathit{max}(0)})$ is uniformly +bounded from below. It now follows from \entryref{3defa9b2209c} that there +is $\kappa_0>0$ such that if $|\mathit{Rm}|$ is bounded by $r^{-2}$ on +the parabolic neighborhood $P(p,t,r,-r^2)$ and $r\le \sqrt{1/4C}$, +then the volume of this neighborhood is at least $\kappa_0r^3$. +Putting all this together we see that there is a universal +$\kappa>0$ such that the standard solution is $\kappa$-non-collapsed +on all scales at most $\sqrt{1/4C}$. + +## Uniqueness + +Now we turn to the proof of uniqueness. The idea is to mimic the +proof of uniqueness in the compact case, by replacing the Ricci flow +by a strictly parabolic flow. The material we present here is +closely related to and derived from the presentation given in +[LuTian]. The presentation here is the analogy in the context +of the standard solution of DeTurck's argument presented in +Section 3.3. + +### From Ricci flow to Ricci-DeTurck flow + +In this subsection we discuss the Ricci-DeTurck flow and the +harmonic map flow. Let $(M^n,g(t)),t \in [t_0,T]$ be a solution of +the Ricci flow and let $\psi_{t}\colon M \rightarrow M,\ t \in +\left[ t_0,T_{1}\right]$ be a solution of the harmonic map +flow + +$$ +\begin{aligned} +\frac{\partial\psi_{t}}{\partial t} & =\Delta_{g\left( t\right) +,g(t_0) }\psi_{t}, \quad \psi_{t_0} =\mathit{Id}. +\end{aligned} +$$ + +Here, $\Delta_{g\left( t\right) ,g(t_0) }$ is the Laplacian for +maps from the Riemannian manifold $(M,g(t))$ to the Riemannian +manifold $(M,g(t_0))$. In local coordinates $(x^i)$ on the domain +$M$ and $(y^\alpha)$ on the target $M$, the harmonic map flow +(12.7) can be written as + +$$ +\left( \frac{\partial}{\partial t}-\Delta_{g\left( t\right) +}\right) \psi^{\alpha}\left( x,t\right) =g^{ij}\left( x,t\right) +\Gamma _{\beta\gamma}^{\alpha}\left( \psi\left( x,t\right) \right) +\frac {\partial\psi^{\beta}\left( x,t\right) }{\partial +x^{i}}\frac{\partial \psi^{\gamma}\left( x,t\right) }{\partial +x^{j}} +$$ + +where $\Gamma_{\beta\gamma}^{\alpha}$ are the Christoffel symbols of +$g(t_0)$. Suppose $\psi\left(x, t\right)$ is a bounded smooth +solution of 12.8 with $\psi_{t_0}=\mathit{Id}$. Then $\psi\left( +t\right),\ t \in[t_0, T_1]$ are diffeomorphisms when $T_{1}>t_0$ is +sufficiently close to $t_0$. For any such $T_1$ and for $t_0 \leq t +\leq T_1$, define $\hat{g}\left( t\right) =\left( +\psi_{t}^{-1}\right) ^{\ast}g\left( t\right)$. Then $\hat g(t)$ +satisfies the following equation: + +$$ +\frac{\partial}{\partial t}\hat{g}_{ij} =-2\widehat{\mathit{Ric}}_{ij}+\hat{\nabla} _{i}W_{j}(t)+\hat{\nabla}_{j}W_{i}(t) \quad +\quad \hat{h}\left( 0\right) =h(0), +$$ + +where $\widehat{\mathit{Ric}}_{ij}$ and $\hat{\nabla}_{i}$ are the +Ricci curvature and Levi-Civita connection of $\hat{g}(t)$ +respectively and $W(t)$ is the time-dependent $1$-form defined by + +$$ +W(t) _{j}=\hat{g}_{jk}(t)\hat{g} ^{pq}(t)\left( +\hat{\Gamma}_{pq}^{k}(t)-\Gamma_{pq}^{k}(t_0) \right). +$$ + +Here, $\hat{\Gamma}_{pq}^{k}(t)$ denotes the Christoffel symbols of +the metric $\hat g(t)$ and $\Gamma_{pq}^k(t_0)$ denotes the +Christoffel symbols of the metric $g(t_0)$. (See, for example, +([Shi1] Lemma 2.1).) + We call a solution to this flow equation a + **Ricci-DeTurck flow** (see [DeTurck], or [ChowKnopf] +Chapter 3 for details). In local coordinates we have + +$$ +\begin{aligned} +& \frac{\partial\hat{g}_{ij}}{\partial t} =\hat{g}^{kl}\nabla +_{k}\nabla_{l}\hat{g}_{ij}-\hat{g}^{kl}g(t_0)_{ ip}\hat{g}^{pq}R_{jk +ql}\left( g(t_0)\right) -\hat{g} ^{kl}g(t_0)_{ +jp}\hat{g}^{pq}R_{ik ql}\left( g(t_0)\right) \nonumber +\\ +& +\frac{1}{2}\hat{g}^{kl}\hat{g}^{pq}\left[ +\begin{array} +[c]{c} \nabla_{i}\hat{g}_{pk}\nabla_{ j}\hat{g}_{ql}+2\nabla +_{k}\hat{g}_{jp}\nabla_{ q}\hat{g}_{il}\\ +-2\nabla_{k}\hat{g}_{jp}\nabla_{l}\hat{g}_{iq}-2\nabla_{ +j}\hat{g}_{pk}\nabla_{l}\hat{g}_{iq}-2\nabla_{ i}\hat +{g}_{pk}\nabla_{l}\hat{g}_{jq} +\end{array} +\right] . +\end{aligned} +$$ + +where $\nabla$ is the Levi-Civita connection of $g(t_0)$. This is a +strictly parabolic equation. + +\entryblock{a8675ea40ba2} + +**Proof.** The first statement follows from the second statement and a standard +Lie derivative computation. For the second statement, we need to +show + +$$ +\triangle_{g(t),g(0)} \psi^\alpha = -\hat +g^{pq}\left(\hat\Gamma_{pq}^\alpha(t)-\Gamma_{pq}^\alpha(t_0)\right). +$$ + +Notice that this equation is a tensor equation, so that we can +choose coordinates in the domain and range so that $\Gamma(t)$ +vanishes at the point $p$ in question and $\Gamma(t_0)$ vanishes at +$\psi_t(p)$. With these assumptions we need to show + +$$ +g^{pq}(t)\frac{\partial^2 \psi^\alpha}{\partial x^p\partial x^q}=-\hat g^{pq}(t)\hat\Gamma_{pq}^\alpha(t). +$$ + +This is a direct computation using the change of variables formula +relating $\hat \Gamma$ and $\Gamma$. + +\entryblock{5b79e248c5cb} + +**Proof.** Since $\psi_{a,t}$ satisfies the equation + +$$ +\frac{\partial \psi_{a,t}}{\partial t}=-\hat g_a^{ij}W(t)_j +$$ + +where the time-dependent vector field $W(t)$ depends only on $\hat +g_a$, we see that $\psi_{1,t}$ and $\psi_{2,t}$ both solve the same +time-dependent ODE and since $\psi_{1,t_0}=\psi_{2,t_0}=\mathit{Id}$, +it follows that $\psi_{1,t}=\psi_{2,t}$ for all $t\in [t_0,T]$. On +the other hand, $g_a(t)=\psi_{a,t}^*\hat g_a(t)$, so that it follows +that $g_1(t)=g_2(t)$ for $t\in [t_0,T]$. + +Our strategy of proof is to begin with a standard solution $g(t)$ +and show that there is a solution to the harmonic map equation for +this Ricci flow with appropriate decay conditions at infinity. It +follows that the solution to the Ricci-DeTurck flow constructed is +well-controlled at infinity. Suppose that we have two standard +solutions $g_1(t)$ and $g_2(t)$ (with the same initial conditions +$g_0$) that agree on the interval $[0,t_0]$ which is a proper +subinterval of the intersection of the intervals of definition of +$g_1(t)$ and $g_2(t)$. We construct solutions to the harmonic map +flow equation from $g_a(t)$ to $g_a(t_0)$ for $a=1,2$. We show that +solutions always exist for some amount of time past $t_0$. The +corresonding Ricci-DeTurck flows $\hat g_a(t)$ starting at $g_{t_0}$ +are well-controlled at infinity. Since the Ricci-DeTurck flow +equation is a purely parabolic equation, it has a unique solution +with appropriate control at infinity and given initial condition +$g_1(t_0)=g_2(t_0)$. This implies that the two Ricci-DeTurck flows +we have constructed are in fact equal. Invoking the above corollary, +we conclude that $g_1(t)$ and $g_2(t)$ agree on a longer interval +extending past $t_0$. From this it follows easily that $g_1(t)$ and +$g_2(t)$ agree on their common domain of definition. Hence, if they +are both maximal flows, they must be equal. + +## Solution of the harmonic map flow + +In order to pass from a solution to the Ricci flow equation to a +solution of the Ricci-DeTurck flow we must prove the existence of a +solution of the harmonic map flow +associated with the Ricci flow. In this section we study the +existence of the harmonic flow (12.7) and its asymptotic +behavior at the space infinity when $h(t)=g(t)$ is a standard +solution. Here we use in an essential way the rotationally symmetric +property and asymptotic property at infinity of $g(t)$. In this +argument there is no reason, and no advantage, to restricting to +dimension three, so we shall consider rotationally symmetric +complete metrics on $\Ar^n$, i.e., complete metrics on $\Ar^n$ +invariant under the standard action of $SO(n)$. Let +$\theta=(\theta^1,\cdots,\theta^{n-1})$ be local coordinates on the +round $(n-1)$-sphere of radius $1$, and let $d\sigma$ be the metric +on the sphere. We denote by $\hat r$ the standard radial coordinate +in $\Ar^n$. Since $g(t)$ is rotationally symmetric and $n \geq 3$, +we can write + +$$ +\begin{aligned} +g(t)=dr^2+f(r,t)^2 d \sigma +\end{aligned} +$$ + +Here $r=r(\hat r,t)$ is the (time-dependent) radial coordinate on +$\mathbb{R}^n$ for the metric $g(t)$. + +\entryblock{19cc95ff1cb9} + +**Proof.** Write the metric $g(t)=g_{ij}dx^idx^j$ and let + +$$ +x^1 = \hat{r} \cos \theta^1, \ \ x^2 = \hat{r} \sin \theta^1 \cos +\theta^2, \cdots, + x^n = \sin \theta^1 \cdots \sin \theta^{n-1}. +$$ + + We compute $f(r,t)$ +by restricting attention to the ray $\hat{r}=x^1$ and $\theta^1 +=\cdots=\theta^{n-1}=0$, i.e., $x^2 =\cdots =x^n=0$. Then + +$$ +g(t)=g_{11}(\hat{r},0,\cdots,0,t)d\hat{r}^2+g_{22}(\hat{r},0,\cdots,0,t) +\hat{r}^2 d \sigma. +$$ + +Both $g_{11}$ and $g_{22}$ are positive smooth and even in $\hat r$. +Clearly,$\sqrt{g_{11}(\hat r,0,\cdots,0,t)}$ is a positive smooth +function defined for all $(\hat r,t)$ and is invariant under the +involution $\hat r\mapsto -\hat r$. Hence its restriction to $\hat +r\ge0$ is an even function. Since + +$$ +r= \int_0^{\hat{r}} \sqrt{g_{11}(\hat{s},0,\cdots,0,t)}d\hat{s} = +\hat{r} \int_0^{1} \sqrt{g_{11}(\hat{r}s,0,\cdots,0,t)}ds, +$$ + +we see that $r(\hat r,t)$ is of the form $\hat r\cdot \phi(\hat +r,t)$ where $\phi(\hat r,t)$ is an even smooth function. This shows +that $r(\hat r,t)$ is an odd function. It is also clear from this +formula that $\partial r/\partial \hat r>0$. + + Since, for each $t_0$, the function $r(\hat +r,t_0)$ is an increasing function of $\hat r$, it can be inverted +to give a function $\hat r(r,t_0)$. In Equation (12.11), we +have chosen to write $f$ as a function of $r$ and $t$, rather than a +function of $\hat r$ and $t$. We look for rotationally symmetric +solutions to Equation (12.7), i.e., solutions of the form: + +$$ +\begin{aligned} +\psi(t)\colon \mathbb{R}^n \rightarrow \mathbb{R}^n \qquad +\psi(t)(r,\theta) & =(\rho(r,t),\theta) \ \ \ \mathit{for\ \} t\ge t_0 \\ +\psi(r,t_0) & = \mathit{Id} \nonumber +\end{aligned} +$$ + +We shall adopt the following conventions: we shall consider +functions $f(w,t)$ defined in the closed half-plane $w\ge 0$. When +we say that such a function is *smooth* we mean that for each +$n,m\ge 0$ we have a continuous function $f_{nm}(w,t)$ defined for +all $w\ge 0$ with the following properties: + +- **(1)** + +$$ +f_{00}=f +$$ + +- **(2)** + +$$ +\frac{\partial f_{nm}}{\partial t}=f_{n(m+1)} +$$ + +- **(3)** + +$$ +\frac{\partial f_{nm}}{\partial w}=f_{(n+1)m}, +$$ + +where in Item (3) the partial derivative along the boundary $w=0$ is +a right-handed derivative only. We say such a function is *even* +if $f_{(2k+1)m}(0,t)=0$ for all $k\ge 0$. + +We have the following elementary lemma: + +\entryblock{8734da813eff} + +**Proof.** Item (a) is obvious, and Item (b) is obvious away from $r=0$. We +establish Item (b) along the line $r=0$. Consider the Taylor theorem +with remainder to order $2N$ in the $r$-direction for $\psi(r,t)$ at +a point $(0,t)$. By hypothesis it takes the form + +$$ +\sum_{i=0}c_i(t)w^{2i}+w^{2N+1}R(w,t). +$$ + +Now we replace $w$ by $\sqrt{r}$ to obtain + +$$ +f(r,t)=\sum_{i=0}c_ir^i+\sqrt{r}^{2N+1}R(\sqrt{r},t). +$$ + +Applying the usual chain rule and taking limits as $r\rightarrow +0^+$ we see that $f(r,t)$ is $N$ times differentiable along the line +$r=0$. Since this is true for every $N<\infty$, the result follows. + +Notice that an even function $f(r,t)$ defined for $r\ge 0$ extends +to a smooth function on the entire plane invariant under $r\mapsto +-r$. When we say a function $f(r,t)$ *defines a smooth family of +smooth functions on $\Ar^n$* we mean that, under the substitution +$\hat f((x^1,\ldots,x^n),t)=f(r(x^1,\ldots,x^n),t)$, the function +$\hat f$ is a smooth function on $\Ar^n$ for each $t$. + +We shall also consider odd functions $f(r,t)$, i.e., smooth +functions defined for $r\ge 0$ whose Taylor expansion in the +$r$-direction along the line $r=0$ involves only odd powers of $r$. +These do not define smooth functions on $\Ar^n$. On the other hand, +by the same argument as above with the Taylor expansion one sees +that they can be written as $rg(r,t)$ where $g$ is even, and hence +define smoothly varying families of smooth functions on $\Ar^n$. +Notice also that the product of two odd functions $f_1(r,t)f_2(r,t)$ +is an even function and hence this product defines a smoothly +varying family of smooth function on $\Ar^n$. + +### The properties of $r$ as a function of $\hat r$ and $t$ + +We shall make a change of variables and write the harmonic map flow +equation in terms of $r$ and $\theta$. For this we need some basic +properties of $r$ as a function of $\hat r$ and $t$. Recall that we +are working on $\Ar^n$ with its usual Euclidean coordinates +$(x^1,\ldots,x^n)$. We shall also employ spherical coordinates $\hat +r,\theta^1,\ldots,\theta^{n-1}$. (We denote the fixed radial +coordinate on $\Ar^n$ by $\hat r$ to distinguish it from the varying +radial function $r=r(t)$ that measures the distance from the tip in +the metric $g(t)$.) + +As a corollary of \entryref{19cc95ff1cb9} we have: + +\entryblock{e353993489e9} + +For future reference we define + +$$ +B(w,t)=\frac{1}{2}\int_0^w\xi(u,t)du. +$$ + + Then $B(r^2,t)$ +is a smooth function even in $r$ and hence, as $t$ varies, defines a +smoothly varying family of smooth functions on $\Ar^n$. Notice that + +$$ +\frac{\partial B(r^2,t)}{\partial r}=2r\frac{\partial B}{\partial +w}(w,t)\left|_{w=r^2}\right.=2r\left(\frac{1}{2}\xi(r^2,t)\right)=\frac{dr}{dt}. +$$ + +Now let us consider $f(r,t)$. + +\entryblock{f917cf2b1576} + +**Proof.** We have + +$$ +f(r,t)=\hat r(r,t)\sqrt{g_{22}(\hat r(r,t),0,\ldots,0,t)}. +$$ + +Since $\sqrt{g_{22}(\hat r,0,\ldots,0,t)}$ is a smooth function of +$(\hat r,t)$ defined for $\hat r\ge 0$ and since it is an even +function of $\hat r$, it follows immediately from the fact that +$\hat r$ is a smooth odd function of $r$, that $f(r,t)$ is a smooth +odd function of $r$. + +\entryblock{9ae0e9dc955c} + +We set $\tilde h(w,t)=\mathit{log}(h(w,t))$, so that $f(r,t)=re^{\tilde +h(r^2,t)}$. Notice that $\tilde h(r^2,t)$ defines a smooth function +of $\hat r^2$ and $t$ and hence is a smoothly varying family of +smooth functions on $\Ar^n$. + +### The harmonic map flow equation +. + +Let $\psi(t)\colon \Ar^n\to \Ar^n$ be a smoothly varying family of +smooth functions as given in Equation (12.13). Using +(12.11) and (12.13) it is easy to calculate the energy +functional using spherical coordinates with $r$ as the radial +coordinate. + +$$ +\begin{aligned} +E(\psi(t)) & = \frac{1}{2} \int_{\mathbb{R}^n} \left \vert \nabla +\psi(t) +\right \vert^2_{g(t),g_{t_0}} dV_{g(t)} \\ +&= \frac{1}{2} \int_{\mathbb{R}^n} \left[ \left( \frac{\partial +\rho}{\partial r} \right)^2 + (n-1) f^2(\rho,t_0) f^{-2}(r,t) +\right] dV_{g(t)}. +\end{aligned} +$$ + +If we have a compactly supported variation $\delta \rho =w$, then +letting $d\mathit{vol}_\sigma$ denote the standard volume element on +$S^{n-1}$, we have + +$$ +\begin{aligned} +& \delta E(\psi(t))(w) = \frac{1}{2} \int_{\mathbb{R}^n} \left[2 +\frac{\partial \rho}{\partial r} \frac{\partial w}{\partial r} ++2(n-1) f(\rho,t_0) +\frac{\partial f(\rho,t_0)}{\partial \rho}f^{-2}(r,t)w \right] dV_g(t) \\ +&= \int_0^{+\infty} \left[ f^{n-1}(r,t)\frac{\partial +\rho}{\partial r} \frac{\partial w}{\partial r} +(n-1) f(\rho,t_0) +\frac{\partial f(\rho,t_0)}{\partial +\rho}f^{n-3}(r,t)w \right] dr \cdot \int_{S^{n-1}} d\mathit{vol}_\sigma \\ +& = \int_0^{+\infty} \left[-\frac{\partial}{\partial +r}\left(f^{n-1}\frac{\partial\rho}{\partial r}\right)w+(n-1) +f(\rho,t_0) \frac{\partial f(\rho,t_0)}{\partial \rho}f^{n-3}(r,t)w +\right] dr \cdot \int_{S^{n-1}} d\mathit{vol}_\sigma\\ &= +\int_{\mathbb{R}^n}\left[ - f^{1-n} \frac{\partial }{\partial r} +\left( \frac{\partial \rho}{\partial r} f^{n-1}\right) +(n-1) +f(\rho,t_0) \frac{\partial f(\rho,t_0)}{\partial \rho}f^{-2}(r,t) +\right] w dV_{g(t)}. +\end{aligned} +$$ + +The usual argument shows that for a compactly supported variation +$w$ we have + +$$ +\delta_w\left(\frac{1}{2}\int_{\Ar^n}|\nabla_{g(t),g(t_0)}\psi|^2d\mathit{vol}=\int_{\Ar^n}\langle +w,-\triangle_{g(t),g(t_0)}\psi\rangle d\mathit{vol}\right). +$$ + + Thus, + +$$ +\triangle_{g(t),g(t_0)}\psi= \left[ +f^{1-n}\frac{\partial }{\partial r} \left( \frac{\partial +\rho}{\partial r} f^{n-1}\right) -(n-1) f(\rho,t_0) \frac{\partial +f(\rho,t_0)}{\partial \rho}f^{-2}(r,t) \right] +\frac{\partial}{\partial r} +$$ + + where we have written this expression +using the coordinates $(r,\theta)$ on the range $\Ar ^n$ (rather +than the fixed coordinates $(\hat r,\theta)$). + +Now let us compute $\partial \psi/\partial t(\hat r,t)$ in these +same coordinates. (We use $\hat r$ for the coordnates for $\psi$ in +the domain to emphasize that this must be the time derivative at a +fixed point in the underlying space.) Of course, by the chain rule, + +$$ +\begin{aligned} +\frac{\partial \psi(\hat r,t)}{\partial t} & = & \frac{\partial +\psi(r,t)}{\partial r}\frac{\partial r}{\partial t}+\frac{\partial +\psi(r,t)}{\partial t} \\ +& = & \frac{\partial \rho(r,t)}{\partial r}\frac{\partial r(\hat +r,t)}{\partial t}+\frac{\partial \rho(r,t)}{\partial +t}. +\end{aligned} +$$ + +Consequently, for rotationally symmetric maps as in +Equation (12.13) the harmonic map flow equation (12.7) +has the following form: + +$$ +\frac{\partial \rho}{\partial t}+\frac{\partial \rho}{\partial r}\frac{\partial r}{\partial t} + = \frac{1}{f^{n-1}(r,t)} +\frac{\partial }{\partial r} \left( f^{n-1}(r,t) \frac{\partial +\rho}{\partial r} \right)-(n-1)f^{-2}(r,t) f(\rho,t_0) +\frac{\partial f(\rho,t_0)}{\partial \rho} +$$ + +or equivalently + +$$ +\frac{\partial \rho}{\partial t}=\frac{1}{f^{n-1}(r,t)} +\frac{\partial }{\partial r} \left( f^{n-1}(r,t) \frac{\partial +\rho}{\partial r} \right)-(n-1)f^{-2}(r,t) f(\rho,t_0) +\frac{\partial f(\rho,t_0)}{\partial \rho} -\frac{\partial +\rho}{\partial r}\frac{\partial r}{\partial t} +$$ + +The point of rewriting the harmonic map equation in this way is to +find an equation for the functions $\rho(r,t), f(r,t)$ defined on +$r\ge 0$. Even though the terms in this rewritten equation involve +odd functions of $r$, as we shall see, solutions to these equations +will be even in $r$ and hence will produce a smooth solution to the +harmonic map flow equation on $\Ar ^n$. + +### An equation equivalent to the harmonic map flow equation + +We will solve (12.15) for solutions of the form + +$$ +\rho(r,t)= re^{\tilde{\rho}(r,t)}, \ \ \ t\ge t_0;\ \ \ \tilde \rho(r,t_0)=0. +$$ + +For $\psi$ as in Equation (12.13) to define a diffeomorphism, +it must be the case that $\rho(r,t)$ is a smooth function for $r\ge +0$ which is odd in $r$. It follows from the above expression that +$\tilde\rho(r,t)$ is a smooth function of $r$ and $t$ defined for +$r\ge 0$ and even in $r$, so that it defines a smoothly varying +family of smooth functions on $\Ar^n$. Then some straightforward +calculation shows that (12.15) becomes + +$$ +\begin{aligned} +\frac{\partial\tilde{\rho} }{\partial t} & = & +\frac{\partial^{2}\tilde{\rho}}{\partial r^{2}}+\frac{n+1}{r}\frac +{\partial\tilde{\rho}}{\partial r} + +\left( n-1\right) \frac{\partial\tilde{h}( +r^{2},t)}{\partial r} \frac{\partial\tilde{\rho} }{\partial r} ++\left( \frac{\partial\tilde{\rho} +}{\partial r}\right) ^{2} \\ +& & +\frac{n-1}{r^{2}}\left[ 1-e^{2\tilde{h}\left( +\rho^{2},t_0\right) -2\tilde{h}\left( r^{2},t\right) }\right] + +2\left( n-1\right) \frac{\partial\tilde{h}}{\partial w}\left( r^{2},t\right) \nonumber \\ +& & -2\left( n-1\right) e^{2\tilde{h}\left( \rho ^{2},t_0\right) ++2\tilde{\rho} -2\tilde{h}\left( r^{2},t\right) +}\frac{\partial\tilde{h}}{\partial w}(\rho^2,t_0) +-\frac{2}{r}\frac{\partial r}{\partial t}-\frac{\partial r}{\partial +t} \frac{\partial\tilde{\rho} }{\partial r} .\nonumber +\end{aligned} +$$ + +Note that from the definition, $\tilde{h}\left( 0,t \right)=0$, we +can write $\tilde{h}\left( w,t \right)=w \tilde{h}^*(w,t)$ where +$\tilde{h}^*(w,t)$ is a smooth function of $w\ge 0$ and $t$. So + +$$ +\frac{n-1}{r^{2}}\left[ 1-e^{2\tilde{h}\left( \rho^{2},t_0\right) +-2\tilde{h}\left( r^{2},t\right) }\right] =\frac{n-1}{r^{2}}\left[ +1-e^{2r^2\left[ e^{2\tilde{\rho}} \tilde{h}^*\left( +\rho^{2},t_0\right) -\tilde{h}^*\left( r^{2},t\right) \right] +}\right] +$$ + +which is a smooth function of $\tilde{\rho},r^2,t$. + +Let + +$$ +\begin{aligned} +G(\tilde{\rho},w,t) = & \frac{n-1}{w}\left[ 1-e^{2\tilde{h}\left( +\rho^{2},t_0\right) -2\tilde{h}\left( w,t\right) }\right] + +2\left( n-1\right) \frac{\partial\tilde{h}}{\partial w}\left( w, + t\right) \nonumber \\ +& -2\left( n-1\right) e^{2\tilde{h}\left( \rho ^{2},t_0\right) ++2\tilde{\rho} -2\tilde{h}\left( w,t\right) }\frac{\partial\tilde +h}{\partial w}(\rho^2,t_0) -2\xi(w,t), +\end{aligned} +$$ + +where $\xi$ is the function from \entryref{e353993489e9}. Then +$G(\tilde{\rho},w,t)$ is a smooth function defined for $w\ge 0$. +Notice that when $r$ and $\tilde \rho$ are the functions associated +with the varying family of metrics $g(t)$ and the solutions to the +harmonic map flow, then $G(\tilde\rho,r^2,t)$ defines a smoothly +varying family of smooth functions on $\Ar^n$. + +We have the following form of equation (12.16): + +$$ +\begin{aligned} +\frac{\partial\tilde{\rho} }{\partial t} = & +\frac{\partial^{2}\tilde{\rho}}{\partial r^{2}}+\frac{n+1}{r}\frac +{\partial\tilde{\rho}}{\partial r} +\left[ ( n-1) +\frac{\partial\tilde{h}}{\partial r} -\frac{\partial B}{\partial r} +\right] \left( r^{2},t\right) \frac{\partial\tilde{\rho} }{\partial +r} +\left( \frac{\partial\tilde{\rho} }{\partial r}\right) ^{2} ++G(\tilde\rho,r^2,t). +\end{aligned} +$$ + +Now we think of $\tilde{\rho}$ as a rotationally symmetric function +defined on $\mathbb{R}^{n+2}$ and let $\widehat +G(\tilde\rho,(x^1,\ldots,x^{n+2}),t)= +G(\tilde\rho,\sum_{i=1}^{n+2}(x^i)^2,t)$ and then the above equation +can be written as + +$$ +\begin{aligned} +\frac{\partial\tilde{\rho} }{\partial t} + = \Delta \tilde{\rho} + \nabla [ (n-1)\tilde{h} -B] \cdot \nabla\tilde{\rho} ++ \left | \nabla \tilde{\rho} \right |^2 +G(\tilde{\rho},x,t) +\end{aligned} +$$ + +where $\nabla$ and $\Delta$ are the Levi-Civita connection and +Laplacian defined by the Euclidean metric on $\mathbb{R}^{n+2}$ +respectively and where $B$ is the function defined in +Equation (12.14). + +\entryblock{9dfe0309dd19} + +It is important to understand the asymptotic behavior of our +functions at spatial infinity. + +\entryblock{ac4ac4fb06f8} + +**Proof.** The first item is immediate from \entryref{77ca4669ffb3}. The +second and third follow immediately from the first. The fourth is a +consequence of the fact that by \entryref{77ca4669ffb3} $dr/dt$ is +asymptotic to a constant at infinity on each time-slice. The fifth +follows immediately from the fourth and the definition of +$B(r^2,t)$. Given all these asymptotic expressions, the last is +clear from the expression for $G$ in terms of $\tilde \rho$, $r^2$, +and $t$. + +### The short time existence + +The purpose of this subsection is to prove the following short-time +existence theorem for the harmonic map flow equation. + +\entryblock{43ea5e1d04cf} + +At this point to simplify the notation we shift time by $-t_0$ so +that our initial time is $0$, though our initial metric is not $g_0$ +but rather is the time $t_0$-slice of the standard solution we are +considering, so that now $t_0=0$ and our initial condition is +$\tilde \rho(r,0)=0$. + +Let $x=(x^1,\cdots,x^{n+2})$ and $y=(y^1,\cdots,y^{n+2})$ be two +points in $\mathbb{R}^{n+1}$ and + +$$ +H(x,y,t)=\frac{1}{(4 \pi t)^{(n+2)/2}} e^{-\frac{|x-y|^2}{4t}} +$$ + +be the heat kernel. We solve (12.18) by successive approximation +[LiTam]. + +Define + +$$ +F(x, \tilde{\rho},\nabla\tilde{\rho},t) += \nabla \left[ (n-1)\tilde{h}-B \right] \cdot \nabla\tilde{\rho} + +\left | \nabla \tilde{\rho} \right |^2 +G( \tilde{\rho},x,t) +$$ + +Let $\tilde{\rho}_0(x,t)=0$ and for $i \geq 1$ we define +$\tilde{\rho}_i$ by + +$$ +\begin{aligned} +\tilde{\rho}_i=\int_0^t \int_{\mathbb{R}^{n+2}} +H(x,y,t-s)F(y,\tilde{\rho}_{i-1}, \nabla\tilde{\rho}_{i-1} ,t)dyds +\end{aligned} +$$ + +which solves + +$$ +\begin{aligned} +\frac{\partial\tilde{\rho}_i}{\partial t} = \Delta \tilde{\rho}_i + +F(x, \tilde{\rho}_{i-1},\nabla\tilde{\rho}_{i-1},t) \qquad +\tilde{\rho}_i(x,0)=0. +\end{aligned} +$$ + +To show the existence of $\tilde{\rho}_i$ by induction, it suffices +to prove the following statement: For any $i \geq 1$, if +$|\tilde{\rho}_{i-1}| , |\nabla \tilde{\rho}_{i-1}| $ are bounded, +then $\tilde{\rho}_i$ exists and $|\tilde{\rho}_{i}|, |\nabla +\tilde{\rho}_{i}|$ are bounded. Assume $|\tilde{\rho}_{i-1}| \leq +C_1, |\nabla \tilde{\rho}_{i-1}| \leq C_2$ are bounded on +$\mathbb{R}^{n+2} \times [0,T]$; then it follows from +\entryref{ac4ac4fb06f8} that $G(\tilde{\rho}_{i-1},\mathbf{x},t)$ is bounded +on $\mathbb{R}^{n+2} \times [0,T]$ + +$$ +|G(\tilde{\rho}_{i-1},x,t)| \leq C_*(C_1,\tilde{h} ), +$$ + +and also because of \entryref{ac4ac4fb06f8} both $|\nabla B|$ and +$|\nabla \tilde h|$ are bounded on all of $\Ar^{n+2}\times [0,T]$, +it follows that $F(x, +\tilde{\rho}_{i-1},\nabla\tilde{\rho}_{i-1},t)$ is bounded: + +$$ +\begin{aligned} +& |F(x, \tilde{\rho}_{i-1},\nabla\tilde{\rho}_{i-1},t)| \\ +\leq & \left[(n-1)\sup |\nabla \tilde{h}| + \sup |\nabla B| +\right]C_2+C_2^2+ C_*(C_1,\tilde{h} ) = C_3 +\end{aligned} +$$ + +Hence $\tilde{\rho}_i$ exists. + +The bounds on $|\tilde{\rho}_{i}|$ and $|\nabla \tilde{\rho}_{i}|$ +follow from the following estimates + +$$ +| \tilde{\rho}_i | \leq \int_0^t \int_{\mathbb{R}^{n+2}} +H(x,y,t-s)C_3 dyds \leq C_3t, +$$ + +and + +$$ +\begin{aligned} +& | \nabla \tilde{\rho}_i | = | \int_0^t \int_{\mathbb{R}^{n+2}} +[\nabla_x H(x,y,t-s)] +F(y,\tilde{\rho}_{i-1}, \nabla\tilde{\rho}_{i-1} ,t)dyds| \\ +& \leq \int_0^t \int_{\mathbb{R}^{n+2}} | \nabla_x H(x,y,t-s)| C_3 dyds \\ +& = \int_0^t \int_{\mathbb{R}^{n+2}} \frac{1}{(4 \pi +{(t-s)})^{(n+2)/2}} +e^{-\frac{|x-y|^2}{4{(t-s)}} } \frac{|x-y|}{2{(t-s)}} C_3 dyds \\ +& \leq \frac{(n+2)C_3}{\sqrt{\pi}} \int_0^t \frac{1}{\sqrt{t-s}} ds += \frac{2(n+2)C_3}{\sqrt{\pi}} \sqrt{t}. +\end{aligned} +$$ + +Assuming, as we shall, that $T\le \min \{\frac{C_3}{C_1},\frac{\pi +C_2^2}{ 4(n+2)^2C_3^2} \}$, then for $0 \leq t \leq T$ we have for +all $i$, + +$$ +\begin{aligned} +|\tilde{\rho}_{i}| \leq C_1 \qquad \mathit{and} \qquad |\nabla +\tilde{\rho}_{i}| \leq C_2. +\end{aligned} +$$ + +We prove the convergence of $\tilde{\rho}_i$ to a solution of +(12.18) via proving that it is a Cauchy sequence in +$C^1$-norm. Note that $\tilde{\rho}_i- \tilde{\rho}_{i-1}$ +satisfies + +$$ +\begin{aligned} +& \frac{\partial (\tilde{\rho}_i- \tilde{\rho}_{i-1})}{\partial t} = +\Delta (\tilde{\rho}_i- \tilde{\rho}_{i-1}) + F(x, +\tilde{\rho}_{i-1},\nabla\tilde{\rho}_{i-1},t)-F(x,\tilde{\rho}_{i-2}, +\nabla\tilde{\rho}_{i-2},t) \nonumber \\ +& (\tilde{\rho}_i- \tilde{\rho}_{i-1})(x,0)= 0. +\end{aligned} +$$ + +where + +$$ +\begin{aligned} +& F(x, +\tilde{\rho}_{i-1},\nabla\tilde{\rho}_{i-1},t)-F(x,\tilde{\rho +}_{i-2},\nabla\tilde{\rho}_{i-2},t) \\ += & [(n-1) \nabla \tilde{h}- \nabla B +\nabla (\tilde{\rho}_{i-1}+ +\tilde{\rho}_{i-2})] + \cdot \nabla(\tilde{\rho}_{i-1} - \tilde{\rho}_{i-2}) \\ +& + G( \tilde{\rho}_{i-1},\mathbf{x},t)-G(\tilde{\rho}_{i-2},\mathbf{x},t) +\end{aligned} +$$ + +By lengthy but straightforward calculations one can verify the +Lipschitz property of $G(\tilde{\rho},\mathbf{x},t)$ + +$$ +|G(\tilde{\rho}_{i-1},\mathbf{x},t)-G(\tilde{\rho}_{i-2},\mathbf{x},t)| +\leq C_\&(C_1,C _2, \tilde{f},\tilde{f}_0) \cdot | +\tilde{\rho}_{i-1}-\tilde{\rho}_{i-2}|. +$$ + +This and (12.21) implies + +$$ +\begin{aligned} +& |F(x, +\tilde{\rho}_{i-1},\nabla\tilde{\rho}_{i-1},t)-F(x,\tilde{\rho}_{i-2}, +\nabla\tilde{\rho}_{i-2},t)| +\nonumber \\ +\leq & C_4 \cdot | \tilde{\rho}_{i-1}-\tilde{\rho}_{i-2}| +C_5 +\cdot |\nabla \tilde{\rho}_{i-1}- \nabla \tilde{\rho}_{i-2} | +\end{aligned} +$$ + +where $C_4 = C_\&(C_1,C_2, \tilde{f},\tilde{f}_0)$ and $C_5 = [(n-1) +\sup |\nabla \tilde{f}| +\sup |\nabla B|+2C_2]$. + +Let + +$$ +\begin{aligned} +& A_i(t)= \sup_{0 \leq s \leq t, x \in \mathbb{R}^{n+2}} +|\tilde{\rho}_i- +\tilde{\rho}_{i-1}|(x,s) \\ +& B_i(t)=\sup_{0 \leq s \leq t,x \in \mathbb{R}^{n+2}} +|\nabla(\tilde{\rho}_i- + \tilde{\rho}_{i-1})|(x,s). +\end{aligned} +$$ + +From Equations (12.22) and (12.23) we can estimate +$|\tilde{\rho}_i- \tilde{\rho}_{i-1}|$ and $ |\nabla(\tilde{\rho}_i- +\tilde{\rho}_{i-1})|$ in the same way as we estimate +$|\tilde{\rho}_{i}|$ and $|\nabla\tilde{\rho}_i|$ above; we conclude + +$$ +\begin{aligned} +& A_i(t)\leq [ C_4 A_{i-1}(t) +C_5 B_{i-1}(t)] \cdot t \\ +& B_i(t) \leq \frac{2(n+2)[C_4 A_{i-1}(t) +C_5 +B_{i-1}(t)]}{\sqrt{\pi}} \cdot \sqrt{t}. +\end{aligned} +$$ + +Let $C_6 = \max \{C_4,C_5 \}$; then we get + +$$ +A_i(t)+B_i(t) \leq \left( C_6t+ \frac{2(n+2)C_6 +\sqrt{t}}{\sqrt{\pi}} \right) \cdot \left( A_{i-1}(t)+B_{i-1}(t) +\right). +$$ + +Now suppose that $T\le T_2$ where $T_2$ satisfies $C_6T_2+ +\frac{2(n+2)C_6 \sqrt{T_2}}{\sqrt{\pi}} =\frac{1}{2}$; then for all +$t\le T$ we have + +$$ +A_i(t)+B_i(t) \leq \frac{1}{2} \left( A_{i-1}(t)+B_{i-1}(t) +\right). +$$ + +This proves that $\tilde{\rho}_{i}$ is a Cauchy sequence in +$C^1(\mathbb{R}^{n+2})$. Let $\lim_{i \rightarrow +\infty} +\tilde{\rho}_{i} =\tilde{\rho}_{\infty}$. Then $\nabla +\tilde{\rho}_{i} \rightarrow \nabla \tilde{\rho}_{\infty}$ and +$F(x,\tilde{\rho}_{i-1},\nabla \tilde{\rho}_{i-1},t) \rightarrow +F(x, \tilde{\rho}_{\infty}, \nabla \tilde{\rho}_{\infty},t)$ +uniformly. Hence we get from (12.19), + +$$ +\begin{aligned} +\tilde{\rho}_\infty=\int_0^t \int_{\mathbb{R}^{n+2}} H(x,y ,t-s)F(y,\tilde{ \rho}_{\infty}, +\nabla\tilde{\rho}_{\infty} ,t)dyds +\end{aligned} +$$ + +The next argument is similar to the argument in [LiTam], p.21. +The function $\tilde{\rho}_{i}$ is a smooth solution of +(12.20) with $\tilde{\rho}_i(x,0)=0$. Also, both + $\tilde{\rho}_{i}$ +and $F(x,\tilde{\rho}_{i_1},\nabla \tilde{\rho}_{i-1},t)$ are +uniformly bounded on $\mathbb{R}^{n+2} \times[0,T]$. Thus, by +Theorem 1.11 [LSU], p.211 and Theorem 12.1 [LSU], p.223, +for any compact $K \subset \mathbb{R}^{n+2}$ and any $0\epsilon_{0}$ for some $\epsilon_{0}>0$. + +We choose a $k_0$ sufficiently large that $x_{0}\in \Omega_{k_0}$ +and for all $t'\in [t_0,T]$ we have + +$$ +\sup_{x\in\partial\Omega_{b}}\left\vert \hat g_1(x, t') -\hat +g_2(x, t') \right\vert _{g}^{2} \leq \epsilon +$$ + +where $\epsilon >0$ is a constant to be chosen later. + +Recall we have the initial condition $\left\vert \hat g_1( 0) -\hat +g_2(0) \right\vert _{g}^{2}=0$. Using Equation (12.28) and applying the maximum principle to $\left\vert \hat g_1( +t) -\hat g_2( t) \right\vert _{g}^{2}$ ) on the domain +$\Omega_{k_0}$, we get + +$$ +e^{-C_{15}t}\left\vert \hat g_1 ( t) -\hat g_2 ( t) \right\vert +_{g}^{2}( x) \leq\epsilon.\text{ for all }x\in\Omega _{k_0}. +$$ + +This is a contradiction if we choose $\epsilon \leq +\epsilon_{0}e^{-C_{15}T}$. This contradiction establishes the +proposition. + + Let $g_1(t),\ 0\le tt_0$ for the Ricci flows +$g_1(t)$ and $g_2(t)$. Let $\hat{g}_1(t)= (\psi^{-1}(t))^*g_1(t)$ +and $\hat{g}_2(t) = (\psi^{-1}(t))^*g_2(t)$. Then $\hat{g}_1(t)$ and +$\hat{g}_2(t)$ are two solutions of the Ricci-DeTurck flow with +$\hat{g}_1(t_0)=\hat{g}_2(t_0)$. Choose $T' \in (t_0,T]$ such that +$\hat{g}_1(t)$ and $\hat{g}_2(t)$ are $\delta$-close to +$\hat{g}_1(t_0)$ as required in \entryref{c5e225b067f1}. It follows +from \entryref{77ca4669ffb3} and the decay estimate in +\entryref{5fbeadbba61c} that $\hat{g}_1(t)$ and $\hat{g}_2(t)$ +are bounded solutions and that they have same sequential asymptotic +behavior at infinity. We can apply \entryref{c5e225b067f1} to +conclude $\hat{g}_1(t)=\hat{g}_2(t)$ on $t_0 \leq t \leq T'$. We +have proved: + +\entryblock{a501760fb0e6} + +## Completion of the proof of uniqueness + +Now we are ready to prove the uniqueness of the standard solution. +Let $g_1(t),\ 0\le tt_0$. Using these harmonic map flows we construct solutions +$\hat g_1(t)$ and $\hat g_2(t)$ to the Ricci-DeTurck flow defined on +the interval $[t_0,T]$. According to \entryref{a501760fb0e6}, there is +a uniqueness theorem for these Ricci-DeTurck flows, which implies +that $\hat g_1(t)=\hat g_2(t)$ for all $t\in [t_0,T']$ for some +$T'>t_0$. Invoking \entryref{5b79e248c5cb} we conclude that +$g_1(t)=g_2(t)$ for all $t\in [0,T']$, contradicting the maximality +of the interval $I$. + +If none of these three cases can occur, then the only remaining +possibility is that $T_1=T_2$ and $I=[0,T_1)$, i.e., the flows are +the same. This then completes the proof of the uniqueness of the +standard flow. + +### $T=1$ and existence of canonical neighborhoods + +At this point we have established all the properties claimed in +\entryref{f44d182ef27e} for the standard flow except for the fact that +$T$, the endpoint of the time-interval of definition, is equal to +$1$. We have shown that $T\le 1$. In order to establish the opposite +inequality, we must show the existence of canonical neighborhoods +for the standard solution. + +Here is the result about the existence of canonical neighborhoods +for the standard solution. + +\entryblock{85e3c925177e} + +**Proof.** Take an increasing sequence of times $t'_n$ converging to $T$. +Since the curvature of $(\Ar^3,g(t))$ is locally bounded in time, +for each $n$, there is a bound on the scalar curvature on +$\Ar^3\times [0,t'_n]$. Hence, there is a finite upper bound $R_n$ +on $R(x,t)$ for all points $(x,t)$ with $t\le t'_n$ for which the +conclusion of the theorem does not hold. (There clearly are such +points since the conclusion of the theorem fails for all $(x,0)$.) +Pick $(x_n,t_n)$ with $t_n\le t'_n$, with $R(x_n,t_n)\ge R_n/2$ and +such that the conclusion of the theorem does not hold for +$(x_n,t_n)$. To prove the theorem we must show that $\overline\mathit{lim}_{n\rightarrow\infty}R(x_n,t_n)<\infty$. Suppose the contrary. +By passing to a subsequence we can suppose that $\mathit{lim}_{n\rightarrow \infty}R(x_n,t_n)=\infty$. We set +$Q_n=R(x_n,t_n)$. We claim that all the hypotheses of +\entryref{a9e023d082dc} apply to the sequence +$(\Ar^3,g(t),(x_n,t_n))$. First, we show that all the hypotheses of +\entryref{d51aab8a8065} (except the last) hold. Since $(\Ar^3,g(t))$ +has non-negative curvature all these flows have curvature pinched +toward positive. By \entryref{ed069175cdf9} there are $r>0$ and +$\kappa>0$ so that all these flows are $\kappa$-non-collapsed on +scales $\le r$. By construction if $t\le t_n$ and $R(y,t)>2Q_n\ge +R_n$ then the point $(y,t)$ has a strong canonical +$(C(\epsilon),\epsilon)$-neighborhood. We are assuming that +$Q_n\rightarrow \infty$ as $n\rightarrow \infty$ in order to achieve +the contradiction. Since all time-slices are complete, all balls of +finite radius have compact closure. + +Lastly, we need to show that the extra hypothesis of +\entryref{a9e023d082dc} (which includes the last hypothesis of +\entryref{d51aab8a8065}) is satisfied. This is clear since +$t_n\rightarrow T$ as $n\rightarrow \infty$ and $Q_n\rightarrow +\infty$ as $n\rightarrow \infty$. Applying \entryref{a9e023d082dc} we +conclude that after passing to a subsequence there is a limiting +flow which is a $\kappa$-solution. Clearly, this and +\entryref{e724298e5bc3} imply that for all sufficiently large +$n$ (in the subsequence) the neighborhood as required by the +theorem exists. This contradicts our assumption that none of the +points $(x_n,t_n)$ have these neighborhoods. This contradiction +proves the result. + +### Completion of the proof of Theorem \entryref{f44d182ef27e} + +The next proposition establishes the last of the conditions claimed +in \entryref{f44d182ef27e}. + +\entryblock{729d9a0428c4} + +**Proof.** We have already seen in \entryref{2db418fba1c1} that $T\le 1$. Suppose now +that $T<1$. Take $T_00$ is the +constant given in \entryref{85e3c925177e}, we have $\partial +R/\partial t(x,t)\le C(\epsilon)R^2(x,t)$. Thus, provided that +$T-T_0$ is sufficiently small, there is a uniform bound to $R(x,t)$ +for all $x\in \Ar^3\setminus X$ and all $t\in [T_0,T)$. Using +\entryref{e0b6a1f646a7} and the fact that the standard flow is +$\kappa$-non-collapsed implies that the restrictions of the metrics +$g(t)$ to $\Ar^3\setminus X$ converge smoothly to a limiting +Riemannian metric $g(T)$ on $\Ar^3\setminus X$. Fix a non-empty open +subset $\Omega\subset \Ar^3\setminus X$ with compact closure. For +each $t\in [0,T)$ let $V(t)$ be the volume of +$(\Omega,g(t)|_{\Omega})$. Of course, $\mathit{lim}_{t\rightarrow +T}V(t)=\mathit{Vol}_{g(T)}(\Omega)>0$. + +Since the metric $g(T)$ exists in a neighborhood of infinity and has +bounded curvature there, if the limit metric $g(T)$ exists on all of +$\Ar^3$, then we can extend the flow keeping the curvature bounded. +This contradicts the maximality of our flow subject to the condition +that the curvature be locally bounded in time. Consequently, there +is a point $x\in \Ar^3$ for which the limit metric $g(T)$ does not +exist. This means that $\overline\mathit{lim}_{t\rightarrow +T}R(x,t)=\infty$. That is to say, there is a sequence of +$t_n\rightarrow T$ such that setting $Q_n=R(x,t_n)$, we have +$Q_n\rightarrow \infty$ as $n$ tends to infinity. By +\entryref{85e3c925177e} the second hypothesis in the statement of +\entryref{d51aab8a8065} holds for the sequence +$(\Ar^3,g(t),(x,t_n))$. All the other hypotheses of this theorem as +well as the extra hypothesis in \entryref{a9e023d082dc} obviously +hold for this sequence. Thus, according to \entryref{a9e023d082dc} +the based flows $(\Ar^3,Q_ng(Q_n^{-1}t'+t_n),(x,0))$ converge +smoothly to a $\kappa$-solution. Since the asymptotic volume of any +$\kappa$-solution is zero (see \entryref{7468282bd017}), we see that +for all $n$ sufficiently large, the following holds: + +\entryblock{cf6eed7b1da4} + +Rescaling, we see that for all $n$ sufficiently large we have + +$$ +\mathit{Vol} B_g(x,t_n,A/\sqrt{Q_n})< \epsilon(A/\sqrt{Q_n})^3. +$$ + +Since the curvature of $g(t_n)$ is non-negative and since the $Q_n$ +tend to $\infty$, it follows from the Bishop-Gromov Inequality +(\entryref{1a1b2c4fdc9b}) that for any $00$, for all $n$ sufficiently large we have + +$$ +\mathit{Vol} B_g(x,t_n,A)<\epsilon A^3. +$$ + +On the other hand, since $\Omega$ has compact closure, there is an +$A_1<\infty$ with $\Omega\subset B(x,0,A_1)$. Since the curvature of +$g(t)$ is non-negative for all $t\in [0,T)$, it follows from +\entryref{51a2afee4465} that the distance is a non-increasing +function of $t$, so that for all $t\in [0,T)$ we have $\Omega\subset +B(x,t,A_1)$. Applying the above, for any $\epsilon>0$ for all $n$ +sufficiently large we have + +$$ +\mathit{Vol} (\Omega,g(t_n))\le \mathit{Vol}_g B(x,t_n,A_1)<\epsilon +(A_1)^3. +$$ + + But this contradicts the fact that + +$$ +\mathit{lim}_{n\rightarrow +\infty}\mathit{Vol} \Omega,g(t_n)=\mathit{Vol} (\Omega,g(T))>0. +$$ + + This +contradiction proves that $T=1$. + +This completes the proof of \entryref{f44d182ef27e}. + +## Some corollaries + +Now let us derive extra properties of the standard solution that +will be important in our applications. + +\entryblock{ffae4504318a} + +**Proof.** First, let us show that there is not a limiting metric $g(1)$ +defined on all of $\Ar^3$. This does not immediately contradict the +maximality of the flow because we are assuming only that the flow is +maximal subject to having curvature locally bounded in time. Assume +that a limiting metric $(\Ar^3,g(1))$ exists. + First, +notice that from the canonical neighborhood assumption and +\entryref{0ddfe578961e} we see that the curvature of $g(T)$ must be +unbounded at spatial infinity. On the other hand, by +\entryref{7c2758f0e2b3} every point of $(\Ar^3,g(1))$ of +curvature greater than $R_0$ has a $(2C,2\epsilon)$-canonical +neighborhood. Hence, since $(\Ar^3,g(1))$ has unbounded curvature, +it then has $2\epsilon$-necks of arbitrarily small scale. This +contradicts \entryref{cb9ca6d6db9f}. (One can also rule this +possibility out by direct computation using the spherical symmetry +of the metric.) This means that there is no limiting metric $g(1)$. + +The next step is to see that for any $p\in \Ar^3$ we have $\mathit{lim}_{t\rightarrow 1}R(p,t)=\infty$. Let $\Omega\subset \Ar^3$ be +the subset of $x\in \Ar^3$ for which $\mathit{liminf}_{t\rightarrow +1}R(x,t)<\infty$. We suppose that $\Omega\not=\emptyset$. According +to \entryref{af00774dbaa9} the subset $\Omega$ is open and the metrics +$g(t)|_\Omega$ converge smoothly to a limiting metric +$g(1)|_\Omega$. On the other hand, we have just seen that there is +not a limit metric $g(1)$ defined everywhere. This means that there +is $p\in \Ar^3$ with $\mathit{lim}_{t\rightarrow 1}R(p,t)=\infty$. +Take a sequence $t_n$ converging to $1$ and set $Q_n=R(p,t_n)$. By +\entryref{a9e023d082dc} we see that, possibly after passing to a +subsequence, the based flows $(\Ar^3,Q_ng(t'-t_n),(p,0))$ converge +to a $\kappa$-solution. Then by \entryref{7468282bd017} for any +$\epsilon>0$ there is $A<\infty$ such that $\mathit{Vol} B_{Q_ng}(p,t_n,A)<\epsilon A^3$, and hence after rescaling we +have $\mathit{Vol} B_g(p,t_n,A/\sqrt{Q_n})<\epsilon (A/\sqrt{Q_n})^3$. +By the Bishop-Gromov inequality (\entryref{1a1b2c4fdc9b}) it +follows that for any $00$ and for all $n$ +sufficiently large, we have $\mathit{Vol} B_g(p,t_n,A)<\epsilon A^3$. +Take a non-empty subset $\Omega'\subset \Omega$ with compact +closure. Of course, $\mathit{Vol} (\Omega',g(t))$ converges to $\mathit{Vol} (\Omega',g(T))>0$ as $t\rightarrow T$. Then there is +$A<\infty$ such that for each $n$, the subset $\Omega'$ is contained +in the ball $B(p_0,t_n,A)$. This is a contradiction since it implies +that for any $\epsilon>0$ for all $n$ sufficiently large we have +$\mathit{Vol} (\Omega',g(t))<\epsilon A^3$. This completes the proof +that for every $p\in \Ar^3$ we have $\mathit{lim}_{t\rightarrow +1}R(p,t)=\infty$. + +Fix $\epsilon>0$ sufficiently small and set $C=C(\epsilon)$. Then +for every $(p,t)$ with $R(p,t)\ge r^{-2}$ we have + +$$ +\left|\frac{dR}{dt}(p,t)\right|\le CR^2(p,t). +$$ + + Fix $t_0=1-1/2r^2C$. Since the flow has curvature locally bounded in + time, there is $2C\le C'<\infty$ such that $R(p,t_0)\le 1/(C'(1-t_0)$ for + all $p\in \Ar ^3$. + Since $R(p,t_0)=1/C'(1-t_0)$, +for all $t\in [t_0,1)$ we have + +$$ +R(p,t)<\mathit{max}\left(\left[(C'-C)(1-t_0)\right]^{-1},\left[r^{-2}-C(1-t_0)\right]^{-1}\right). +$$ + +This means that $R(p,t)$ is uniformly bounded as $t\rightarrow 1$, +contradicting what we just established. This shows that for $t\ge +1-1/2r^2C$ the result holds. For $t\le 1-1/2r^2C$ there is a +positive lower bound on the scalar curvature, and hence the result +is immediate for these $t$ as well. + +\entryblock{6ff28dba4e56} + +\entryblock{246771506ee5} + +**Proof.** By \entryref{85e3c925177e}, there is $r_0$ such that if $R(x,t)\ge +r_0^{-2}$, then $(x,t)$ has a $(C,\epsilon)$-canonical neighborhood +and if this canonical neighborhood is a strong $\epsilon$-neck +centered at $x$, then that neck extends to an evolving neck defined +for rescaled time $(1+\epsilon)$. By \entryref{ffae4504318a}, there +is $\theta<1$ such that if $R(x,t)\le r_0^{-2}$ then $t\le \theta$. +By \entryref{77ca4669ffb3}, there is a compact subset $X\subset \Ar +^3$ such that if $t\le \theta$ and $x\notin X$, then there is an +evolving $\epsilon$-neck centered at $x$ whose initial time is zero +and whose initial time-slice is at distance at least one from the +surgery cap. Lastly, by compactness there is $C'<\infty$ such that +every $(x,t)$ for $x\in X$ and every $t\le \theta$ is contained in +the core of a $(C',\epsilon)$-cap. + +\entryblock{20ae3cd37b93} + +\entryblock{fac15ebeba31} + +\entryblock{096bebd3ecff} + +**Proof.** This follows immediately from \entryref{6ff28dba4e56} and +\entryref{7c2758f0e2b3}. + +There is one property that we shall use later in proving the +finite-time extinction of Ricci flows with surgery for manifolds +with finite fundamental group (among others). This is a distance +decreasing property which we record here. + +Notice that for the standard initial metric constructed in +\entryref{f1d3021d32b8} we have the following: + +\entryblock{545ca0a10566} + +**Proof.** Clearly, the metric $\rho^*g_0$ is rotationally symmetric and its +component in the $s$-direction is $ds^2$. On the other hand, since +each cross section $\{s\}\times S^2$ maps conformally onto a sphere +of radius $\le\sqrt{2}$ the result follows. diff --git a/projects/poincare-conjecture/.astrolabe/docs/14-surgery-on-a-delta-neck.mdx b/projects/poincare-conjecture/.astrolabe/docs/14-surgery-on-a-delta-neck.mdx new file mode 100644 index 00000000..3bb8b468 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/docs/14-surgery-on-a-delta-neck.mdx @@ -0,0 +1,502 @@ + + +# Chapter 13 -- Surgery on a $\delta$-neck + +## Notation and the Statement of the Result + +In this chapter we describe the surgery process. For this chapter we +fix: + +- **(1)** A $\delta$-neck $(N,g)$ centered at a point $x_0$. +We denote by $\rho\colon S^2\times (-\delta^{-1},\delta^{-1})\to N$ +the diffeomorphism that gives the $\delta$-neck structure. +- **(2)** Standard initial conditions $(\Ar^3,g_0)$. + +We denote by $h_0\times ds^2$ the metric on $S^2\times \Ar$ which is +the product of the round metric $h_0$ on $S^2$ of scalar curvature +$1$ and the Euclidean metric $ds^2$ on $\Ar$. We denote by +$N^-\subset N$ the image $\rho((-\delta^{-1},0]\times S^2)$ and we +denote by $s\colon N^-\to (-\delta^{-1},0]$ the composition +$\rho^{-1}$ followed by the projection to the second factor. + +Recall that the standard initial metric $(\Ar^3,g_0)$ is invariant +under the standard $SO(3)$-action on $\Ar^3$. We let $p_0$ denote +the origin in $\Ar^3$. It is the fixed point of this action and is +called the *tip* +of the standard initial metric. Recall from \entryref{0b07beec84c0} that +there are $A_0>0$ and an isometry + +$$ +\psi\colon (S^2\times +(-\infty,4],h_0\times ds^2)\to (\Ar^3\setminus B(p_0,A_0),g_0). +$$ + + The +composition of $\psi^{-1}$ followed by projection onto the second +factor defines a map $s_1\colon \Ar^3\setminus B(p_0,A_0)\to +(-\infty,4]$. Lastly, there is $00,\end{cases} +$$ + +and then we define the metric $\tilde g$ on ${\mathcal S}$ by first +defining a metric: + +$$ +\hat g=\begin{cases} \mathit{exp}(-2f(s))R(x_0)\rho^*g & \ \ \mathit{on} \ \ +s^{-1}(-\infty,1] \\ +\mathit{exp}(-2f(s))\left(\alpha(s)R(x_0)\rho^*g+(1-\alpha(s))\eta +g_0\right) & \ \ \mathit{on}\ \ s^{-1}([1,2]) \\ +\mathit{exp}(-2f(s))\eta g_0 & \ \ \mathit{on}\ \ s^{-1}([2,A_{r_0}]\\ +\left[\beta(s)\mathit{exp}(-2f(s))+(1-\beta(s))\mathit{exp}(-2f(4+A_0))\right]\eta +g_0 & \ \ \mathit{on} \ \ s^{-1}([A_{r_0},A']),\end{cases} +$$ + + where +$A_{r_0}=4+A_0-r_0$ and $A'=A_0+4$. + Then we define + +$$ +\widetilde g=R(x_0)^{-1}\hat g. +$$ + +See Fig. 13.1. + +\entryblock{cf61b731c6f2} + +The rest of this chapter is devoted to the proof of this theorem. + +Before starting the curvature computations let us make a remark about the surgery cap. + +\entryblock{443843fff58d} + +The following is immediate from the definitions provided that +$\delta>0$ is sufficiently small. + +\entryblock{752133843e1e} + +## Preliminary computations + +We shall compute in a slightly more general setup. + Let $I$ be an open interval contained in $(-\delta^{-1},4+A_0)$ and let +$h$ be a metric on $S^2\times I$ within $\delta$ in the +$C^{[1/\delta]}$-topology of the restriction to this open +submanifold of the standard metric $h_0\times ds^2$. We let $\hat +h=e^{-2f}h$. Fix local coordinates near a point $y\in S^2\times I$. +We denote by $\nabla$ the covariant derivative for $h$ and by +$\widehat\nabla$ the covariant derivative for $\hat h$. + We also denote by +$(R_{ijkl})$ the matrix of the Riemann curvature operator of $h$ +in the associated basis of $\wedge^2T(S^2\times I)$ and by $(\hat +R_{ijkl})$ the matrix of the Riemann curvature operator of $\hat +h$ with respect to the same basis. Recall the formula for the +curvature of a conformal change of metric (see, (3.34) on p.51 of +[Sakai]): + +$$ +\begin{aligned} +\hat R_{ijkl} & = & +e^{-2f}\left(R_{ijkl}-f_jf_kh_{il}+f_jf_lh_{ik}+f_if_kh_{jl}-f_if_lh_{jk}\right. +\\ & & \left.- +(\wedge^2h)_{ijkl}|\nabla +f|^2-f_{jk}h_{il}+f_{ik}h_{jl}+f_{jl}h_{ik}-f_{il}h_{jk}\right).\nonumber +\end{aligned} +$$ + +Here, $f_i$ means $\partial_if$, + +$$ +f_{ij}=\mathit{Hess}_{ij}(f)=\partial_if_j-f_l\Gamma_{ij}^l, +$$ + +and $\wedge^2h$ is the metric induced by $h$ on $\wedge^2TN$, so +that + +$$ +\wedge^2h_{ijkl}=h_{ik}h_{jl}-h_{il}h_{jk}. +$$ + +Now we introduce the notation $O(\delta)$. When we say that a +quantity is $O(\delta)$ we mean that there is some universal +constant $C$ such that, provided that $\delta>0$ is sufficiently +small, the absolute value of the quantity is $\le C\delta$. The +universal constant is allowed to change from inequality to +inequality. + +In our case we take local coordinates adapted to the $\delta$-neck: +$(x^0,x^1,x^2)$ where $x^0$ agrees with the $s$-coordinate and +$(x^1,x^2)$ are Gaussian local coordinates on the $S^2$ such that +$dx^1$ and $dx^2$ are orthonormal at the point in question in the +round metric $h_0$. The function $f$ is a function only of $x^0$. +Hence $f_i=0$ for $i=1,2$. Also, $f_0=\frac{q}{s^2}f$. It follows +that + +$$ +|\nabla f|_h=\frac{q}{s^2}f\cdot(1+O(\delta)), +$$ + + so that + +$$ +|\nabla +f|_h^2=\frac{q^2}{s^4}f^2\cdot (1+O(\delta)). +$$ + +Because the metric $h$ is $\delta$-close to the product $h_0\times +ds^2$, we see that $h_{ij}(y)=(h_0)_{ij}(y)+O(\delta)$ and the +Christoffel symbols $\Gamma_{ij}^k(y)$ of $h$ are within $\delta$ in +the $C^{([1/\delta]-1)}$-topology of those of the product metric +$h_0\times ds^2$. In particular, $\Gamma_{ij}^0=O(\delta)$ for all +$ij$. + The components $f_{ij}$ of the Hessian with +respect to $h$ are given by + +$$ +f_{00}=\left(\frac{q^2}{s^4}-\frac{2q}{s^3}\right)f+\frac{q}{s^2}fO(\delta), +$$ + +$$ +f_{i0}=\frac{q}{s^2}fO(\delta)\ \ \ \mathit{for}\ \ 1\le i\le 2, +$$ + +$$ +f_{ij}=\frac{q}{s^2}fO(\delta)\ \ \ \mathit{for}\ \ 1\le i,j\le 2. +$$ + +In the following $a,b,c,d$ are indices taking values $1$ and $2$. +Substituting in Equation (13.1) yields + +$$ +\begin{aligned} +\hat +R_{0a0b} & = & +e^{-2f}\left(R_{0a0b}+\frac{q^2}{s^4}f^2h_{ab}-h_{ab}(\frac{q^2}{s^4})f^2(1+O(\delta)) ++\left(\frac{q^2}{s^4}-\frac{2q}{s^3}\right)fh_{ab} \right. \\ +& & \left. +\frac{q}{s^2}fO(\delta)\right)\\ +& = & +e^{-2f}\left(R_{0a0b}+\left(\frac{q^2}{s^4}-\frac{2q}{s^3}\right)fh_{ab}+ +h_{ab}(\frac{q^2}{s^4})f^2O(\delta) ++\frac{q}{s^2}fO(\delta)\right) +\end{aligned} +$$ + + Also, we have + +$$ +\begin{aligned} +\hat +R_{ab0c} & = & +e^{-2f}\left(R_{ab0c}-(\wedge^2h)_{ab0c}(\frac{q^2}{s^4})f^2(1+O(\delta)) ++\frac{q}{s^2}fO(\delta)\right) \\ +& = & e^{-2f}\left(R_{ab0c}+(\frac{q^2}{s^4})f^2O(\delta) ++\frac{q}{s^2}fO(\delta)\right). +\end{aligned} +$$ + + Lastly, + +$$ +\begin{aligned} +\hat R_{1212} & = & +e^{-2f}\left(R_{1212}-(\wedge^2h)_{1212}\frac{q^2}{s^4}f^2(1+O(\delta))+\frac{q}{s^2}fO(\delta)\right) +\\ +& = & +e^{-2f}\left(R_{1212}-\frac{q^2}{s^4}f^2(1+O(\delta))+\frac{q}{s^2}fO(\delta)\right). +\end{aligned} +$$ + +Now we are ready to fix the constant $q$. We fix it so that for all $s\in +[0,4+A_0]$ we have + +$$ +q \gg (4+A_0)^2\ \ \ \mathit{and}\ \ \ \frac{q^2}{s^4}e^{-q/s} \ll 1. +$$ + +It follows immediately that $q^2/s^4\gg q/s^3$ for all $s\in [0,4+A_0]$. + We are not yet ready +to fix the constant $C_0$, but once we do we shall always require $\delta$ to +satisfy +$\delta\ll C_0^{-1}$ so that for all $s\in [0,4+A_0]$ we have + +$$ +\frac{q}{s^2}f^2\ll \frac{q^2}{s^4}f^2\ll \frac{q}{s^2}f\ll 1. +$$ + +(These requirements are not circular, since $C_0$ and $q$ are chosen +independent of $\delta$.) + + Using +these inequalities and putting our computations in matrix form show +the following. + +\entryblock{33f705095319} + +Similarly, we have the equation relating scalar curvatures + +$$ +\hat R=e^{2f}\left(R+4\triangle f-2|\nabla f|^2\right), +$$ + +and hence + +$$ +\hat R=e^{2f}\left(R+4\left(\frac{q^2}{s^4}-\frac{2q}{s^3}\right)f-2\frac{q^2}{s^4}f^2+ +\frac{q}{s^2}fO(\delta)\right). +$$ + +\entryblock{7f0b4a692c5e} + +**Proof.** By our choice of $q$, since $C_0\delta<1$, then $f^2\ll f$ and $q^2/s^4\gg \mathit{max}(q/s^3,q/s^2)$ so that the result follows immediately from the above +formula. + +Now let us compute the eigenvalues of the curvature $ R_{ijkl}(y)$ +for any $y\in S^2\times I$. + +\entryblock{52d3cb562bdc} + +**Proof.** Since $h$ is within $\delta$ of $h_0\times ds^2$ in the +$C^{[1/\delta]}$-topology, it follows that the matrix for $h(y)$ in +$\{e_0,e_1,e_2\}$ is within $O(\delta)$ of the identity matrix, and the matrix +for the curvature of $h$ in the associated basis of $\wedge^2T_y(S^2\times I)$ +is within $O(\delta)$ of the curvature matrix for $h_0\times ds^2$, the latter +being the diagonal matrix with diagonal entries $\{1/2,0,0\}$. Thus, the usual +Gram-Schmidt orthonormalization process constructs the basis $\{f_0,f_1,f_2\}$ +satisfying the first two items. Let $A=(A^{ab})$ be the change of basis matrix +expressing the $\{f_a\}$ in terms of the $\{e_b\}$, so that $A=\mathit{Id}+O(\delta)$. The curvature of $h$ in this basis is then given by +$B^\mathit{tr}(R_{ijkl})B$ where $B=\wedge^2A$ is the induced change of basis matrix +expressing the basis $\{f_0\wedge f_1,f_1\wedge f_2,f_2\wedge f_0\}$ in terms +of $\{e_0\wedge e_1,e_1\wedge e_2,e_2\wedge e_0\}$. Hence, in the basis +$\{f_0\wedge f_1,f_1\wedge f_2,,f_2\wedge f_0\}$ the curvature matrix for $h$ +is within $O(\delta)$ of the same diagonal matrix. For $\delta$ sufficiently +small then the eigenvalues of the curvature matrix for $h$ are within +$O(\delta)$ of $(1/2,0,0)$. + +\entryblock{ad2042196c95} + +**Proof.** We have an $h$-orthonormal basis $\{f_0\wedge f_1,f_1\wedge +f_2,f_2\wedge f_0\}$ for $\wedge^2T_y(S^2\times \Ar)$ in which the +quadratic form $(R_{ijkl}(y)$ is + +$$ +\begin{pmatrix} 1/2 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{pmatrix} ++O(\delta). +$$ + + It follows that the restriction to the $h$-unit sphere +in $\wedge^2T_y(S^2\times \Ar)$ of this quadratic form achieves its +maximum value at some vector $v$, which, when written out in this +basis, is given by $(x,y,z)$ with $|y|,|z|\le O(\delta)$ and +$|x-1|\le O(\delta)$. Of course, this maximum value is within +$O(\delta)$ of $1/2$. Clearly, on the $h$-orthogonal subspace to +$v$, the quadratic form is given by a matrix all of whose entries +are $O(\delta)$ in absolute value. This gives us a new basis of +$\wedge^2T_y(S^2\times I)$ within $O(\delta)$ of the given basis in +which $(R_{ijkl}(y))$ is diagonal. The corresponding basis for +$T_y(S^2\times\Ar)$ is as required. + +Now we consider the expression $(\hat R_{ijkl}(y))$ in this basis. + +\entryblock{0a41b5c61dd2} + +**Proof.** We simply conjugate the expression in Equation (13.3) by the change of +basis matrix and use the fact that by our choice of $q$ and the fact that +$C_0\delta<1$, we have $f\gg f^2$ and $q/s^3\ll q^2/s^4$. + +\entryblock{347896f0385b} + +**Proof.** To diagonalize $(R_{ijkl}(y))$ we need only rotate in the +$\{f_1\wedge f_2,f_2\wedge f_3\}$-plane. Applying this rotation to +the expression in \entryref{52d3cb562bdc} gives the result. + +\entryblock{41a5f172b2dc} + +**Proof.** Let $\{f_0,f_1,f_2\}$ be the $h$-orthonormal basis given in +\entryref{347896f0385b}. Then $\{e^ff_0,e^ff_1,e^ff_2\}$ is orthonormal for $\hat +h=e^{-2f}h$. This change multiplies the curvature matrix by $e^{4f}$. Since +$f\ll 1$, $e^{4f}<2$ so that the expression for $(\hat R_{ijkl}(y))$ in this +basis is exactly the same as in \entryref{0a41b5c61dd2} except that the factor +in front is $e^{2f}$ instead of $e^{-2f}$. Now, it is easy to see that since +$((q^2/s^4)fA\delta)^2\ll (q^2/s^4)fA\delta$ the eigenvalues will differ from +the diagonal entries by at most a constant multiple of $(q^2/s^4)fA\delta$. + + The first three inequalities are immediate +from the previous corollary. The last two follow since $q^2/s^4\gg q/s^3$ and +$\delta\ll 1$. + +One important consequence of this computation is the following: + +\entryblock{e70bacf8aab0} + +**Proof.** Since $|\lambda-1/2|, |\mu|, |\nu|$ are all $O(\delta)$ and since +$\frac{q^2}{s^4}f\ll 1$, it follows that the smallest eigenvalue of $(\hat +R_{ijkl}(y))$ is either $\mu'$ or $\nu'$. But it is immediate from the above +expressions that $\mu'>\mu$ and $\nu'>\nu$. This completes the proof. + +Now we are ready to fix $C_0$. +There is a universal constant $K$ such that for all $\delta>0$ sufficiently small +and for any $\delta$-neck $(N,h)$ of +scale one, every eigenvalue of $\mathit{Rm}_h$ is at least $-K\delta$. +We set + +$$ +C_0=2Ke^{q}. +$$ + +\entryblock{cc9110b4d2fe} + +**Proof.** Then by the previous result we have + +$$ +\nu'\ge e^{2f}\left(\nu+\frac{q^2}{2s^4}f\right). +$$ + +It is easy to see that since $q\gg (4+A_0)$ the function $(q^2/2s^4)f$ is an +increasing function on $[1,4+A_0]$. Its value at $s=1$ is +$(q^2/2)e^{-q}C_0\delta>K\delta$. Hence $\nu+\frac{q^2}{2s^4}f>0$ for all $s\in +[1,4+A_0]$ and consequently $\nu'>0$ on this submanifold. The same argument +shows $\mu'>0$. Since $q^2/s^4 f^2\ll 1$ and $02X_{\hat p}(p)(\mathit{log}(X_{\hat h}(p)+\mathit{log}(1+t)-3). +$$ + +This completes the proof in both cases. + +This establishes the first item in the conclusion of +\entryref{cf61b731c6f2} for $\delta>0$ sufficiently small on +$s^{-1}(-\delta^{-1},4)$. As we have seen in \entryref{cc9110b4d2fe}, +the curvature is positive on $s^{-1}[1,4)$. + +### Proof of the first two items for $s\ge 4$ + +Now let us show that the curvature on $\tilde g$ is positive in the region +$s^{-1}([4,4+A_0])$. First of all in the preimage of the +interval $[4,4+A_0,-r_0]$ this follows +from \entryref{e70bacf8aab0} and the fact that $\eta g_0$ has non-negative +curvature. As for the region $s^{-1}([4+A_0-r_0,4+A_0])$, as $\delta$ tends to +zero, the metric here tends smoothly to the restriction of the metric $g_0$ to +that subset. The metric $g_0$ has positive curvature on +$s^{-1}([4+A_0-r_0,4+A_0])$. Thus, for all $\delta>0$ sufficiently small the +metric $\tilde g$ has positive curvature on all of $s^{-1}([4+A_0-r_0,4+A_0])$. +This completes the proof of the first two items. + +### Proof of the third item + + By +construction the restriction of the metric $\widetilde g$ to +$s^{-1}((-\delta^{-1},0])$ is equal to the metric $\rho^*g$. Hence, +in this region the mapping is an isometry. In the region +$s^{-1}([0,4])$ we have $R(x_0)\rho^*g\ge \eta g_0$ so that by +construction in this region $\rho^*g\ge \tilde g$. Lastly, in the +region $s^{-1}([4,A_0+4])$ we have $R(x_0)^{-1}\eta g_0\ge \tilde +g$. On the other hand, it follows from \entryref{6168ffcb3bed} that +the map from $([0,\delta^{-1}]\times S^2,R(x_0)\rho^* g)$ to +$(B(p_0,4+A_0),\eta g)$ is distance decreasing. This completes the +proof of the third item. + +### Completion of the proof + +As $\delta$ goes to zero, $f$ tends to zero in the +$C^\infty$-topology and $\eta$ limits to $1$. From this the fourth +item is clear. + +This completes the proof of \entryref{cf61b731c6f2}. + +## Other properties of the result of surgery + +\entryblock{d3d4ce2e821e} + +**Proof.** The Riemannian manifold $({\mathcal S},\tilde g)$ is identified by a +diffeomorphism with the union of $s_{N}^{-1}(-\delta^{-1},0]$ to +$B_{g_0}(p_0,A_0+4)$ glued along their boundaries. Thus, we have a natural +identification of ${\mathcal S}$ with the ball $B_{g_0}(p,A_0+4+\delta^{-1})$ +in the standard solution. This identification pulls back the metric $\tilde g$ +to be within $2\delta$ of the standard initial metric. The result then follows +immediately for $\delta$ sufficiently small. diff --git a/projects/poincare-conjecture/.astrolabe/docs/15-ricci-flow-with-surgery-the-definition.mdx b/projects/poincare-conjecture/.astrolabe/docs/15-ricci-flow-with-surgery-the-definition.mdx new file mode 100644 index 00000000..e5f57e1f --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/docs/15-ricci-flow-with-surgery-the-definition.mdx @@ -0,0 +1,329 @@ + + +# Chapter 14 -- Ricci Flow with surgery: the definition + +In this chapter we introduce Ricci flows with surgery. These objects +are closely related to generalized Ricci flows but they differ +slightly. The space-time of a Ricci flow with surgery has an open +dense subset that is a manifold, and the restriction of the Ricci +flow with surgery to this open subset is a generalized Ricci flow. +Still there are other, more singular points allowed in a Ricci flow +with surgery. + +## Surgery space-time + +\entryblock{41da6871e864} + +We are interested in a certain class of space-times, which we call +*surgery space-times*. These objects have a 'smooth structure' +(even though they are not smooth manifolds). As in the case of a +smooth manifold, this smooth structure is given by local coordinate +charts with appropriate overlap functions. + +### An exotic chart + +There is one exotic chart, and we begin with its description. To +define this chart we consider the open unit square $(-1,1)\times +(-1,1)$. We shall define a new topology, denoted by ${\mathcal P}$, +on this square. The open subsets of ${\mathcal P}$ are the open +subsets of the usual topology on the open square together with open +subsets of $(0,1)\times [0,1)$. Of course, with this topology the +'identity' map $\iota\colon {\mathcal P}\to (-1,1)\times (-1,1)$ is +a continuous map. Notice that the restriction of the topology of +${\mathcal P}$ to the complement of the closed subset $[0,1)\times +\{0\}$ is a homeomorphism onto the corresponding subset of the open +unit square. Notice that the complement of $(0,0)$ in ${\mathcal P}$ +is a manifold with boundary, the boundary being $(0,1)\times\{0\}$. +(See Fig. 0.5 in the Introduction.) + +Next, we define a 'smooth structure' on ${\mathcal P}$ by defining a +sheaf of germs of 'smooth' functions. The restriction of this sheaf +of germs of 'smooth functions' to the complement of $(0,1)\times +\{0\}$ in ${\mathcal P}$ is the usual sheaf of germs of smooth +functions on the corresponding subset of the open unit square. +In particular, a function is smooth near $(0,0)$ if and only if its restriction +to some neighborhood of $(0,0)$ is the pullback under $\iota$ +of a usual smooth function on a +neighborhood of the origin in the square. + Now +let us consider the situation near a point of the form $x=(a,0)$ for +some $00$ and an embedding $i_U\colon +U\times[t,t+\epsilon)\subset {\mathcal M}$ compatible with time and +the vector field. Of course, two such embeddings agree on their +common domain of definition. Notice also that for each $t'\in +[t,t+\epsilon)$ the restriction of the map $i_U$ to $U\times \{t'\}$ +induces an diffeomorphism from $U$ to an open subset $U_{t'}$ of +$M_{t'}$. It follows that the local flow generated by the vector +field $\chi$ preserves the horizontal subbundle. Hence, the vector +field $\chi$ acts by Lie derivative on the sections of ${\mathcal +HT}({\mathcal M})$ and on all associated bundles (for example the +symmetric square of the dual bundle). + +### The equation + +\entryblock{842f5a7c856b} + +\entryblock{21f9e3c12b4c} + +### Examples of Ricci flows with surgery + +\entryblock{7e213cc8aad9} + +The next lemma gives an example of a Ricci flow with surgery where +the topology of the time-slices changes. + +\entryblock{b186908cc3b4} + +**Proof.** As the union of Hausdorff spaces along closed subsets, ${\mathcal +M}$ is a Hausdorff topological space. The time function is the one +induced from the projections onto the second factor. For any point +outside the $b$ time-slice there is the usual smooth coordinate +coming from the smooth manifold $M_1\times (a,b)$ (if $tb$). At any point of +$(M_2\setminus\Omega_2)\times \{b\}$ there is the smooth manifold +with boundary coordinate patch coming from $M_2\times[b,c)$. For any +point in $\mathit{int}(\Omega_1)\times\{b\}$ we have the smooth +manifold structure obtained from gluing $(\mathit{int}(\Omega_1))\times +(a,b]$ to $\mathit{int}(\Omega_2)\times [b,c)$ along the $b$ time-slice +by $\psi$. Thus, at all these points we have neighborhoods on which +our data determine a smooth manifold structure. Lastly, let us +consider a point $x\in +\partial \Omega_1\times \{b\}$. +Choose local coordinates $(x^1,\ldots,x^n)$ for a neighborhood $V_1$ +of $x$ such that $\Omega_1\cap V_1=\{x^n\le 0\}$. We can assume that +$\psi$ is defined on all of $V_1$. Let $V_2=\psi(V_1)$ and take the + local coordinates on $V_2$ induced from the $x^i$ on $V_1$. + Were we to identify $V_1\times (a,b]$ +with $V_2\times [b,c)$ along the $b$ time-slice using this map, then +this union would be a smooth manifold. There is a +neighborhood of the point $(x,b)\in {\mathcal M}$ which is obtained +from the smooth manifold $V_1\times (a,b]\cup_\psi V_2\times [b,c)$ +by inducing a new topology where the open subsets are, in addition +to the usual ones, any open subset of the form $\{x^n>0\}\times +[b,b')$ where $b + +# Chapter 15 -- Controlled Ricci flows with surgery + + We do not +wish to consider all Ricci flows with surgery. Rather we shall +concentrate on $3$-dimensional flows (that is to say $4$-dimensional +space-times) whose singularities are closely controlled both +topologically and geometrically. We introduce the hypotheses that we +require these evolutions to satisfy. Then main result, which is +stated in this chapter and proved in the next two, is that these +controlled $3$-dimensional Ricci flows with surgery always exist for +all time with any compact $3$-manifold as initial metric. + +### Normalized initial conditions + +Consider a compact connected Riemannian $3$-manifold $(M,g(0))$ +satisfying + +- **(1)** $|\mathit{Rm}(x,0)|\le 1$ for all $x\in M$ and +- **(2)** for every $x\in M$ we have $\mathit{Vol} B(x,0,1)\ge \omega/2$ where +$\omega$ is the volume of the unit ball in $\Ar^3$. + +Under these conditions we say that $(M,g(0))$ is *normalized*. +Also, if $(M,g(0))$ is the initial manifold of a Ricci flow with +surgery then we say that it is a *normalized initial metric*. Of +course, given any compact Riemannian $3$-manifold $(M,g(0))$ there +is a positive constant $Q<\infty$ such that $(M,Qg(0))$ is +normalized. + +Starting with a normalized initial metric implies that the flow exists +and has uniformly bounded curvature for a fixed amount of time. This is the +content of the following claim which is an immediate corollary of +\entryref{e6b1bd9fa0d7}, \entryref{b6bc786f7f44}, \entryref{ed4b72caf9f9}, +and \entryref{c1330c5a8be3}. + +\entryblock{f5e180ecfd18} + +## Gluing together evolving necks + +\entryblock{a992d6a89479} + +**Proof.** Suppose that the result does not hold. Take a sequence of $\beta_n$ +tending to zero and counterexamples $(N_n\times +[-t_{0,n},0],g_{1,n}(t));\ (N'_n\times +(-t_{1,n},-t_{0,n}],g_{2,n}(t))$. Pass to a subsequence so that the +$t_{0,n}$ tend to a limit $t_{0,\infty}\ge 0$. Since $\beta_n$ tends +to zero, we can take a smooth limit of a subsequence and this limit +is an evolving cylinder $(S^2\times \Ar, h(t)\times ds^2)$, where +$h(t)$ is the round metric of scalar curvature $1/(1-t)$ defined for +some amount of backward time. Notice that, for all $\beta$ +sufficiently small, on a $\beta\epsilon$-neck the derivative of the +scalar curvature is positive. Thus, $R_{g_{1,n}}(x,-t_{0,n})<1$. +Since we have a strong neck structure on $N'_n$ centered at +$(x,-t_{0,n})$, this implies that $t_{1,n}>1$ so that the limit is +defined for at least time $t\in [0, 1+t_{0,\infty})$. If +$t_{0,\infty}>0$, then, restricting to the appropriate subset of +this limit, a subset with compact closure in space-time, it follows +immediately that for all $n$ sufficiently large there is a strong +$\epsilon$-neck centered at $(x,0)$. This contradicts the assumption +that we began with a sequence of counterexamples to the +proposition. + +Let us consider the case when $t_{0,\infty}=0$. In this case the +smooth limit is an evolving round cylinder defined for time +$(-1,0]$. + Since $t_{1,n}>1$ we +see that for any $A<\infty$ for all $n$ sufficiently large the ball +$B(x_n,0,A)$ has compact closure in every time-slice and there are +uniform bounds to the curvature on $B(x_n,0,A)\times (-1,0]$. This +means that the limit is uniform for time $(-1,0]$ on all these +balls. Thus, once again for all $n$ sufficiently large we see that +$(x,0)$ is the center of a strong $\epsilon$-neck in the union. In +either case we have obtained a contradiction, and hence we have +proved the result. See Fig. 15.1. + +### First assumptions + +**Choice of $C$ and $\epsilon$:** The first thing we need +to do is fix for the rest of the argument $C<\infty$ and +$\epsilon>0$. We do this in the following way. We fix $0<\epsilon\le +\mathit{min}(1/200,\left(\sqrt{D}(A_0+5)\right)^{-1},\bar\epsilon_1/2,\bar +\epsilon'/2,\epsilon_0)$ where $\bar\epsilon_1$ is the constant from +\entryref{cb9ca6d6db9f}, $\bar\epsilon'$ is the constant from +\entryref{c2be59f6090c}, $\epsilon_0$ is the constant from Section 10.1, +and $A_0$ and $D$ are the constants from +\entryref{0b07beec84c0}. + We fix $\beta<1/2$, the constant from +\entryref{a992d6a89479}. Then we let $C$ be the maximum of the +constant $C(\epsilon)$ as in \entryref{fc2d2036a96c} and +$C'(\beta\epsilon/3)+1$ as in \entryref{6ff28dba4e56}. + +For all such $\epsilon$, \entryref{ecd62fc2ea2b} holds for $\epsilon$ and +\entryref{cb9ca6d6db9f}, \entryref{7c2758f0e2b3} and +Corollaries \entryref{fc2d2036a96c} and \entryref{e724298e5bc3} and Theorems \entryref{d51aab8a8065} and \entryref{a9e023d082dc} + hold for $2\epsilon$. Also, all the topological results of the Appendix hold +for $2\epsilon$ and $\alpha=10^{-2}$. + +Now let us turn to the assumptions we shall make on the Ricci flows +with surgery that we shall consider. Let ${\mathcal M}$ be a +space-time. Our first set of assumptions are basically topological +in nature. They are: + +i̱gskip + + **Assumption (1). Compactness and dimension:** *Each +time-slice $M_t$ of space-time is a compact $3$-manifold +containing no embedded $\Ar P^2$ with trivial normal bundle.* + +i̱gskip + +**Assumption (2). Discrete singularities:** *The set of +singular times is a discrete subset of $\Ar$.* + +i̱gskip + + **Assumption (3). Normalized initial conditions:** *$0$ is the initial time of the Ricci flow with surgery and the +initial metric $(M_0,G(0))$ is normalized.* + +i̱gskip + + It follows +from Assumption (2) that for any time $t$ in the time-interval of +definition of a Ricci flow with surgery, with $t$ being distinct +from the initial and final times (if these exist), for all +$\delta>0$ sufficiently small, the only possible singular time in +$[t-\delta,t+\delta]$ is $t$. Suppose that $t$ is a singular time. +The singular locus at time $t$ is a closed, smooth subsurface +$\Sigma_{t}\subset M_{t}$. From the local +model, near every point of $x\in \Sigma_{t}$ we see that this +surface separates $M_{t}$ into two pieces: + +$$ +M_{t}=C_{t}\cup_{\Sigma_{t}}E_{t}, +$$ + +where $E_{t}$ is the exposed region at time $t$ and $C_{t}$ is the +complement of the interior of $E_{t}$ in $M_{t}$. We call $C_t$ the +*continuing region*. $C_t\subset M_t$ is the maximal subset of +$M_t$ for which there is $\delta>0$ and an embedding $C_t\times +(t-\delta,t]\to {\mathcal M}$ compatible with time and the vector +field. + + **Assumption (4). Topology of the exposed regions:** *At all singular times $t$ we require that $E_{t}$ be a finite +disjoint union of $3$-balls. In particular, $\Sigma_{t}$ is a +finite disjoint union of $2$-spheres.* + +i̱gskip + +The next assumptions are geometric in nature. +Suppose that $t$ is a surgery time. Let ${\mathcal M}_{(-\infty,t)}$ be $\mathbf{t}^{-1}((-\infty,t))$ +and let $(\widehat {\mathcal M}_{(-\infty,t)},\widehat G)$ be the maximal extension of +$({\mathcal M}_{(-\infty,t)},G)$ to time $t$, as given in \entryref{8651cf20bd2d}. + +**Assumption (5). Boundary components of the exposed +regions:** *There is a surgery control parameter +function, +$\delta(t)>0$, a non-increasing function of +$t$, such that each component of $\Sigma_{t}\subset M_{t}$ is the +central $2$-sphere of a strong $\delta(t)$-neck in $(\widehat{\mathcal +M}_{(-\infty,t)},\widehat G)$.* + +i̱gskip + +Suppose that $t$ is a singular time. Then for all $t^-0$ also decays to zero rapidly as time goes to infinity. +Here then are the functions that will play the crucial role in defining +the surgery process. + +\entryblock{91876608df01} + +We shall consider Ricci flows with surgery $({\mathcal M},G)$ that +satisfy Assumptions (1) -- (7) and also satisfy: + + **For any singular time $t$ the surgery at time $t$ is +performed with control $\overline\delta(t)$ and at scale +$h(t)=h(\rho(t),\overline\delta(t))$, where +$\rho(t)=\overline\delta(t)r(t)$, in the sense that each boundary +component of $C_t$ is the central $2$-sphere of a strong +$\overline\delta(t)$-neck centered at a point $y$ with +$R(y)=h(t)^{-2}$.** + +There is quite a bit of freedom in the choice of these parameters. +But it is not complete freedom. They must decay rapidly enough as +functions of $t$. We choose to make $r(t)$ and $\kappa(t)$ step +functions, and we require $\overline\delta(t)$ to be bounded above +by a step function of $t$. Let us fix the step sizes. + +\entryblock{1f2e2e449b34} + +The steps we consider are $[0,T_0]$ and then $[T_i,T_{i+1}]$ for +every $i\ge 0$. The first step is somewhat special. Suppose that +$({\mathcal M},G)$ is a Ricci flow with surgery with normalized +initial conditions. Then + according to +\entryref{f5e180ecfd18} the flow exists on $[0,T_1]$ and the norm +of the Riemann curvature is bounded by $2$ on $[0,T_1]$, so that +by Assumption (7) there are no surgeries in this time interval. Also, +by \entryref{f5e180ecfd18} there is a $\kappa_0>0$ so +that $\mathit{Vol} B(x,t,r)\le \kappa_0r^3$ for every $t\le T_1$ and +$x\in M_t$ and every $r\le \epsilon$. + +\entryblock{22c40a5664ac} + +We let $r(t)$ be the step function whose value on $[T_i,T_{i+1})$ is $r_{i+1}$ +and whose value on $[0,T_0)$ is $r_0$. We say that a Ricci flow with surgery +satisfies the strong $(C,\epsilon)$-canonical neighborhood assumption with parameter $\mathbf{r}$ if it satisfies this condition with respect to the step function $r(t)$ +associated with $\mathbf{r}$. This means that any $x\in {\mathcal M}$ with +$R(x)\ge r^{-2}(\mathbf{t}(x))$ has a strong $(C,\epsilon)$-canonical neighborhood in +${\mathcal M}$. Let $\kappa(t)$ be the step function whose value on +$[T_i,T_{i+1})$ is $\kappa_{i+1}$ and whose value on $[0,T_0)$ is $\kappa_0$. +Given $\kappa>0$, we say that a Ricci flow defined on $[0,t]$ is $\kappa$-non-collapsed +on scales $\le \epsilon$ provided that for every point $x$ not contained in a +component of its time-slice with positive sectional curvature, +if for some $r\le \epsilon$, the parabolic neighborhood $P(x,\mathbf{t}(x),r,-r^2)$ exists in ${\mathcal M}$ +and the norm of the Riemann curvature is bounded on this backward +parabolic neighborhood by $r^{-2}$, then $\mathit{Vol} B(x,\mathbf{t}(x),r)\ge +\kappa r^3$. +We say that a Ricci flow with surgery is $\mathbf{K}$-non-collapsed on scales +$\epsilon$ if for every $t\in [0,\infty)$ the restriction of the flow to +$[0,t]$ is $\kappa(t)$-non-collapsed on scales $\le \epsilon$. + Lastly, we fix a non-increasing function $\overline\delta(t)>0$ +with $\overline\delta(t)\le \delta_{i+1}$ if $t\in [T_i,T_{i+1})$ for all $i\ge +0$ and $\overline\delta(t)\le \delta_0$ for $t\in [0,T_0)$. We denote the fact +that such inequalities hold for all $t$ by saying $\overline\delta(t)\le +\Delta$. + +Having fixed surgery parameter sequences $\mathbf{K}$, $\mathbf{r}$ and +$\Delta$, defined step functions $r(t)$ and $\kappa(t)$, and fixed +$\overline\delta(t)\le \Delta$ as above, we shall consider only +Ricci flows with surgery where the surgery at time $t$ is defined +using the surgery parameter functions $r(t)$ and +$\overline\delta(t)$. In addition, we require that these Ricci +flows with surgery satisfy Assumptions (1) -- (7). + +What we shall show is that there are surgery parameter +sequences $\mathbf{r}$, $\mathbf{K}$ and $\Delta$ with the property that +for any normalized initial metric and any positive, +non-increasing function $\overline\delta(t)\le \Delta$, it is +possible to construct a Ricci flow with surgery using the surgery +parameters $r(t)$ and $\overline\delta(t)$ with the given initial +conditions and furthermore that this Ricci flow with surgery +satisfies the Assumptions (1) -- (7), has curvature pinched toward +positive, satisfies the canonical neighborhood assumption, and satisfies the +non-collapsing assumption using these parameters. + +In fact we shall prove this inductively, constructing the step +functions inductively one step at a time. Thus, given surgery +parameter sequences indexed by $0,\ldots,i$ we show that there are appropriate choices of +$r_{i+1}, \kappa_{i+1}$ and $\delta_{i+1}$ such that the following is true. +Given a Ricci flow with +surgery defined on time $[0,T_i)$ satisfying all the properties with +respect to the first set of data, that Ricci +flow with surgery extends to one defined for time $[0,T_{i+1})$ and satisfies +Assumptions (1) -- (7), the canonical neighborhood assumption and the +non-collapsing assumption with respect to the extended surgery +parameter sequences, and has curvature pinched toward positive. +As stated this is not quite true; there is a slight twist: we must also assume +that $\overline\delta(t)\le \delta_{i+1}$ for all $t\in [T_{i-1},T_{i+1})$. +It is for this reason that we consider +pairs consisting of sequences $\Delta$ and a surgery control +parameter $\overline\delta(t)$ bounded above by $\Delta$. + +## The process of surgery + +Suppose given surgery parameter sequences $\{r_0,\ldots,r_{i+1}\}$, +$\{\kappa_0,\ldots,\kappa_{i+1}\}$ and +$\Delta_i=\{\delta_0,\ldots,\delta_i\}$ and also given a positive, +decreasing function $\overline\delta(t)\le \Delta_i$, defined for +$t\le T_{i+1}$ with $\delta_0=\mathit{min}(\alpha\epsilon/3,\delta_0',K^{-1},D^{-1})$ as above. Suppose +that $({\mathcal M},G)$ is a Ricci flow with surgery defined for +$t\in [0,T)$ that goes singular at time $T\in (T_i,T_{i+1}]$. We +suppose that it satisfies Assumptions (1) -- (7). Since the flow has +normalized initial conditions and goes singular at time $T$, it +follows that $i\ge 1$. We suppose that $({\mathcal M},G)$ satisfies +the $(C,\epsilon)$-canonical neighborhood assumption with parameter +$r_{i+1}$ and that its curvature is pinched toward positive. By +\entryref{af00774dbaa9} we know that there is a maximal extension +$(\widehat{\mathcal M},\widehat G)$ of this generalized flow to time +$T$ with the $T$ time-slice being $(\Omega(T),G(T))$. Set +$\rho=\overline\delta(T)r_{i+1}$, and set +$h(T)=h(\rho(T),\overline\delta(T))$ as in \entryref{074e02b067b8}. +Since $\overline\delta(T)\le \delta_0<1$, we see that +$\rho\rho^{-2}>r_i^{-2}. +$$ + +Since $M_{t'}\setminus \psi(\mathit{int} C_T\times \{t'\})$ is compact +for every $t'$, there is $T_1r_i^{-2}$ for +all $p\in M_{t'}\setminus \psi(\mathit{int}C_T\times \{t'\})$ and all +$t\in [T_1,T)$. Hence, by our assumptions all these points have +strong $(C,\epsilon)$-canonical neighborhoods. This establishes that +Assumption (6) holds at the singular time $T$. By hypothesis +Assumption (6) holds at all earlier singular times. Clearly, from the +construction the Ricci flow on $[T,T')$ is maximal. Hence, +Assumption (7) holds for the new Ricci flow with surgery. + + From \entryref{cf61b731c6f2} the +fact that $\delta(T)\le \delta_i\le \delta_0\le\delta'_0$ and +$h(T)\le R_0^{-1/2}$ imply that the Riemannian manifold $(M_T,G(T))$ +has curvature pinched toward positive for time $T$. It then follows +from \entryref{fc44d8653f35} that the Ricci flow defined on $[T,T')$ +with $(M_T,G(T))$ as initial conditions has curvature pinched toward +positive. The inductive hypothesis is that on the time-interval +$[0,T)$ the Ricci flow with surgery has curvature pinched toward +positive. This completes the proof of the lemma. + +\entryblock{9021a4eca03f} + +**Proof.** This is immediate from the third item in \entryref{cf61b731c6f2}. + +\entryblock{289ab251dd83} + +The other two inductive properties in \entryref{703e4cfeece7} -- that the result is +$\mathbf{K}$-non-collapsed and also that it satisfies the strong +$(C,\epsilon)$-canonical neighborhood assumption with parameter $\mathbf{r}$ - +require appropriate inductive choices of the sequences. The arguments +establishing these are quite delicate and intricate. They are given in the next +two sections. + +## Outline of the proof of Theorem \entryref{703e4cfeece7} + +Before giving the proof proper of \entryref{703e4cfeece7} let us outline +how the argument goes. We shall construct the surgery parameter +sequences $\Delta$, $\mathbf{r}$, and $\mathbf{K}$ inductively. Because of +\entryref{c1330c5a8be3} we have the beginning of the inductive +process. We suppose that we have defined sequences as required up to +index $i$ for some $i\ge 1$. Then we shall extend them one more step +to sequences defined up to $(i+1)$, though there is a twist: to do +this we must redefine $\delta_i$ in order to make sure that the +extension is possible. In Chapter 16 we establish the +non-collapsing result assuming the strong canonical neighborhood +result. More precisely, suppose that we have a Ricci flow with +surgery $({\mathcal M},G)$ defined for time $0\le t0$. +Then there is $\delta(r_{i+1})>0$ and $\kappa_{i+1}>0$ such that, +provided that $\overline\delta(t)\le \delta(r_{i+1})$ for all $t\in +[T_{i-1},T)$, the Ricci flow with surgery $({\mathcal M},G)$ is +$\kappa_{i+1}$ non-collapsed on scales $\le \epsilon$. + +In Section 17.1 we show that the strong $(C,\epsilon)$-canonical neighborhood +assumption extends for some parameter $r_{i+1}$, assuming again that +$\overline\delta(t)\le \delta(r_{i+1})$ for all $t\in [T_{i-1},T)$. + +Lastly, in Section 17.2 we complete the proof by +showing that the number of surgeries possible in $[0,T_{i+1})$ is +bounded in terms of the initial conditions and $\overline\delta(T)$. +The argument for this is a simple volume comparison argument -- +under Ricci flow with normalized initial conditions, the volume +grows at most at a fixed exponential rate and under each surgery an +amount of volume, bounded below by a positive constant depending +only on $\overline \delta(T_{i+1})$, is removed. diff --git a/projects/poincare-conjecture/.astrolabe/docs/17-proof-of-the-non-collapsing.mdx b/projects/poincare-conjecture/.astrolabe/docs/17-proof-of-the-non-collapsing.mdx new file mode 100644 index 00000000..89c08b34 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/docs/17-proof-of-the-non-collapsing.mdx @@ -0,0 +1,1417 @@ + + +# Chapter 16 -- Proof of the non-collapsing + +The precise statement of the non-collapsing result is given in the +next section. Essentially, the proof of non-collapsing in the +context of Ricci flow with surgery is the same as the proof in the +case of ordinary Ricci flows. Given a point $x\in {\mathcal M}$, one +finds a parabolic neighborhood whose size, $r'$, is determined by +the constants $r_i$, $C$ and $\epsilon$, contained in $\mathbf{t}^{-1}([T_{i-1},T_i))$ and on which the curvature is bounded by +$(r')^{-2}$. Hence, by the inductive hypothesis, the final +time-slice of this neighborhood is $\kappa_i$-non-collapsed. +Furthermore, we can choose this neighborhood so that the reduced +${\mathcal L}$-length of its central point from $x$ is bounded by +$3/2$. This allows us to produce an open subset at an earlier time +whose reduced volume is bounded away from zero. Then using +\entryref{3defa9b2209c} we transfer this conclusion to a non-collapsing +conclusion for $x$. The main issue in this argument is to show that +there is a point in each earlier time-slice whose reduced length +from $x$ is at most $3/2$. We can argue as in the case of a Ricci +flow if we can show that any curve parameterized by backward time +starting at $x$ (a point where the hypothesis of +$\kappa$-non-collapsing holds) that comes close to a surgery cap +either from above or below must have large ${\mathcal L}$-length. In +establishing the relevant estimates we are forced to require that +$\delta_i$ be sufficiently small. + +## The statement of the non-collapsing result + +Here, we shall assume that after surgery the strong canonical +neighborhood assumption holds, and we shall establish the +non-collapsing result. + +\entryblock{4ce8b5da9c58} + +\entryblock{d93b1988d45d} + +## The proof of non-collapsing when $R(x)=r^{-2}$ with $r\le r_{i+1}$ + +Let us begin with an easy case of the non-collapsing result, where +non-collapsing follows easily from the strong canonical neighborhood +assumption, rather than from using ${\mathcal L}$-length and +monotonicity along ${\mathcal L}$-geodesics. We suppose that we have +a Ricci flow with surgery $({\mathcal M},G)$ defined for $0\le t0$ be such that the +supremum of $|\mathit{Rm}|$ on $B(x,t,r')$ is $(r')^{-2}$. +Then, by the +definition + of a $(C,\epsilon)$-cap, +$\mathit{vol} B(x,t,r')\ge C^{-1}(r')^3$. +Clearly, $r'\le r$ and there is a point $y\in \overline{B(x,t,r')}$ +with $R(y)=(r')^{-2}$. On the other hand, by the +definition of a $(C,\epsilon)$-cap, we have $R(y)/R(x)\le C$, so that $r'/r\ge C^{-1/2}$. +Thus, the +volume of $B(x,t,r)$ is at least $C^{-5/2}r^3$. + +This completes an examination of all cases and establishes the +proposition. + +## Minimizing ${\mathcal L}$-geodesics exist when $R(x)\le r^{-2}_{i+1}$: the statement + +The proof of the non-collapsing result when $R(x)=r^{-2}$ with +$r_{i+1}0$ sufficiently small and also +independent of the strong $\delta'$-neck $N$. Gluing in the cap with +a $C^\infty$-metric that converges smoothly to the standard initial +metric $g_0$ as $\delta'$ tends to zero using a fixed +$C^\infty$-partition of unity produces a family of manifolds +uniformly bounded in the $C^\infty$-topology. + +This leads immediately to: + +\entryblock{6cc16821de1e} + +**Proof.** That there is a smooth limit of a subsequence is immediate from the +previous claim. Since the $\delta_n$ tend to zero, it is clear that the +limiting metric is the standard initial metric. + +\entryblock{4b6ee3b7a329} + +**Proof.** Let $Q<\infty$ be an upper bound for the scalar curvature of the +standard flow on the time interval $[0,\theta)$. Since +$\delta'_n\rightarrow 0$, according to the previous corollary, there is +a smooth limit at time $0$ for a subsequence, and this limit is the +standard initial metric. Suppose that, for some $0\le +\theta'<\theta$, we have established that there is a smooth limiting +flow on $[0,\theta']$. Since the initial conditions are the standard +solution, it follows from the uniqueness statement in +\entryref{f44d182ef27e} that in fact the limiting flow is +isomorphic to the restriction of the standard flow to this time +interval. Then the scalar curvature of the limiting flow is bounded +by $Q$. Hence, for any $A<\infty$, for all $n$ sufficiently large, +the scalar curvature of the restriction of $G'_n$ to the image of +$B_{G'_n}(p_n,0,2A)\times[0,\theta']$ is bounded by $2Q$. According +to \entryref{0ddfe578961e} there is an $\eta>0$ and a constant +$Q'<\infty$, each depending only on $Q$, $r_{i+1}$, $C$ and +$\epsilon$, such that for all $n$ sufficiently large, the scalar +curvature of the restriction of $G_n'$ to $B_{G_n'}(p_n,0,A)\times +[0,\mathit{min}(\theta'+\eta,\theta_n))$ is bounded by $Q'$. Because of +the fact that the curvature is pinched toward positive, this implies +that on the same set the sectional curvatures are uniformly bounded. +Hence, by Shi's theorem with derivatives +(\entryref{e0b6a1f646a7}), it follows that there are uniform bounds +for the curvature in the $C^\infty$-topology. Thus, passing to a +subsequence we can extend the smooth limit to the time interval +$[0,\theta'+\eta/2]$ unless $\theta'+\eta/2\ge \theta$. Since $\eta$ +depends on $\theta$ (through $Q$), but is independent of $\theta'$, +we can repeat this process extending the time-interval of definition +of the limiting flow by $\eta/2$ until $\theta'+\eta/2\ge \theta$. +Now suppose that $\theta'+\eta/2\ge \theta$. Then the argument shows +that by passing to a subsequence we can extend the limit to any +compact subinterval of $[0,\theta)$. Taking a diagonal sequence +allows us to extend it to all of $[0,\theta)$. By the uniqueness of +the standard flow, this limit is the standard flow. + +\entryblock{712124d7ea02} + +**Proof.** Let $\eta>0$ depending on $\theta$ (though $Q$) as well as +$r_{i+1}$, $C$ and $\epsilon$ be as in the proof of the previous +lemma, and take $0<\eta'<\eta$. For all $n$ sufficiently large +$\theta_n>\theta-\eta'$, and consequently for all $n$ sufficiently +large there is an embedding $B_{G_n}(p_n,t_n,Ah_n)\times +[t_n,t_n+h_n^2(\theta-\eta')]$ into ${\mathcal M}_n$ compatible with +time and with the vector field. For all $n$ sufficiently large, we +consider the restriction of $G_n'$ to $B_{G_n'}(p_n,0,A)\times +[0,\theta-\eta']$. These converge smoothly to the restriction of the +standard flow to the ball of radius $A$ on the time interval +$[0,\theta-\eta']$. In particular, for all $n$ sufficiently large, +the restrictions to these time intervals are within $\delta''$ in +the $C^{[1/\delta'']}$-topology of the standard flow. Also, for all $n$ sufficiently +large, $\theta_n-(\theta-\eta')<\eta$. Thus, by +\entryref{0ddfe578961e}, we see that the scalar curvature of $G_n'$ +is uniformly bounded (independent of $n$) on +$B_{G_n'}(p_n,0,A)\times [0,\theta_n)$. By the assumption that the +curvature is pinched toward positive, this means that the sectional +curvatures of the $G_n'$ are also uniformly bounded on these sets, +and hence so are the Ricci curvatures. (Notice that these bounds are +independent of $\eta'>0$.) By Shi's theorem with derivatives +(\entryref{e0b6a1f646a7}), we see that there are uniform bounds on +the curvatures in the $C^\infty$-topology on these subsets, and +hence bounds in the $C^\infty$-topology on the Ricci curvature. +These bounds are independent of both $n$ and $\eta'$. Thus, choosing +$\eta'$ sufficiently close to zero, so that $\theta_n-\eta'$ is also +close to $\theta$ for all $n$ sufficiently large, we see that for all such large $n$ +and all +$t\in [\theta-\eta',\theta)$, the +restriction of $G_n'$ to $B_{G_n'}(p_n,0,A)\times \{t\}$ is +arbitrarily close in the $C^{[1/\delta'']}$-topology to +$G_n'(\theta-\eta')$. The same is of course true of the standard +flow. This completes the proof of the corollary. + +Now we turn to the proof proper of \entryref{ad4f67ea75bc}. We fix +$A<\infty$, $\delta''>0$ and $\theta<1$. We are free to make $A$ +larger so we can assume by \entryref{77ca4669ffb3} that for the +standard flow the restriction of the flow to $B(p_0,0,A)\setminus +B(p_0,0,A/2)$ remains close to a standard evolving $S^2\times +[A/2,A]$ for time $[0,\theta]$. Let $K<\infty$ be a constant with +the property that $R(x,t)\le K$ for all $x\in B(p_0,0,A)$ in the +standard flow and all $t\in [0,\theta]$. If there is no +$\delta''_0>0$ as required, then we can find a sequence +$\delta'_n\rightarrow 0$ as $n\rightarrow\infty$ and Ricci flows with +surgery $({\mathcal M}_n,G_n)$ with surgeries at time $t_n$ with +surgery control parameter $\delta_n(t_n)\le \delta_n'$ and surgery +scale parameter $ h_n=h(r_{i+1}\delta_n( t_n),\delta_n(t_n))$ +satisfying the hypothesis of the lemma but not the conclusion. Let +$T'_n$ be the final time of $({\mathcal M}_n,G_n)$. Let $\theta_n\le +\theta$ be maximal subject to the condition that there is an +embedding $\rho_n\colon B_{G_n}(x, t_n,Ah_n)\times [ t_n, +t_n+h_n^2\theta_n)\to {\mathcal M}_n$ compatible with time and the +vector field. Let $G'_n$ be the result of shifting the time by +$-t_n$ and scaling the result by $h_n^{-2}$. According to +\entryref{712124d7ea02}, for all $n$ sufficiently large, the +restriction of $G_n'$ to the image of $\rho_n$ is within $\delta''$ +in the $C^{[1/\delta'']}$-topology of the standard flow restricted +to the ball of radius $A$ about the tip of the standard solution on +the time interval $[0,\theta_n)$. If $\theta_n=\mathit{min}(\theta,(T_n'-t_n)/h_n^2)$, then the first conclusion of +\entryref{ad4f67ea75bc} holds for $({\mathcal M}_n,G_n)$ for all +$n$ sufficiently large which contradicts our assumption that the +conclusion of this proposition holds for none of the $({\mathcal +M}_n,G_n)$. If on the other hand $\theta_n<\mathit{min}(\theta,(T_n'-t_n)/h_n^2)$, we need only show that all of +$B(x_n,t_n,Ah_n)$ disappears at time $t_n+h_n^2\theta_n$ in order to +show that the second conclusion of \entryref{ad4f67ea75bc} holds +provided that $n$ is sufficiently large. Again this would contradict +the fact that the conclusion of this proposition holds for none of +the $({\mathcal M}_n,G_n)$. + +So now let us suppose that $\theta_n<\mathit{min}(\theta,(T'_n-t_n)/h_n^2)$. Since there is no further extension +in forward time for $B(p_n,t_n,A h_n)$, it must be the case that $ +t_n+ h_n^2\theta_n$ is a surgery time and there is some flow line +starting at a point of $B(p_n,t_n,A h_n)$ that does not continue to +time $t_n+h_n^2\theta_n$. It remains to show that in this case that +for any $t< t_n+ h_n^2\theta_n$ sufficiently close to $t_n+ +h_n^2\theta_n$ we have $\rho_n\left(B_{G_n}(x, t_n,Ah_n)\times +\{t\}\right)\subset D_t$, the region in $M_t$ that disappears at +time $t_n+h_n^2\theta_n$. + +\entryblock{683bb06c1059} + +**Proof.** There is a constant $K'<\infty$ depending on $\theta$ such that for +the standard flow we have $R(x,t)\le K'$ for all $x\in B(p_0,0,A)$ +and all $t\in [0,\theta)$ for the standard solution. Consider the +embedding $\rho_n\left(B(p_n,t_n,Ah_n)\times [t_n,t_n+h_n^2 +\theta_n)\right)$. After time shifting by $-t_n$ and rescaling by +$h_n^{-2}$, the flow $G_n'$ on the image of $\rho_n$ is within +$\delta''$ of the standard flow. Thus, we see that for all $n$ sufficiently +large and for every point $x$ in the image of $\rho_n$ we have $ +R_{G_n'}(x)\le 2K'$ and hence $R_{G_n}(x)\le 2K'h_n^{-2}$. + +Let $h_n'$ be the scale of the surgery at time $t_n+ h_n^2\theta_n$. +(Recall that $h_n$ is the scale of the surgery at time $t_n$.) +Suppose that $\rho_n(B(p_n,t_n,Ah_n)\times \{t'\})$ meets one of the +surgery $2$-spheres $\Sigma_{i}(t')$ at time $t'$ at a point +$y(t')$. Then, for all $t\in [t',t_n+h_n^2\theta_n)$ we have + the image $y(t)$ of $y(t')$ under the flow. All these points +$y(t)$ are points of intersection of $\rho_n(B(p,t_n,Ah_n)\times +\{t\})$ with $\Sigma_{i}(t)$. Since $y(t)\in \rho_n(B(p,t_n,A +h_n)\times\{t\})$, we have $ R(y(t))\le 2K' h_n^{-2}$. On the other +hand $R(y(t)) (h'_n)^2$ is within $O(\delta)$ of $1$ as $t$ tends +to $t_n+h_n^2\theta_n$. This means that $h_n/h_n'\le \sqrt{3K'}$ for +all $n$ sufficiently large. Since the standard solution has +non-negative curvature, the metric is a decreasing function of $t$, +and hence the diameter of $B(p_0,t,A)$ is at most $ 2A$ in the standard +solution. Using \entryref{712124d7ea02} we see that for all $n$ +sufficiently large, the diameter of $\rho_n\left(B(p,t_n,Ah_n)\times +\{t\}\right)$ is at most $Ah_n\le 4\sqrt{K'} Ah_n'$. + This +means that for $\delta'_n$ sufficiently small the distance at time +$t$ from $\Sigma_{i}(t)$ to the complement of the $t$ time-slice +of the strong $\delta_n( t_n+ h_n^2\theta_n)$-neck $N_{i}(t)$ +centered at $\Sigma_{i}(t)$ (which is at least +$(\delta'_n)^{-1}h_n'/2$) is much larger than the diameter of + +$$ +\rho_n(B(p_n,t_n,Ah_n)\times \{t\}). +$$ + + Consequently, for all $n$ +sufficiently large, the image $\rho_n(B(p_n,t_n,Ah_n)\times \{t\})$ +is contained in $N_{i}(t)$. But by our choice of $A$, and +\entryref{712124d7ea02} there is an $\epsilon$-neck of rescaled +diameter approximately $Ah_n/2$ contained in $\rho_n(B(p_n,t_n,A +h_n)\times \{t\})$. By \entryref{98c6a95f8556} the spheres coming +from the neck structure in + +$$ +\rho_n(B(p_n,t_n,Ah_n)\times \{t\}) +$$ + + are + isotopic in $N_{i}(t)$ to the central $2$-sphere of this neck. +This is a contradiction because in $N_{i}(t)$ the central +$2$-sphere is homotopically non-trivial whereas the spheres in +$\rho_n(B(p_n,t_n,A\bar h_n)\times \{t\})$ clearly bound +$3$-disks. + +Since $\rho_n(B(p_n,t_n,Ah_n)\times\{t\})$ is disjoint from the +backward flow to time $t$ of all the surgery $2$-spheres +$\Sigma_{i}(t)$ and since $\rho_n(B(p_n,t_n,Ah_n)\times \{t\})$ is +connected, + if there is a flow line starting at some point $z\in +B(p,t_n,Ah_n)$ that disappears at time $t_n+ h_n^2\theta_n$, then +the flow from every point of $B(p,t_n,Ah_n)$ disappears at time $ +t_n+ h_n^2\theta_n$. This shows that if $\theta_n<\mathit{min}(\theta,T'_n- t_n/h_n^{2})$, and if there is no extension of +$\rho_n$ to an embedding defined at time $t_n+h^2_n\theta_n$, then +all forward flow lines beginning at points of $B(p,t_n,A h_n)$ +disappear at time $ t_n+h_n^2\theta_n$, which of course means that +for all $t \ell$. +This fixes the value of $\theta_0$. + +\entryblock{e3c80c436275} + +**Proof.** This is immediate from \entryref{77ca4669ffb3} and the fact that +$\theta_0<1$. + +Now fix $A_0=\mathit{max}(A_0', 10\sqrt{\ell})$ and let $A\ge A_0$. + +Since $\overline\delta_0<1/4$ and since $T'\le \theta_0$, for $\overline\delta(\bar t)\le +\delta''_0(A,\theta_0,\overline\delta_0)$ +by \entryref{ad4f67ea75bc} +the $g(T')$-distance between $B(p,0,A/2)$ and $\partial +B(p,0,A)$ is at least $A/5$. + + Since the +flow on $B(p,0,A)\times[0,T']$ is within $\overline\delta_0$ of +the standard solution, and since the curvature of the standard +solution is non-negative, for any horizontal tangent vector $X$ at +any point of $B(p,0,A)\times [0,T']$ we have that + +$$ +\mathit{Ric}_g(X,X)\ge -\frac{1}{4}|X|_{g_0}^2\ge -\frac{1}{2}|X|^2_g, +$$ + + and hence + +$$ +\frac{d}{dt}|X|_g^2\le |X|_g^2. +$$ + +Because $T'\le 1$, we see that + +$$ +|X|^2_{g(T')}\le e\cdot|X|^2_{g(t)}<3|X|^2_{g(t)} +$$ + +for any $t\in [0,T']$. + +Now suppose that $\gamma(0)\in \partial B(p,0,A)\times\{T'\}$. Since +the image of $\gamma$ is contained in the closure of $P(p,0,A,T')$ +for every $\tau\in [0,T']$ we have +$\sqrt{3}|X_\gamma(\tau)|_{g(T'-\tau)}\ge |X_\gamma(\tau)|_{g(T')}$. +Since the flow $g(t)$ on $P(p,0,A, T')$ is within +$\overline\delta_0$ in the $C^{[1/\overline\delta_0]}$-topology of +the standard flow on the corresponding parabolic neighborhood, +$R(\gamma(t))\ge 0$ for all $t\in [0,T']$. Thus, because of these +two estimates we have + +$$ +\int_0^{T'-t'}\left(R(\gamma(\tau))+|X_\gamma(\tau)|^2\right)d\tau\ge +\int_0^{T'-t'} +\frac{1}{3}|X_\gamma(\tau)|_{g(T')}^2d\tau. +$$ + + Since $\gamma(0)\in +\partial B(p,0,A)\times \{T'\}$ and $\gamma(T')\in B(p,0,A/2)$, it follows from +Cauchy-Schwarz that + +$$ +\begin{aligned} +(T'-t')^2\int_0^{T'}|X_\gamma(\tau)|_{g(T')}^2d\tau & +\ge & \left(\int_0^{T'-t'}|X_\gamma(\tau)|_{g(T')}d\tau\right)^2 +\\ +& \ge & \left(d_{g(T')}(B(p,0,A/2),\partial +B(p,0,A))\right)^2\ge \frac{A^2}{25}. +\end{aligned} +$$ + + Since $T'-t'<1$, it +immediately follows from this and Equation (16.1) that + +$$ +\int_0^{T'-t'}\left(R(\gamma(\tau))+|X_\gamma(\tau)|^2\right)d\tau\ge +\frac{A^2}{75}. +$$ + + Since $A\ge A_0\ge 10\sqrt{ \ell}$, this +expression is $>\ell$. This completes the proof of +\entryref{0a4d6f17810d} + +### Paths with short ${\mathcal L}_+$-length avoid the surgery caps + +Here we show that a path parameterized by backward time that ends in +a surgery cap (or comes close to it) must have +long ${\mathcal L}$-length. Let $({\mathcal M},G)$ be a Ricci flow +with surgery, and let $x\in {\mathcal M}$ be a point with $\mathbf{t}(x)=T\in (T_i,T_{i+1}]$. We suppose that these data satisfy the +hypothesis of \entryref{89a60eb69c3f} with respect to the given +sequences and $r\ge r_{i+1}>0$. In particular, the parabolic +neighborhood $P(x,T,r,-r^2)$ exists in ${\mathcal M}$ and $|\mathit{Rm}|$ is +bounded on this parabolic neighborhood by $r^{-2}$. + +Actually, here we do not work directly with the length function +${\mathcal L}$ defined from $x$, but rather with a closely related +function. We set $R_+(y)=\mathit{max}(R(y),0)$. + +\entryblock{86043191a3c2} + +\entryblock{4ee1bc9acc48} + +**Proof.** We define $\ell=L_0/\sqrt{\tau'}$, then define $A=\mathit{max}(A_0(\ell),2(50+A_0))$ and $\theta=\theta_0(\ell)$. Here, +$A_0(\ell)$ and $\theta_0(\ell)$ are the constants in +\entryref{0a4d6f17810d}. + Lastly, we require $\overline\delta_1\le \delta''(A)$ from +\entryref{0a4d6f17810d}. Notice that, by construction, + $\delta''(A)=\delta''_0(A,\theta,\overline\delta_0)$ from +\entryref{ad4f67ea75bc}. Thus, if $p$ is the tip of a surgery cap at time $\bar t$ +with the scale of the surgery being $\bar h$, then it follows that +for any $\Delta t\le \theta$, if there is an embedding + +$$ +\rho\colon B(p,\bar t,A\bar h)\times [\bar t,\bar t+\bar h^2\Delta t)\to {\mathcal M} +$$ + + compatible with time and the vector field, then the induced flow (after time shifting by $-\bar t$ + and scaling by $(\bar h)^{-2}$ is within $\overline\delta_0$ in the + $C^{[1/\overline\delta_0]}$-topology of the standard solution. + In particular, the scalar curvature +at any point of the image of $\rho$ is positive and is within a +multiplicative factor of two of the scalar curvature at the +corresponding point of the standard flow. + + Recall that we have $r\ge r_{i+1}$ and +that $P(x,T,r,-r^2)$ exists in ${\mathcal M}$ and that $|\mathit{Rm}|\le +r^{-2}$ on this parabolic neighborhood. We begin by proving by +contradiction that there is no $\tau\le \tau'$ with the property +that $\gamma(\tau)\not\in P(x,T,r/2,-r^2)$. Suppose there is such a +$\tau\le \tau'$. Notice that by construction +$\tau' +\frac{1}{\sqrt{2}}\int_0^{\tau''}|X_\gamma(\tau)|_{G(T)}\ge +\frac{r}{2\sqrt{2}}, +$$ + + where we use the fact that +$d_T(\gamma(0),\gamma(\tau''))=r/2$. + +Applying Cauchy-Schwarz to $\tau^{1/4}|X_\gamma|$ and $\tau^{-1/4}$ on the +interval $[0,\tau'']$ yields + +$$ +\begin{aligned} +\int_0^{\tau''}\sqrt{\tau}\left(R_+(\gamma(\tau))+|X_\gamma(\tau)|^2\right)d\tau & +\ge & \int_0^{\tau''}\sqrt{\tau}|X_\gamma(\tau)|^2d\tau +\\ +& \ge & +\frac{\left(\int_0^{\tau''}|X_\gamma(\tau)|d\tau\right)^2}{\int_0^{\tau''}\tau^{-1/2}d\tau} +\\ & > & +\frac{r^2}{16\sqrt{\tau''}}\ge L_0. +\end{aligned} +$$ + +Of course, the integral from $0$ to $\tau''$ is less than or equal +the entire integral from $0$ to $\tau_0$ since the integrand is +non-negative, contradicting the assumption that ${\mathcal +L}_+(\gamma)\le L_0$. This completes the proof of the first +numbered statement. + +We turn now to the second statement. We impose a further condition +on $\overline\delta_1$. Namely, require that +$\overline\delta_1^22r_{i+1}^{-2}$). Thus, if +$\tau'\ge \tau_0$ we have completed the proof. Suppose that +$\tau'<\tau_0$. It suffices to establish that for every $\tau_1\in +[\tau',\tau_0]$ the point $\gamma(\tau_1)$ is not contained in the +image of $\rho$ for any surgery cap and any $t'$ as in the +statement. Suppose that in fact there is $\tau_1\in [\tau', \tau_0]$ +with $\gamma(\tau_1)$ contained in the image of $\rho(B(p,\bar +t,(A_0+50)\bar h)\times [\bar t,t'])$ where $\bar t\le t'\le \bar +t+(\bar h)^2/2$ and where $p$ is the tip + of some surgery cap at time $\bar t$. We estimate + +$$ +\begin{aligned} +\nonumber +\int_0^{\tau_0}\sqrt{\tau}\left(R_+(\gamma(\tau))+|X_\gamma(\tau)|^2\right)d\tau & & \\ +& \ge & +\int_{\tau'}^{\tau_0}\sqrt{\tau}\left(R_+(\gamma(\tau))+|X_\gamma(\tau)|^2\right)d\tau +\nonumber +\\ & \ge & +\sqrt{\tau'}\int_{\tau'}^{\tau_1}\left(R_+(\gamma(\tau))+|X_\gamma(\tau)|^2\right)d\tau. +\end{aligned} +$$ + + Let $\Delta t\le +T-\bar t$ be the supremum of the set of $s$ for which there is a +parabolic neighborhood $P(p,\bar t,A\bar h,s)$ embedded in $\mathbf{t}^{-1}((-\infty,T])\subset {\mathcal M}$. Let $\Delta t_1=\mathit{min}(\theta \bar h^2,\Delta t)$. We consider $P(p,\bar t,A\bar +h,\Delta t_1)$. First, notice that since $\bar h\le +\overline\delta_1^2r_{i}r_{i+1}^{-2}\ge r^{-2}$. In particular, the parabolic +neighborhood $P(x,T,r,-r^2)$ is disjoint from $P(p,\bar t,A\bar +h,\Delta t_1)$. This means that there is some $\tau''\ge \tau'$ such +that $\gamma(\tau'')\in +\partial P(p,\bar t,A\bar h,\Delta t_1)$ and $\gamma|_{[\tau'',\tau_1]}\subset +P(p,\bar t,A\bar h,\Delta t_1)$. There are two cases to consider. +The first is when $\Delta t_1=\theta \bar h^2$, $\tau''=T-(\bar +t+\Delta t_1)$ and $\gamma(\tau'')\in B(p,\bar t,A\bar +h)\times\{\bar t+\Delta t_1\}$. Then, according to +\entryref{0a4d6f17810d}, + +$$ +\int_{\tau''}^{\tau_1}R_+(\gamma(\tau))d\tau> \ell. +$$ + +Now let us consider the other case. If $\Delta t_1<\theta \bar h^2$, +this means that either $\bar t+\Delta t_1=T$ or, according to +\entryref{ad4f67ea75bc}, at the time $\bar t+\Delta t_1$ there is a +surgery that removes all of $B(p,\bar t,A\bar h)$. Hence, under +either possibility it must be the case that $\gamma(\tau'')\in +\partial B(p,\bar t,A\bar h)\times \{T-\tau''\}$. Thus, the remaining +case to consider is when, whatever $\Delta t_1$ is, +$\gamma(\tau'')\subset +\partial B(p,\bar t,A\bar h)\times \{T-\tau''\}$. +\entryref{0a4d6f17810d} and the fact that $R\ge 0$ on $P(p,\bar t,A\bar +h,\Delta t_1)$ imply that + +$$ +\ell<\int_{\tau''}^{\tau_1}\left(R(\gamma(\tau))+|X_\gamma(\tau)|^2\right)d\tau = +\int_{\tau''}^{\tau_1}\left(R_+(\gamma(\tau))+|X_\gamma(\tau)|^2\right)d\tau. +$$ + +Since $\ell=L_0/\sqrt{\tau'}$ and $\tau''\ge \tau'$, it follows from +Equation (16.2) that in both +cases + +$$ +{\mathcal L}_+(\gamma)\ge +\int_{\tau''}^{\tau_1}\sqrt{\tau}\left(R_+(\gamma(\tau))+|X_\gamma(\tau)|^2\right)d\tau> +\ell\sqrt{\tau'}=L_0, +$$ + + which contradicts our hypothesis. This +completes the proof of \entryref{86043191a3c2}. + +### Paths with small energy avoid the disappearing regions + +At this point we have shown that paths of small energy do not approach +the surgery caps from above. We also need to rule out that they can be arbitrarily +close from below. That is to say, + we need to see that paths whose ${\mathcal L}$-length is not +too large avoid neighborhoods of the disappearing regions at all +times just before the surgery time at which they disappear. Unlike +the previous estimates which were universal for all $({\mathcal +M},G)$ satisfying the hypothesis of \entryref{89a60eb69c3f}, in +this case the estimates will depend on the Ricci flow with surgery. +First, let us fix some notation. + +\entryblock{0ea74aba0c14} + +\entryblock{5ed7c639bd97} + +**Proof.** Suppose that the lemma is false and let $\gamma\colon [0,\bar +\tau]\to {\mathcal M}$ be a path satisfying the hypothesis of the +lemma with $\gamma(\bar \tau)\in J(\bar t-\tau_1,\bar t)$. + Since + +$$ +\gamma(0)\in +M_{\bar t}\setminus \cup_iB(p_i,\bar t,(50+A_0)\bar h), +$$ + + if follows +that $\gamma(0)$ is separated from the boundary of +$s_{N_i}^{-1}(-25,0]$ by distance at least $20\bar h$. Since the $ +J_0(t)$ are contained in the disjoint union of strong $\delta$-necks +$N_i$ centered at the $2$-spheres along which we do surgery, and +since $\tau_1\le \bar h^2/\ell<\bar h^2$, it follows that, provided +that $\delta$ is sufficiently small, for every $t\in [\bar +t-\tau_1,\bar t)$, the metric on $J_0(t)$ is at least +$1/2$ the metric on $ J_0(\bar t)$. + It follows that, for $\delta$ sufficiently small, if there is a $\tau\in [0,\tau_1]$ +with $\gamma(\tau)\in J(t)$ then $\int_0^{\tau_1}|X_\gamma|d\tau> 10\bar h$. +Applying Cauchy-Schwarz we see that + +$$ +\int_0^{\tau_1}|X_\gamma(\tau)|^2d\tau\ge (10\bar h)^2/\tau_1. +$$ + +Since $\tau_1\le \ell^{-1}(\bar h)^2$, we see that + +$$ +\int_0^{\tau'}|X_\gamma(\tau)|^2d\tau>\ell, +$$ + +contradicting our hypothesis. + +### Limits of a sequence of paths with short ${\mathcal L}$-length + +Now using Lemmas \entryref{86043191a3c2} and \entryref{5ed7c639bd97} we show that it is possible to take limits +of certain types of sequences of paths parameterized by backward time to create +minimizing ${\mathcal L}$-geodesics. + +We shall work with a compact subset of $\mathbf{t}^{-1}([T_{i-1},T])$ that is +obtained by removing appropriate open neighborhoods of the exposed regions. + +\entryblock{1ff58113d89b} + +\entryblock{d98ad25d9bc3} + +**Proof.** Given $L_0$, we set + +$$ +\tau'=\mathit{min}\left(\frac{r_{i+1}^4}{(256)L_0^2},\mathit{ln}(\sqrt[3]{2})r_{i+1}^2\right) +$$ + +as in \entryref{86043191a3c2} and then define $\ell=L_0/\sqrt{\tau'}$. We +also let $A=\mathit{min}(2(50+A_0),A_0(\ell))$ and +$\theta_0=\theta_0(\ell)$ as in \entryref{0a4d6f17810d}. Lastly, we +let +$\overline\delta_1(L_0,r_{i+1})=\delta''(A)=\delta''(A,\theta_0,\overline\delta_0)$ +from Propositions \entryref{0a4d6f17810d} and \entryref{ad4f67ea75bc}. We suppose that +$\delta(t)\le \overline\delta_1(L_0,r_{i+1})$ for all $t\in +[T_{i-1},T]$. + +Let $\bar t\in [T_{i-1},T]$ be a surgery time, and let $\bar h$ be +the scale of the surgery at this time. For each surgery cap +${\mathcal C}$ with tip $p$ at a time $\bar t\in [T_{i-1},T]$ let +$\Delta t({\mathcal C})$ be the supremum of those $s$ with $0\le s\le \theta_0\bar +h^2$ for which there is an embedding + +$$ +\rho_{\mathcal C}\colon +B(p,\bar t,2(A_0+50)\bar h)\times[\bar t,\bar t+s)\to{\mathcal M} +$$ + +compatible with time and the vector field. We set + +$$ +P_0({\mathcal C})=\rho_{\mathcal C}\left(B(p,\bar t,(A+50)\bar h)\times +[\bar t,\bar t+\mathit{min}(\bar h^2/2,\Delta t({\mathcal C})))\right). +$$ + +\entryblock{cfb464e3cf35} + +**Proof.** Set $\tau_0=T-\bar t$. Of course, $\tau_0\le T-T_{i-1}\le T_{i+1}-T_{i-1}$. +Consider the restriction of $\gamma$ to $[0,\tau_0]$. +We have + +$$ +\begin{aligned} +\int_0^{\tau_0}\sqrt{\tau}\left(R_+(\gamma_n(\tau))+|X_{\gamma_n}(\tau)|^2\right)d\tau + \\ +& \le & +\int_0^{T-T_{i-1}}\sqrt{\tau}\left(R_+(\gamma_n(\tau))+|X_{\gamma_n}(\tau)|^2\right)d\tau \\ +& \le & \int_0^{T-T_{i-1}}\sqrt{\tau}\left(R(\gamma_n(\tau))+|X_{\gamma_n}(\tau)|^2\right)d\tau+ +\int_0^{T-T_{i-1}} +6\sqrt{\tau}d\tau \\ +& = & +\int_0^{T-T_{i-1}}\sqrt{\tau}\left(R(\gamma_n(\tau))+|X_{\gamma_n}(\tau)|^2\right)d\tau ++4(T-T_{i-1})^{3/2} +\\ +& \le & +\int_0^{\tau_0}\sqrt{\tau}\left(R(\gamma_n(\tau))+|X_{\gamma_n}(\tau)|^2\right)d\tau+4(T_{i+1})^{3/2} +\end{aligned} +$$ + +Thus, the hypothesis that ${\mathcal L}(\gamma_n)\le L$ implies that + +$$ +\int_0^{\tau_0}\sqrt{\tau}\left(R_+(\gamma_n(\tau))+|X_{\gamma_n}(\tau)|^2\right)d\tau\le +L_0. +$$ + +The claim now follows immediately from \entryref{86043191a3c2}. + +Now set $t'$ equal to the last surgery time before $\bar t$ or set $t'=0$ if $\bar t$ is +the first surgery time. We set $\tau_1(\bar t)$ equal to the minimum of $\bar t-t'$ and +$\bar h^2/\ell$. + +Assume that $\gamma(0)=x$ and that ${\mathcal L}(\gamma)\le L$. It follows from +\entryref{86043191a3c2} that the restriction of the path $\gamma$ to $[0,\tau']$ lies +in a region where the Riemann curvature is bounded above by $r^{-2}\le +r_{i+1}^{-2}$. Hence, since $\bar h<\bar\delta(t)^2r_{i+1}\ll r_{i+1}$, this +part of the path is disjoint from all strong $\delta$-necks (evolving +backward for rescaled time $(-1,0]$). That is to say, $\gamma|_{[0,\tau']}$ is +disjoint from $J_0(t)$ for every $t\in (\bar t-\tau_1(\bar t),\bar t)$ for any +surgery time $\bar t\le T$. It follows immediately that $\gamma|_{[0,\tau']}$ +is disjoint from $J(\bar t-\tau_1(\bar t),\bar t)$. + +\entryblock{5ba0ec2c68db} + +**Proof.** By the remarks above, it suffices to consider surgery times $\bar +t\le T-\tau'$. It follows immediately from the previous claim that +for any surgery time $\bar t$, with the scale of the surgery being +$\bar h$ and with $p$ being the tip of a surgery cap at this time, +we have $\gamma$ is disjoint from $B(p,\bar t,(50+A_0)\bar h)$. +Also, + +$$ +\int_{T-\bar t}^{T-\bar t+\tau_1(\bar t)}\sqrt{\tau}|X_\gamma(\tau)|^2d\tau +\le {\mathcal L}_+(\gamma)\le L_0. +$$ + +Since we can assume $T-\bar t\ge \tau'$ this implies that + +$$ +\int_{T-\bar t}^{T-\bar t-\tau_1(\bar t)}|X_\gamma(\tau)|^2d\tau\le L_0/\sqrt{\tau'}=\ell. +$$ + +The claim is now immediate from \entryref{5ed7c639bd97}. + +From these two claims we see immediately that $\gamma$ is contained in the +compact subset $Y(\ell)$ which is contained in the open subset of smooth points +of ${\mathcal M}$. This proves the third item in the statement of the +proposition. Now let us turn to the limit statements. + +Take a sequence of paths $\gamma_n$ as in the statement of +\entryref{d98ad25d9bc3}. + By \entryref{86043191a3c2} the +restriction of each $\gamma_n$ to the interval $[0,\mathit{min}(\bar\tau,\tau')]$ is contained in $P(x,T,r/2,-r^2)$. The +arguments in the proof of \entryref{cd062526b91a} (which involve +changing variables to $s=\sqrt{\tau}$) show that, after passing to a +subsequence, the restrictions of the $\gamma_n$ to $[0,\mathit{min}(\bar\tau,\tau')]$ converge uniformly to a path $\gamma$ defined +on the same interval. Furthermore, + +$$ +\int_0^{\mathit{min}(\bar\tau,\tau')}\sqrt{\tau}|X_\gamma(\tau)|^2d\tau\le \mathit{liminf}_{n\rightarrow\infty}\int_0^{\mathit{min}(\bar\tau,\tau')}\sqrt{\tau}|X_{\gamma_n}(\tau)|^2d\tau, +$$ + +so that + +$$ +\begin{aligned} +\int_0^{\mathit{min}(\bar\tau,\tau')}\sqrt{\tau}\left(R(\gamma(\tau)+|X_\gamma(\tau)|^2\right)d\tau\le \\ +& & \mathit{liminf}_{n\rightarrow\infty}\int_0^{\mathit{min}(\bar\tau,\tau')}\sqrt{\tau}\left(R(\gamma_n(\tau))+|X_{\gamma_n}(\tau)|^2\right)d\tau.\nonumber +\end{aligned} +$$ + +If $\bar\tau\le \tau'$, then we have established the existence of a +limit as required. Suppose now that $\bar\tau>\tau'$. We turn our +attention to the paths $\gamma_n|_{[\tau',\bar\tau]}$. Let +$T_{i-1}<\bar t\le T-\tau'$ be either a surgery time or $T-\tau'$, and let $t'$ be the +maximum of the last surgery time before $\bar t$ and $T_{i-1}$. We consider the +restriction of the $\gamma_n$ to the interval $[T-\bar t,T-t']$. As +we have seen, these restrictions are disjoint from $J(\bar +t-\tau_1(\bar t),\bar t)$ and also from the exposed region at time +$\bar t$, which is denoted $E(\bar t)$, and from $J_0(\bar t)$. Let + +$$ +Y=\mathbf{t}^{-1}([T-t',T-\bar t])\setminus +\left(J(\bar t-\tau_1(\bar t),\bar t)\cup (E(\bar t)\cup J_0(\bar +t))\right). +$$ + + This is a compact subset with the property that any +point $y\in Y$ is connected by a backward flow line lying entirely +in $Y$ to a point $y(t')$ contained in $M_{t'}$. + + Since $Y$ is compact there is a finite upper bound on the +Ricci curvature on $Y$, and hence to ${\mathcal L}_\chi(G)$ at any +point of $Y$. Since all backward flow lines from points of $Y$ +extend all the way to $M_{t'}$, it follows that there is a constant +$C'$ such that + +$$ +|X_{\gamma_n}(\tau)|_{G(t')}\le C'|X_{\gamma_n}(\tau)|_{G(t)} +$$ + +for all $t\in [t',\bar t]$. Our hypothesis that the ${\mathcal +L}(\gamma_n)$ are uniformly bounded, the fact that the curvature is +pinched toward positive and the fact that there is a uniform bound +on the lengths of the $\tau$-intervals implies that the + +$$ +\int_{T-\bar t}^{T-t'} \sqrt{\tau}|X_{\gamma_n}(\tau)|^2d\tau +$$ + +are uniformly bounded. Because $T-\bar t$ is at least $\tau'>0$, it +follows that the $\int_{T-\bar t}^{T-t'}|X_{\gamma_n}|^2d\tau$ have +a uniform upper bound. This then implies that there is a constant +$C_1$ such that for all $n$ we have + +$$ +\int_{T-\bar t}^{T-t'}|X_{\gamma_n}(\tau)|_{G(t')}^2d\tau\le +C_1. +$$ + + Thus, after passing to a subsequence, the $\gamma_n$ converge +uniformly to a continuous $\gamma$ defined on $[T-\bar t,T-t']$. +Furthermore, we can arrange that the convergence is a weak +convergence in $W^{1,2}$. This means that $\gamma$ has a derivative +in $L^2$ and + +$$ +\int_{T-\bar t}^{T-t'}|X_{\gamma}(\tau)|^2d\tau\le \mathit{liminf}_{n\rightarrow +\infty}\int_{T-\bar t}^{T-t'}|X_{\gamma_n}(\tau)|^2d\tau. +$$ + +Now we do this simultaneously for all $\bar t=T-\tau'$ and for all the finite number of surgery times +in $[T_{i-1},T-\tau']$. This +gives a limiting path $\gamma\colon [\tau',\bar\tau]\to {\mathcal M}$. +Putting together the above inequalities we see that the limit +satisfies + +$$ +\int_{\tau'}^{\bar\tau}\sqrt{\tau}\left(R(\gamma(\tau))+|X_\gamma(\tau)|^2\right)d\tau\le +\mathit{liminf}_{n\rightarrow\infty}\int_{\tau'}^{\bar\tau}\sqrt{\tau}\left(R(\gamma_n(\tau))+ +|X_{\gamma_n}(\tau)|^2\right)d\tau. +$$ + + Since we have already +arranged that there is a limit on $[0,\tau']$, this produces a +limiting path $\gamma\colon [0,\tau_0]\to {\mathcal M}$. By +Inequalities 16.5 and 16.6 we see that + +$$ +{\mathcal L}(\gamma)\le \mathit{liminf}_{i\rightarrow \infty}{\mathcal +L}(\gamma_n). +$$ + +The limit lies in the compact subset $Y(\ell)$ and hence is contained +in the open subset of smooth points of ${\mathcal M}$. + This completes the proof of the first statement of the proposition. + +Now suppose, in addition to the above, that all the $\gamma_n$ +have the same endpoint $y\in M_{T-\tau_0}$ and that $\mathit{lim}_{n\rightarrow\infty}{\mathcal L}(\gamma_n)$ is less than or equal +to the ${\mathcal L}$-length of any path parameterized by backward +time connecting $x$ to $y$. Let $\gamma$ be the limit of a +subsequence as constructed in the proof of the first part of this +result. Clearly, by what we have just established, $\gamma$ is a path +parameterized by backward time from $x$ to $y$ and ${\mathcal +L}(\gamma)\le \mathit{lim}_{n\rightarrow\infty}{\mathcal L}(\gamma_n)$. +This means that $\gamma$ is a minimizing ${\mathcal L}$-geodesic +connecting $x$ to $y$, an ${\mathcal L}$-geodesic contained in the open subset of smooth +points of ${\mathcal M}$. + +This completes the proof of the proposition. + +\entryblock{ae8837c505cc} + +**Proof.** Choose an ${\mathcal L}$-minimizing sequence of paths from $x$ to +$y$ and apply the previous proposition. + +### Completion of the proof of Proposition \entryref{89a60eb69c3f} + +Having found a compact subset of the open subset of smooth points of ${\mathcal M}$ +that contains all paths parameterized by backward time whose ${\mathcal L}$-length is not +too large, we are in a position to prove \entryref{89a60eb69c3f}, which states +the existence of a minimizing ${\mathcal L}$-geodesics in ${\mathcal M}$ from $x$ +and gives estimates on their ${\mathcal L}$-lengths. + +**Proof.** (of \entryref{89a60eb69c3f}). +Fix $r\ge r_{i+1}>0$. Let $({\mathcal M},G)$ and $x\in {\mathcal M}$ be +as in the statement of \entryref{89a60eb69c3f}. We set +$L=8\sqrt{T_{i+1}}(1+T_{i+1})$, and we set + +$$ +\delta=\mathit{min}\left(\delta_i,\overline \delta_1(L+4(T_{i+1})^{3/2},r_{i+1})\right), +$$ + +where $\overline\delta_1$ is as given in \entryref{86043191a3c2}. +Suppose that $\overline\delta(t)\le +\delta$ for all $t\in [T_{i-1},T_{i+1})$. We set $U$ equal to the +subset of $\mathbf{t}^{-1}([T_{i-1},T))$ consisting of all points $y$ +for which there is a path $\gamma$ from $x$ to $y$, parameterized by +backward time, with ${\mathcal L}(\gamma)0$ sufficiently small. Also, +extending $\gamma$ by adding a backward vertical flow line from $z$ +shows that the minimum value of ${\mathcal L}_x$ on $U_{t-\mu}$ is +at most $(D'+D)/2$ for all $\mu>0$ sufficiently small. (Such a +vertical flow line backward in time exists since $z\in U$ and hence +$z$ is contained in the smooth part of ${\mathcal M}$.) This +contradicts the fact that limit of the minimum values of ${\mathcal +L}_x$ on $U_{t_n}$ converge to $D$ as $t_n$ converges to $t$. This +contradiction proves that the minimum value of ${\mathcal L}_x$ on +$U_t$ is $D$, and consequently the point $y\in Z'$. This proves that +$Z'\cap \mathbf{t}^{-1}(I)$ is compact, establishing the claim. + +At this point we have established that Properties (1),(2), and (4); +So it remains only to prove Property (3) of \entryref{89a60eb69c3f}. + To do this we define the reduced length function +$l_x\colon U\to \Ar$ by + +$$ +l_x(q)=\frac{{\mathcal L}_x(q)}{2\sqrt{T-\mathbf{t}(q)}} \ \ \mathit{and} \ \ +l_x^\mathit{min}(\tau)=\mathit{min}_{q\in M_t}l_x(q). +$$ + + We consider the +subset ${\mathcal S}$ of $\tau'\in (0,T-T_{i-1}]$ with $l_x^\mathit{min}(\tau)\le L/2$ for all $\tau\le \tau'$. Recall that by the +choice of $L$, we have $3\sqrt{T-T_{i-1}}< L/2$. Clearly, the +minimum value of $l_x$ on $U_{T-\tau}$ converges to $0$ +as $\tau\rightarrow 0$, implying that this set is non-empty. Also, from +its definition, ${\mathcal S}$ is an interval with $0$ being one +endpoint. + +\entryblock{8cae73a6a02c} + +**Proof.** Given that we have already established Properties 1,2 and 4 of +\entryref{89a60eb69c3f}, this is immediate from +\entryref{71669e7fd5ec}. + +Now let us establish that ${\mathcal S}=(0,T-T_{i-1}]$. As we remarked above, +${\mathcal S}$ is a non-empty interval with $0$ as one endpoint. Suppose that +it is of the form $(0,\tau]$ for some $\tau\tau$ but sufficiently close to +$\tau$, there is a point $y(\tau')\in U_{T-\tau'}$ with ${\mathcal +L}_x(y(\tau'))0.85$ and +$(2/3)^{3/2}\le 0.6$, the first inequality follows. + +The second inequality is clear since $T'<(T-T_{i-1})$. + +The last inequality is clear from the fact that +$T''=T-T_{i-1}-\epsilon^2$ and $\epsilon\le \sqrt{t_0/50}$. + + Putting these together yields + +$$ +{\mathcal L}(\gamma)\ge + \left[\left(\frac{1}{6}r_i^{-2}-4\right)\left(T-T_{i-1}\right)-\frac{2t_0}{25}\right] + \sqrt{T-T_{i-1}}. +$$ + +Since + +$$ +r_i^{-2}\ge r_0^{-2}\ge \epsilon^{-2}\ge 50/t_0, +$$ + +and $T-T_{i-1}\ge t_0$ we see that + +$$ +\begin{aligned} +{\mathcal +L}(\gamma) & \ge & +\left[\left(\frac{50}{6t_0}-4\right)t_0-\frac{2t_0}{25}\right]\sqrt{T-T_{i-1}} +\\ +& \ge & (8-5t_0)\sqrt{T-T_{i-1}} \\ +& \ge & 4\sqrt{T-T_{i-1}}. +\end{aligned} +$$ + +(The last inequality uses the fact that $t_0=2^{-5}$.) But this +contradicts the fact that ${\mathcal L}(\gamma)\le +3\sqrt{T-T_{i-1}}$. + +Now fix $\tau_0$ satisfying \entryref{8aac9c08c058}. + Let $\gamma_1$ be the +restriction of $\gamma$ to the subinterval $[0,\tau_0]$, and let +$y=\gamma_1(\tau_0)$. Again using the fact that $R(\gamma(\tau))\ge +-6$ for all $\tau$, we see that + +$$ +{\mathcal L}(\gamma_1)\le {\mathcal L}(\gamma)+4(T-T_{i-1})^{3/2}\le +3(T_{i+1})^{1/2}+4(T_{i+1})^{3/2}. +$$ + +Set $t'=T-\tau_0$. Notice that from the definition we +have $t'\le T_i$. Consider $B=B(y,t',\frac{r_i}{2C})$, and +define $\Delta=\mathit{min}(r_i^2/16C,\epsilon^2)$. According to +\entryref{0ddfe578961e} every point $z$ on a backward flow line +starting in $B$ and defined for time at most $ \Delta$ has the +property that $R(z)\le 2r_i^{-2}$. For any surgery time $\bar t$ in +$[t'-\Delta,t')\subset [T_{i-1},T)$ the scale $\bar h$ of the +surgery at time $\bar t$ is $\le \delta(\bar t)^2 r_i$, and hence +every point of the surgery cap has scalar curvature at least +$D^{-1}\delta(\bar t)^{-4}r_i^{-2}$. Since $\overline\delta(\bar t)\le +\overline \delta\le \delta_0\le \mathit{min}(D^{-1},1/10)$, it follows +that every point of the surgery cap has curvature at least +$\delta_0^{-3}r_i^{-2}\ge 1000r_i^{-2}$. Thus, + no point +$z$ as above can lie in a surgery cap. This means that the entire +backward parabolic neighborhood $P(y,t',\frac{r_i}{2C},-\Delta)$ +exists in ${\mathcal M}$, and the scalar curvature is bounded by $2 +r_i^{-2}$ on this backward parabolic neighborhood. Because of the +curvature pinching toward positive assumption, there is $C'<\infty$ +depending only on $r_i$ and such that the Riemann curvature is +bounded by $C'$ on $P(y,t',\frac{r_i}{2C},-\Delta)$. + +Consider the one-parameter family of metrics $g(\tau),\ 0\le \tau\le +\Delta$, on $B(y,t',\frac{r_i}{2C})$ obtained by restricting the +horizontal metric $G$ to the backward parabolic neighborhood. +There is $0<\Delta_1\le \Delta /2$ +depending only on $C'$ such that for every $\tau\in [0,\Delta_1]$ and +every non-zero tangent vector $v$ at a point of +$B(y,t',\frac{r_i}{2C})$ we have + +$$ +\frac{1}{2}\le \frac{|v|^2_{g(\tau)}}{|v|^2_{g(0)}}\le 2. +$$ + +Set $\hat r=\mathit{min}(\frac{r_i}{32C},\Delta_1/2)$, so that $\hat r$ depends +only on $r_i$, $C$, and $\epsilon$. Set $t''=t'-\Delta_1$. Clearly, +$B(y, t'',\hat r)\subset B(y,t',\frac{r_i}{2C})$ so that $B(y,t'',\hat r)\subset +P(y,t',\frac{r_i}{2C},-\Delta)$. Of course, it then follows that the parabolic neighborhood +$P(y,t'',\hat r,-\Delta_1)$ exists in ${\mathcal M}$ and + +$$ +P(y,t'',\hat r,-\Delta_1)\subset P(y,t',\frac{r_i}{2 C},-\Delta), +$$ + + so that the +Riemann curvature is bounded above by $C'$ on the parabolic +neighborhood $P(y,t'',\hat r,-\Delta_1)$. We set $r'=\mathit{min}(\hat r,(C')^{-1/2},\sqrt{\Delta_1}/2)$, so that $r'$ depends only on +$r_i$, $C$, and $\epsilon$. Then the parabolic neighborhood +$P(y,t'',r',-(r')^2)$ exists in ${\mathcal M}$ and $|\mathit{Rm}|\le +(r')^{-2}$ on $P(y,t'',r',-(r')^2)$. Hence, by the inductive +non-collapsing assumption either $y$ is contained in a component of $M_{t''}$ of positive sectional curvature or + +$$ +\mathit{Vol} B(y,t'',r')\ge +\kappa_i(r')^3. +$$ + +If $y$ is contained in a component of $M_{t''}$ of positive sectional curvature, then by + Hamilton's result, \entryref{86e5a5766bda}, +under Ricci flow the component of $M_{t''}$ containing $y$ flows forward as +a family of components of positive sectional curvature until it disappears. Since there is path moving backwards in time +from $x$ to $y$, this means that the original point $x$ is contained +in a component of its time-slice with positive sectional curvature. + +Let us consider the other possibility when $\mathit{Vol} B(y,t'',r')\ge +\kappa_i(r')^3$. + For each $z\in B(y,t'',r')$ let + +$$ +\mu_z\colon [T-t',T-t'']\to B(y,t',\frac{r_i}{2C}) +$$ + + be + the $G(t')$-geodesic connecting $y$ to $z$. Of course + +$$ +|X_{\mu_z}(\tau)|_{G(t')}\le \frac{r'}{\Delta_1} +$$ + + for every $\tau\in [0,\Delta_1]$. Thus, + +$$ +|X_{\mu_z}(\tau)|_{G(T-\tau)}\le + \frac{\sqrt{2}r'}{\Delta_1} +$$ + + for all $\tau\in [T-t',T-t'']$. + Now we let $\tilde \mu_z$ be the resulting path parameterized by backward time + on the time-interval $[T-t',T-t'']$. + We estimate + +$$ +\begin{aligned} +{\mathcal + L}(\tilde \mu_z) & = & + \int_{T-t'}^{T-t''}\sqrt{\tau}\left(R(\tilde\mu_z(\tau))+|X_{\tilde\mu_z}(\tau)|^2\right)d\tau + \\ +& \le & \sqrt{T-t''}\int_{T-t'}^{T-t''}\left(2 r_i^{-2}+\frac{2(r')^2}{\Delta_1^2}\right)d\tau +\\ +& \le & \sqrt{T-t''}(2r_i^{-2}\Delta_1+\frac{1}{2}) \le +\sqrt{T}\left(\frac{1}{16C}+\frac{1}{2}\right). +\end{aligned} +$$ + +In passing to the last inequality we use the fact, from the +definitions that $r'\le \sqrt{\Delta_1}/2$ and $\Delta\le +r_i^2/16C$, whereas $\Delta_1\le \Delta/2$. + +Since $C>1$, we see that + +$$ +{\mathcal L}(\tilde \mu_z)\le \sqrt{T}. +$$ + +Putting this together with the estimate, Equation (16.7), +for ${\mathcal L}(\gamma_1)$ tells us that for each $z\in +B(y,t'',r')$ we have + +$$ +{\mathcal L}(\gamma_1*\tilde\mu_z)\le +4(T_{i+1})^{1/2}+4(T_{i+1})^{3/2}\le L/2. +$$ + + Hence, by +\entryref{89a60eb69c3f} and the choice of $L$, there is a +minimizing ${\mathcal L}$-geodesic from $x$ to each point of +$B(y,t'',r')$ of length $\le L/2$, and these geodesics are contained +in the smooth part of ${\mathcal M}$. In fact, by +\entryref{d98ad25d9bc3} there is a compact subset $Y$ of +the open subset of smooth points of ${\mathcal M}$ that contains all +the minimizing ${\mathcal L}$-geodesics from $x$ to points of +$B(y,t'',r')$. + +Then, by \entryref{7aad50e49af2} (see also, +\entryref{db4ced24b05f}), the intersection, $B'$, of ${\mathcal U}_x$ +with $B(y,t'',r')$ + is an open subset of full measure in $B(y,t'',r')$. +Of course, $\mathit{Vol} B'=\mathit{Vol} B(y,t'',r')\ge \kappa_i(r')^3$ +and the function $l_x$ is bounded by $L/2$ on $B'$. It now follows +from \entryref{3defa9b2209c} that there is $\kappa>0$ depending only on +$\kappa_i$, $r'$, $\epsilon$ and $L$ such that $x$ is $\kappa$ +non-collapsed on scales $\le \epsilon$. Recall that $L$ depends only +on $T_{i+1}$, and $r'$ depends only on $r_i,C,C'$ and $\epsilon$, +whereas $C'$ depends only on $r_i$. Thus, in the final analysis, +$\kappa$ depends only on $\kappa_i$ and $r_i$ (and $C$ and +$\epsilon$ which are fixed). This entire analysis assumed that for +all $t\in [T_{i-1},T_{i+1})$ we have the inequality +$\overline\delta(t)\le +\overline\delta_1(L+4(t_{i+1})^{3/2},r_{i+1})$ as in +\entryref{86043191a3c2}. Since $L$ depends only on $i$ and $t_0$, this shows +that the upper bound for $\delta$ depends only on $r_{i+1}$ (and on +$i$, $t_0$, $C$, and $\epsilon$). This completes the proof of +\entryref{4ce8b5da9c58}. diff --git a/projects/poincare-conjecture/.astrolabe/docs/18-completion-of-the-proof-of-theorem.mdx b/projects/poincare-conjecture/.astrolabe/docs/18-completion-of-the-proof-of-theorem.mdx new file mode 100644 index 00000000..24bb82db --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/docs/18-completion-of-the-proof-of-theorem.mdx @@ -0,0 +1,907 @@ + + +# Chapter 17 -- Completion of the proof of Theorem \entryref{703e4cfeece7} + +We have established the requisite non-collapsing result assuming the +existence of strong canonical neighborhoods. In order to complete +the proof of \entryref{703e4cfeece7} it remains for us to show the +existence of strong canonical neighborhoods. This is the result of +the next section. + +## Proof of the strong canonical neighborhood assumption + +\entryblock{254e8dae97e6} + +**Proof.** Suppose that the result does not hold. + Then we can take a sequence of $r_a\rightarrow 0$ as + $a\rightarrow \infty$, all less than $r_i$, and for each $a$ a sequence + $\delta_{a,b}\rightarrow 0$ as $b\rightarrow \infty$ with each + $\delta_{a,b}\le \delta(r_a)$, where $\delta(r_a)\le \delta_i$ is the constant in \entryref{4ce8b5da9c58} + associated to the three sequences given in the statement of this proposition and $r_a$, such that for + each $a,b$ there is a Ricci flow with surgery $({\mathcal + M}_{(a,b)},G_{(a,b)})$ defined for $0\le t0$ and for any counterexample point +$x\in {\mathcal M}_{(a,b)}$ eventually $\mathbf{t}(x_{n})< \mathbf{t}(x)+\xi$. Let $t'=t_{(a,b)}'=\mathit{lim}_{n\rightarrow\infty}\mathbf{t}(x_{n})$. Clearly, $t'\in [T_i,T_{(a,b)})$, and by construction +the restriction of $({\mathcal M}_{(a,b)},G_{(a,b)})$ to $\mathbf{t}^{-1}([0,t'))$ satisfies the $(C,\epsilon)$-canonical neighborhood assumption +with parameter $r_a$. Since the surgery times are discrete, there is +$t''=t_{(a,b)}''$ with $t'0$ for all $b$ sufficiently large, we +have $\delta_{(a,b)}\le \delta''(A,\theta_1,\overline\delta)$, where +$\delta''(A,\theta_1,\overline\delta)$ +is the constant given in \entryref{0a4d6f17810d}. + +\entryblock{d130b5adaed6} + +**Proof.** In this proof we shall fix $(a,b)$, so we drop these indices from +the notation. + Consider +$s\le \theta_1$ maximal so that there is an embedding + +$$ +\psi=\psi_{(a,b)}\colon +B(p_0,0,A)\times [0,s)\to \widetilde{\mathcal +M}_{(a,b)} +$$ + + compatible with time and the vector field. First +suppose that $s<\theta_1$. Then according to +\entryref{ad4f67ea75bc} either the entire ball $B(p,0,A)$ +disappears at time $s$ or $s$ is the final time of the +time interval of definition for the flow $(\widetilde{\mathcal +M}_{(a,b)},\widetilde G_{(a,b)})$. Since we have the flow line from $z\in +B(p_0,0,A)$ extending to time $\tilde t'=\tilde t'_{(a,b)}$, in either case this +implies that $\tilde t'0$, for +all $b$ sufficiently large the pullback under this map of $\widetilde G_{(a,b)}$ is within $\delta$ +in the $C^{[1/\delta]}$-topology of the restriction of the standard solution. +Let $w_{(a,b)}$ be the result of flowing $x_{(a,b)}$ backward to time $0$. + +\entryblock{9b87a4e6674a} + +**Proof.** First notice that, by our choice of $\epsilon$, + every point in the $0$ time-slice of the closure of the strong $\epsilon$-neck +centered at $x_{(a,b)}$ is within distance $(1.1)\tilde Q_{(a,b)}^{-1}\epsilon^{-1}$ +of $w_{(a,b)}$. In particular, + +$$ +d_{\widetilde G_{(a,b)}}(w_{(a,b)},y_{(a,b)})<(1.1)\tilde Q_{(a,b)}^{-1/2}\epsilon^{-1}. +$$ + +Since $y_{(a,b)}$ is contained in the surgery cap and the scale of the surgery +at this time is $1$, $y_{(a,b)}$ is within distance $A_0+5$ of $p_{(a,b)}$. +Hence, by the triangle inequality and Inequality (17.1), we have + +$$ +\begin{aligned} +d_{\widetilde G_{(a,b)}}(w_{(a,b)},p_{(a,b)}) & < & (1.1)\tilde Q_{(a,b)}^{-1/2}\epsilon^{-1}+(A_0+5) \\ +& < & (1.1)\sqrt{5D}\epsilon^{-1}+(A_0+5). +\end{aligned} +$$ + +For $b$ sufficiently large, the image $\psi_{(a,b)}(B(p_0,0,A))$ +contains the ball of radius $(0.95)A$ +centered at $p_{(a,b)}$. Since by our choice of $A$ we have $(0.95)A>(1.1)\sqrt{5D}\epsilon^{-1}+(A_0+5)$, +the claim follows. + +We define $q_{(a,b)}\in B(p_0,0,A)$ so that +$\psi_{(a,b)}(q_{(a,b)},0)=w_{(a,b)}$. Of course, + +$$ +\psi_{(a,b)}(q_{(a,b)},\tilde t'_{(a,b)})=x_{(a,b)}. +$$ + + If follows from the +above computation that for all $b$ sufficiently large we have + +$$ +d_0(q_{(a,b)},p_0)<(1.15)\tilde Q_{(a,b)}^{-1/2}\epsilon^{-1}+(1.05)(A_0+5). +$$ + +Since the standard flow has non-negative curvature, it is a distance non-increasing flow. Therefore, + +$$ +d_{\tilde t'_{(a,b)}}(q_{(a,b)},p_0)<(1.15)\tilde Q_{(a,b)}^{-1/2}\epsilon^{-1}+(1.05)(A_0+5). +$$ + +Suppose that a point $(q,\tilde t'_{(a,b)})$ in the standard solution were the center of a +$\beta\epsilon/3$-neck, where $\beta$ is the constant from \entryref{a992d6a89479}. + Of course, for all $b$ sufficiently large, $R(q,\tilde +t'_{(a,b)})>(0.99)\tilde Q_{(a,b)}$. Since $\beta<1/2$ and +$\epsilon<\sqrt{5D}(A_0+5)/2$ and $\tilde Q_{(a,b)}\le 5D$, it follows from the +above distance estimate that this neck would contain $(p_0,\tilde t'_{(a,b)})$. +But this is impossible: since $(p_0,\tilde t'_{(a,b)})$ is an isolated fixed +point of an isometric $SO(3)$-action on the standard flow, all the sectional +curvatures at $(p_0,\tilde t'_{(a,b)})$ are equal, and this is in contradiction +with estimates on the sectional curvatures at any point of an $\epsilon$-neck +given in \entryref{dc5709757703}. We can then conclude from \entryref{6ff28dba4e56} +that for all $b$ sufficiently large, the point $(p_0,\tilde t'_{(a,b)})$ is +contained in the core of a $(C(\beta\epsilon/3),\beta\epsilon/3)$-cap +$Y_{(a,b)}$ in the $\tilde t'_{(a,b)}$ time-slice of the standard solution. +Now note that for +all $b$ sufficiently large, the scalar curvature of $(q_{(a,b)},\tilde +t'_{(a,b)})$ is at least $(0.99)\tilde Q_{(a,b)}$, since the scalar curvature of $x_{(a,b)}$ +is equal to $Q_{(a,b)}$. This implies that the +diameter of $Y_{(a,b)}$ is at most + +$$ +(1.01)\tilde Q^{-1/2}_{(a,b)}C(\beta\epsilon/3)<(1.1)\sqrt{5D}C(\beta\epsilon/3). +$$ + +Since $B(p_0,0,A)$ contains $B(p_0,\tilde t'_{(a,b)},A/2)$, and since +$C>C(\beta\epsilon/3)$, it follows from the definition of $A$, the above +distance estimate, and the triangle inequality that for all $b$ sufficiently +large $B(p_0,0,A)\times\{\tilde t'_{(a,b)}\}$ contains $Y_{(a,b)}$. + + Since $C> +C(\beta\epsilon/3)+1$ and since for $b$ sufficiently large +$\psi_{(a,b)}^*\widetilde G_{(a,b)}$ is arbitrarily close to the +restriction of the standard solution metric, it follows from +\entryref{7c2758f0e2b3} that for all $b$ sufficiently large, the image $\psi_{(a,b)}(Y_{(a,b)})$ + is a $(C,\epsilon)$-cap whose core contains $x_{(a,b)}$. +As we have already remarked, this contradicts the assumption that no $x_n$ +has a strong $(C,\epsilon)$-canonical neighborhood. + + This completes the proof in the last case and establishes +\entryref{778d690c4e7c}. + +\entryblock{1596df6ade35} + +Now we return to the proof of \entryref{254e8dae97e6}. For each $a$, we pass +to a subsequence (in $b$) so that \entryref{778d690c4e7c} holds for all $(a,b)$. For +each $(a,b)$, let $t_{(a,b)}$ be as in that lemma. We fix a point +$x_{(a,b)}\in \mathbf{t}^{-1}(t_{(a,b)})\subset {\mathcal M}_{(a,b)}$ at which the +canonical neighborhood assumption with parameter $r_a$ fails. For each $a$ +choose $b(a)$ such that $\delta_{b(a)}\rightarrow 0$ as $a\rightarrow \infty$. For each +$a$ we set $({\mathcal M}_a,G_a)=({\mathcal M}_{(a,b(a))},G_{(a,b(a))})$, we +set $t_a=t_{(a,b(a))}$, and we let $x_a=x_{(a,b(a))}\in {\mathcal M}_a$. Let +$(\widetilde {\mathcal M}_a,\widetilde G_a)$ be the Ricci flow with surgery +obtained from $({\mathcal M}_a,G_a)$ by shifting $t_a$ to $0$ and rescaling the +metric and time by $R(x_a)$. We have the points $\tilde x_a$ in the $0$ +time-slice of $\widetilde {\mathcal M}_a$ corresponding to $x_a\in {\mathcal +M}_a$. Of course, by construction $R_{\widetilde G_a}(\tilde x_a)=1$ for all +$a$. + +We shall take limits of a subsequence of this sequence of based +Ricci flows with surgery. Since $r_a\rightarrow 0$ and $R(x_a)\ge +r_a^{-2}$, it follows that $R(x_a)\rightarrow \infty$. By +\entryref{4ce8b5da9c58}, since $\delta_{b(a)}\le \delta(r_a)$ +it follows that the restriction of $(\widetilde {\mathcal +M}_a,\widetilde G_a)$ to $\mathbf{t}^{-1}(-\infty,0)$ is +$\kappa$-non-collapsed on scales $\le \epsilon R_{G_a}^{1/2}(x_a)$. +By passing to a subsequence we arrange that one of the following two +possibilities holds: + +- **(i)** There is $A<\infty$ and $t'<\infty$ such that, +for each $a$ there is a flow line through a point $y_a$ of $B_{\widetilde +G_a}(\tilde x_a,0,A)$ that is not defined on all of $[-t',0]$. (See +Fig. 17.2.) +- **(ii)** For every $A<\infty$ and every $t'<\infty$, for all $a$ +sufficiently large all flow lines through points of $B_{\widetilde +G_a}(\tilde x_a,0,A)$ are defined on the interval $[-t', 0]$. + +Let us consider the second case. By \entryref{4ce8b5da9c58} +these rescaled solutions are $\kappa$-non collapsed on scales $\le +\epsilon R_{G_a}(x_a)^{1/2}$ for all $t<0$. Since this condition is +a closed constraint, the same is true if $t=0$. Since $R(x_a)\ge +r_a^{-2}$, by construction every point $\tilde +x\in(\widetilde{\mathcal M}_a,\widetilde G_a)$ with $R(\tilde x)\ge +1$ and $\mathbf{t}(\tilde x)<0$ has a strong $(C,\epsilon)$-canonical +neighborhood. + +\entryblock{8de850960905} + +**Proof.** Assume that $\tilde x\in \widetilde{\mathcal M}$ has $R(\tilde +x)>1$. Suppose that $\tilde x$ is an exposed point. If $a$ is +sufficiently large, then $\delta_{b(a)}$ is arbitrarily close to +zero and hence by the last item in \entryref{cf61b731c6f2} and +the structure of the standard initial condition, we see that +$\tilde x$ is contained in the core of a $(2C,2\epsilon)$-cap. + +Suppose now that $\tilde x$ is not an exposed point. + Then we can take a sequence of points +$\tilde y_n\in \widetilde{\mathcal M}_a$ all lying on the flow line for the vector field through $\tilde x$ +converging to $\tilde x$ with $\mathbf{t}(\tilde y_n)<0$. Of course, for all $n$ sufficiently large +$R(\tilde y_n)>1$, +which implies that for all $n$ sufficiently large $\tilde y_n$ has a strong $(C,\epsilon)$-canonical +neighborhood. Passing to a subsequence, we can arrange that all of these canonical neighborhoods +are of the same type. If they are all $\epsilon$-round components, all $C$-components, or all +$(C,\epsilon)$-caps whose cores contain $y_n$, then by taking limits and arguing as in the proof of +\entryref{0760630d3990} we see that $\tilde x$ has a strong +$(2C,2\epsilon)$-canonical neighborhood of the same type. On the other hand, if $\tilde y_n$ is the center +of a strong $\epsilon$-neck for all $n$, then according to \entryref{d74749898b25}, +the limit point $\tilde x$ is the center +of a strong $2\epsilon$-neck. + +Since we have chosen $\epsilon>0$ sufficiently small so that \entryref{a9e023d082dc} +applies with $\epsilon$ replaced by $2\epsilon$, applying this theorem shows +that we can pass to a subsequence and take a smooth limiting +flow of a subsequence of the rescaled flows $(\widetilde {\mathcal +M}_a,\widetilde G_a)$ based at $\tilde x_a$ and defined for all +$t\in (-\infty,0]$. Because the $({\mathcal M}_a,G_a)$ all have +curvature pinched toward positive and since $R(x_a)\rightarrow \infty$ +as $a$ tends to infinity, this result says that the limiting flow +has non-negative, bounded curvature and is $\kappa$-non-collapsed on +all scales. That is to say, the limiting flow is a +$\kappa$-solution. By \entryref{e724298e5bc3} this contradicts +the fact that the strong $(C,\epsilon)$-canonical neighborhood +assumption fails at $x_a$ for every $a$. This contradiction shows +that in the second case there is a subsequence of the $a$ such that +$x_a$ has a strong canonical neighborhood and completes the proof of the second case. + +Let us consider the first case. In this case we will arrive at a +contradiction by showing that for all $a$ sufficiently large, the +point $x_a$ lies in a strong $(C,\epsilon)$-canonical neighborhood +coming from a surgery cap. Here is the basic result we use to find +that canonical neighborhood. + +\entryblock{111551d7831b} + +**Proof.** The proof is by contradiction. Suppose +the result does not hold. Then there are $A',D',t'<\infty $ and we +can pass to a subsequence (in $a$) such that the hypotheses of the +lemma hold for every $a$ but no $x_a$ has a strong +$(C,\epsilon)$-canonical neighborhood. The essential point of the +argument is to show that **in the units of the surgery scale** the +elapsed time between the surgery time and $0$ is less than $1$ and +the distance from the point $z_a$ to the tip of the surgery cap is +bounded independent of $a$. + + By \entryref{0b07beec84c0}, the fact +that the scalar curvature at $z_a$ is bounded by $D'$ implies that +for all $a$ sufficiently large the scale $\bar h_a$ of the surgery +at time $-t_a$ satisfies + +$$ +\bar h_a^2\ge (2D'D)^{-1}. +$$ + +(Recall that we are working in the rescaled flow $(\widetilde {\mathcal M}_a,\widetilde G_a)$.) + +Now we are ready to show that the elapsed time is bounded less than +one in the surgery scale. + +\entryblock{a01b63baabde} + +**Proof.** We consider two cases: either $t_a\le \bar h_a^2/2$ or $\bar +h_a^2/2D'$ as + $t_a\ge \bar h_a^2/2$. Hence, +$R(y_a,(-t_a+(2\theta_1-1)\bar h_a^2))\le 2t_aD'\bar +h_a^{-2}\le 2t'D'\bar h_a^{-2}$. This contradiction +shows that if $a$ is sufficiently large then $t_a<\theta_1\bar +h_a^2$. + +We pass to a subsequence so that $t_a\bar h_a^{-2}$ converges to +some $\theta\le \theta_1$. We define $\widetilde C$ to be the +maximum of $C$ and $3\epsilon^{-1}\beta^{-1}$. Now, using Part 5 of +\entryref{f44d182ef27e} we set $A''\ge (9\widetilde +C+3A')\sqrt{2DD'}+6(A_0+5)$ sufficiently large so that in the + standard flow $B(p_0,0,A'')$ contains +$B(p_0,t,A''/2)$ for any $t\le (\theta_1+1)/2$. This constant is chosen only to +depend on $\theta_1$, $A'$, and $C$. As $a$ tends to infinity, $\delta_a$ tends +to zero which means, by \entryref{ad4f67ea75bc}, that for all $a$ sufficiently +large there is an embedding $\rho_a\colon B(p_0,-t_a,A''\bar h_a)\times +[-t_a,0]\to \widetilde {\mathcal M}_a$ compatible with time and the vector +field such that (after translating by $t_a$ to make the flow start at time $0$ +and scaling by $\bar h_a^{-2}$) the restriction of $\widetilde G_a$ to this +image is close in the $C^\infty$-topology to the restriction of the standard +flow to $B(p_0,0,A'')\times [0,\bar h_a^{-2}t_a]$. The image $\rho_a(p_0,-t_a)$ +is the tip $p_a$ of the surgery cap ${\mathcal C}_a$ in $\widetilde {\mathcal +M}_a$. In particular, for all $a$ sufficiently large the image +$\rho_a\left(B(p_0,-t_a,A''\bar h_a)\times\{0\}\right)$ contains the $A''\bar +h_a/3$-neighborhood of the image $\rho_a(p_0,0)$ of the tip of the surgery cap +under the flow forward to time $0$. By our choice of $A''$, and +Equation (17.2), this means that for all $a$ sufficiently large +$\rho_a\left(B(p_0,-t_a,A''\bar h_a)\times\{-t_a\}\right)$ contains the +$(3\widetilde C+A') +2(A_0+5)\bar h_a$ neighborhood of $p_a=\rho_a(p_0,-t_a)$. +Notice also that, since the standard solution has positive curvature and hence +the distance between points is non-increasing in time by +\entryref{51a2afee4465}, the distance at time $0$ between $\rho_a(p_0,0)$ and +$y_a$ is less than $2(A_0+5)\bar h_a$. By the triangle inequality, we conclude +that for all $a$ sufficiently large, $\rho_a\left(B(p_0,-t_a,A''\bar h_a)\times +\{0\}\right)$ contains the $3\widetilde C$-neighborhood of $x_a$. Since the +family of metrics on $\rho_a\left(B(p_0,-t_a,A''\bar h_a)\times +[-t_a,0]\}\right)$ (after time-shifting by $t_a$ and rescaling by $\bar +h_a^{-2}$) are converging smoothly to the ball $B(p_0,0,A'')\times [0,\theta]$ +in the standard flow, + for all $a$ sufficiently large then the flow from time $-t_a$ to $0$ on the +$3\widetilde C$-neighborhood of $x_a$ is, after rescaling by $\bar +h_a^{-2}$, + very nearly isometric to the restriction of the standard flow from time $0$ to $\bar h^{-2}_at_a$ on + the $3\widetilde C\bar h_a^{-1}$-neighborhood of some +point $q_a$ in the standard flow. Of course, since the scalar +curvature of $x_a$ is one, $R(q_a,\bar h_a^{-2}t_a)$ in the standard flow is +close to $\bar h_a^{-2}$. Hence, by \entryref{6ff28dba4e56} +there is a neighborhood $X$ of $(q_a,\bar h_a^{-2}t_a)$ in the standard solution +that either is a $(C,\epsilon)$-cap, or is an evolving +$\beta\epsilon/3$-neck centered at $(q_a,\bar h_a^{-2}t_a)$. In the latter case +either the evolving neck is defined for backward time +$(1+\beta\epsilon/3)$ or its initial time-slice is the zero +time-slice and this initial time-slice lies at distance at least $1$ +from the surgery cap. Of course, $X$ is contained in the +$CR(q_a,\bar h_a^{-2}t_a)^{-1/2}$ neighborhood of $(q_a,\bar h_a^{-2}t_a)$ in the standard +solution. Since $\widetilde C\ge C$ and $R(q_a,\bar h_a^{-2}t_a)$ is close to +$\bar h_a^{-2}$, the neighborhood $X$ is contained in the +$2\widetilde C\bar h_a^{-1}$-neighborhood of $(q_a,\bar h_a^{-2}t_a)$ in the +standard solution. Hence, after rescaling, the corresponding +neighborhood of $x_a$ is contained in the $3\widetilde +C$-neighborhood of $x_a$. If either of the first two cases in \entryref{6ff28dba4e56} occurs +for a subsequence of $a$ tending to infinity, then by +\entryref{7c2758f0e2b3} and the fact that $\widetilde C>\mathit{max}(C,\epsilon^{-1})$, we see that there is a subsequence of $a$ +for which $x_a$ either is contained in the core of a +$(C,\epsilon)$-cap or is the center of a strong $\epsilon$-neck. + +We must examine further the last case. We suppose that for every $a$ +this last case holds. Then for all $a$ sufficiently large we have an +$\beta\epsilon/3$-neck $N_a$ in the zero time-slice of $\widetilde{\mathcal +M}_a$ centered at $x_a$. It is an evolving neck and there is an +embedding $\psi\colon N_a\times [-t_a,0]\to \widetilde{\mathcal +M}_a$ compatible with time and the vector field so that the initial +time-slice $\psi(N_a\times \{-t_a\})$ is in the surgery time-slice +$M_{-t_a}$ and is disjoint from the surgery cap, so in fact it is +contained in the continuing region at time $-t_a$. As we saw above, +the image of the central $2$-sphere $\psi(S^2_a\times\{-t_a\})$ lies +at distance at most $ A''\bar h_a$ from the tip of the surgery cap +$p_a$ (where, recall, $A''$ is a constant independent of $a$). The +$2$-sphere, $\Sigma_a$, along which we do surgery, creating the surgery +cap with $p_a$ as its tip, is the central $2$-sphere of a strong +$\delta_{b(a)}$-neck. As $a$ tends to infinity the surgery control +parameter $\delta_{b(a)}$ tends to zero. Thus, for $a$ sufficiently +large this strong $\delta_{b(a)}$-neck will contain a strong +$\beta\epsilon/2$- neck $N'$ centered at $\psi(x_a,-t_a)$. Since we +know that the continuing region at time $-t_a$ contains a +$\beta\epsilon/3$-neck centered at $(x_a,-t_a)$, it follows that +$N'$ is also contained in $C_{-t_a}$. That is to say, $N'$ is +contained in the negative half of the $\delta_{b(a)}$-neck centered +at $\Sigma_a$. Now we are in the situation of +\entryref{a992d6a89479}. Applying this result tells us that $x_a$ +is the center of a strong $\epsilon$-neck. + +This completes the proof that for all $a$ sufficiently large, $x_a$ +has a $(C,\epsilon)$-canonical neighborhood in contradition to our +assumption. This contradiction completes the proof of +\entryref{111551d7831b}. + +There are several steps required to complete the proof of +\entryref{254e8dae97e6}. The first step helps us apply the previous claim to +find strong $(C,\epsilon)$-canonical neighborhoods. + +\entryblock{7549bace9302} + +**Proof.** Since all points $y\in ({\mathcal M}_a,G_a)$ with $R_{G_a}(y)\ge +r^{-2}_a$ and $\mathbf{t}(y)<\mathbf{t}(x_a)$ have strong +$(C,\epsilon)$-canonical neighborhoods, and since $R(x_a)=r^{-2}_a$, +we see that all points $y\in (\widetilde{\mathcal M}_a,\widetilde +G_a)$ with $\mathbf{t}(y_a)<0$ and with $R_{\tilde G_a}(y_a)\ge 1$ have +strong $(C,\epsilon)$-canonical neighborhoods. It follows that all +points in $(\widetilde {\mathcal M}_a,\widetilde G_a)$ with $\mathbf{t}(y)\le 0$ and $R(y)>1$ have strong $(2C,2\epsilon)$-canonical +neighborhoods. Also, since $\delta_a\le \delta(r_a)$, where +$\delta(r_a)$ is the constant given in \entryref{4ce8b5da9c58}, +and since the condition of being $\kappa$-non-collapsed is a closed +constraint, it follows from \entryref{4ce8b5da9c58} that these +Ricci flows with surgery are $\kappa$-non-collapsed for a fixed +$\kappa>0$. It is now immediate from \entryref{ecd62fc2ea2b} that there +is a constant $D_0(A)$ such that $R$ is bounded above on +$B_{\widetilde G_a}(\tilde x_a,0,A)$ by $D_0(A)$. Since every point +$y\in ({\mathcal M}_a,G_a)$ with $R(y)>1$ of the sequence of with +scalar curvature at least $1$ has a $(C,\epsilon)$ canonical +neighborhood, it follows from the definition that for every such +point $y$ we have $\left|\partial R(y)/\partial t\right|0$ and a bound $D'(A)$, both depending only in +$D_0(A)$, + for the scalar curvature at all points +of backward flow lines beginning in $B_{\widetilde +G_a}(\tilde x_a,0,A)$ and defined for backward time at most $\delta(A)$. Since the curvature is pinched toward positive, +it follows that there is a bound $D(A)$ depending only on $D'(A)$ to $|\mathit{Rm}|$ on the same flow lines. + +\entryblock{3b7da22bf357} + +**Proof.** First notice that if there is $t(A)>0$ for which the backwards parabolic neighborhood +$P=P_{\widetilde G_a}(\tilde x_a,0,A,-t(A))$ exists, +then, by \entryref{7549bace9302}, +there are constants $D(A)<\infty$ and $\delta(A)>0$ such that, replacing $t(A)$ by $\mathit{min}(t(A),\delta(A))$, + $|\mathit{Rm}|$ is bounded by $D(A)$ on $P$. +Thus, either Item (1) holds or +passing to a subsequence, we can suppose that there is some $A<\infty$ for which no $t(A)>0$ as +required by Item (1) exists. +Then, for each $a$ we find a point +$y_a\in B_{\widetilde G_a}(\tilde x_a,0,A)$ such that the backwards +flow line from $y_a$ meets a surgery cap at a time $-t_a$ where +$\mathit{lim}_{a\rightarrow \infty}(t_a)=0$. Then, by the previous claim, for all $a$ sufficiently large, the +sectional curvature along any backward flow line beginning in $B_{\widetilde G_a}(\widetilde x_a,0,A)$ +and defined for backward time $t_a$ is bounded by +a constant $D(A)$ independent of $a$. Under our assumption this +means that for all $a$ sufficiently large, there is a point $y_a\in +B_{\widetilde G_a}(\tilde x_a,0,A)$ and a backwards flow line +starting at $y_a$ ending at a point $z_a$ of a surgery cap, and the +sectional curvature along this entire flow line is bounded by $D(A)<\infty$. Thus, +applying \entryref{111551d7831b} produces the strong +$(C,\epsilon)$-canonical neighborhood around $x_a$, proving the claim. + +But we are assuming that no $x_a$ has a strong $(C,\epsilon)$-canonical +neighborhood. Thus, the consequence of the previous claim is that for each +$A<\infty$ there is a $t(A)>0$ such that for all $a$ sufficiently large +$P_{\widetilde G_a}(\tilde x_a,0,A,-t(A))$ exists in $\widetilde {\mathcal +M}_a$ and there is a bound, depending only on $A$ for $|\mathit{Rm}|$ on +this backward parabolic neighborhood. Applying \entryref{d9f3522dec12} we see that, after passing to a +subsequence, there is a smooth limit $(M_\infty,g_\infty,x_\infty)$ to the zero +time-slices $(\widetilde{\mathcal M}_a,\widetilde G_a,\tilde x_a)$. +Clearly, since the curvatures of the sequence are pinched toward positive, this +limit has non-negative curvature. + +Lastly, we show that $(M_\infty,g_\infty)$ has bounded curvature. By Part 3 of \entryref{7c2758f0e2b3} +each point of $(M_\infty,g_\infty)$ with scalar curvature +greater than one has a $(2C,2\epsilon)$-canonical neighborhood. If a point lies in an $2\epsilon$-component +or in a $2C$-component, then $M_\infty$ is compact, and hence clearly has bounded curvature. +Thus, we can assume that each $y\in M_\infty$ with $R(y)>1$ is either the center of a $2\epsilon$-neck +or is contained in the core of a $(2C,2\epsilon)$-cap. According to \entryref{cb9ca6d6db9f} + $(M_\infty,g_\infty)$ does not contain $2\epsilon$-necks of arbitrarily high curvature. + It now follows then that $(M_\infty,g_\infty)$ there is a bound to the scalar curvature of any + $2\epsilon$-neck and of any $(2C,2\epsilon)$-cap, and hence it follows that $(M_\infty,g_\infty)$ + has bounded curvature. + +\entryblock{8dafe9c95b7f} + +**Proof.** Let $Q$ be the bound of the scalar curvature of +$(M_\infty,g_\infty,x_\infty)$. Then by \entryref{0ddfe578961e} + there is a constant +$\Delta t>0$ such that if $R_{\widetilde G_a}(y,0)\le 2Q$, then the +scalar curvature is bounded by $16Q$ on the backward flow line from +$y$ defined for any time $\le \Delta t$. Suppose that there is +$A<\infty$ and a subsequence of $a$ for which there is a flow line +beginning at a point $y_a \in B_{\widetilde G_a}(\tilde x_a,0,A)$ +defined for backward time at most $\Delta t$ and ending at a point +$z_a$ of a surgery cap. Of course, the fact that the scalar +curvature of $(M_\infty,g_\infty)$ is at most $Q$ implies that for +all $a$ sufficiently large, the scalar curvature of $B_{\widetilde +G_a}(\tilde x_a,0,A)$ is less than $2Q$. This implies that for all +$a$ sufficiently large the scalar curvature along the flow line +from $y_a$ to $z_a$ in a surgery cap is $\le 16Q$. Now invoking +\entryref{111551d7831b} we see that for all $a$ sufficiently large the +point $\widetilde x_a$ has a strong $(C,\epsilon)$-canonical +neighborhood. This is a contradiction, and this contradiction proves +that we can choose $t(A)>0$ independent of $A$. + +Since we are assuming that no $x_a$ has a strong +$(C,\epsilon)$-canonical neighborhood, this means that it is +possible to find a constant $t'>0$ such that $t(A)\ge t'$ for all +$A<\infty$. Now let $0T'$, after passing to a subsequence, for all +$a$ sufficiently large there is $A<\infty$ and a backwards flow +line, defined for a time less than $t$, starting at a point $y_a$ of +$B_{\widetilde G_a}(\tilde x_a,0,A)$ and ending at a point $z_a$ of +a surgery cap. Invoking \entryref{0ddfe578961e} again, we see that +for all $a$ sufficiently large, the scalar curvature is bounded on +the flow line from $y_a$ to $z_a$ by a constant independent of $a$. +Hence, as before, we see that for all $a$ sufficiently large $x_a$ +has a strong $(C,\epsilon)$-canonical neighborhood; again this is a contradiction. + +Hence, we have now shown that our assumption that the strong +$(C,\epsilon)$-canonical neighborhood assumption fails for all $r_a$ +and all $\delta_{a,b}$ leads to a contradiction and hence is false. + +This completes the proof of \entryref{254e8dae97e6}. + +## Surgery times don't accumulate + +Now we turn to the proof of \entryref{703e4cfeece7}. + Given surgery parameter +sequences + +$$ +\Delta_i=\{\delta_0,\ldots,\delta_i\} +$$ + +$$ +\mathbf{r_i}=\{r_0,\ldots,r_i\} +$$ + +$$ +\mathbf{K_i}=\{\kappa_0,\ldots,\kappa_i\}, +$$ + + we let +$r_{i+1}$ and $\delta_{i+1}$ be as in \entryref{254e8dae97e6} and +then set $\kappa_{i+1}=\kappa(r_{i+1})$ as in +\entryref{4ce8b5da9c58}. + Set + +$$ +{\mathbf{r}_{i+1}}=\{\mathbf{r_i},r_{i+1}\} +$$ + +$$ +{\mathbf{K}_{i+1}}=\{\mathbf{K_i},\kappa_{i+1}\} +$$ + +$$ +\Delta_{i+1}=\{\delta_0,\ldots,\delta_{i-1},\delta_{i+1},\delta_{i+1}\}. +$$ + +Of course, these are also surgery parameter sequences. + +Let $\overline \delta\colon [0,T]\to \Ar^+$ be any non-increasing positive +function and let $({\mathcal M},G)$ be a Ricci flow with surgery defined on +$[0,T)$ for some $T\in [T_i,T_{i+1})$ with surgery control parameter +$\overline\delta$. Suppose $\overline \delta\le \Delta_{i+1}$ and that this +Ricci flow with surgery satisfies the conclusion of \entryref{703e4cfeece7} with +respect to these sequences on its entire interval of definition. We wish to +extend this Ricci flow with surgery to one defined on $[0,T')$ for some $T'$ +with $T0$ and the canonical +neighborhood parameter $r$ at time $t_0$ is at least $r_{i+1}>0$, it follows +that $h(t_0)\ge h(T_{i+1})>0$. Thus, each surgery at time $ t_0\le T_{i+1}$ +along a $2$-sphere removes at least a fixed amount of volume depending on +$\overline \delta(T_{i+1})$ and $r_{i+1}$. Since under Ricci flow the volume +grows at most exponentially, we see that there is a bound depending only on +$\overline\delta(T_{i+1})$, $T_{i+1}$, $r_{i+1}$ and $\mathit{Vol}(M_0,g(0))$ to +the number of $2$-sphere surgeries that we can do in this time interval. On the +other hand, the number of components at any time $t$ is at most $N_0+S(t)-D(t)$ +where $N_0$ is the number of connected components of $M_0$, $S(t)$ is the +number of $2$-sphere surgeries performed in the time interval $[0,t)$ and +$D(t)$ is the number of connected components removed by surgeries at times in +the interval $[0,t)$. Hence, there is a bound on the number of components in +terms of $N_0$ and $S(T)$ that can be removed by surgery in the interval +$[0,T)$. Since the initial conditions are normalized, $N_0$ is bounded by the +volume of $(M_0,g(0))$. This completes the proof of the result. + +This lemma completes the proof of the fact that for any $T\le T_{i+1}$, we +encounter only a fixed bounded number surgeries in the Ricci flow with +surgery from $0$ to $T$. The bound depends on the volume of the initial +manifold as well as the surgery constants up to time $T_{i+1}$. In particular, +for a given initial metric $(M_0,g(0))$ there is a uniform bound, depending +only on the surgery constants up to time $T_{i+1}$, on the number of surgeries +in any Ricci flow with surgery defined on a subinterval of $[0,T_{i+1})$. It +follows that the surgery times cannot accumulate in any finite interval. This +completes the proof of \entryref{703e4cfeece7}. + +To sum up, we have sequences $\Delta$, $\mathbf{K}$ and $\mathbf{r}$ as given in +\entryref{703e4cfeece7}. Let $\overline \delta\colon [0,\infty)\to \Ar$ be a +positive, non-increasing function with $\overline \delta\le \Delta$. Let $M$ be +a compact $3$-manifold that contains no embedded $\Ar P^2$ with trivial normal +bundle. We have proved that for any normalized initial Riemannian metric +$(M_0,g_0)$ there is a Ricci flow with surgery with time-interval of definition +$[0,\infty)$ and with $(M_0,g_0)$ as initial conditions. This Ricci flow with +surgery is $\mathbf{K}$-non-collapsed and satisfies the strong +$(C,\epsilon)$-canonical neighborhood theorem with respect to the parameter +$\mathbf{r}$. It also has curvature pinched toward positive. Lastly, for any $T\in +[0,\infty)$ if there is a surgery at time $T$ then this surgery is performed +using the surgery parameters $\overline\delta(T)$ and $r(T)$, where if $T\in +[T_i,T_{i+1})$ then $r(T)=r_{i+1}$. In this Ricci flow with surgery, there are +only finitely many surgeries on each finite time interval. As far as we know +there may be infinitely many surgeries in all. diff --git a/projects/poincare-conjecture/.astrolabe/docs/19-finite-time-extinction.mdx b/projects/poincare-conjecture/.astrolabe/docs/19-finite-time-extinction.mdx new file mode 100644 index 00000000..e4bccb80 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/docs/19-finite-time-extinction.mdx @@ -0,0 +1,4499 @@ + + +# Chapter 18 -- Finite-time extinction + +Our purpose in this chapter is to prove the following finite-time +extinction theorem for certain Ricci flows with surgery which, as we +shall show below, when combined with the theorem on the existence of +Ricci flows with surgery defined for all $t\in [0,\infty)$ +(\entryref{703e4cfeece7}), immediately yields \entryref{9999efa2be2c}, +thus completing the proof of the Poincaré Conjecture and the +$3$-dimensional space-form conjecture. + +## The result + +\entryblock{4b8a44aa316f} + +Let us quickly show how this theorem implies our main result +\entryref{9999efa2be2c}. + +**Proof.** (of \entryref{9999efa2be2c} assuming +\entryref{4b8a44aa316f}). Fix a normalized metric $g(0)$ on $M$, and let +$({\mathcal M},G)$ be the Ricci flow with surgery defined for all $t\in +[0,\infty)$ produced by \entryref{703e4cfeece7} with initial conditions $(M,g(0))$. +According to \entryref{4b8a44aa316f} there is $T>0$ for which the time-slice +$M_T$ is empty. By \entryref{7822d06083e0}, if there is $T$ for which $M_T$ +is empty, then for any $T'T_0$ the time-slice $M_{T'}$ is empty. The argument +here is similar in spirit. There are two approaches. The first +approach is due to Perelman [Perelman3]. Here, one represents a +non-trivial element in $\pi_3(M_{T_0},x_0)$ by a non-trivial +element in $\pi_2(\Lambda M,*)$, where $\Lambda M$ is the free loop +space on $M$ and $*$ is the trivial loop at $x_0$. For any compact +family $\Gamma$ of homotopically trivial loops in $M$ we consider +the areas of minimal spanning disks for each of the loops in the +family and set $W(\Gamma)$ equal to the maximal area of these +minimal spanning disks. For a given element in $\gamma\in +\pi_2(\Lambda M)$ we set $W(\gamma)$ equal to + the infimum over all representative $2$-sphere families $\Gamma$ +for $\gamma$ of $W(\Gamma)$. Under Ricci flow, the forward +difference quotient of this invariant satisfies an inequality and +the distance-decreasing property of surgery +(\entryref{9021a4eca03f}) says that the inequality remains +valid for Ricci flow with surgery. The inequality implies that the +value $W(\gamma)$ goes negative in finite time, which is impossible. + +The other approach, by Colding-Minicozzi [ColdingMinicozzi], is +to represent a non-trivial element in $\pi_3(M_T)$ as a non-trivial +element in $\pi_1(\mathit{Maps}(S^2,M))$, and associate to such an +element the infimum over all representative families of the maximal +energy of the $2$-spheres in the family. Again, one shows that under +Ricci flow the forward difference quotient of this minimax satisfies +an inequality that implies that it goes negative in finite time. As +before, the distance-decreasing property of surgery +(\entryref{9021a4eca03f}) implies that this inequality is +valid for Ricci flows with surgery. This tells us that the manifold +must completely disappear in finite time. + +Our first reaction was that, of the two approaches, the one +considered by Colding-Minicozzi was preferable since it seemed more +natural and it had the advantage of avoiding the boundary issues +that occupy most of Perelman's analysis in [Perelman3]. + In the Colding-Minicozzi approach one must construct paths of $2$-spheres with the + property that when the energy of the $2$-sphere is close to the + maximum value along the path, then the $2$-sphere in question + represents a point in the space $\mathit{Maps}(S^2,M)$ that is + close to a (usually) non-minimal critical point for the energy + functional on this space. Such paths are needed in order to + establish the forward difference quotient result alluded to above. +In Perelman's approach, one deals only with area-minimizing disks so +that one avoids having to deal with non-minimal critical points at +the expense of dealing with the technical issues related to the +boundary. Since the latter are one-dimensional in nature, they are +much easier to handle. In the end we decided to follow Perelman's +approach, and that is the one we present here. In [Perelman3] +there were two points that we felt required quite a bit of argument +beyond what Perelman presented. In Section 2.2 on page 4 of +[Perelman3], Perelman asserts that there is a local, pointwise +curvature estimate that can be obtained by adapting arguments in the +literature; see Lemmas \entryref{38712e8cea3d} and \entryref{ff5d51a483d9} for the precise +statement. To implement this adaption required further non-trivial +arguments. We present these arguments in Section 18.7. In Section +2.5 on page 5 of [Perelman3] Perelman asserts that an +elementary argument establishes a lower bound on the length of a +boundary curve of a minimal annulus; see +\entryref{6e9739b0c59b} for a precise statement. While the +statement seems intuitively clear, we found the argument, while +elementary, was quite intricate. We present this argument in +Section 18.6. + +The first use of these types of ideas to show that geometric objects +must disappear in finite time under Ricci flow is due to Hamilton +[HamiltonNSRF3M]. He was considering a situation where a +time-slice $(M,g(t_0))$ of a $3$-dimensional Ricci flow had +submanifolds on which the metric was close to (a truncated version) +of a hyperbolic metric of finite volume. He wished to show that +eventually the boundary tori of the truncation were incompressible +in the $3$-manifold. If not, then there would be an immersed minimal +disk in $M$ whose boundary was a non-trivial loop on the torus. He +represented this relative homotopy class by a minimal energy disk in +$(M,g(t_0))$ and proved the same sort of forward difference quotient +estimate for the area of the minimal disk in the relative homotopy +class. The same contradiction -- the forward difference quotient +implies that after a finite time the area would go negative if the +disk continued to exist --- implies that after a finite amount of +time this compressing disk must disappear. Using this he showed that +for sufficiently large time all the boundary tori of almost +hyperbolic submanifolds in $(M,g(t))$ were incompressible. + +In the next section we deal with $\pi_2$ and, using $W_2$, we show +that given a Ricci flow with surgery as in \entryref{703e4cfeece7} there +is $T_1<\infty$ such that for all $T\ge T_1$ every connected +component of $M_T$ has trivial $\pi_2$. Then in the section after +that, by analyzing $W_3$, we show that, under the group-theoretic +hypothesis of \entryref{4b8a44aa316f}, there is a $T_2<\infty$ such +that $M_T=\emptyset$ for all $T\ge T_2$. In both these arguments we +need the same type of results -- a forward difference inequality for +the energy function; the statement that away from surgery times +this function is continuous; and lastly, the statement that the +value of the energy function at a surgery time is at most the liminf +of its values at a sequence of times approaching the surgery time +from below. + +### Existence of the Ricci flow with surgery + +Let $(M,g(0))$ be as in the statement of \entryref{4b8a44aa316f}, so that $M$ is +a compact, connected $3$-manifold whose fundamental group is a free product of +finite groups and infinite cyclic groups. By scaling $g(0)$ by a sufficiently +large constant, we can assume that $g(0)$ is normalized. Let us show that such +a manifold cannot contain an embedded $\RR P^2$ with trivial normal bundle. +First note that since $\RR P^2$ has Euler characteristic one, it is not the +boundary of a compact $3$-manifold. Hence, an $\RR P^2$ embedded with trivial +normal bundle does not separate the connected component of $M$ containing it. +Also, any non-trivial loop in $\Ar P^2$ has non-trivial normal bundle in $M$ so +that inclusion of $\RR P^2$ into $M$ induces an injection on fundamental +groups. Under the fundamental group hypotheses, $M$ decomposes as a connected +sum of $3$-manifolds with finite fundamental groups and $2$-sphere bundles over +$S^1$, see [Hempel]. + Given +an $\RR P^2$ with trivial normal bundle embedded in a connected sum, +it can be replaced by one contained in one of the connected factors. +[Proof: Let $\Sigma=\Sigma_1\cup\cdots\cup\Sigma_n$ be the spheres +giving the connected sum decomposition of $M$. Deform the $\Ar P^2$ +until it is transverse to $\Sigma$ and let $\gamma$ be a circle of +intersection of $\Ar P^2$ with one of the $\Sigma_i$ that is +innermost on $\Sigma_i$ in the sense that $\gamma$ bounds a disk $D$ +in $\Sigma_i$ disjoint from all other components of intersection of +$\Sigma_i$ and $\Ar P^2$. The loop $\gamma$ also bounds a disk $D'$ +in $\Ar P^2$. Replace $D'$ by $D$ and push $D$ slightly off to the +correct side of $\Sigma_i$. This will produce a new embedded $\Ar +P^2$ with trivial normal bundle in $M$ and at least one fewer +component of intersection with $\Sigma$. Continue inductively until +all components of intersection with $\Sigma$ are removed.] + +Now suppose that we have an $\Ar P^2$ with trivial normal bundle +embedded disjointly from $\Sigma$, and hence embedded in one of the +prime factors of $M$. + Since it +does not separate this factor, by the Mayer-Vietoris sequence (see p. 149 of +[Hatcher]) the first homology of the factor in question maps onto $\Zee$ +and hence the factor in question has infinite fundamental group. But this group +also contains the cyclic subgroup of order two, namely the image of $\pi_1(\RR +P^2)$ under the map induced by the inclusion. Thus, the fundamental group of +this prime factor is not finite and is not infinite cyclic. This is a +contradiction. (We have chosen to give a topological argument for this result. +There is also an argument using the theory of groups acting on trees which is +more elementary in the sense that it uses no $3$-manifold topology. Since it is +a more complicated, and to us, a less illuminating argument, we decided to +present the topological argument.) + +Thus, by \entryref{703e4cfeece7}, for any compact $3$-manifold $M$ whose +fundamental group is a free product of finite groups and infinite cyclic groups +and for any normalized metric $g(0)$ on $M$ there is a Ricci flow with surgery +$({\mathcal M},G)$ defined for all time $t\in [0,\infty)$ satisfying the +conclusion of \entryref{703e4cfeece7} with $(M,g(0))$ as the initial conditions. + +\entryblock{5b649fe97568} + +Let ${\mathcal X}$ be a path of components in a Ricci flow with surgery +$({\mathcal M},G)$, a path defined for all $t\in I$. Let $I'$ be a subinterval +of $I$ with the property that no point of $I'$ except possibly its initial +point is a surgery time. Then the intersection of ${\mathcal X}$ with $\mathbf{t}^{-1}(I')$ is the Ricci flow on the time interval $I'$ applied to ${\mathcal +X}(t)$ for any $t\in I'$. Thus, for such intervals $I'$ the intersection, +${\mathcal X}(I')$, of ${\mathcal X}$ with $\mathbf{t}^{-1}(I')$ is determined by +the time-slice ${\mathcal X}(t)$ for any $t\in I'$. That is no longer +necessarily the case if some point of $I'$ besides its initial point is a +surgery time. Let $t\in I$ be a surgery time, distinct from the initial point +of $I$ (if there is one), and let $I'\subset I$ be an interval of the form +$[t',t)$ for some $t'1$ we +consider the perturbed energy $E_\alpha$ given by + +$$ +E_\alpha(s)=\int_{S^2}\left(1+|ds|^2\right)^\alpha da. +$$ + + According to +[SacksUhlenbeck] this energy functional is Palais-Smale on the +space of $H^{1,2\alpha}$ maps and has an absolute minimum among +homotopically non-trivial maps, realized by a map $s_\alpha\colon +S^2\to X$. We consider a decreasing sequence of $\alpha$ tending to +$1$ and the minimizers $s_\alpha$ among homotopically non-trivial +maps. According to [SacksUhlenbeck], after passing to a +subsequence, there is a weak limit which is a strong limit on the +complement of a finite set of points in $S^2$. This limit extends to +a harmonic map of $S^2\to M$, and its energy is less than or equal +to the limit of the $\alpha$-energies of $s_\alpha$. If the result +is homotopically non-trivial then it realizes a minimum value of the +usual energy among all homotopically non-trivial maps, for were +there a homotopically non-trivial map of smaller energy, it would +have smaller $E_\alpha$ energy than $s_\alpha$ for all $\alpha$ +sufficiently close to $1$. Of course if the limit is a strong limit, +then the map is homotopically non-trivial, and the proof is +complete. + +We must examine the case when the limit is truly a weak limit. Let +$s_n$ be a sequence as above with a weak limit $s$. If the limit is +truly a weak limit, then there is bubbling. Let $x\in S^2$ be a +point where the limit $s$ is not a strong limit. Then according to +[SacksUhlenbeck] pre-composing with a sequence of conformal +dilations $\rho_n$ centered at this point leads to a sequence of +maps $s_n'$ converging uniformly on compact subsets of $\Ar ^2$ to a +non-constant harmonic map $s'$ that extends over the one-point +compactification $S^2$. The energy of this limiting map $s'$ is at +most the limit of the $\alpha$-energies of the $s_\alpha$. If $s'$ +is homotopically non-trivial, then, arguing as before, we see that +it realizes the minimum energy among all homotopically non-trivial +maps, and once again we have completed the proof. We rule out the +possibility that $s'$ is homotopically trivial. Let $\alpha$ be the +area, or equivalently the energy, of $s'$. Let $D\subset \Ar^2$ be a +disk centered at the origin which contains three-quarters of the +energy of $s'$ (or equivalently three-quarters of the area of $s'$), +and let $D'$ be the complementary disk to $D$ in $S^2$. For all $n$ +sufficiently large the area of $s_n'|D$ minus the area of $s_n'|D'$ +is at least $\alpha/3$. The restrictions of $s_n'$ on $\partial D$ +are converging smoothly to $s'|\partial D'$. Let $D_n\subset S^2$ be +$\rho_n^{-1}(D)$. Then the area of $s_n|D_n$ equals the area of +$s_n'|D$ and hence is at least the area of $s'|D'$ plus $\alpha/4$ +for all $n$ sufficiently large. Also, as $n$ tends to infinity the +image $s_n(D_n)$ converges smoothly, after reparameterization, to +$s'(\partial D)$. Thus, for all $n$ large, we can connect +$s_n(\partial D_n)$ to $s'(\partial D')$ by an annulus $A_n$ +contained in a small neighborhood of $s'(\partial D')$ and whose +area tends to $0$ as $n$ goes to infinity. For all $n$ sufficiently +large, the resulting $2$-sphere $\Sigma_n$ made out of +$s_n|(S^2\setminus D_n)\cup A_n\cup S'(D')$ is homotopic to $s(S^2)$ +since $s'$ is homotopically trivial. Also, for all $n$ sufficiently +large, the area of $\Sigma_n$ is less than the area of $s_n$ minus +$\alpha/5$. Reparameterizing this $2$-sphere by a conformal map +leads to a homotopically non-trivial map of energy less than the +area of $s_n$ minus $\alpha/5$. Since as $n$ tends to infinity, the +limsup of the areas of the $s_n$ converge to at most $e_0$, for all +$n$ sufficiently large we have constructed a homotopically +non-trivial map of energy less than $e_0$, which contradicts the +fact that the minimal $\alpha$ energy for a homotopically +non-trivial map tends to $e_0$ as $\alpha$ tends to $1$. + +Of course, any minimal energy map of $S^2$ into $M$ is conformal because there +is no non-trivial holomorphic quadratic differential on $S^2$. It follows that +such a map is a branched immersion. + +Now suppose that ${\mathcal X}$ is a path of components defined for +all $t\in [T_0,\infty)$ with $\pi_2({\mathcal X}(t))\not= 0$ for all +$t\in [T_0,\infty)$. + For +each $t\ge T_0$ we define $W_2(t)$ to be $e_0({\mathcal X}(t))$, where $e_0$ is +the invariant given in the previous lemma. Our assumption on ${\mathcal X}$ +means that $W_2(t)$ is defined and positive for all $t\in [T_0,\infty)$. + +\entryblock{7bb0fa0c9412} + +Let us show how this lemma implies \entryref{4415092c2079}. Because +the curvature is pinched toward positive, we have + +$$ +R_\mathit{min}(t)\ge (-6)/(1+4t). +$$ + +Let $w_2(t)$ be the function satisfying the differential equation + +$$ +\frac{dw_2}{dt}=-4\pi+\frac{3w_2}{1+4t} +$$ + +and $w_2(T_0)=W_2(T_0)$. Then by \entryref{67c378f87457} and +\entryref{7bb0fa0c9412} we have $W_2(t)\le w_2(t)$ for all $t\ge T_0$. On +the other hand, we can integrate to find + +$$ +w_2(t)=w_2(T_0)\frac{(4t+1)^{3/4}}{(4T_0+1)^{3/4}}+4\pi +(4T_0+1)^{1/4}(4t+1)^{3/4}-4\pi (4t+1). +$$ + + Thus, for $t$ sufficiently +large, $w_2(t)<0$. This is a contradiction since $W_2(t)$ is always +positive, and $W_2(t)\le w_2(t)$. + +This shows that to complete the proof of \entryref{4415092c2079} we +need only establish \entryref{7bb0fa0c9412}. + +**Proof.** (of \entryref{7bb0fa0c9412}) +Let $f\colon S^2\to (X(t_0),g(t_0))$ be a minimal $2$-sphere. + +\entryblock{db9ef2c8d5c1} + +**Proof.** Recall that, for any immersed surface $f\colon S^2\to (M,g(t_0))$, we +have ([Ha]) + +$$ +\begin{aligned} +\frac{d}{dt}\mathit{Area}_{g(t)}(f(S^2))\bigl|_{t=t_0}\bigr. & = & + \int_{S^2}\frac{1}{2}\mathit{Tr}|_{S^2}\Bigl(\frac{\partial g}{\partial t}\Bigr)\Bigl|_{t=t_0}\Bigr.da \\ + & = & + -\int_{S^2}(R-\mathit{Ric}(\mathbf{n},\mathbf{n}))da \nonumber +\end{aligned} +$$ + + where $R$ denotes the scalar curvature of $M$, $\mathit{Ric}$ is the +Ricci curvature of $M$, and $\mathbf{n}$ is the unit normal vector field of $\Sigma$ +in $M$. Now suppose that $f(S^2)$ is minimal. We can rewrite this as + +$$ +\begin{aligned} +\frac{d}{dt}\mathit{Area}_{g(t)}(f(S^2))\big |_{t=0} + &= & + -\int_{S^2}K_{S^2}da-\frac{1}{2}\int_{S^2}(|A|^2+R)da, +\end{aligned} +$$ + +where $K_{S^2}$ is the Gaussian curvature of $S^2$ and $A$ is the +second fundamental form of $f(S^2)$ in $M$. (Of course, since +$f(S^2)$ is minimal, the determinant of its second fundamental form +is $-|A|^2/2$.) Even if $f$ is only a branched minimal surface, +(18.2) still holds when the integral on the right is replaced +by the integral over the immersed part of $f(S^2)$. Then by the +Gauss-Bonnet theorem we have + +$$ +\begin{aligned} +\frac{d}{dt}\mathit{Area}_{g(t)}(f(S^2))\bigl|_{t=t_0}\bigr. & \leq & +-4\pi-\frac{1}{2}\mathit{Area}_{g(t_0)}(S^2)\min_{x\in M}\{R_{g}(x,t_0)\}. +\end{aligned} +$$ + +Since $f(S^2)$ is a homotopically non-trivial sphere in ${\mathcal X}(t)$ for +all $t$ sufficiently close to $t_0$ we see that $W_2(t)\le \mathit{Area}_{g(t)}f(S^2)$. Since $\mathit{Area}_{g(t)}f(S^2))$ is a smooth function of +$t$, the forward difference quotient statement in \entryref{7bb0fa0c9412} follows +immediately from \entryref{db9ef2c8d5c1}. + +We turn now to continuity at non-surgery times. Fix $t'\ge T_0$ +distinct from all surgery times. We show that the function +$e_0(t')$ is continuous at $t'$. If $f\colon S^2\to {\mathcal +X}(t')$ is the minimal area, homotopically non-trivial sphere, then +the area of $f(S^2)$ with respect to a nearby metric $g(t)$ is close +to the area of $f(S^2)$ in the metric $g(t')$. Of course, the area +of $f(S^2)$ in the metric $g(t)$ is greater than or equal to +$W_2(t)$. This proves that $W_2(t)$ is upper semi-continuous at +$t'$. Let us show that it is lower semi-continuous at $t'$. + +\entryblock{1554c9624fa7} + +**Proof.** The rate of change of the area of $f(S^2)$ at time $t$ is + +$$ +\int_{f(S^2)}\frac{\partial g}{\partial t}(t)da=-2\int_{f(S^2)}\mathit{Tr}|_{TS^2}(\mathit{Ric}_{g(t)})da\le +4D\mathit{Area}_{g(t)}f(S^2) . +$$ + + Integrating from $t_0$ to $t_1$ gives +the result. + +Now suppose that we have a family of times $t_n$ converging to a +time $t'$ that is not a surgery time. Let $f_n\colon S^2\to +{\mathcal X}(t_n)$ be the minimal area non-homotopically trivial +$2$-sphere in ${\mathcal X}(t_n)$, so that the area of $f_n(S^2)$ in +${\mathcal X}(t_n)$ is $e_0(t_n)$. Since $t'$ is not a surgery time, +for all $n$ sufficiently large we can view the maps $f_n$ as +homotopically non-trivial maps of $S^2$ into ${\mathcal X}(t')$. By +the above claim, for any $\delta>0$ for all $n$ sufficiently large, +the area of $f_n(S^2)$ with respect to the metric $g(t')$ is at most +the area of $f_n(S^2)$ plus $\delta$. This shows that for any +$\delta>0$ we have $W_2(t')\le W_2(t_n)+\delta$ for all $n$ +sufficiently large, and hence $W_2(t')\le \mathit{liminf}_{n\rightarrow\infty}W_2(t_n)$. This is the lower +semi-continuity. + +The last thing to check is the behavior of $W_2$ near a surgery time +$t$. According to the description of the surgery process given in +Section 15.4, we write ${\mathcal X}(t)$ as the union +of a compact subset $C(t)$ and a finite number of surgery caps. For +every $t'0$ for all $t'0$ for all $t'0$ for all $t'0$, it follows that + +$$ +W_2(t)\le \mathit{liminf}_{t'\rightarrow t^-}W_2(t'). +$$ + +This establishes all three statements in \entryref{4415092c2079} and +completes the proof of the proposition. + +As an immediate corollary of \entryref{4415092c2079}, we obtain the +sphere theorem for closed $3$-manifolds. + +\entryblock{a679e0086677} + +**Proof.** Let $M$ be as in the statement of the corollary. Let $g$ be a +normalized metric on $M$, and let $({\mathcal M},G)$ be the Ricci +flow with surgery defined for all time with $(M,g)$ as initial +conditions. According to \entryref{4415092c2079} there is $T<\infty$ +such that every component of $M_T$ has trivial $\pi_2$. Thus, by the +analysis above, we see that there must be surgeries that kill +elements in $\pi_2$: either the removal of a component with +non-trivial $\pi_2$ or surgery along a homotopically non-trivial +$2$-sphere. We consider the first such surgery in $M$. + The only +components with non-trivial $\pi_2$ that can be removed by surgery +are $S^2$-bundles over $S^1$ and $\Ar P^3\#\Ar P^3$. Since each of +these has homotopically non-trivially embedded $2$-spheres, if the +first surgery killing an element in $\pi_2$ is removal of such a +component, then, because all the earlier $2$-sphere surgeries are +along homotopically trivial $2$-spheres, the homotopically +non-trivial embedded $2$-sphere in this component deforms back to an +embedded, homotopically non-trivial $2$-sphere in $M$. The other +possibility is that the first time an element in $\pi_2(M)$ is +killed it is by surgery along a homotopically non-trivial +$2$-sphere. Once again, using the fact that all previous surgeries +are along homotopically trivial $2$-spheres, deform this $2$-sphere +back to $M$ producing a homotopically non-trivial $2$-sphere in $M$. + +\entryblock{e800ffebd2ce} + +## Extinction + +Now we assume that the Ricci flow with surgery $({\mathcal M},G)$ satisfies the +conclusion of \entryref{703e4cfeece7} and also has initial condition $M$ that is a +connected $3$-manifold whose fundamental group satisfies the hypothesis of +\entryref{4b8a44aa316f}. The argument showing that components with non-trivial +$\pi_3$ disappear after a finite time is, in spirit, very similar to the +arguments above, though the technical details are more intricate in this case. + +### Forward difference quotient for $\pi_3$ + +Let $M$ be a compact, connected $3$-manifold. Fix a base point +$x_0\in M$. Denote by $\Lambda M$ the free loop space of $M$. By +this we mean the space of $C^1$-maps of $S^1$ to $M$ with the +$C^1$-topology. The components of $\Lambda M$ are the conjugacy +classes of elements in $\pi_1(M,x_0)$. The connected component of +the identity of $\Lambda M$ consists of all homotopically trivial +loops in $M$. Let $*$ be the trivial loop at $x_0$. + +\entryblock{39199318fe64} + +**Proof.** An element in $\pi_2(\Lambda M,*)$ is represented by a map +$S^2\times S^1\to M$ that sends $\{\mathit{pt}\}\times S^1$ to $x_0$. +Hence, this map factors through the quotient of $S^2\times S^1$ +obtained by collapsing $\{\mathit{pt}\}\times S^1$ to a point. The +resulting quotient space is homotopy equivalent to $S^2\vee S^3$, +and a map of this space into $M$ sending the wedge point to $x_0$ +is, up to homotopy, the same as an element of $\pi_2(M,x_0)\oplus +\pi_3(M,x_0)$. But we are assuming that $\pi_2(M,x_0)=0$. The first +statement follows. For the second, notice that since $\pi_2(M,x_0)$ +is trivial, $\pi_3(M,x_0)$ is identified with $H_3$ of the universal +covering $\widetilde M$ of $M$. Hence, for any map of $S^2$ into +the component of $\Lambda M$ containing the trivial loops, the +resulting map $S^2\times S^1\to M$ lifts to $\widetilde M$. The +corresponding element in $\pi_3(M,x_0)$ is the image of the +fundamental class of $S^2\times S^1$ in $H_3(\widetilde +M)=\pi_3(M)$. + +\entryblock{d88ab31ec11f} + +Now let us formulate the analogue of \entryref{4415092c2079} for +$\pi_3$. Suppose that ${\mathcal X}$ is a path of components of the +Ricci flow with surgery $({\mathcal M},G)$ defined for $t\in +[t_0,t_1]$. Suppose that $\pi_2({\mathcal X}(t_0),x_0)=0$ and that +$\pi_3({\mathcal X}(t_0),x_0)\not=0$. Then, the same two conditions +hold for ${\mathcal X}(t)$ for each $t\in [t_0,t_1]$. The reason is +that at a surgery time $t$, since all the $2$-spheres in ${\mathcal +X}(t')$ ($t'0$ for every $t'0$ be given. Then by \entryref{eaf31795ca89} +and \entryref{37b374f0934d}, there is $0<\zeta<\eta/2$ such that the +following two conditions hold: + +- **(a)** Any loop in ${\mathcal X}(t_1)$ of length less than $\zeta$ +bounds a disk of area less than $\eta$. +- **(b)** For every $a\in [0,W_\xi(t_0)+2\zeta]$ +the solution $w_{a}$ satisfies $w_{a}(t_1)0$ we can find a family $\widetilde\Gamma(t)$ with +$\widetilde\Gamma(t_0)$ representing $\xi$ and with +$W(\widetilde\Gamma(t_1))< w(t_1)+\eta$. This completes the proof of +\entryref{fd85732525c1} when $w(t_1)\ge 0$. + +Now suppose that $w(t_1)<0$. In this case, we must derive a contradiction since +clearly it must be the case that for any one-parameter family +$\widetilde\Gamma(t)$ we have $W(\widetilde\Gamma(t_1))\ge 0$. We fix $\eta>0$ +such that $w(t_1)+\eta<0$. Then using \entryref{ed0e92ce8f57} and +\entryref{eaf31795ca89}, we fix $\zeta$ with $0<\zeta<\eta/2$ such that: + +- **(i)** If $\Gamma\colon S^2\to \Lambda {\mathcal X}(t_1)$ is a family +of loops and each loop in the family is of length less than $\zeta$, +then the family is homotopically trivial. +- **(ii)** For any $a\in [0,W_\xi(t_0)+2\zeta]$ we have $w_a(t_1)0$ we define a loop $c^\lambda$ +in $M\times S^1_\lambda$. The loop $c^\lambda$ is obtained by +setting $c^\lambda(x)=(c(x),x)$ where we use a standard +identification of the domain circle (the unit circle) for the free +loop space with $S^1_\lambda$, an identification that defines a loop +in $S^1_\lambda$ of constant speed $\lambda/2\pi$. + +\entryblock{6bb9851ce528} + +Before proving this lemma we need some preliminary definitions and +constructions. + +\entryblock{3b2ad0db7f6e} + +The following is immediate from the definition. + +\entryblock{981d63c19e50} + +**Proof.** The length of $c$ is the limit of the lengths of the $n$-polygonal +approximations as $n$ goes to infinity. The first item is immediate +from this. As to the second, for $n$ sufficiently large, the +distance between the maps $c$ and $c_n$ will be arbitrarily small in +the $C^0$-topology, and in particular will be much smaller than the +injectivity radius of $M$. Thus, for each $k$ we can connect $A_k$ +to the corresponding part of $c$ by a family of short geodesics. +Together, these form an annulus, and it is clear that for $n$ +sufficiently large the area of this annulus is arbitrarily small. + +As the next result shows, for $\zeta>0$, the integer $n(c)$ +associated by the previous claim to a $C^1$-map $c$ can be made +uniform as $c$ varies over a compact subset of $\Lambda M$. + +\entryblock{028d3b5182b5} + +**Proof.** Suppose the result is false. Then for each $N$ there is $c_N\in X$ +and $n\ge N$ so that the lemma does not hold for $c_N$ and $n$. +Passing to a subsequence, we can suppose that the $c_N$ converge to +$c_\infty\in X$. Applying \entryref{981d63c19e50} we see that there is $N$ +such that the conclusion of \entryref{981d63c19e50} holds with $\zeta$ +replaced by $\zeta/2$ for $c_\infty$ and all $n\ge N$. Clearly, then +by continuity for all $n\ge N$ the conclusion of \entryref{981d63c19e50} +holds for the $n$-polygonal approximation for every $c_l$ for all +$l$ sufficiently large. This is a contradiction. + +\entryblock{7a8f55be175e} + +**Proof.** Given $\Gamma$ there is a uniform bound over all $c\in S^2$ on the +maximal speed of $\Gamma(c)$. Hence, for all $n$ sufficiently large, +the lengths of the sides in the $n$-polygonal approximation to +$\Gamma(c)$ will be uniformly small. Once this length is less than +the injectivity radius of $M$, the minimal geodesics between the +endpoints are unique and vary continuously with the endpoints. This +implies that for $n$ sufficiently large the family $\Gamma_n$ is +uniquely determined and itself forms a continuous family of loops in +$M$. This proves the first item. We have already seen that, for $n$ +sufficiently large, for all $c\in S^2$ there is an annulus +connecting $\Gamma(c)$ and $\Gamma_n(c)$. Hence, these loops are +homotopic in $M$. The first statement in the second item follows +immediately. The last statement in the second item and third item +follow immediately from \entryref{028d3b5182b5}. + +The next step is to turn these $n$-polygonal approximations into +$C^2$-curves. We fix, once and for all, a $C^\infty$ function +$\psi_n$ from the unit circle to $[0,\infty]$ with the following +properties: + +- **(1)** $\psi_n$ is non-negative and vanishes to infinite order at the point $1$ on the unit circle. +- **(2)** $\psi_n$ is periodic with period $2\pi i/n$. +- **(3)** $\psi_n$ is positive on the interior of the interval $[1,\xi_n]$ on the unit circle, +and the restriction of $\psi_n$ to this interval is symmetric about +$\mathit{exp}(\pi i/n)$, and is increasing from $1$ to $\mathit{exp}(\pi +i/n)$. +- **(4)** $\int_1^{\xi_n}\psi_n(s)ds =2\pi/n$. + + Now we define a map $\tilde\psi_n\colon S^1\to S^1$ by + +$$ +\tilde \psi_n(x)= \int_{1}^x\psi_n(y)dy. +$$ + + It is easy to see that +the conditions on $\psi_n$ imply that this defines a $C^\infty$-map +from $S^1$ to $S^1$ which is a homeomorphism and is a diffeomorphism +on the complement of the $n^{th}$ roots of unity. + +Now given an $n$-polygonal loop $c_n$ we define the smoothing +$\tilde c_n$ of $c_n$ by $\tilde c_n=c_n\circ \tilde \psi_n$. This +smoothing $\tilde c_n$ is a $C^\infty$-loop in $M$ with the same +length as the original polygonal loop $c_n$. Notice that the +curvature of $\tilde c_n$ is not itself a continuous function: just +like the polygonal map it replaces, it has a $\delta$-function at +the 'corners' of $c_n$. + +**Proof.** (of \entryref{6bb9851ce528}) +Given a continuous map $\Gamma\colon S^2\to \Lambda M$ and $\zeta>0$ +we fix $n$ sufficiently large so that \entryref{7a8f55be175e} +holds for these choices of $\Gamma$ and $\zeta$. Let $\widetilde +\Gamma=\widetilde \Gamma_n$ be the family of smoothings of the +family $\Gamma_n$ of $n$-polygonal loops. Since this smoothing +operation changes neither the length nor the area of a minimal +spanning disk, it follows immediately from the construction and +\entryref{7a8f55be175e} that $\widetilde \Gamma$ satisfies the +conclusions of \entryref{6bb9851ce528} except possibly the last one. + +To establish the last conclusion we must examine the lengths and +total curvatures of the ramps $\widetilde \Gamma(c)^\lambda$ +associated to this family of $C^2$-loops. Fix $\lambda$ with +$0<\lambda<1$, and consider the product Ricci flow $(M,g(t))\times +(S^1_\lambda,ds^2)$ where the metric on $S^1_\lambda$ has length +$\lambda$. + +\entryblock{cb69804a2166} + +**Proof.** The arc length element for $\widetilde \Gamma(c)^\lambda$ is +$\sqrt{a(x)^2+(\lambda/2\pi)^2}dx\le (a(x)+\lambda/2\pi)dx$ where +$a(x)dx$ is the arc length element for $\widetilde\Gamma(c)$. +Integrating gives the length estimate. + +The total curvature of $\widetilde \Gamma(c)^\lambda$ is the sum +over the intervals $[\xi_n^k,\xi_n^{k+1}]$ of the total curvature on +these intervals. On any one of these intervals we have a curve in a +totally geodesic, flat surface: the curve lies in the product of a +geodesic arc in $M$ times $S^1_\lambda$. Let $u$ and $v$ be unit +tangent vectors to this surface, $u$ along the geodesic (in the +direction of increasing $x$) and $v$ along the $S^1_\lambda$ factor. +These are parallel vector fields on the flat surface. + The tangent +vector $X(x)$ to the restriction of $\widetilde \Gamma(c)^\lambda$ +to this interval is $L\psi_n(x) u+(\lambda/2\pi)v$, where $L$ is the +length of the geodesic segment we are considering. Consider the +first-half subinterval $[\xi_n^k,\xi_n^k\cdot\mathit{exp}(\pi i/n)]$. +The tangent vector $X(x)$ is $(\lambda/2\pi) v$ at the initial point +of this subinterval and is $L\psi_n(\xi_n^k\cdot\mathit{exp}(\pi +i/2))u+(\lambda/2\pi) v$ at the final point. Throughout this +interval the vector is of the form $a(x)u+(\lambda/2\pi) v$ where +$a(x)$ is an increasing function of $x$. Hence, the tangent vector +is always turning in the same direction and always lies in the first +quadrant (using $u$ and $v$ as the coordinates). Consequently, the +total turning (the integral of $k$ against arc-length) over this +interval is the absolute value of the difference of the angles at +the endpoints. This difference is less than $\pi/2$ and tends to +$\pi/2$ as $\lambda$ tends to zero, unless $L=0$ in which case there +is zero turning for any $\lambda>0$. By symmetry, the total turning +on the second-half subinterval $[\xi_n^k\cdot\mathit{exp}(\pi +i/n),\xi_n^{k+1}]$ is also bounded above by $\pi/2$. Thus, for any +$\lambda>0$, the total turning on one of the segments is bounded +above by $\pi$. Since there are $n$ segments this gives the upper +bound of $n\pi$ on the total turning of $\widetilde +\Gamma(c)^\lambda$ as required. + +This claim completes the proof of the last property required of +$\widetilde\Gamma=\widetilde \Gamma_n$ and hence completes the proof +of \entryref{6bb9851ce528}. + +Having fixed $\Gamma$ and $\zeta>0$, we fix $n$ and set +$\widetilde\Gamma=\widetilde \Gamma_n$. We choose $n$ sufficiently +large so that $\widetilde \Gamma$ satisfies \entryref{6bb9851ce528}. + Fix $\lambda\in (0,1)$ and +define $\widetilde \Gamma^\lambda\colon S^2\to (M\times S^1_\lambda)$, by +setting $\widetilde \Gamma^\lambda(c)=\widetilde \Gamma(c)^\lambda$. + +Fix $c\in S^2$, and let $\widetilde\Gamma_c^\lambda(t),\ t_0\le +t\le t_1$, be the curve-shrinking flow given in +\entryref{11ec367eae93} with initial data the ramp +$\widetilde\Gamma^\lambda(c)$ . As $c$ varies over $S^2$ these fit +together to produce a one-parameter family +$\widetilde\Gamma^\lambda(t)$ of maps $S^2\to\Lambda(M\times +S^1_\lambda)$. Let $p_1$ denote the projection of $M\times +S^1_\lambda$ to $M$. Notice that for any $\lambda$ we have +$\widetilde\Gamma^\lambda_c(t_0)=\widetilde\Gamma^\lambda(c)$, so +that $p_1\widetilde \Gamma_c^\lambda(t_0)=\widetilde \Gamma(c)$. We +shall show that for $\lambda>0$ sufficiently small, the family +$p_1\widetilde\Gamma^\lambda(t)$ satisfies the conclusion of +\entryref{fd85732525c1} for the fixed $\Gamma$ and $\zeta>0$. We do +this in steps. First, we show that fixing one $c\in S^2$, for +$\lambda$ sufficiently small (depending on $c$) an analogue of +\entryref{fd85732525c1} holds for the one-parameter family of loops +$p_1\widetilde\Gamma_c^\lambda(t)$. By this we mean that either +$p_1\widetilde\Gamma^\lambda_c(t_1)$ has length less than $\zeta$ or +$A(p_1\widetilde\Gamma^\lambda_c(t_1))$ is at most the value +$v(t_1)+\zeta$, where $v$ is the solution to the +Equation (18.4) with initial condition +$v(t_0)=A(\widetilde\Gamma(c))$. (Actually, we establish a slightly +stronger result, see \entryref{9df99ae6f4cb}.) The next step in the +argument is to take a finite subset ${\mathcal S}\subset S^2$ so +that for every $c\in S^2$ there is $\hat c\in {\mathcal S}$ such +that $\widetilde\Gamma(c)$ and $\widetilde\Gamma(\hat c)$ are +sufficiently close. Then, using the result of a single $c$, we fix +$\lambda>0$ sufficiently small so that the analogue of +\entryref{fd85732525c1} for individual curves (or rather the slightly +stronger version of it) holds for every $\hat c\in {\mathcal S}$. +Then we complete the proof of \entryref{fd85732525c1} using the fact +that for every $c$ the curve $\widetilde\Gamma(c)$ is sufficiently +close to a curve $\widetilde\Gamma(\hat c)$ associated to an element +$\hat c\in {\mathcal S}$. + +### The case of a single $c\in S^2$ + +According to \entryref{6bb9851ce528}, for all $\lambda\in (0,1)$ the +lengths and total curvatures of the $\widetilde\Gamma^\lambda(c)$ +are uniformly bounded for all $c\in S^2$. Hence, by +\entryref{156a3da68fbd} the same is true for +$\widetilde\Gamma_c^\lambda(t)$ for all $c\in S^2$ and all $t\in +[t_0,t_1]$. + +\entryblock{52123e3c5a27} + +**Proof.** All the constants in this argument are allowed to depend on +$t_1-t_0$, on the curvature bound of the sectional curvature of the +Ricci flow $(M,g(t)),\ t_0\le t\le t_1$, on the original family +$\Gamma$ and on $\zeta$ but are independent of $\lambda$, $c\in +S^2$, and $t't'$ +we have a spanning disk for $p_1\widetilde\Gamma^\lambda_c(t'')$ +defined by taking the union of a minimal spanning disk for +$p_1\widetilde \Gamma^\lambda_c(t')$ and the annulus +$p_1S^\lambda_c[t',t'']$. As before, the derivative of the area of +this family of disks has two contributions, one coming from the +change in the metric over the minimal spanning disk at time $t'$ and +the other which we computed above to be at most $C_1'$. Thus, the +derivative is bounded above by +$C'_2A(p_1\widetilde\Gamma^\lambda_c(t'))+C_1'$. This implies that +the forward difference quotient of $A(p_1\widetilde +\Gamma^\lambda_c(t'))$ is bounded above by the same quantity. It +follows immediately that the areas of all the minimal spanning +surfaces are bounded by a constant depending only on the areas of +the minimal spanning surfaces at time $t_0$, the sectional curvature +of the ambient Ricci flow and $t_1-t_0$. Hence, there is a constant +$C_4$ such that the forward difference quotient of +$A(p_1\widetilde\Gamma^\lambda_c(t))$ is bounded above by $C_4$. +This proves the claim. + +Next, by the uniform bounds on total length of all the curves +$\widetilde\Gamma_c^\lambda(t)$, it follows from +Equation (18.11) that there is a constant $C_5$ (we take +$C_5>1$) depending only on the curvature bound of the ambient +manifolds and the family $\Gamma$ such that for any $c\in S^2$ we +have + +$$ +\int_{t_0}^{t_1}\int_{\widetilde\Gamma_c^\lambda(t)} +k^2dsdt\le C_5. +$$ + + Thus, for any constant $10$ (and also $\delta<1$) as described in the last +lemma for the Ricci flow $(M,g(t)),\ t_0\le t\le t_1$. By +Cauchy-Schwarz it follows that for every $t\in I_B(c,\lambda)$, + and for any arc $J$ in $\widetilde\Gamma_c^\lambda(\cdot,t)$ of +length at most $\delta^2B^{-1}$ we have + +$$ +\int_{J\times\{t\}}k\le \delta. +$$ + +Applying the previous lemma, for each $a\in I_B(c,\lambda)$ with +$a\le t_1-B^{-1}-\delta^5B^{-2}$ we set +$J(a)=[a+\delta^5B^{-2}/2,a+\delta^5B^{-2}]\subset +[t_0,t_1-B^{-1}]$. Then for all $t\in \cup_{a\in +I_B(c,\lambda)}J(a)$ for which the length of $\widetilde +\Gamma_c^\lambda(\cdot,t)$ is at least $\delta^2B^{-1}$ we have that +$k$ and all the norms of spatial derivatives of $H$ are pointwise +uniformly bounded. Since $I_B(c,\lambda)$ covers all of $[t_0,t_1]$ +except a subset of measure at most $C_5B^{-1}$, it follows that the +union $\widehat J_B(c,\lambda)$ of intervals $J(a)$ for $a\in +I_B(c,\lambda)\cap [t_0,t_1-B^{-1}-\delta^5B^{-2}]$ cover all of +$[t_0,t_1]$ except a subset of measure at most +$C_5B^{-1}+B^{-1}+\delta^5B^{-1}<3C_5B^{-1}$. Now it is +straightforward to pass to a finite subset of these intervals +$J(a_i)$ that cover all of $[t_0,t_1]$ except a subset of measure at +most $3C_5B^{-1}$. + Once we +have a finite number of $J(a_i)$, we order them along the interval +$[t_0,t_1]$ so that their initial points form an increasing +sequence. (Recall that they all have the same length.) Then if we +have $J_i\cap J_{i+2}\not=\emptyset$, then $J_{i+1}$ is contained in +the union of $J_i$ and $J_{i+2}$ and hence can be removed from the +collection without changing the union. In this way we reduce to a +finite collection of intervals $J_i$, with the same union, where +every point of $[t_0,t_1]$ is contained in at most $2$ of the +intervals in the collection. Once we have arranged this we have a +uniform bound, independent of $\lambda$ and $c\in S^2$, on the +number of these intervals. We let $J_B(c,\lambda)$ be the union of +these intervals. According to the construction and \entryref{38712e8cea3d} +these sets $J_B(c,\lambda)$ satisfy the following: + +- **(1)** $J_B(c,\lambda)\subset [t_0,t_1-B^{-1}]$ is a union of a bounded number of intervals +(the bound being independent of $c\in S^2$ and of $\lambda$) of +length $\delta^5B^{-2}/2$. +- **(2)** The measure of $J_B(c,\lambda)$ is at least +$t_1-t_0-3C_5B^{-1}$. +- **(3)** For every $t\in J_B(c,\lambda)$ either the length of $\widetilde\Gamma_c^\lambda(t)$ is +less than $\delta^2 B^{-1}$ or there are uniform bounds, depending +only on the curvature bounds of the ambient Ricci flow and the +initial family $\Gamma$, on the curvature and its higher spatial +derivatives of $\widetilde\Gamma_c^\lambda(t)$. + +Now we fix $c\in S^2$ and $1C_2/(\mathit{log}4-\mathit{log}3)$. + +The first step in the proof of \entryref{9df99ae6f4cb} is the +following: + +\entryblock{07ac984fbe88} + +**Proof.** Suppose that the first case does not hold for any subsequence. Fix a +component $J_i$ of $J_B(c)$. Then, by passing to a subsequence, by +the fact that $J_B(c)\subset J_B(c,\lambda_n)$ for all $n$, the +curvatures and all the derivatives of the curvatures of +$\widetilde\Gamma_c^{\lambda_n}(t)$ are uniformly bounded +independent of $n$ for all $t\in J_B(c)$. We reparameterize the +domain circle so that the $\widetilde\Gamma_c^{\lambda_n}(t_i^-)$ +have constant speed. By passing to a subsequence we can suppose that +the lengths of the $\widetilde\Gamma_c^{\lambda_n}(t_i^-)$ converge. +The limit is automatically positive since we are assuming that the +first case does not hold for any subsequence. Denote by +$S_n=S_c^{\lambda_n}(t_i^-)$ the unit tangent vector to $\widetilde +\Gamma_c^{\lambda_n}(t_i^-)$ and by $u_n$ the inner product $\langle +S_n,U\rangle$. Now we have a family of loops with tangent vectors +and all higher derivatives bounded. Since $u_n$ is everywhere +positive, since $\int u_nds=\lambda_n$, since the length of the loop +$\widetilde \Gamma_c^{\lambda_n}(t_i^-)$ is bounded away from $0$ +independent of $n$, and since +$|(u_n)'|=|\langle\nabla_{S_n}S_n,U\rangle|$ is bounded above +independent of $n$, we see that $u_n$ tends uniformly to zero as $n$ +tends to infinity. This means that the $|p_1(S_n)|$ converge +uniformly to one as $n$ goes to infinity. + Since +the ambient manifold is compact, passing to a further subsequence we +have a smooth limit of the +$p_1\widetilde\Gamma_c^{\lambda_n}(t_i^-)$. The result is an +immersed curve in $(M,g(t_i^-))$ parameterized at unit speed. Since +all the spatial and time derivatives of the $p_1\widetilde +\Gamma_c^{\lambda_n}(t)$ are uniformly bounded, by passing to a +further subsequence, there is a smooth map $f\colon S^1\times +[t_i^-,t_i^+]\to M$ which is a smooth limit of the sequence +$\widetilde\Gamma_c^{\lambda_n}(t),\ t_i^-\le t\le t_i^+$. If for +some $t\in [t_i^-,t_i^+]$ the curve $f|_{S^1\times\{t\}}$ is +immersed, then this limiting map along this curve agrees to first +order with the curve-shrinking flow. Thus, for some $t>t_i^-$ the +restriction of $f$ to the interval $[t_i^-,t]$ is a curve-shrinking +flow. We claim that $f$ is a curve-shrinking flow on the entire +interval $[t_i^-,t_i^+]$. Suppose not. Then there is a first $t'\le +t_i^+$ for which $f|_{S^1\times \{t'\}}$ is not an immersion. +According to \entryref{656c8fbce129} the maximum of the norms of +the curvature of the curves $f(t)$ must tend to infinity as $t$ +approaches $t'$ from below. But the curvatures of $f(t)$ are the +limits of the curvatures of the family $p_1\widetilde +\Gamma_c^{\lambda_n}(t)$ and hence are uniformly bounded on the +entire interval $[t_i^-,t_i^+]$. + This contradiction shows that the +entire limiting surface + +$$ +f\colon S^1\times [t_i^-,t_i^+]\to (M,g(t)) +$$ + +is a curve-shrinking flow of immersions. + +\entryblock{012eefbbc44a} + +For each $n$, the family of curves +$p_1\widetilde\Gamma_c^{\lambda_n}(t)$ in $M$ all have +$p_1\widetilde\Gamma_c^{\lambda_n}(t_0)=\widetilde\Gamma(c)$ as +their initial member. Thus, these curves are all homotopically +trivial. Hence, for each $t\in J_B(c)$ the limiting curve +$\widetilde\Gamma(c)(t)$ of the +$p_1\widetilde\Gamma_c^{\lambda_n}(t)$ is then also homotopically +trivial. It now follows from \entryref{27b452d6cb48}, +\entryref{07ac984fbe88} and \entryref{012eefbbc44a} that one of the +following two conditions holds: + +- **(1)** for some $t\in J_B(c)$ the length of $\widetilde\Gamma(c)(t)$ is less than or equal to + $e^{-C_2(t_1-t_0)}\zeta/3$ or +- **(2)** the function $A(t)$ that assigns to each $t\in +J_B(c)$ the area of the minimal spanning disk for +$p_1\widetilde\Gamma(c)(t)$ satisfies + +$$ +\frac{dA(t)}{dt}\le -2\pi-\frac{1}{2}R_\mathit{min}(t)A(t) +$$ + +in the sense of forward difference quotients. + +By continuity, for any $\delta''>0$ then for all $n$ + sufficiently large one of the following two conditions holds: + +- **(1)** there is $t_n\in J_B(c)$ such that the length of $\widetilde\Gamma_c^{\lambda_n}(t_n)$ +is less than $e^{-C_2(t_1-t_0)}\zeta/2$, or +- **(2)** for every $t\in J_B(c)$, the areas of the minimal +spanning disks for $p_1(\widetilde\Gamma_c^{\lambda_n}(t))$ satisfy + +$$ +\frac{dA(p_1\widetilde\Gamma_c^{\lambda_n}(t))}{dt}\le -2\pi-\frac{1}{2}R_\mathit{min}(t)A(p_1\widetilde\Gamma_c^{\lambda_n}(t))+\delta'' +$$ + + in the +sense of forward difference quotients. + +Suppose that for every $n$ sufficiently large, for every $t\in +J_B(c)$ the length of $\widetilde\Gamma_c^{\lambda_n}(t)$ is at +least $e^{-C_2(t_1-t_0)}\zeta/2$. We have already seen in +\entryref{52123e3c5a27} that for every $t'0$ sufficiently small so that \entryref{10dace7fb165} and +\entryref{52f36e456355} hold. Then we choose a $\mu/2$-net $X$ for +$\widetilde\Gamma$. We take $\lambda$ sufficiently small so that +\entryref{9df99ae6f4cb} holds for every $\hat c\in {\mathcal S}$. We +also choose $\lambda$ sufficiently small so that $X$ is a $\mu$-net +for $\widetilde\Gamma^\lambda$. Let $c\in S^2$. Then there is $\hat +c\in {\mathcal S}$ and an arc $\omega$ connecting $c$ and $\hat c$ +such that the area of $\widetilde\Gamma^\lambda(\omega)<\mu$. Let +$v_{\hat c}$, resp., $v_c$ be the solution to +Equation 18.4 with initial condition $v_{\hat +c}(t_0)=A(\widetilde\Gamma({\hat c}))$, resp., +$v_c(t_0)=A(\widetilde\Gamma(c))$. According to +\entryref{9df99ae6f4cb} either $A(p_1\widetilde\Gamma_{\hat +c}^\lambda(t_1))0$. + +Next, let us deal with the singularities of the pulled back metric on $A$ +caused by the interior branch points. As the next claim shows, it is an easy +matter to deform the metric slightly near each branch point without increasing +the area much and without changing the upper bound on the Gaussian curvature +too much. Here is the result: + +\entryblock{d91a6d9473a7} + +**Proof.** Fix an interior branch point $p$. + Since $\psi$ is smooth and +conformal onto its image, there is a disk in $A$ centered at $p$ in +which $h=f(z,\bar z)|dz|^2$ for a smooth function $f$ on the disk. +The function $f$ vanishes at the origin and is positive on the +complement of the origin. Direct computation shows that the Gaussian +curvature $K(h)$ of $h$ in this disk is given by + +$$ +K(h)=\frac{-\triangle f}{2f^2}+\frac{|\nabla f|^2}{2f^3}\le C, +$$ + +where $\triangle$ is the usual Euclidean Laplacian on the disk and $|\nabla +f|^2=(\partial f/\partial x)^2+(\partial f/\partial y)^2$. Now consider the +metric $(f+\epsilon)|dz|^2$ on the disk. Its Gaussian curvature is + +$$ +\frac{-\triangle f}{2(f+\epsilon)^2}+\frac{|\nabla f|^2}{2(f+\epsilon)^3}. +$$ + +\entryblock{541ad02dd5f8} + +**Proof.** We see that $-\triangle f\le C''f^2$, so that + +$$ +\begin{aligned} +\frac{-\triangle +f}{(f+\epsilon)^2}+\frac{|\nabla f|^2}{(f+\epsilon)^3} & = & +\frac{(f+\epsilon)(-\triangle +f)+|\nabla f|^2}{(f+\epsilon)^3} \\ +& \le & \frac{C''f^3-\epsilon\triangle f}{(f+\epsilon)^3}\le +C''+\frac{\epsilon f^2C''}{(f+\epsilon)^3}\le 2C''. +\end{aligned} +$$ + +Now we fix a smooth function $\rho(r)$ which is identically one on a subdisk +$D'$ of $D$ and vanishes near $\partial D$ and we replace the metric $h$ on the +disk by + +$$ +h_\epsilon=(f+\epsilon \rho(r))|dz|^2. +$$ + + The above computation shows that the +Gaussian curvature of $h_\epsilon$ on $D'$ is bounded above by +$2C''$. As $\epsilon$ tends to zero the restriction of the metric +$h_\epsilon$ to $D\setminus D'$ converges uniformly in the +$C^\infty$-topology to $h$. Thus, for all $\epsilon>0$ sufficiently +small the Gaussian curvature of $h_\epsilon$ on $D\setminus D'$ will +also be bounded by $2C''$. Clearly, as $\epsilon$ tends to zero the +area of the metric $h_\epsilon$ on $D$ tends to the area of $h$ on +$D$. + +Performing this construction near each of the finite number of interior branch +points and taking $\epsilon$ sufficiently small gives the perturbation $\tilde +h$ as required. + +Thus, if $\gamma$ and $\hat \gamma$ are ramps as in +\entryref{8e814eab5ddf}, then replacing $\hat \gamma$ by a close $C^2$ +approximation we have an abstract smooth annulus with a Riemannian +metric connecting $\gamma$ and $\hat \gamma$. Taking limits shows +that establishing the conclusion of \entryref{8e814eab5ddf} for a +sequence of better and better approximations to $\hat \gamma$ will +also establish it for $\hat \gamma$. This allows us to assume that +$\gamma$ and $\hat \gamma$ are disjoint. The area of this annulus is +bounded above by a constant arbitrarily close to $\mu$. The Gaussian +curvature of the Riemannian metric is bounded above by a constant +depending only on the curvature bounds of the ambient Ricci flow. +Finally, the integral of the absolute value of the geodesic +curvature over any interval of length $r$ of $\gamma$ is at most +$\delta$. + +With all these preliminary remarks, we see that \entryref{8e814eab5ddf} +follows from: + +\entryblock{6e9739b0c59b} + +To us, this statement was intuitively extremely reasonable but we +could not find a result along these lines stated in the literature. +Also, in the end, the argument we constructed is quite involved, +though elementary. + +The intuition is that we exponentiate in from the boundary component +$c_0$ using the family of geodesics perpendicular to the boundary. +The bounds on the Gaussian curvature and local bounds on the +geodesic curvature of $c_0$ imply that the exponential mapping will +be an immersion out to some fixed distance $\delta$ or until the +geodesics meet the other boundary, whichever comes first. +Furthermore, the metric induced by this immersion will be close to +the product metric. Thus, if there is not much area, it must be the +case that, in the measure sense, most of the geodesics in this +family must meet the other boundary before distance $\delta$. One +then deduces the length inequality. There are two main difficulties +with this argument that must be dealt with. The first is due to the +fact that we do not have a pointwise bound on the geodesic curvature +of $c_0$, only an integral bound of the absolute value over all +curves of short length. There may be points of arbitrarily high +geodesic curvature. Of course, the length of the boundary where the +geodesic curvature is large is very small. On these small intervals +the exponential mapping will not be an immersion out to any fixed +distance. We could of course, simply omit these regions from +consideration and work on the complement. But these small regions of +high geodesic curvature on the boundary can cause focusing (i.e., +crossing of the nearby geodesics). We must estimate out to what +length along the boundary this happens. Our first impression was +that the length along the boundary where focusing occurred would be +bounded in terms of the total turning along the arc in $c_0$. We +were not able to establish this. Rather we found a weaker estimate +where this focusing length is bounded in terms of the total turning + and the area bounded by the triangle cut out by the two geodesics +that meet. This is a strong enough result for our application. Since +the area is small and the turning on any interval of length $r$ is +small, a maximal collection of focusing regions will meet each +interval of length $r$ in $c_0$ in a subset of small total length. +Thus, on the complement (which is most of the length of $c_0$) the +exponential mapping will be an immersion out to length $\delta$ and +will be an embedding when restricted to each interval of length one. +The second issue to face is to show that the exponential mapping on +this set is in fact an embedding, not just an immersion. Here one +uses standard arguments invoking the Gauss-Bonnet theorem to rule +out various types of pathologies, e.g., that the individual +geodesics are not embedded or geodesics that end on $c_0$ rather +than $c_1$, etc. Once these are ruled out, one has established that +the exponential map on this subset is an embedding and the argument +finishes as indicated above. + +### First reductions + +Of course, if the hypothesis of the proposition holds for $r>0$ then +it holds for any $00$ such that +the proposition holds provided that the area of the annulus is less +than $\mu$. + +The function $k_\mathit{geod}\colon c_0\to \Ar$ is smooth. We choose a +regular value $\alpha$ for $k_\mathit{geod}$ with $1<\alpha<1.1$. In +this way we divide $c_0$ into two disjoint subsets, $Y$ where +$k_\mathit{geod}>\alpha$, and $X$ where $k_\mathit{geod}\le \alpha$. The +subset $Y$ is a union of finitely many disjoint open intervals and +$X$ is a disjoint union of finitely many closed intervals. + +\entryblock{bb88301ceeda} + +Fix $\delta'>0$. For each $x\in X$ there is a geodesic $D_x$ in $A$ +whose initial point is $x$ and whose initial direction is orthogonal +to $c_0$. Let $f(x)$ be the minimum of $\delta$ and the distance +along $D_x$ to the first point (excluding $x$) of its intersection +with $\partial A$. We set + +$$ +S_X(\delta') =\{(x,t)\in X\times [0,\delta']\bigl|\bigr.\ t\le f(x)\}. +$$ + +The subset $S_X(\delta')$ inherits a Riemannian metric from the +product of the metric on $X$ induced by the embedding $X\subset +c_0$ and the standard metric on the interval $[0,\delta']$. + +\entryblock{47e16059fb12} + +**Proof.** This is a standard computation using the Gaussian curvature upper +bound and the geodesic curvature bound. + +Now we fix $0<\delta'<1/10$ so that \entryref{47e16059fb12} holds, and +we set $S_X=S_X(\delta')$. We define + +$$ +\partial_+S_X=\{(x,t)\in S_X \bigl|\bigr. t=f(x)\}. +$$ + +Then the boundary of $S_X$ is made up of $X$, the arcs $\{x\}\times +[0,f(x)]$ for $x\in \partial X$ and $\partial_+(S_X)$. For any +subset $Z\subset X$ we denote by $S_Z$ the intersection $(Z\times +[0,\delta])\cap S_X$, and we denote by $\partial_+(S_Z)$ the +intersection of $S_Z\cap \partial_+S_X$. + +Lastly, we fix $\mu>0$ with $\mu<(1-\delta)^2(\delta')/10$. Notice +that this implies that $\mu<1/100$. We now assume that the area of +$A$ is less than this value of $\mu$ (and recall that $r=1$, $C''=1$ +and $l(c_0)\ge 2$). We must show that $l(c_1)>3l(c_0)/4$. + +### Focusing triangles + +By a *focusing triangle* we mean the following. We have distinct +points $x,y\in X$ and sub-geodesics $D'_x\subset D_x$ and +$D'_y\subset D_y$ that are embedded arcs with $x$, respectively $y$, +as an endpoint. The intersection $D'_x\cap D'_y$ is a single point +which is the other endpoint of each of $D'_x$ and $D'_y$. Notice +that since $D'_x\subset D_x$ and $D'_y\subset D_y$, by construction +both $D'_x$ and $D'_y$ have lengths at most $\delta'$. We have an +arc $\xi$ in $c_0$ with endpoints $x$ and $y$ and the loop +$\xi*D'_y*(D'_x)^{-1}$ bounds a disk $B$ in $A$. The arc $\xi$ is +called the *base* of the focusing triangle and with, respect to +an orientation of $c_0$, if $x$ is the initial point of $\xi$ then +$D'_x$ is called the *left-hand side* of the focusing triangle +and $D'_y$ is called its *right-hand side*. See Fig. 18.1. + +Our main goal here is the following lemma which gives an upper bound +for the length of the base, $\xi$, of a focusing triangle in terms +of the turning along the base and the area of the region $B$ +enclosed by the triangle. + +\entryblock{45c783d384c1} + +**Proof.** We begin with a preliminary computation. We denote by $a(B)$ the +area of $B$. We define + +$$ +t_{\xi}=\int_{\xi}k_\mathit{geod}ds \ \ \ \mathit{and} \ \ \ T_\xi=\int_{\xi}|k_\mathit{geod}|ds. +$$ + +Recall that given a piecewise smooth curve, its *total turning* +is the integral of the geodesic curvature over the smooth part of +the boundary plus the sum over the break points of $\pi$ minus the +interior angle at the break point. The Gauss-Bonnet theorem tells +us that for any compact surface with piecewise smooth boundary the +integral of the Gaussian curvature over the interior of the surface +plus the total turning around the boundary equals $2\pi$ times the +Euler characteristic of the surface. + +\entryblock{512c74b9ad1f} + +**Proof.** Since $D'_x$ and +$D'_y$ meet $\partial A$ in right angles, the total turning around +the boundary of $B$ is + +$$ +t_\xi+2\pi-\theta_B. +$$ + +Thus, by Gauss-Bonnet, we have + +$$ +\theta_B=\int_BKda+ t_\xi. +$$ + +But $K\le 1$, giving the first stated inequality. On the other hand +$\int_BKda=\int_BK^+da+\int_BK^-da$, where $K^+=\mathit{max}(K,0)$ and +$K^-=K-K^+$. Since $0\le \int_BK^+da\le a(B)$ and $\int_BK^-\le 0$, +the second string of inequalities follows. + +In order to make the computation we need to know that this triangle +is the image under the exponential mapping of a spray of geodesics +out of the vertex $v$. Establishing that requires some work. + +\entryblock{ea7c58d04fc7} + +**Proof.** The length estimate is obvious: Since $\xi$ has length at most $1$, +a path along $\partial A\cap B $ from $a$ to the closest of $x$ and +$y$ has length at most $1/2$. The corresponding side has length at +most $\delta'$. Thus, there is a path from $a$ to $v$ in $B$ of +length at most $(1/2)+\delta'$. + +Standard convergence arguments show that there is a shortest path in +$B$ from $a$ to $v$. Fix $a\in \mathit{int}(\partial A\cap B)$. It is +clear that the shortest path cannot meet either of the 'sides' +$D'_x$ and $D'_y$ at any point other than $v$. If it did, then there +would be an angle at this point and a local shortcut, cutting off a +small piece of the angle, would provide a shorter path. We must rule +out that the shortest path from $a$ to $v$ meets $\partial A\cap B$ +in another point. If it does, let $a'$ be the last such point +(parameterizing the geodesic starting at $a$). The shortest path +from $a$ then leaves $\partial A$ at $a'$ in the direction tangent +at $a'$ to $\partial A$. (Otherwise, we would have an angle which +would allow us to shorten the path just as before.) This means that +we have a geodesic $\gamma$ from $v$ to $a'$ whose interior is +contained in the interior of $B$ and which is tangent to $\partial +A$ at $a'$. We label the endpoints of $\partial A\cap B$ so that the +union of $\gamma$ and the interval on $\partial A\cap B$ from $a'$ +to $y$ gives a $C^1$-curve. Consider the disc $B'$ bounded + by $\gamma$, the arc of $\partial A$ from $a'$ +to $y$, and $D'_y$. The total turning around the boundary is at most +$3\pi/2+\delta$, and the integral of the Gaussian curvature over +$B'$ is at most the area of $B$, which is less than +$\mu<1/20<(\pi/4)-\delta$. This contradicts the Gauss-Bonnet +theorem. + +\entryblock{a9617f70db9f} + +**Proof.** Suppose not; suppose there are two $\gamma$ and $\gamma'$ from $v$ +to $a$. Since they are both minimal in $B$, each is embedded, and +they must be disjoint except for their endpoints. The upper bound on +the curvature and the Gauss-Bonnet theorem implies that the angles +that they make at each endpoint are less than $\mu<\pi/2$. Thus, +there is a spray of geodesics (i.e. geodesics determined by an +interval $\beta$ in the circle of directions at $v$) coming out of +$v$ and moving into $B$ with extremal members of the spray being +$\gamma$ and $\gamma'$. The geodesics $\gamma$ and $\gamma'$ have +length at most $(1/2)+\delta'$, and hence the exponential mapping +from $v$ is a local diffeomorphism on all geodesics of length at +most the length of $\gamma$. Since the angle they make at $a$ is +less than $\pi/2$ and since the exponential mapping is a local +diffeomorphism near $\gamma$, as we move in from the $\gamma$ end of +the spray we find geodesics from $v$ of length less than the length +of $\gamma$ ending on points + of $\gamma'$. The same +Gauss-Bonnet argument shows that the angles that each of these +shorter geodesics makes with $\gamma'$ is at most $\mu$. Consider +the subset $\beta'$ of $\beta$ which are directions of geodesics in +$B$ of length $<(1/2)+\delta'$ that end on points of $\gamma'$ and +make an angle less than $\mu$ with $\gamma'$. We have just seen that +$\beta'$ contains an open neighborhood of the end of $\beta$ +corresponding to $\gamma$. Since the Gaussian curvature is bounded +above by $1$, and these geodesics all have length at most +$1/2+\delta$, it follows that the exponential map is a local +diffeomorphism near all such geodesics. Thus, $\beta'$ is an open +subset of $\beta$. On the other, hand if the direction of +$\gamma''\not=\gamma'$ is a point $b''\in \beta$ which is an +endpoint of an open interval $ \beta'$, and if this interval +separates $b''$ from the direction of $\gamma$ then the length of +$\gamma''$ is less than the length of each point in the interval. +Hence, the length of $\gamma''$ is less than $(1/2)+\delta'$. +Invoking Gauss-Bonnet again we see that the angle between $\gamma''$ +and $\gamma'$ is $<\mu$. + +This proves that if $U$ is an open interval in $\beta'$ then the +endpoint of $U$ closest to the direction of $\gamma'$ is also +contained in $\beta'$ (unless that endpoint is the direction of +$\gamma'$). It is now elementary to see that $\beta'$ is all of +$\beta$ except the endpoint corresponding to $\gamma'$. But this is +impossible. Since the exponential mapping is a local diffeomorphism +out to distance $(1/2)+\delta'$, and since $\gamma'$ is embedded, +any geodesic from $v$ whose initial direction is sufficiently close +to that of $\gamma'$ and whose length is at most $(1/2)+\delta'$ +will not cross $\gamma'$. + +See Fig. 18.2 + +\entryblock{cbf568c0a596} + +Let $E$ be the sub-interval of the circle of tangent directions at +$v$ consisting of all tangent directions of geodesics pointing into +$B$ at $v$. The endpoints of $E$ are the tangent directions for +$D'_x$ and $D'_y$. We define a function from $\xi$ to the interval +$E$ by assigning to each $a\in \xi$ the direction at $v$ of the +unique minimal geodesic in $B$ from $v$ to $a$. Since the minimal +geodesic is unique, this function is continuous and, by the above +remark, associates to $x$ and $y$ the endpoints of $E$. Since +geodesics are determined by their initial directions, this function +is one-to-one. Hence it is a homeomorphism from $\xi$ to $E$. That +is to say the spray of geodesics coming out of $v$ determined by the +interval $E$ produces a diffeomorphism between a wedge-shaped subset +of the tangent space at $v$ and $B$. Each of the geodesics in +question ends when it meets $\xi$. + +Now that we have shown that the region enclosed by the triangle is +the image under the exponential map from the vertex $v$ of a +wedge-shaped region in the tangent space at $v$, we can make the +usual computation relating length and geodesic curvature. To do this +we pull back to the tangent space at $v$, and, using polar +coordinates, we write $\xi$ as $\{s=h(\psi); \psi\in E\}$ where $s$ +is the radial coordinate and $\psi$ is the angular coordinate. +Notice that $h(\psi)\le (1/2)+\delta'$ for all $\psi\in E$. (In +fact, because the angles of intersection at the boundary are all +close to $\pi/2$ we can give a much better estimate on $h$ but we do +not need it.) + We consider the one-parameter family of arcs $\lambda(t)$ defined + to be the graph of the function $t\mapsto s(t)=th(\psi)$, for $0\le t\le + 1$. We set $l(t)$ equal to the length of + $\lambda(t)$. + +\entryblock{e94b6de05447} + +**Proof.** First of all notice that, by construction, the curve $\xi$, which +is defined by $\{s=h(\psi)\}$, is orthogonal to the radial geodesics +to the endpoints. As a consequence, $h'(\psi)=0$ at the endpoints. +Thus, each of the curves $\lambda(t)$ is orthogonal to the radial +geodesics through its end points. Therefore, as we vary the family +$\lambda(t)$ the formula for the derivative of the length is + +$$ +l'(t)=\int_{\lambda(t)}k_\mathit{geod}(\psi)h(\psi)|\mathit{cos}(\theta(\psi,t))|ds +$$ + + where $\theta(\psi,t)$ is the angle at +$(th(\psi),\psi)$ between the curve $s=th(\psi)$ and the radial +geodesic. The result follows immediately. + +Next, we must bound the turning of $\lambda(t)$. For this we invoke the +Gauss-Bonnet theorem once again. Applying this to the wedge-shaped disk $W(t)$ +cut out by $\lambda(t)$ gives + +$$ +\int_{W(t)}Kda+\int_{\lambda(t)}k_\mathit{geod}ds=\theta_B. +$$ + +From \entryref{512c74b9ad1f} we conclude that + +$$ +\int_{\lambda(t)}k_\mathit{geod}ds\le t_\xi+a(B). +$$ + +Of course, by \entryref{ea7c58d04fc7} we have $\mathit{max}_{\psi\in +E}h(\psi)\le (1/2)+\delta'$. Since $l(0)=0$, this implies that + +$$ +l(\xi)=l(1)\le (a(B)+t_B)((1/2)+\delta')1$. + +Let us show that the first case is not possible. Since $D_x$ is a +geodesic and $D_x$ is perpendicular to $c_0$ at one end, the total +turning around the boundary of $B$ is at most + +$$ +3\pi/2+\int_{c_0'}k_\mathit{geod}ds<3\pi/2+\delta, +$$ + +where the last inequality uses the fact that the length of $c_0'$ is +at most one. On the other hand, $\int_{B}Kda<\mu$, and +$\mu<1/20<(\pi/2)-\delta$. This contradicts the Gauss-Bonnet +theorem. + +Now let us consider the second case. Let $J$ be the subinterval of +$c_0'$ with one end point being $x$ and with the length of $J$ being +one. We orient $J$ so that $x$ is its initial point. We set +$X_J=J\cap X$. We define $S_{X_J}(B)\subset S_{X_J}$ as follows. For +each $y\in X_J$ we let $f_B(y)$ be the minimum of $\delta'$ and the +distance along $D_y$ to the first point (excluding $y$) of $D_y$ +contained in $\partial B$ and let $D_y(B)$ be the sub-geodesic of +$D_y$ of this length starting at $y$. Then $S_{X_J}(B)\subset +S_{X_J}$ is the union over $y\in X_J$ of $[0,f_B(y)]$. Clearly, the +exponential mapping defines an immersion of $S_{X_J}(B)$ into $B$. +We need to replace $X_ J$ by a slightly smaller subset in order to +make the exponential mapping be an embedding. To do this we shall +remove bases of a maximal focusing triangles in $B$. + +First notice that for each $y\in X_J$ the exponential mapping is an +embedding on $D_y(B)$. The reason is that the image of $D_y(B)$ is a +geodesic contained in the ball $B$. \entryref{b6fb2b9c364b} then shows +that this geodesic is embedded. This leads to: + +\entryblock{14aeee986b49} + +**Proof.** Since the geodesics that make up $S_c(B)$ have length at most +$\delta'<1/10$ and since the curvature of the annulus is bounded +above by $1$, the restriction of the exponential mapping to $S_c(B)$ +is a local diffeomorphism. The restriction to each $\{y\}\times +[0,f_B(y)]$ is an embedding onto $D_y(B)$. If the restriction of the +exponential mapping to $S_c(B)$ is not an embedding, then there are +$y\not= y'$ in $c$ such that the geodesics $D_y(B)$ and $D_{y'}(B)$ +meet. When they meet, they meet at a positive angle and by the +Gauss-Bonnet theorem this angle is less than $\mu+\delta$. Thus, +all the geodesics starting at points sufficiently close to $y'$ and +between $y$ and $y'$ along $c$ must also meet $D_y(B)$. Of course, +if a sequence of $D_{y_i}(B)$ meet $D_y(B)$, then the same is true +for the limit. It now follows that $D_{y''}(B)$ meets $D_y(B)$ for +all $y''$ between $y$ and $y'$. This contradicts the fact that +$D_y(B)$ is embedded. + +\entryblock{2ff35cef8589} + +**Proof.** The first statement is immediate from \entryref{14aeee986b49}. + Since $X\cap J$ is compact, it is clear that after passing to a + subsequence each of the sequence of left-hand sides and the + sequence of right-hand sides converge to a geodesic arc orthogonal to + $J$ at points of $X$. Furthermore, these limiting geodesics meet in + a point at distance at most $\delta'$ from the end of each. The only + thing remaining to show is that the limiting left- and right-hand + sides do not begin at the same point of $X$. +This is clear since each focusing triangle contains one of the finitely many +components of $J\setminus X_J$. + +Using \entryref{2ff35cef8589} we see that if there is a focusing +triangle for $J$ there is a first point $x_1$ in $ X_J$ whose +associated geodesic contains the left-hand side of a focusing +triangle for $J$. Then since the base length of any focusing +triangle is bounded by a fixed constant, invoking again +\entryref{2ff35cef8589}, that there is a focusing triangle ${\mathcal +T}_1$ for $J$ that has left-hand side contained in the geodesic +$D_{x_1}$ and has a maximal base among all such focusing triangles, +Maximal in the sense that the base of this focusing triangle +contains the base of any other focusing triangle with left-hand side +contained in $D_{x_1}$. Denote its base by $\xi_1$ and denote the +right-hand endpoint of $\xi_1$ by $y_1$. For the triangle we take +the geodesic arcs to the first point of intersection measured along +$D_{y_1}$. Set $J_1=\overline{J\setminus \xi}$, and repeat the +process for $J_1$. If there is a focusing triangle for $J_1$ we find +the first left-hand side of such focusing triangles and then find +the maximal focusing triangle ${\mathcal T}_2$ with this left-hand +side. + +\entryblock{cf4b9ce8f8e3} + +**Proof.** Since by construction the interiors of the bases of ${\mathcal T}_1$ +and ${\mathcal T}_2$ are disjoint, if the interior of ${\mathcal +T}_2$ meets ${\mathcal T}_1$, then one of the sides of ${\mathcal +T}_2$ crosses the interior of one of the sides of ${\mathcal T}_1$. +But since ${\mathcal T}_1$ is a maximal focusing triangle with its +left-hand side, neither of the sides of ${\mathcal T}_2$ can cross +the interior of the left-hand side of ${\mathcal T}_1$. If one of +the sides of ${\mathcal T}_2$ crosses the interior of the right-hand +side of ${\mathcal T}_1$, then the right-hand side of ${\mathcal +T}_1$ is the left-hand side of a focusing triangle for $J_1$. Since +by construction the left-hand side of ${\mathcal T}_2$ is the first +such, this means that the left-hand side of ${\mathcal T}_2$ is the +right-hand side of ${\mathcal T}_1$. This means that the right-hand +side of ${\mathcal T}_2$ terminates when it meets the right-hand +side of ${\mathcal T}_1$ and hence the right-hand side of ${\mathcal +T}_2$ ends the first time that it meets the right-hand side of +${\mathcal T}_1$ and hence does not cross it. + +We continue in this way constructing focusing triangles for $J$ with +disjoint interiors. Since each focusing triangle for $J$ contains a +component of $J\setminus X_J$, and as there are only finitely many +such components, this process must terminate after a finite number +of steps. Let ${\mathcal T}_1,\ldots,{\mathcal T}_k$ be the focusing +triangles so constructed, and denote by $\xi_i$ the base of +${\mathcal T}_i$. Let $X_J'= X_J\setminus \cup_{i=1}^k\xi_i$. + +\entryblock{2b50ea294b54} + +\entryblock{e967f00df0a8} + +**Proof.** Since the interiors of the ${\mathcal T}_i$ are disjoint, according +to \entryref{9c8d4163f700}, we have $\sum_il(\xi_i)<\delta+\mu$. We +also know by \entryref{bb88301ceeda} that the length of $X_J$ is at +least $(1-\delta)$. Putting these together gives the result. + +We define $S_{X'_J}(B)$ to be the intersection of $S_{X_J}(B)$ with +$S_{X'_J}$. + +\entryblock{5f77e346e12a} + +**Proof.** Suppose that we have distinct points $x',y'$ in $X'_J$ such that +$D_{x'}(B)\cap D_{y'}(B)\not=\emptyset$. We assume that $x'(0.99)l(X'')>(0.83)l(c_0). +$$ + +It follows that the length of $c_1$ is at least +$(0.83)l(c_0)>3(l(c_0))/4$. + +This completes the proof. + +## Proof of the first inequality in Lemma \entryref{38712e8cea3d} + +Here is the statement that we wish to establish when the manifold +$(W,h(t))$ is the product of $(M,g(t))\times (S^1_\lambda,ds^2)$. + +\entryblock{ff5d51a483d9} + +The rest of this section is devoted to the proof of this lemma. In +[AG] such a local estimate was established when the ambient +manifold was Euclidean space and the curve in question is a graph. A +related result for hypersurfaces that are graphs appears in +[EckerHuisken]. The passage from Euclidean space to a general +Ricci flow is straightforward, but it is more delicate to use the +bound on total curvature on initial sub-arcs of length $r$ to show +that in appropriate coordinates the evolving curve can be written as +an evolving graph, so that the analysis in [2] can be applied. + +We fix $\delta>0$ sufficiently small. We fix $t'\in [t_0,t_1-\delta +r^2]$ for which the hypotheses of the lemma hold. The strategy of +the proof is to first restrict to the maximum subinterval of +$[t',t_2]$ of $[t',t'+\delta r^2]$ on which $k$ is bounded by +$\sqrt{2/(t-t')}$. If $t_20$ is sufficiently +small, the length of $\gamma_t$ is at least $(0.9)r$ for all $t\in +[t',t_2]$, and more generally for any subinterval $\gamma'_{t'}$ of +$\gamma_{t'}$ and for any $t\in [t',t_2]$ the length of the +corresponding interval $\gamma'_t$ is at least $(0.9)$ times the +length of $\gamma'_{t'}$. We introduce a cut-off function on +$\gamma_{t'}\times [t',t_2]$ as follows. First, fix a smooth +function $\psi\colon [-1/2,1/2]\to [0,1]$ which is identically zero +on $[-0.50,-0.45]$ and on $[0.45,0.50]$, and is identically $1$ on +$[-3/8,3/8]$. There is a constant $D'$ such that $|\psi'|\le D'$ and +$|\psi''|\le D'$. Now we fix the midpoint $x_0\in \gamma_{t'}$ and +define the signed distance from $(x_0,t)$, denoted + +$$ +s\colon \gamma_{t'}\times [t',t_2]\to \Ar, +$$ + +as follows: + +$$ +s(x,t)=\int _{x_0}^x|X(y,t)|dy. +$$ + +We define the cut-off function + +$$ +\varphi(x,t)=\psi\left(\frac{s(x,t)}{r}\right). +$$ + +\entryblock{80e5fc6759b2} + +**Proof.** Clearly, + +$$ +\frac{\partial \varphi(x,t)}{\partial +t}=\psi'\left(\frac{s(x,t)}{r}\right)\cdot \frac{1}{r}\frac{\partial +s(x,t)}{\partial t}. +$$ + + We know that $|\psi'|\le D'$ so that + +$$ +\left|\frac{\partial \varphi(x,t)}{\partial +t}\right|\le \frac{D'}{r}\left|\frac{\partial s(x,t)}{\partial +t}\right|. +$$ + + On the other hand, + +$$ +s(x,t)=\int_{x_0}^x|X(y,t)|dy, +$$ + +so that + +$$ +\left|\frac{\partial s(x,t)}{\partial t}\right| = +\left|\int_{x_0}^x\frac{\partial |X(y,t)|}{\partial t}dy\right|, +$$ + +By \entryref{4fc24fa1d31e} we have + +$$ +\frac{\partial +|X(y,t)|}{\partial y}dy = \left(-\mathit{Ric}(S(y,t),S(y,t))-k^2(y,t)\right)ds, +$$ + + so that there is a constant $D$ depending only on the bound of the +sectional curvatures of the ambient Ricci flow with + +$$ +\begin{aligned} +\left|\frac{\partial s(x,t)}{\partial t}\right| & +\le & \int_{x_0}^x(D+k^2)ds\le +Dl(\gamma_t)+\int_{x_0}^xk^2(y,t)ds(y,t), +\end{aligned} +$$ + + and hence +by \entryref{1b8d34cae3a8} + +$$ +\left|\frac{\partial s(x,t)}{\partial t}\right|\le + DD_0r+ +\int_{x_0}^xk^2(y,t)ds(y,t). +$$ + + Using +the fact that $k^2\le 2/(t-t')$, we have + +$$ +\int_{x_0}^xk^2(y,t)ds(y,t)\le\sqrt{\frac{2}{t-t'}}\int_{x_0}^xkds\le +\frac{\sqrt{2}D_0}{\sqrt{t-t'}}. +$$ + + Putting all this together, we see +that there is a constant $D_1$ such that + +$$ +\left|\frac{\partial \varphi(x,t)}{\partial t}\right|\le +D_1(\frac{1}{r\sqrt{t-t'}}+1). +$$ + +\entryblock{1f7d2e438fef} + +**Proof.** We have + +$$ +\left|\frac{d}{dt}\int_{\gamma_t}\varphi +kds\right|\le\left|\int_{\gamma_t}\frac{\partial +\varphi(x,t)}{\partial +t}kds\right|+\left|\int_{\gamma_t}\varphi\frac{\partial +(kds)}{\partial t}\right|. +$$ + + Using \entryref{80e5fc6759b2} for the first +term and \entryref{67b855d0d6b5} and arguing as in the proof of +\entryref{88b8b3366b19} for the second term, we have + +$$ +\left|\frac{d}{dt}\int_{\gamma_t}\varphi +kds\right|\le +D_1\left(1+\frac{1}{r\sqrt{t-t'}}\right)\int_{\gamma_t}kds+\left|\int_{\gamma_t} +\varphi k''ds\right|+\int_{\gamma_t}C'_1\varphi kds, +$$ + + where $C_1'$ +depends only on the ambient curvature bound. We bound the first term +by + +$$ +D_1D_0\left(\frac{1}{r\sqrt{t-t'}}+1\right), +$$ + +where $D_0$ is the constant depending on $\Theta$ and the ambient +curvature bound from \entryref{1b8d34cae3a8}. Since the ends of +$\gamma_t$ are at distance at least $(0.45)r$ from $x_0$ all $t\in +[t',t_2]$, we see that for all $t\in [t',t_2]$ + +$$ +\int_{\gamma_t}\varphi k''=\int_{c(\cdot,t)}\varphi k''. +$$ + +Integrating by parts we have + +$$ +\int_{c(\cdot,t)} \varphi k'' ds=\int_{c(\cdot,t)}\varphi'' k ds, +$$ + +where the prime here refers to the derivative along $c(\cdot,t)$ +with respect to arc length. Of course $|\varphi''|\le +\frac{D'}{r^2}$. Thus, we see that + +$$ +\left|\int_{\gamma_t}\varphi k''ds\right|\le +\frac{D'}{r^2}\int_{c(\cdot,t)}kds\le \frac{D'D_0}{r^2}. +$$ + + Putting +all this together, we have + +$$ +\left|\frac{d}{dt}\int_{\gamma_t}\varphi +kds\right|\le +D_2\left(1+\frac{1}{r\sqrt{t-t'}}\right)+\frac{D_2}{r^2}+D_3\int_{\gamma_t}\varphi +kds +$$ + + for $D_2=D_0 \mathit{max}(D',D_1)$ and $D_3=C_1'$. This gives +the required estimate. + +\entryblock{f7b6896c9e7d} + +**Proof.** This is immediate from the previous result by integrating from $t'$ +to $t_2\le t'+\delta r^2$, and using the fact that $\delta<1$ and +$r<1$, and using the fact that + +$$ +\int_{\gamma_{t'}}\varphi kds\le +\int_{\gamma_{t'}}kds<\delta +$$ + + since $\gamma_{t'}$ has length at +most $r$. + +This gives: + +\entryblock{55b7b71de6e4} + +**Proof.** We divide an interval $\gamma_{t'}\subset c(\cdot,t')$ of length at +most $r$ into two subintervals $\gamma'_{t'}$ and $\gamma''_{t'}$ of +lengths at most $r/2$. Let $\hat \gamma'_{t'}$ and $\hat +\gamma''_{t'}$ be intervals of length $r$ containing $\gamma'_{t'}$ +and $\gamma''_{t'}$ respectively as middle subintervals. We then +apply the previous corollary to $\hat \gamma'_{t'}$ and $\hat +\gamma''_{t'}$ using the fact that $\varphi k\ge 0$ everywhere and +$\varphi k=k$ on the middle subintervals of $\hat \gamma'_{t'}$ and +$\hat \gamma ''_{t'}$. For an interval $J\subset \gamma_t$ of length +$r/2$, according to \entryref{156a3da68fbd} the length of +$\gamma_{t'}|_J$ with respect to the metric $h(t')$ is at most $r$, +and hence this case follows from the previous case. + +### Writing the curve flow as a graph + +Now we restrict attention to $[t',t_2]$, the maximal interval in +$[t',t'+\delta r^2]$ where $k^2\le 2/(t-t')$. Let $\gamma_{t'}$ be +an arc of length $r$ in $c(\cdot,t')$ and let $x_0$ be the central +point of $\gamma_{t'}$. Denote $\gamma_{t'}(x_0)=p\in W$. We take +the $h(t')$-exponential mapping from $T_pW\to W$. This map will be a +local diffeomorphism out to a distance determined by the curvature +of $h(t')$. For an appropriate choice of the ball (depending on +the ambient curvature bound) the metric on the ball induced by +pulling back $h(t)$ for all $t\in [t',t_2]$ will be within $\delta$ +in the $C^1$-topology to the Euclidean metric $h'=h(t')_p$. By this +we mean that + +- **(1)** $\left|\langle X,Y\rangle_{h(t)}-\langle +X,Y\rangle_{h'}\right|<\delta |X|_{h'}|Y|_{h'}$ for all tangent vectors in the +coordinate system, and +- **(2)** viewing the connection $\Gamma$ as a bilinear +map on the coordinate space with values in the coordinate space we have $ +|\Gamma(X,Y)|_{h'}<\delta |X|_{h'}|Y|_{h'}$. + +We choose $00$ is sufficiently small, for every $t\in [t',t_3]$ there is +a subinterval of $\gamma_t$ that is the graph of a function defined +on all of $I'$. We now restrict attention to the family of curves +parameterized by $I'\times [t',t_3]$. For every $t\in[t',t_3]$ the +curve $\gamma_t|_{I'}$ has length between $(0.8)r$ and $r$. The +curve-shrinking flow is not defined on this product because under +the curve-shrinking flow the $z$-coordinate of any given point is +not constant. But the graph flow defined above, and studied in +[AG] (in the case of Euclidean background metric), is defined +on $I'\times [t',t_3]$ since this flow preserves the $z$-coordinate. +The time partial derivative in the curve-shrinking flow is given by + +$$ +\nabla_SS=\frac{\nabla_ZZ}{|Z|^2}-\frac{1}{|Z|^4}\langle \nabla_ZZ,Z\rangle Z. +$$ + + The time partial derivative in the graph-flow is given by +$Y=\partial\widetilde \gamma/\partial t$. The tangent vector $Y$ is +characterized by being $h'$-orthogonal to the $z$-axis and differing from +$\nabla_SS$ by a functional multiple of $Z$. + +\entryblock{188acafae968} + +**Proof.** In our Euclidean coordinates, $Z=(1,f_z)$ so that +$\nabla_ZZ=(0,f_{zz})+\Gamma(Z,Z)$. Thus, + +$$ +\langle \nabla_ZZ,\partial_z\rangle_{h'}=\langle +\Gamma(Z,Z),\partial_z\rangle_{h'}. +$$ + + Since $\langle +Z,\partial_z\rangle_{h'}=1$, it follows that + +$$ +\frac{ \nabla_ZZ-\langle\Gamma(Z,Z),\partial_z\rangle_{h'} Z}{|Z|^2} +$$ + + is +$h'$-orthogonal to the $z$-axis and hence is a multiple of $Y$. Since it +differs by a multiple of $Z$ from $\nabla_SS$, it follows that it is $Y$. This +gives the first equation; the second follows from this and +Equation (18.14). + +To simplify the notation we set + +$$ +\psi(Z)=\frac{\langle\Gamma(Z,Z),\partial_z\rangle_{h'}}{|Z|^2}. +$$ + +Notice that from the conditions on $\Gamma$ and $h'$ it follows +immediately that $|\psi(Z)|<(1.5)\delta$. + +### Proof that $t_3=t_2$ + +At this point we have a product coordinate system on which the +metric is almost the Euclidean metric in the $C^1$-sense, and we +have the graph flow given by + +$$ +Y=\frac{\partial\widetilde \gamma}{\partial +t}=\frac{\nabla_ZZ}{|Z|^2}-\psi(Z)Z +$$ + + defined on $[t',t_3]$ with +image always contained in the given coordinate patch and written as +a graph over a fixed interval $I'$ in the $z$-axis. For every $t\in +[t',t_3]$ the length of $\gamma_{t'}|_{I'}$ in the metric $h(t')$ is +between $(0.8)r$ and $r$. The function $f(z,t)$ whose graphs give +the flow satisfies $|f_z|_{h'}\le 1/10$. Our next goal is to +estimate $|f_z|_{h'}$ and show that it is always less than $1/10$ +as long as $k^2\le 2/(t-t')$ and $t-t'\le \delta r^2$ for a +sufficiently small $\delta$, i.e., for all $t\in [t',t_2]$; that is +to say, our next goal is to prove that $t_3=t_2$. In all the +arguments that follow $C'$ is a constant that depends only on the +curvature bound for the ambient Ricci flow, but the value of $C'$ is +allowed to change from line to line. + +The first step in doing this is to consider the angle between +$\nabla_ZZ$ and $Z$. + +\entryblock{8973f9acd2ae} + +**Proof.** Under the hypothesis that $|f_z|_{h'}\le 1/10$, it is easy to see +that the Euclidean angle between $(0,f_{zz})$ and $(1,f_z)$ is at +most $\pi/2-\pi/5$. From this, the first statement follows +immediately provided that $\delta$ is sufficiently small. Since $Y$ +is the sum of $\nabla_SS$ and a multiple of $Z$ and since +$\nabla_SS$ is $h(t)$-orthogonal to $Z$, it follows that +$|Y|=|\nabla_SS|\left(\mathit{cos}(\theta)\right)^{-1}$, where $\theta$ +is the angle between $\nabla_SS$ and $Y$. Since $Y$ is a multiple of +$(0,f_{zz})$, it follows from the first part of the claim that the +$h(t)$-angle between $Y$ and $\nabla_SS$ is less than $\pi/4$. Item +(1) of the claim then follows from the fact that by definition +$|\nabla_SS|=k$. + +Since + +$$ +\frac{\nabla_ZZ}{|Z|^2}=Y+\psi(Z)Z, +$$ + + and $|Z|^2\le (1.01)(1+\delta)$, the third item is +immediate. For Item (4), since $Y$ is $h'$-orthogonal to the $z$-axis, we have + +$$ +\left|\langle Y,Z\rangle_{h'}\right|=\left|\langle Y,(0,f_z)\rangle_{h'}\right|\le |Y|_{h'}|f_z|_{h'}. +$$ + +From this and the comparison of $h(t)$ and $h'$, the Item (4) is +immediate. Lastly, let us consider Item (2). We have + +$$ +\langle Y,Z\rangle =\frac{\langle \nabla_ZZ,Z\rangle}{|Z|^2}-\langle +\Gamma(Z,Z),\partial_z\rangle_{h'}. +$$ + + Thus, from Item (4) we have + +$$ +\frac{\langle \nabla_ZZ,Z\rangle}{|Z|^2}\le |Y||f_z|(1+3\delta)+(1.5)\delta. +$$ + +Since $Y<\sqrt{2}k$ and $|f_z|<1/10$, the Item (2) follows. + +\entryblock{d3f828726e2b} + +**Proof.** For the first item, we write $Z(\psi(Z))$ as a sum of terms where the +differentiation by $Z$ acts on the various. When the $Z$-derivative acts on +$\Gamma$ the resulting term has norm bounded by a constant depending only on +the curvature of the ambient Ricci flow. When the $Z$-derivative acts on one of +the $Z$-terms in $\Gamma(Z,Z)$ the norm of the result is bounded by +$2\delta|\nabla_ZZ||Z|$. Action on each of the other $Z$-terms gives a term +bounded in norm by the same expression. Lastly, when the $Z$-derivative acts on +the constant metric $h'$ the norm of the result is bounded by $2\delta^2$. +Since $|\nabla_ZZ|\le 2(|Y|+\delta)$, the first item follows. + +We compute $Y(\psi(Z))$ in a similar fashion. When the $Y$-derivative acts on +the $\Gamma$, the norm of the result is bounded by $C'|Y|$. When the +$Y$-derivative acts on one of the $Z$-terms the norm of the result is bounded +by $2\delta|\nabla_YZ|$. Lastly, when the $Y$-derivative acts on the constant +metric $h'$, the norm of the result is bounded by $\delta^2|Y|$. Putting all +these terms together establishes the second inequality above. + +Now we wish to compute $\int_{I'\times\{t\}}|Z|^2dz.$ To do this we +first note that using the definition of $Y$, and arguing as in the +proof of the first equation in of \entryref{4fc24fa1d31e} we have we +have + +$$ +\begin{aligned} +\frac{\partial}{\partial t}|Z|^2 +& = & -2\mathit{Ric}(Z,Z)+2\langle \nabla_YZ,Z\rangle \\ +& = & -2\mathit{Ric}(Z,Z)+2\langle \nabla_ZY,Z\rangle +\end{aligned} +$$ + +Direct computation shows that + +$$ +2\langle +\nabla_Z\left(\frac{\nabla_ZZ}{|Z|^2}\right),Z\rangle +=Z\left(\frac{Z(|Z|^2)}{|Z|^2}\right)-2\frac{|\nabla_ZZ|^2}{|Z|^4}|Z|^2. +$$ + + Thus +from the \entryref{188acafae968}, we have + +$$ +\begin{aligned} +\frac{\partial}{\partial t}|Z|^2 & = & 2\langle\nabla_ZY,Z\rangle -2\mathit{Ric}(Z,Z)\nonumber \\ +& = & +Z\left(\frac{Z(|Z|^2)}{|Z|^2}\right)-2\frac{|\nabla_ZZ|^2}{|Z|^4}|Z|^2-2\langle +\nabla_Z(\psi(Z)Z),Z\rangle-2\mathit{Ric}(Z,Z) \nonumber \\ +& = & Z\left(\frac{Z(|Z|^2)}{|Z|^2}\right)-2|Y|^2|Z|^2 +V, +\end{aligned} +$$ + +where + +$$ +V=-4|Z|^2\langle Y,\psi(Z)Z\rangle-2\psi^2(Z)|Z|^4-2\langle +\nabla_Z(\psi(Z)Z),Z\rangle-2\mathit{Ric}(Z,Z). +$$ + + By Item (1) in +\entryref{d3f828726e2b} and Item (4) in \entryref{8973f9acd2ae} we have + +$$ +|V|0$ sufficiently small, using the fact that +$1-\delta<|Z|^2<(1+\delta)(1.01)$ we can rewrite this as + +$$ +\begin{aligned} +Q_t & \le & \frac{Q_{zz}}{|Z|^2}+\frac{8(1+3\delta)|Y|^2|f_z| +|\nabla_ZY|}{(2-|Z|^2)^2} +\frac{4|Y|^2|(1+\delta) + |\nabla_ZY||f_z|}{|Z|^2(2-|Z|^2)} \\ + & & -\frac{2|\nabla_ZY|^2}{|Z|^2(2-|Z|^2)}-\frac{(1.95)|Y|^4}{(2-|Z|^2)^2} +\tilde + A, +\end{aligned} +$$ + +where $\tilde A\le C'(|Y|^2+\delta|Y||\nabla_ZY|+\delta|Y|^3)$. Of +course, $|Y||\nabla_ZY|+|Y|^3\le 2|Y|^2+|\nabla_ZY|^2+|Y|^4$. Using +this, provided that $\delta$ is sufficiently small, we can rewrite +this as + +$$ +\begin{aligned} +Q_t & \le & \frac{Q_{zz}}{|Z|^2} +\frac{1}{(2-|Z|^2)}\cdot +\left[\frac{8(1+3\delta)|\nabla_ZY||f_z||Y|^2-(0.9)|Y|^4}{(2-|Z|^2)} \right.\\& +& + +\left. \frac{4|Y|^2|(1+\delta) + |\nabla_ZY||f_z|-(1.9)|\nabla_ZY|^2}{|Z|^2}\right]-Q^2+\tilde A'' +\end{aligned} +$$ + +where $\tilde A''\le C'(|Y|^2)$. We denote the quantity within the +brackets by $B$ and we estimate + +$$ +\begin{aligned} +B & \le & +8(1+3\delta)\frac{|Y|^2|\nabla_ZY|(1/10)(1+\delta)}{(2-(1.01)(1+\delta))} ++\frac{4(1/10)(1+\delta)|Y|^2|\nabla_ZY|}{(1-\delta)} \\ +& & -\frac{(1.9)}{(1.01)(1+\delta)}|\nabla_ZY|^2 +-\frac{(0.9)|Y|^4}{2-(1.01)(1+\delta)} \\ +& \le & (1.6)|Y|^2|\nabla_ZY|-(0.8)|\nabla_ZY|^2-(0.8)|Y|^4 \\ +& \le & 0. +\end{aligned} +$$ + +Therefore, + +$$ +Q_t\le \frac{Q_{zz}}{|Z|^2}-Q^2+|\tilde A|\le \frac{Q_{zz}}{|Z|^2}-(Q-C_1')^2+(C_1')^2, +$$ + +for some constant $C'_1>1$ depending only on the curvature bound for the +ambient Ricci flow. + +Denote by $l$ the length of $I'$ under $h'$. As we have already +seen, $(0.8)r\le l\le r$. We translate the $z$-coordinate so that +$z=0$ is one endpoint of $I'$ and $z=l$ is the other endpoint; the +point $x_0$ then corresponds to $z=l/2$. Consider the function +$g=l^2/(z^2(l-z)^2)$ on $I'\times [t',t_2]$. Direct computation +shows that $g_{zz}\le 12g^2$. + Now set + +$$ +\widetilde Q=Q-C_1' +$$ + + and + +$$ +h=\frac{1}{t-t'}+\frac{4(1-\delta)^{-1}l^2}{z^2(l-z)^2}+C_1'. +$$ + + Then + +$$ +-h_t+(1-\delta)^{-1}h_{zz}+(C_1')^2\le h^2, +$$ + +so that + +$$ +(\widetilde Q-h)_t\le \frac{\widetilde Q_{zz}}{|Z|^2}-\frac{h_{zz}}{1-\delta}-\widetilde +Q^2+h^2. +$$ + + Since both $h$ and $h_{zz}$ are positive, at any point +where $\widetilde Q -h\ge 0$ and $\widetilde Q_{zz}<0$, we have +$(\widetilde Q-h)_t<0$. At any point where $Q_{zz}\ge 0$, using the +fact that $|Z|^2\ge (1-\delta)$ we have + +$$ +(\widetilde Q-h)_t\le (1-\delta)^{-1}(\widetilde Q-h)_{zz}-\widetilde +Q^2+h^2. +$$ + + Thus, for any fixed $t$, at any local maximum for +$(\widetilde Q-h)(\cdot,t)$ at which $(\widetilde Q-h)$ is $\ge 0$ +we have $(\widetilde Q-h)_t\le 0$. Since $\widetilde Q-h$ equals +$-\infty$ at the end points of $I'$ for all times, there is a +continuous function $f(t)=\mathit{max}_{z\in I'}(\widetilde Q-h)(z,t)$, +defined for all $t\in (t',t_2]$ approaching $-\infty$ uniformly as +$t$ approaches $t'$ from above. By the previous discussion, at any +point where $f(t)\ge 0$ we have $f'(t)\le 0$ in the sense of forward +difference quotients. It now follows that $f(t)\le 0$ for all $t\in +(t',t_2]$. + This means that for all $t\in (t',t_2]$ at the $h'$-midpoint $x_0$ of +$I'$ (the point where $z=l/2$) we have + +$$ +Q(x_0,t)\le \frac{1}{t-t'}+\frac{16\cdot 4(1-\delta)^{-1}}{l^2}+C_1'. +$$ + +Since $l\ge (0.8)r$ and since $t-t'\le \delta r^2$, we see that +provided $\delta$ is sufficiently small (depending on the bound of +the curvature of the ambient flow) we have + +$$ +Q(x_0,t)< \frac{3}{2(t-t')} +$$ + + for all $t\in [t',t_2].$ +Of course, since $|Z|^2\ge 1-\delta$ everywhere, this shows that + +$$ +k^2(x_0,t)\le |Y(x_0,t)|^2= +(2-|Z(x_0,t)|^2)Q(x_0,t)< \frac{2}{(t-t')} +$$ + + for all $t\in +[t',t_2]$. Since $x_0$ was an arbitrary point of $c(\cdot,t')$, + this shows that + $k(x,t)<\sqrt{\frac{2}{t-t'}}$ for all $x\in c(\cdot,t)$ and all $t\in + [t',t_2]$. By the definition of $t_2$ this implies that $t_2=t'+\delta r^2$ and completes the + proof of \entryref{ff5d51a483d9}. diff --git a/projects/poincare-conjecture/.astrolabe/docs/20-appendix-canonical-neighborhoods.mdx b/projects/poincare-conjecture/.astrolabe/docs/20-appendix-canonical-neighborhoods.mdx new file mode 100644 index 00000000..192157df --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/docs/20-appendix-canonical-neighborhoods.mdx @@ -0,0 +1,727 @@ + + +# Chapter 19 -- Appendix: Canonical neighborhoods + +Recall that an $\epsilon$-neck structure on a Riemannian manifold +$(N,g)$ centered at a point $x\in N$ is a diffeomorphism $\psi\colon +S^2\times (-\epsilon^{-1},\epsilon^{-1})\to N$ with the property +that $x\in \psi(S^2\times\{0\})$ and the property that $R(x)\psi^*g$ +is within $\epsilon$ in the $C^{[1/\epsilon]}$-topology of the +product metric $h_0\times ds^2$, where $h_0$ is the round metric on +$S^2$ of scalar curvature $1$ and $ds^2$ is the Euclidean metric on +the interval. Recall that the *scale* of the $\epsilon$-neck is +$R(x)^{-1/2}$. We define $s=s_N\colon N\to +(-\epsilon^{-1},\epsilon^{-1})$ as the composition of $\psi^{-1}$ +followed by the projection to the second factor. + +## Shortening curves + +\entryblock{fc6c6ae605d2} + +**Proof.** Since $\gamma$ contains a point on the negative side of $S(x)$ and +it ends on the positive side of $S(x)$, there is a $c\in (0,a)$ such +that $\gamma(c)\in S(x)$ and $\gamma|_{(c,a]}$ is disjoint from +$S(x)$. Since $\gamma|_{[0,c]}$ has both endpoints in $S(x)$ and +also contains a point of $s^{-1}(-\epsilon^{-1},-\epsilon^{-1}/2)$, +it follows that for $\epsilon$ sufficiently small, the length of +$\gamma|_{[0,c]}$ is at least $3\epsilon^{-1}R(x)^{-1/2}/4$. On the +other hand, there is a path $\mu$ in $S(x)$ connecting $x$ to +$\gamma(c)$ of length at most $2\sqrt{2}\pi(1+\epsilon)$. Thus, if +$\epsilon$ is sufficiently small, the concatenation of $\mu$ +followed by $\gamma|_{[c,a]}$ is the required shorter path. + +## The geometry of an $\epsilon$-neck + +\entryblock{dc5709757703} + +**Proof.** The principal curvatures and their directions are continuous +functions of the metric $g$ in the space of metrics with the +$C^2$-topology. The statements follow immediately. + +\entryblock{98c6a95f8556} + +**Proof.** By the previous lemma, if $\epsilon$ is sufficiently small for every +$n\in N$ there is a unique two-plane, $P_n$, at each point on which +the sectional curvature is maximal. The sectional curvature on this +two-plane is close to $1/2$ and the other two eigenvalues of the +curvature operator at $n$ are close to zero. Furthermore, $P_n$ +makes small $g$-angle with the tangent planes to the $S^2$-factors +in the neck structure. Under the condition that the restriction of +the metric to $f(S^2)$ is close to the round metric $h_0$ and the +norm of the second fundamental form is small, we see that for every +$p\in S^2$ the two-plane $df(T_pS^2)$ makes a small $g$-angle with +$P_n$ and hence with the tangent planes to the family of two-spheres +coming from the neck structure. Since $g$ is close to the product +metric, this means that the angle between $df (T_nS^2)$ and the +tangents to the family of two-spheres coming from the neck +structure, measured in the product metric, is also small. Hence, the +composition of $f$ followed by the projection mapping $N\to S^2$ +induced by the neck structure determines a submersion of $S^2$ onto +itself. Since $S^2$ is compact and simply connected, any submersion +of $S^2$ onto itself is a diffeomorphism. This means that $f(S^2)$ +crosses each line $\{x\}\times (-\epsilon^{-1},\epsilon^{-1})$ +transversely and in exactly one point. Clearly then, it is isotopic +in $N$ to any two-sphere of the form $S^2\times \{s\}$. + +\entryblock{88a5a2233418} + +**Proof.** Let us consider a geodesic $\mu$ in the product Riemannian manifold +$S^2\times \Ar$ with the metric on $S^2$ being of constant Gaussian +curvature $1/2$, i.e., radius $\sqrt{2}$. Its projections, $\mu_1$ +and $\mu_2$, to $S^2$ and to $\Ar$, respectively, + are also geodesics, and $|\mu|=\sqrt{|\mu_1|^2+|\mu_2|^2}$. + For $\mu$ to be a minimal geodesic, the same is true of each of its projections. +In particular, when $\mu$ is minimal, the length of $\mu_1$ is at +most $\sqrt{2}\pi$. Hence, for any $\alpha'>0$, if $\mu$ is +sufficiently long and if the final endpoint has a larger $s$-value +than the initial point, then the angle between the tangent vectors +$\mu'(s)$ and $\partial/\partial s$ is less than $\alpha'$. This +establishes the result for the standard metric on the model for +$\epsilon$-necks. + +The first statement now follows for all $\epsilon$ sufficiently +small and all $\epsilon$-necks because minimal geodesics between a +pair of points in a manifold vary continuously in the $C^1$-topology +as a function of the space of metrics with the $C^k$-topology, since +$k\ge 2$. The second statement is obvious since the diameter of any +member of the family of two-spheres in the standard metric is +$\sqrt{2}\pi$. + +\entryblock{b74cd360f72f} + +\entryblock{82f43e137897} + +There is a closely related lemma. + +\entryblock{bb69d9bc83dc} + +**Proof.** We parameterize $\gamma$ as a map from $[a,b]\to M$. By +\entryref{82f43e137897}, provided that $\epsilon>0$ is sufficiently +small, each component of $\gamma\cap N$ crosses each two-sphere of +the neck structure at most once. Suppose that there is some +two-sphere $S^2\times\{x\}$ that is crossed by two different +components of $\gamma$. Let $c0$ +as close to zero as we want by making $\epsilon$ smaller. Clearly, +then in this case as well, replacing $\gamma([c,d])$ with a path of +length less than $2\pi R(x)^{-1/2}$ on $S^2\times\{s\}$ will shorten +the length of $\gamma$. + +\entryblock{5ef5906bb370} + +**Proof.** If $\gamma$ is completely contained in $\widehat Y$ then the result +is clear. Suppose that the path is $\gamma\colon [a,b]\to M$ and +$\gamma(d)\in Y$ for some $d\in [a,b]$. Suppose that there are +$a'0$ sufficiently small. + +**N.B.** It is important that the central two-sphere of +$N$ separates the ambient manifold $M$. Otherwise, there may be +shorter geodesics from $z$ to $q$ entering the other end of $N$. + +\entryblock{04dd3c3ea1c8} + +**Proof.** The statement is scale-invariant, so we can assume that $R(x)=1$. + Denote by $S(z,d)$ the boundary of the metric ball $B(z,d)$. We +orient $s$ so that $z$ lies to the negative side of the central +two-sphere of $N$. It follows immediately from the previous result +that, provided that $\epsilon>0$ is sufficiently small, $S(z,d)$ +intersects any line ${y}\times (-\epsilon^{-1}/3,\epsilon^{-1}/3)$ +in at most one point. To complete the proof we need only show that +$S(z,d)$ is contained +$s^{-1}((s(p)-\alpha\epsilon^{-1},s(p)+\alpha\epsilon^{-1}))$. The +distance from $d$ to any point in the two-sphere factor of $N$ +containing $p$ is contained in the interval $[d-2\pi,d+2\pi]$. +Provided that $\epsilon$ is sufficiently small depending on +$\alpha$, the result follows immediately from +\entryref{6cf1cbc39ada}. + +## Overlapping $\epsilon$-necks + +The subject of this section is the internal geometric properties of +$\epsilon$-necks and of intersections of $\epsilon$-necks. We assume that +$\epsilon\le 1/200$. + +\entryblock{a46d72b87e07} + +**Proof.** Fix $0<\epsilon_5(\alpha)\le\mathit{min}(\epsilon_1(\alpha_1),\epsilon_2(\alpha/3),\epsilon_3(\alpha),\alpha/3)$ +sufficiently small so that \entryref{98c6a95f8556} holds. The +first two items are then immediate from \entryref{dc5709757703}. The +third statement is immediate from \entryref{88a5a2233418}, and the +fourth statement from \entryref{98c6a95f8556}. Let us consider +the last statement. Let $y\in N\cap N'$ have $-(0.9)\epsilon^{-1}\le +s_N(y)\le (0.9)\epsilon^{-1}$. By reversing the $s$-directions of +$N$ and/or $N'$ if necessary, we can assume that $0\le s_N(y)\le +(0.9)\epsilon^{-1}$ and that $\partial_{s_N}$ and +$\partial_{s_{N'}}$ almost agree at $y$. If $-(0.96)\epsilon^{-1}\le +s_{N'}(y)\le (0.96)\epsilon^{-1}$, we set $y'=y$. Suppose that +$s_{N'}(y)>(0.96)\epsilon^{-1}$. We move along the straight line +though $y$ in the neck structure on $N$ in the negative direction to +a point $y'$ with $(0.96)\epsilon^{-1}=s_{N'}(y')$ According to Item +3 of this result we have $-(0.96)\epsilon^{-1}\le s_{N}(x')\le +(0.96)\epsilon^{-1}$. There is a similar construction when +$s_{N'}(y)<-(0.96)\epsilon^{-1}$. In all cases this allows us to +find $y'$ such that both the following hold: + +$$ +-(0.96)\epsilon^{-1}\le s_N(y')\le (0.96)\epsilon^{-1} +$$ + +$$ +-(0.96)\epsilon^{-1}\le s_{N'}(y')\le (0.96)\epsilon^{-1}. +$$ + +Let $y'$ be any point satisfying both these inequalities. According +to \entryref{88a5a2233418} and Part (1) of this result, the diameter +of $S(y')$ is at most $2\pi h$, where $h$ is the scale of $N$ and +$N'$. Since $\epsilon^{-1}\ge 200$, it follows from +\entryref{b74cd360f72f} that $S(y')$ is contained in $N'$. +Symmetrically $S'(y')$ is contained in $N$. + +Now consider the intersection of any straight line in the neck +structure on $N$ with $N'$. According to Part (3), this intersection +is connected. Thus, $N\cap N'$ is a union of open arcs in the +$s_N$-directions thought the points of $S(y')$. These arcs can be +used to define a diffeomorphism from $N\cap N'$ to $S^2\times +(0,1)$ sending $S(y')$ to $S^2\times \{1/2\}$. Also, we have the +straight line isotopy from $S'(y')$ to $S(y')$ contained in $N\cap +N'$. + +## Regions covered by $\epsilon$-necks and $(C,\epsilon)$-caps + +Here we fix $0<\epsilon\le 1/200$ sufficiently small so that all the +results in the previous two sections hold with $\alpha=10^{-2}$. + +### Chains of $\epsilon$-necks + +\entryblock{36d5883d0459} + +Notice that in an $\epsilon$-chain $N_i\cap N_j=\emptyset$ if +$|i-j|\ge 5$. + +\entryblock{035b5e6ca144} + +**Proof.** Let us first prove the result for finite chains. The proof that $U$ +is diffeomorphic to $S^2\times (0,1)$ is by induction on $b-a+1$. If +$b=a$, then the result is clear. Suppose that we know the result for +chains of smaller cardinality. Then $N_a\cup\cdots\cup N_{b-1}$ is +diffeomorphic to $S^2\times (0,1)$. Hence by Part (5) of +\entryref{a46d72b87e07}, $U$ is the union of two manifolds +each diffeomorphic to $S^2\times (0,1)$ meeting in an open subset +diffeomorphic to $S^2\times (0,1)$. Furthermore, by the same result + in the intersection there is a two-sphere isotopic to each of the +two-sphere factors from the two pieces. It now follows easily that +the union is diffeomorphic to $S^2\times (0,1)$. Now consider an +infinite chain. It is an increasing union of finite chains each +diffeomorphic to $S^2\times (0,1)$ and with the two-spheres of one +isotopic to the two-spheres of any larger one. It is then immediate +that the union is diffeomorphic to $S^2\times (0,1)$. + +Notice that the frontier of the union of the necks in a finite +chain, $U=\cup_{a\le i\le b}N_i$, in $M$ is equal to the frontier of +the positive end of $N_b$ union the frontier of the negative end of +$N_a$. Thus, we have: + +\entryblock{809fbb8ba32a} + +The next result shows there is no frontier at an infinite end. + +\entryblock{f6496966d395} + +**Proof.** Suppose that $x$ is a point of the frontier of $U$. Let $x_i\in U$ +be a sequence converging to $x$. If the $x_i$ were contained in a +finite union of the $N_k$, say $N_0\cup\cdots\cup N_\ell$, then $x$ +would be in the closure of this union and hence by the previous +comment would be either be in the frontier of the negative end of +$N_0$ or the frontier of the positive end of $N_\ell$. But the +frontier of the positive end of $N_\ell$ is contained in +$N_{\ell+1}$ and hence contains no points of the frontier of $U$. +Thus, in this case $x$ is a point of the frontier of the negative +end of $N_0$. If $\{x_i\}$ is not contained in any finite union, +then after passing to a subsequence, we can suppose that $x_i\in +N_{k(i)}$ where $k(i)$ is an increasing sequence tending to +infinity. Clearly $R(x_i)$ converges to $R(x)<\infty$. Hence, there +is a uniform lower bound to the scales of the $N_{k(i)}$. For all +$i$ sufficiently large $x_i\not\in N_0$. Thus, for such $i$ any path +from $x_i$ to $x$ must traverse either $N_0$ or $N_{k(j)}$ for all +$j\ge i+5$. The length of such a path is at least the minimum of the +width of $N_0$ and the width of $N_{k(j)}$ for some $j$ sufficiently +large. But we have just seen that there is a positive lower bound to +the scales of the $N_{k(j)}$ independent of $j$, and hence by +\entryref{b74cd360f72f} there is a positive lower bound, independent +of $j$, to the widths of the $N_{k(j)}$. This shows that there is a +positive lower bound, independent of $i$, to the distance from $x_i$ +to $x$ .This is impossible since $x_i$ converges to $x$. + + In fact, there is a geometric version of \entryref{035b5e6ca144}. + +\entryblock{d5a1d88909aa} + +**Proof.** Since according to \entryref{dc5709757703} the two-spheres for $N_j$ +and $N_{j+1}$ almost line up, it is an easy matter to interpolate +between the projection maps to the interval to construct a fibration +of $U$ by two-spheres with the given property. The interval $I$ is +simply the base space of this fibration. + +A finite or infinite chain $\{N_j\}_{j\in J}$ of $\epsilon$-necks is +*balanced* provided that for every $j\in J$, not the largest +element of $J$, we have + +$$ +(0.99)R(x_j)^{-1/2}\epsilon^{-1}\le d(x_j,x_{j+1})\le +(1.01)R(x_j)^{-1/2}\epsilon^{-1}, +$$ + +where, for each $j$, $x_j$ is the central point of $N_j$. + +Notice that in a balanced chain $N_j\cap N_{j'}=\emptyset$ if +$|j-j'|\ge 3$. + +\entryblock{58804e22eff2} + +**Proof.** By \entryref{b74cd360f72f}, +Inequality (19.1) holds for $d(x,x')$. Once we have this +inequality, it follows immediately from the same corollary that, +possible after reversing, the $s$-directions $\{N,N'\}$ makes a +balanced chain of $\epsilon$-necks. (It is not possible for the +positive end of $N_b$ to meet $N_a$ for this would allow us to +create a loop meeting the central two-sphere of $N_b$ transversely +in a single point, so that this two-sphere would not separate $M$.) + +\entryblock{6dc87ad116fc} + +**Proof.** The two cases are +symmetric; we consider only the first. Since $x$ is contained in the +closure of $N_b$, clearly $N_b\cap N\not=\emptyset$. Also, clearly, +provided that $\epsilon>0$ is sufficiently small, $d(x_b,x)$ +satisfies Inequality (19.1) so that \entryref{58804e22eff2} +the pair $\{N_b,N\}$ forms an $\epsilon$-chain, and hence a balanced +$\epsilon$-chain. It is not possible for $N$ to meet the negative +end of $N_a$ since the central two-sphere of $N_a$ separates $M$. +Hence $\{N_a,\ldots,N_b,N\}$ is a balanced chain of +$\epsilon$-necks. + +\entryblock{ba681e15e1a6} + +**Proof.** According to \entryref{6dc87ad116fc} for $\epsilon>0$ sufficiently +small the following holds. Suppose that we have a balanced chain of +$\epsilon$-necks $N_a\ldots,N_{b}$, with $N_i$ centered at $x_i\in +X$, whose union $U$ does not contain $X$. + Then one of the following holds: + +- **(1)** It is possible to find an $\epsilon$-neck $N_{b+1}$ centered +at a point of the intersection of $X$ with the closure of the +positive end of $N_{b}$ so that $N_a,\ldots,N_{b+1}$ is a balanced +$\epsilon$-chain. +- **(2)** It is possible to find an $\epsilon$-neck $N_{a-1}$ centered at a +point of the intersection of $X$ with the closure of the negative +end of $N_a$ so that $N_{a-1},N_a,\ldots,N_b$ is a balanced +$\epsilon$-chain. + +Now assume that there is no finite balanced chain of +$\epsilon$-necks $N(x_i)$ containing $X$. Then we can repeatedly +lengthen a balanced chain of $\epsilon$-necks centered at points of +$X$ by adding necks at one end or the other. Suppose that we have a +half-infinite balanced chain $\{N_0,N_1,\ldots,\}$. By +\entryref{f6496966d395} the frontier of this union is the frontier +of the negative end of $N_0$. Thus, if we can construct a balanced +chain which is infinite in both directions, then the union of the +necks in this chain is a component of $M$ and hence contains the +connected set $X$. If we can construct a balanced chain that is +infinite at one end but not the other that cannot be further +extended, then the connected set is disjoint from the frontier of +the negative end of the first neck in the chain and, as we have see +above, the 'infinite' end of the chain has no frontier. Thus, $X$ is +disjoint from the frontier of $U$ in $M$ and hence is contained in +$U$. Thus, in all cases we construct a balanced chain of +$\epsilon$-necks containing $X$. By \entryref{035b5e6ca144} the union +of the necks in this chain is diffeomorphic to $S^2\times (0,1)$ and +hence is an $\epsilon$-tube. + +\entryblock{aa8a682b4513} + +**Proof.** If the two-spheres of the $\epsilon$-necks do not separate $M$, then +it follows from the previous result that $M$ is an $\epsilon$-tube. +If one of the two-spheres does separate, then take the universal +covering $\widetilde M$ of $M$. Every point of $\widetilde M$ is +the center of an $\epsilon$-neck (lifting an $\epsilon$-neck in $M$) +and the two-spheres of these necks separate $\widetilde M$. Thus the +first case applies, showing that $\widetilde M$ is diffeomorphic to +$S^2\times (0,1)$. Every point is the center of an $\epsilon$-neck +that is disjoint from all its non-trivial translates under the +fundamental group. This means that the quotient is fibered by +$S^2$'s over $S^1$, and the fibers of this fibration are isotopic to +the central two-spheres of the $\epsilon$-necks. + +## Subsets of the union of cores of $(C,\epsilon)$-caps and $\epsilon$-necks. + In this section we fix $0<\epsilon\le 1/200$ +so that all the results of this section hold with $\alpha=0.01$. + +\entryblock{0e9a1a167caa} + +**Proof.** We divide the proof into two cases: Case I: There is a point of $X$ +contained in the core of a $(C,\epsilon)$-cap. Case II: Every point +of $X$ is the center of an $\epsilon$-neck. + +**Case I:** We begin the study of this case with a claim. + +\entryblock{9737d037f97b} + +**Proof.** We argue by contradiction. Suppose there is such an infinite chain. +Fix a point $x_0\in C_0$ and let $Q_0=R(x_0)$. For each $i\ge 1$ let +$x_i$ be a point in the frontier of $C_{i-1}$ that is contained in +the closure of the core of $C_i$. For each $i$ let $N_i$ be the +$\epsilon$-neck in $C_{i}$ that is the complement of the closure of +its core. We orient the $s_{N_{i}}$-direction so that the core of +$C_{i}$ lies off the negative end of $N_{i}$. Let $S'_i$ be the +boundary of the core of $C_i$. It is the central two-sphere of an +$\epsilon$-neck $N'_i$ in $C_i$. We orient the $s$-direction of +$N'_i$ so that the non-compact end of $C_i$ lies off the positive +end of $N'_i$. We denote by $h_{i-1}$ the scale of $N_{i-1}$ and by +$h_i'$ the scale of $N'_i$. By \entryref{dc5709757703} the ratio +$h_{i-1}/h'_i$ is between $0.99$ and $1.01$. Suppose that $S'_i$ is +disjoint from $C_{i-1}$. Then one of the complementary components of +$S'_i$ in $M$ contains $C_{i-i}$, and of course, one of the +complementary components of $S'_i$ is the core of $C_i$. These +complementary components must be the same, for otherwise $C_{i-1}$ +would be disjoint from the core of $C_i$ and hence the intersection +of $C_{i-1}$ and $C_i$ would be contained in $N_i$. This cannot +happen since $C_{i-1}$ is contained in $C_i$. Thus, if $S'_i$ is +disjoint from $C_{i-1}$, then the core of $C_i$ contains $C_{i-1}$. + This means that the distance from $x_0$ to the +complement of $C_i$ is greater than the distance of $x_0$ to the +complement of $C_{i-1}$ by an amount equal to the width of $N_i$. +Since the scale of $N_i$ is at least $C^{-1/2}R(x_0)^{-1/2}$ (see +(5) of \entryref{757e83881e62}), it follows from +\entryref{b74cd360f72f} that this width is at least +$2(0.99)C^{-1/2}R(x_0)^{-1/2}\epsilon^{-1}$. + +Next suppose that $S'_i$ is contained in $C_{i-1}$. Then one of the +complementary components $A$ of $S'_i$ in $M$ has closure contained +in $C_{i-1}$. This component cannot be the core of $C_i$ since the +closure of the core of $C_i$ contains a point of the frontier of +$C_{i-1}$ in $M$. Thus, $A$ contains $N_i$. Of course, $A\not= N_i$ +since the frontier of $A$ in $M$ is $S'_i$ whereas $N_i$ has two +components to its frontier in $M$. This means that $C_i$ does not +contain $A$, which is a contradiction since $C_i$ contains $C_{i-1}$ +and $A\subset C_{i-1}$. + +Lastly, we suppose that $S'_i$ is neither contained in $C_{i-1}$ nor +in its complement. Then $S'_i$ must meet $N_{i-1}$. According to +\entryref{a46d72b87e07} the $s$-directions in $N_{i-1}$ and +$N'_i$ either almost agree or are almost opposite. Let $x\in +S'_i\cap \partial N_{i-1}$ so that $s_{N'_i}(x)=0$. Move from $x$ +along the $s_{N'_i}$-direction that moves into $N_{i-1}$ to a point +$x'$ with $|s_{N_i}(x')|=(0.05)\epsilon^{-1}$. According to +\entryref{a46d72b87e07} +$(0.94)\epsilon^{-1}0 + refSpan: + coordinateSpace: comment-stripped-tex + end: 10512 + file: stdsoln.tex + sourceTransform: tex2mdx.strip_comments + start: 10502 + sentence: By Corollary~\ref{rm>0}, for each $n$ the sectional curvature of the + flow $(S^3_{R_n},g_{R_n}(t)),\ 0\le t0 + mtref: '4.14' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 16848 + sourceTransform: tex2mdx.strip_comments + start: 16607 + type: prose-mention-nearest-previous-statement + via: + - prose +generator: tools/poincare_tex_extract.py +inference: weak +kind: mention +ref: +- f44d182ef27e +- 0db415314964 +rel: mentions +reviewStatus: candidate +scope: prose +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- prose +--- +Morgan--Tian prose mention. The source is the nearest preceding statement in the same chapter segment. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/05bfd7c1a79d.md b/projects/poincare-conjecture/.astrolabe/edges/05bfd7c1a79d.md new file mode 100644 index 00000000..2f295f89 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/05bfd7c1a79d.md @@ -0,0 +1,84 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: narrows + proofSpan: + bodyEnd: 103549 + bodyStart: 101913 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 103560 + file: temp2kappa.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 101900 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 103271 + file: temp2kappa.tex + sourceTransform: tex2mdx.strip_comments + start: 103258 + via: proof + - context: proof + label: narrows + proofSpan: + bodyEnd: 103549 + bodyStart: 101913 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 103560 + file: temp2kappa.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 101900 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 103474 + file: temp2kappa.tex + sourceTransform: tex2mdx.strip_comments + start: 103461 + via: proof + sourceStatement: + chapter: 9 + file: temp2kappa.tex + label: 3DGSSkappa + mtref: '9.53' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 101898 + sourceTransform: tex2mdx.strip_comments + start: 101535 + targetStatement: + chapter: 2 + file: prelim.tex + label: narrows + mtref: '2.19' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 76622 + sourceTransform: tex2mdx.strip_comments + start: 76377 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 351e4ac14e4e +- cb9ca6d6db9f +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/05ccfd9b086b.md b/projects/poincare-conjecture/.astrolabe/edges/05ccfd9b086b.md new file mode 100644 index 00000000..f28bb6dd --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/05ccfd9b086b.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 16 + file: surgery.tex + label: '' + mtref: '16.22' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 156009 + sourceTransform: tex2mdx.strip_comments + start: 155858 + targetStatement: + chapter: 16 + file: surgery.tex + label: '' + mtref: '16.23' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 158095 + sourceTransform: tex2mdx.strip_comments + start: 157903 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- cfb464e3cf35 +- 5ba0ec2c68db +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/05d94b8c616d.md b/projects/poincare-conjecture/.astrolabe/edges/05d94b8c616d.md new file mode 100644 index 00000000..a497e734 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/05d94b8c616d.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 4 + file: maxprin.tex + label: '' + mtref: '4.6' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 7829 + sourceTransform: tex2mdx.strip_comments + start: 7225 + targetStatement: + chapter: 4 + file: maxprin.tex + label: MP + mtref: '4.7' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 8808 + sourceTransform: tex2mdx.strip_comments + start: 7939 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- f3441849c78b +- c954e95b5d89 +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/05e4579eda46.md b/projects/poincare-conjecture/.astrolabe/edges/05e4579eda46.md new file mode 100644 index 00000000..4d38fc55 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/05e4579eda46.md @@ -0,0 +1,79 @@ +--- +confidence: 0.35 +edgeClass: prose +evidence: + mentionTriggers: + - label: KAPPALIMIT + refSpan: + coordinateSpace: comment-stripped-tex + end: 173427 + file: surgery.tex + sourceTransform: tex2mdx.strip_comments + start: 173411 + sentence: \section{Completion of the proof of Proposition~\protect{\ref{KAPPALIMIT}}} + sentenceSpan: + coordinateSpace: comment-stripped-tex + end: 173429 + file: surgery.tex + sourceTransform: tex2mdx.strip_comments + start: 173353 + sourceStrategy: nearest-previous-statement + via: prose + - label: KAPPALIMIT + refSpan: + coordinateSpace: comment-stripped-tex + end: 173489 + file: surgery.tex + sourceTransform: tex2mdx.strip_comments + start: 173473 + sentence: Now we are ready to establish Proposition~\ref{KAPPALIMIT}, the non-collapsing + result + sentenceSpan: + coordinateSpace: comment-stripped-tex + end: 173516 + file: surgery.tex + sourceTransform: tex2mdx.strip_comments + start: 173430 + sourceStrategy: nearest-previous-statement + via: prose + sourceStatement: + chapter: 16 + file: surgery.tex + label: lvalue + mtref: '16.26' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 171497 + sourceTransform: tex2mdx.strip_comments + start: 171318 + targetStatement: + chapter: 16 + file: surgery.tex + label: KAPPALIMIT + mtref: '16.1' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 103576 + sourceTransform: tex2mdx.strip_comments + start: 102336 + type: prose-mention-nearest-previous-statement + via: + - prose +generator: tools/poincare_tex_extract.py +inference: weak +kind: mention +ref: +- 8cae73a6a02c +- 4ce8b5da9c58 +rel: mentions +reviewStatus: candidate +scope: prose +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- prose +--- +Morgan--Tian prose mention. The source is the nearest preceding statement in the same chapter segment. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/063196cd5014.md b/projects/poincare-conjecture/.astrolabe/edges/063196cd5014.md new file mode 100644 index 00000000..85debb95 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/063196cd5014.md @@ -0,0 +1,62 @@ +--- +confidence: 0.9 +edgeClass: structural +evidence: + containmentDepth: 1 + proofSpan: + bodyEnd: 74813 + bodyStart: 59556 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 74824 + file: singlimit2.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 59543 + sourceStatement: + chapter: 11 + file: singlimit2.tex + label: hexist + mtref: '11.31' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 59267 + sourceTransform: tex2mdx.strip_comments + start: 58375 + targetContainedSpan: + coordinateSpace: comment-stripped-tex + end: 60461 + file: singlimit2.tex + sourceTransform: tex2mdx.strip_comments + start: 60310 + targetStatement: + chapter: 11 + file: singlimit2.tex + label: hyp + mtref: '11.32' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 60461 + sourceTransform: tex2mdx.strip_comments + start: 60310 + type: proof-containment + via: + - proof-contains +generator: tools/poincare_tex_extract.py +inference: structural +kind: containment +ref: +- 074e02b067b8 +- dff170186455 +rel: contains +reviewStatus: accepted +scope: proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof-contains +--- +Morgan--Tian proof containment. The source is the statement whose proof contains the target statement. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/063ff2372f87.md b/projects/poincare-conjecture/.astrolabe/edges/063ff2372f87.md new file mode 100644 index 00000000..93b3efae --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/063ff2372f87.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: S2intersection + proofSpan: + bodyEnd: 63099 + bodyStart: 61619 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 63110 + file: bddcurvbdddist.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 61606 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 61722 + file: bddcurvbdddist.tex + sourceTransform: tex2mdx.strip_comments + start: 61702 + via: proof + sourceStatement: + chapter: 10 + file: bddcurvbdddist.tex + label: '' + mtref: '10.31' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 61603 + sourceTransform: tex2mdx.strip_comments + start: 61488 + targetStatement: + chapter: 19 + file: canonnbhd.tex + label: S2intersection + mtref: '19.11' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 19482 + sourceTransform: tex2mdx.strip_comments + start: 17256 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 6d6b9e050054 +- a46d72b87e07 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/065b5e410b8e.md b/projects/poincare-conjecture/.astrolabe/edges/065b5e410b8e.md new file mode 100644 index 00000000..a57ccf55 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/065b5e410b8e.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: localprod + proofSpan: + bodyEnd: 34465 + bodyStart: 32850 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 34476 + file: maxprin.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 32837 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 33948 + file: maxprin.tex + sourceTransform: tex2mdx.strip_comments + start: 33933 + via: proof + sourceStatement: + chapter: 4 + file: maxprin.tex + label: nocones + mtref: '4.22' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 32834 + sourceTransform: tex2mdx.strip_comments + start: 32471 + targetStatement: + chapter: 4 + file: maxprin.tex + label: localprod + mtref: '4.19' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 25705 + sourceTransform: tex2mdx.strip_comments + start: 25065 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 92b5c664399e +- 8ec855a2aab5 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/065d1cde4fcf.md b/projects/poincare-conjecture/.astrolabe/edges/065d1cde4fcf.md new file mode 100644 index 00000000..673be142 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/065d1cde4fcf.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: triL + proofSpan: + bodyEnd: 79076 + bodyStart: 78416 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 79087 + file: newcompar.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 78403 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 78889 + file: newcompar.tex + sourceTransform: tex2mdx.strip_comments + start: 78879 + via: proof + sourceStatement: + chapter: 6 + file: newcompar.tex + label: linequal + mtref: '6.49' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 78401 + sourceTransform: tex2mdx.strip_comments + start: 77391 + targetStatement: + chapter: 6 + file: newcompar.tex + label: triL + mtref: '6.43' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 70273 + sourceTransform: tex2mdx.strip_comments + start: 68899 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 2b8ab39364e7 +- 882677eecdc1 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/06684041e803.md b/projects/poincare-conjecture/.astrolabe/edges/06684041e803.md new file mode 100644 index 00000000..00db3d67 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/06684041e803.md @@ -0,0 +1,62 @@ +--- +confidence: 0.9 +edgeClass: structural +evidence: + containmentDepth: 3 + proofSpan: + bodyEnd: 23189 + bodyStart: 20006 + coordinateSpace: comment-stripped-tex + depth: 3 + end: 23200 + file: singlimit2.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 19993 + sourceStatement: + chapter: 11 + file: singlimit2.tex + label: compactbd + mtref: '11.11' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 19990 + sourceTransform: tex2mdx.strip_comments + start: 19475 + targetContainedSpan: + coordinateSpace: comment-stripped-tex + end: 20248 + file: singlimit2.tex + sourceTransform: tex2mdx.strip_comments + start: 20027 + targetStatement: + chapter: 11 + file: singlimit2.tex + label: distch + mtref: '11.12' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 20248 + sourceTransform: tex2mdx.strip_comments + start: 20027 + type: proof-containment + via: + - proof-contains +generator: tools/poincare_tex_extract.py +inference: structural +kind: containment +ref: +- cc3a26ca3809 +- 002f3496c54c +rel: contains +reviewStatus: accepted +scope: proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof-contains +--- +Morgan--Tian proof containment. The source is the statement whose proof contains the target statement. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/0681ef6daa97.md b/projects/poincare-conjecture/.astrolabe/edges/0681ef6daa97.md new file mode 100644 index 00000000..0580b26e --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/0681ef6daa97.md @@ -0,0 +1,63 @@ +--- +confidence: 0.35 +edgeClass: prose +evidence: + mentionTriggers: + - label: csshi1 + refSpan: + coordinateSpace: comment-stripped-tex + end: 231514 + file: energy1.tex + sourceTransform: tex2mdx.strip_comments + start: 231502 + sentence: By the definition of $t_2$ this implies that $t_2=t'+\delta r^2$ and + completes the proof of Lemma~\ref{csshi1}. + sentenceSpan: + coordinateSpace: comment-stripped-tex + end: 231516 + file: energy1.tex + sourceTransform: tex2mdx.strip_comments + start: 231401 + sourceStrategy: nearest-previous-statement + via: prose + sourceStatement: + chapter: 18 + file: energy1.tex + label: '' + mtref: '18.95' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 227168 + sourceTransform: tex2mdx.strip_comments + start: 227049 + targetStatement: + chapter: 18 + file: energy1.tex + label: csshi1 + mtref: '18.86' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 195723 + sourceTransform: tex2mdx.strip_comments + start: 194943 + type: prose-mention-nearest-previous-statement + via: + - prose +generator: tools/poincare_tex_extract.py +inference: weak +kind: mention +ref: +- 25dda5cb31e0 +- ff5d51a483d9 +rel: mentions +reviewStatus: candidate +scope: prose +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- prose +--- +Morgan--Tian prose mention. The source is the nearest preceding statement in the same chapter segment. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/06acf0306186.md b/projects/poincare-conjecture/.astrolabe/edges/06acf0306186.md new file mode 100644 index 00000000..a89d7652 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/06acf0306186.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 9 + file: temp2kappa.tex + label: '' + mtref: '9.92' + sort: remark + span: + coordinateSpace: comment-stripped-tex + end: 177845 + sourceTransform: tex2mdx.strip_comments + start: 177446 + targetStatement: + chapter: 9 + file: temp2kappa.tex + label: kappasummary + mtref: '9.93' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 179958 + sourceTransform: tex2mdx.strip_comments + start: 177878 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 744167bbcef7 +- c2be59f6090c +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/06b4538844bd.md b/projects/poincare-conjecture/.astrolabe/edges/06b4538844bd.md new file mode 100644 index 00000000..3f0fc66a --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/06b4538844bd.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: Rcompar + proofSpan: + bodyEnd: 24647 + bodyStart: 23636 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 24658 + file: surgery.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 23623 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 23851 + file: surgery.tex + sourceTransform: tex2mdx.strip_comments + start: 23838 + via: proof + sourceStatement: + chapter: 13 + file: surgery.tex + label: '' + mtref: '13.14' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 23621 + sourceTransform: tex2mdx.strip_comments + start: 23086 + targetStatement: + chapter: 13 + file: surgery.tex + label: Rcompar + mtref: '13.6' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 12879 + sourceTransform: tex2mdx.strip_comments + start: 12743 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- c9d8cb9d34c3 +- 7f0b4a692c5e +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/06dc58ddea46.md b/projects/poincare-conjecture/.astrolabe/edges/06dc58ddea46.md new file mode 100644 index 00000000..1defac5b --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/06dc58ddea46.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 18 + file: energy1.tex + label: avgeo + mtref: '18.68' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 163185 + sourceTransform: tex2mdx.strip_comments + start: 162959 + targetStatement: + chapter: 18 + file: energy1.tex + label: '' + mtref: '18.69' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 165064 + sourceTransform: tex2mdx.strip_comments + start: 164946 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- ea7c58d04fc7 +- a9617f70db9f +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/070da2b1a203.md b/projects/poincare-conjecture/.astrolabe/edges/070da2b1a203.md new file mode 100644 index 00000000..cf58ffea --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/070da2b1a203.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 18 + file: energy1.tex + label: C1C4delta + mtref: '18.54' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 124884 + sourceTransform: tex2mdx.strip_comments + start: 124119 + targetStatement: + chapter: 18 + file: energy1.tex + label: deltaprime + mtref: '18.55' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 128310 + sourceTransform: tex2mdx.strip_comments + start: 127628 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 7e0b4b1362c0 +- b87df0b67468 +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/073d0b3391ac.md b/projects/poincare-conjecture/.astrolabe/edges/073d0b3391ac.md new file mode 100644 index 00000000..d5189903 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/073d0b3391ac.md @@ -0,0 +1,56 @@ +--- +confidence: 0.85 +edgeClass: semantic +evidence: + containmentReason: target theorem-like statement occurs inside source proof + proofSpan: + bodyEnd: 97366 + bodyStart: 94222 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 97377 + file: newcompar.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 94209 + sourceStatement: + chapter: 6 + file: newcompar.tex + label: upperbd + mtref: '6.62' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 94207 + sourceTransform: tex2mdx.strip_comments + start: 92964 + targetStatement: + chapter: 6 + file: newcompar.tex + label: '' + mtref: '6.64' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 96436 + sourceTransform: tex2mdx.strip_comments + start: 96257 + type: proof-contained-theorem-like-statement + via: + - proof-contained-statement +generator: tools/poincare_tex_extract.py +inference: inferred +kind: dependency +ref: +- 7d130fba9693 +- 30414a5a14a9 +rel: depends +reviewStatus: accepted +scope: proof-containment +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof-contained-statement +--- +Morgan--Tian proof-contained statement dependency. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/076cd8f87f98.md b/projects/poincare-conjecture/.astrolabe/edges/076cd8f87f98.md new file mode 100644 index 00000000..3ae41182 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/076cd8f87f98.md @@ -0,0 +1,45 @@ +--- +confidence: 0.2 +edgeClass: structural +evidence: + sourceStatement: + chapter: 18 + file: energy1.tex + label: extinct + mtref: '18.1' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 1107 + sourceTransform: tex2mdx.strip_comments + start: 511 + targetStatement: + chapter: 18 + file: energy1.tex + label: '' + mtref: '18.2' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 14028 + sourceTransform: tex2mdx.strip_comments + start: 13563 + type: chapter-reading-order-adjacency + via: + - chapter-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 4b8a44aa316f +- 5b649fe97568 +rel: in-chapter +reviewStatus: accepted +scope: chapter +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- chapter-sequence +--- +Morgan--Tian reading-order adjacency across sections in a chapter. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/078da6d6f77b.md b/projects/poincare-conjecture/.astrolabe/edges/078da6d6f77b.md new file mode 100644 index 00000000..2cf0ef0a --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/078da6d6f77b.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 18 + file: energy1.tex + label: '1891' + mtref: '18.93' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 214989 + sourceTransform: tex2mdx.strip_comments + start: 214556 + targetStatement: + chapter: 18 + file: energy1.tex + label: psiclaim + mtref: '18.94' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 216766 + sourceTransform: tex2mdx.strip_comments + start: 216425 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 8973f9acd2ae +- d3f828726e2b +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/07d5b9c5db8d.md b/projects/poincare-conjecture/.astrolabe/edges/07d5b9c5db8d.md new file mode 100644 index 00000000..07c94f10 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/07d5b9c5db8d.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: volinj + proofSpan: + bodyEnd: 20196 + bodyStart: 18617 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 20207 + file: converge2.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 18604 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 19723 + file: converge2.tex + sourceTransform: tex2mdx.strip_comments + start: 19711 + via: proof + sourceStatement: + chapter: 5 + file: converge2.tex + label: 2ndmfdconv + mtref: '5.10' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 18602 + sourceTransform: tex2mdx.strip_comments + start: 18149 + targetStatement: + chapter: 1 + file: prelim.tex + label: volinj + mtref: '1.36' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 46059 + sourceTransform: tex2mdx.strip_comments + start: 45642 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- b45765a3070e +- f90cce212e88 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/07d88e9278cb.md b/projects/poincare-conjecture/.astrolabe/edges/07d88e9278cb.md new file mode 100644 index 00000000..9626e142 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/07d88e9278cb.md @@ -0,0 +1,45 @@ +--- +confidence: 0.2 +edgeClass: structural +evidence: + sourceStatement: + chapter: 18 + file: energy1.tex + label: lncurvgrowth + mtref: '18.38' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 87232 + sourceTransform: tex2mdx.strip_comments + start: 86861 + targetStatement: + chapter: 18 + file: energy1.tex + label: rampcomp + mtref: '18.39' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 88812 + sourceTransform: tex2mdx.strip_comments + start: 88619 + type: chapter-reading-order-adjacency + via: + - chapter-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 156a3da68fbd +- 2f115406780a +rel: in-chapter +reviewStatus: accepted +scope: chapter +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- chapter-sequence +--- +Morgan--Tian reading-order adjacency across sections in a chapter. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/07da8e234204.md b/projects/poincare-conjecture/.astrolabe/edges/07da8e234204.md new file mode 100644 index 00000000..170467c7 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/07da8e234204.md @@ -0,0 +1,62 @@ +--- +confidence: 0.9 +edgeClass: structural +evidence: + containmentDepth: 1 + proofSpan: + bodyEnd: 34517 + bodyStart: 26920 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 34528 + file: bddcurvbdddist.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 26907 + sourceStatement: + chapter: 10 + file: bddcurvbdddist.tex + label: Uinfty + mtref: '10.7' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 26826 + sourceTransform: tex2mdx.strip_comments + start: 25406 + targetContainedSpan: + coordinateSpace: comment-stripped-tex + end: 32796 + file: bddcurvbdddist.tex + sourceTransform: tex2mdx.strip_comments + start: 32551 + targetStatement: + chapter: 10 + file: bddcurvbdddist.tex + label: '' + mtref: '10.10' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 32796 + sourceTransform: tex2mdx.strip_comments + start: 32551 + type: proof-containment + via: + - proof-contains +generator: tools/poincare_tex_extract.py +inference: structural +kind: containment +ref: +- e2e64cdb0b75 +- 1418d3c286c7 +rel: contains +reviewStatus: accepted +scope: proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof-contains +--- +Morgan--Tian proof containment. The source is the statement whose proof contains the target statement. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/07e45877b57d.md b/projects/poincare-conjecture/.astrolabe/edges/07e45877b57d.md new file mode 100644 index 00000000..7d08134e --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/07e45877b57d.md @@ -0,0 +1,63 @@ +--- +confidence: 0.35 +edgeClass: prose +evidence: + mentionTriggers: + - label: LOCALSURGERY + refSpan: + coordinateSpace: comment-stripped-tex + end: 22985 + file: surgery.tex + sourceTransform: tex2mdx.strip_comments + start: 22967 + sentence: The first conclusion of Theorem~\ref{LOCALSURGERY} follows by applying + the above considerations to the case of $h=R_g(x_0)\rho^*g$ + sentenceSpan: + coordinateSpace: comment-stripped-tex + end: 23066 + file: surgery.tex + sourceTransform: tex2mdx.strip_comments + start: 22934 + sourceStrategy: nearest-previous-statement + via: prose + sourceStatement: + chapter: 13 + file: surgery.tex + label: nuprime>0 + mtref: '13.13' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 21433 + sourceTransform: tex2mdx.strip_comments + start: 21250 + targetStatement: + chapter: 13 + file: surgery.tex + label: LOCALSURGERY + mtref: '13.2' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 6442 + sourceTransform: tex2mdx.strip_comments + start: 4875 + type: prose-mention-nearest-previous-statement + via: + - prose +generator: tools/poincare_tex_extract.py +inference: weak +kind: mention +ref: +- cc9110b4d2fe +- cf61b731c6f2 +rel: mentions +reviewStatus: candidate +scope: prose +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- prose +--- +Morgan--Tian prose mention. The source is the nearest preceding statement in the same chapter segment. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/07f68b1b148f.md b/projects/poincare-conjecture/.astrolabe/edges/07f68b1b148f.md new file mode 100644 index 00000000..4ada0153 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/07f68b1b148f.md @@ -0,0 +1,67 @@ +--- +confidence: 0.72 +edgeClass: semantic +evidence: + anaphoraTriggers: + - context: proof + direction: previous + phrase: previous lemma + proofSpan: + bodyEnd: 27391 + bodyStart: 22887 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 27402 + file: energy1.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 22874 + sort: lemma + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 27389 + file: energy1.tex + sourceTransform: tex2mdx.strip_comments + start: 27375 + via: local-anaphora-proof + sourceStatement: + chapter: 18 + file: energy1.tex + label: homess + mtref: '18.7' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 22871 + sourceTransform: tex2mdx.strip_comments + start: 22605 + targetStatement: + chapter: 18 + file: energy1.tex + label: '' + mtref: '18.5' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 20812 + sourceTransform: tex2mdx.strip_comments + start: 20598 + type: local-anaphora-reference + via: + - local-anaphora-proof +generator: tools/poincare_tex_extract.py +inference: inferred +kind: dependency +ref: +- 46f9d27654be +- a86000e8dd35 +rel: depends +reviewStatus: accepted +scope: local-context +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- local-anaphora-proof +--- +Morgan--Tian local anaphora dependency. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/0804aebb5ebb.md b/projects/poincare-conjecture/.astrolabe/edges/0804aebb5ebb.md new file mode 100644 index 00000000..eb895796 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/0804aebb5ebb.md @@ -0,0 +1,84 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: '9.1' + proofSpan: + bodyEnd: 54348 + bodyStart: 52734 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 54359 + file: temp2kappa.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 52721 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 53755 + file: temp2kappa.tex + sourceTransform: tex2mdx.strip_comments + start: 53746 + via: proof + - context: proof + label: '9.1' + proofSpan: + bodyEnd: 54348 + bodyStart: 52734 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 54359 + file: temp2kappa.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 52721 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 53871 + file: temp2kappa.tex + sourceTransform: tex2mdx.strip_comments + start: 53862 + via: proof + sourceStatement: + chapter: 9 + file: temp2kappa.tex + label: gradshrink + mtref: '9.32' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 52719 + sourceTransform: tex2mdx.strip_comments + start: 52484 + targetStatement: + chapter: 9 + file: temp2kappa.tex + label: '9.1' + mtref: '9.33' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 53478 + sourceTransform: tex2mdx.strip_comments + start: 52780 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 07232fc42f3a +- ba1adbac0a75 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/080d782219c7.md b/projects/poincare-conjecture/.astrolabe/edges/080d782219c7.md new file mode 100644 index 00000000..58f8cea9 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/080d782219c7.md @@ -0,0 +1,65 @@ +--- +confidence: 0.45 +edgeClass: semantic +evidence: + endpointSemanticDegreeBeforeTermBackfill: + source: 0 + target: 0 + selectionReason: semantic-connectivity-backfill + sourceMatchSpan: + coordinateSpace: comment-stripped-tex + end: 9767 + file: flowbasics.tex + sourceTransform: tex2mdx.strip_comments + start: 9741 + sourceStatement: + chapter: 3 + file: flowbasics.tex + label: GSSgeneration + mtref: '3.9' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 9814 + sourceTransform: tex2mdx.strip_comments + start: 9457 + targetStatement: + chapter: 3 + file: flowbasics.tex + label: '' + mtref: '3.8' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 9455 + sourceTransform: tex2mdx.strip_comments + start: 9202 + targetTermSpan: + coordinateSpace: comment-stripped-tex + end: 9305 + file: flowbasics.tex + sourceTransform: tex2mdx.strip_comments + start: 9279 + term: gradient shrinking soliton + termAmbiguity: unique-in-chapter + termSource: italic + termSpecificity: no-longer-defined-term + type: same-chapter-definition-term-match + via: + - definition-term:gradient shrinking soliton +generator: tools/poincare_tex_extract.py +inference: weak +kind: definition-use +ref: +- 8a667675febd +- 6ace86bb0b70 +rel: uses +reviewStatus: candidate +scope: same-chapter-term-match +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- definition-term:gradient shrinking soliton +--- +Morgan--Tian same-chapter definition term match. This is a low-confidence candidate edge for review. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/08839803db86.md b/projects/poincare-conjecture/.astrolabe/edges/08839803db86.md new file mode 100644 index 00000000..0a917612 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/08839803db86.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 9 + file: temp2kappa.tex + label: '' + mtref: '9.41' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 74254 + sourceTransform: tex2mdx.strip_comments + start: 73978 + targetStatement: + chapter: 9 + file: temp2kappa.tex + label: GSSclass + mtref: '9.42' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 75321 + sourceTransform: tex2mdx.strip_comments + start: 74299 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 2104e6f56e28 +- 2d68880771ff +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/0889e4cb2d33.md b/projects/poincare-conjecture/.astrolabe/edges/0889e4cb2d33.md new file mode 100644 index 00000000..8572e003 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/0889e4cb2d33.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: weak + proofSpan: + bodyEnd: 32942 + bodyStart: 30411 + coordinateSpace: comment-stripped-tex + depth: 2 + end: 32953 + file: temp2kappa.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 30398 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 30518 + file: temp2kappa.tex + sourceTransform: tex2mdx.strip_comments + start: 30508 + via: proof + sourceStatement: + chapter: 9 + file: temp2kappa.tex + label: '' + mtref: '9.22' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 30396 + sourceTransform: tex2mdx.strip_comments + start: 30212 + targetStatement: + chapter: 7 + file: newcomp2.tex + label: weak + mtref: '7.13' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 23409 + sourceTransform: tex2mdx.strip_comments + start: 22180 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 598f6059280c +- 7d5d94efe4e9 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/08903356ac57.md b/projects/poincare-conjecture/.astrolabe/edges/08903356ac57.md new file mode 100644 index 00000000..3cd472b4 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/08903356ac57.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: diffeo + proofSpan: + bodyEnd: 10257 + bodyStart: 9452 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 10268 + file: newcomp2.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 9439 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 9938 + file: newcomp2.tex + sourceTransform: tex2mdx.strip_comments + start: 9926 + via: proof + sourceStatement: + chapter: 7 + file: newcomp2.tex + label: lipcomplete + mtref: '7.5' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 9437 + sourceTransform: tex2mdx.strip_comments + start: 8692 + targetStatement: + chapter: 6 + file: newcompar.tex + label: diffeo + mtref: '6.28' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 38046 + sourceTransform: tex2mdx.strip_comments + start: 37549 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 16fbe263a7c8 +- 17254b50b27a +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/08948f889a01.md b/projects/poincare-conjecture/.astrolabe/edges/08948f889a01.md new file mode 100644 index 00000000..002b84cd --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/08948f889a01.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 5 + file: converge2.tex + label: '' + mtref: '5.13' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 23788 + sourceTransform: tex2mdx.strip_comments + start: 23341 + targetStatement: + chapter: 5 + file: converge2.tex + label: partialflowlimit + mtref: '5.14' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 25227 + sourceTransform: tex2mdx.strip_comments + start: 23995 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 59dc7d470f3e +- 4673eb9f4424 +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/08b8fa33fa3a.md b/projects/poincare-conjecture/.astrolabe/edges/08b8fa33fa3a.md new file mode 100644 index 00000000..97e8a0ab --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/08b8fa33fa3a.md @@ -0,0 +1,62 @@ +--- +confidence: 0.9 +edgeClass: structural +evidence: + containmentDepth: 1 + proofSpan: + bodyEnd: 137790 + bodyStart: 127828 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 137801 + file: temp2kappa.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 127815 + sourceStatement: + chapter: 9 + file: temp2kappa.tex + label: proposition + mtref: '9.65' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 127812 + sourceTransform: tex2mdx.strip_comments + start: 127549 + targetContainedSpan: + coordinateSpace: comment-stripped-tex + end: 134931 + file: temp2kappa.tex + sourceTransform: tex2mdx.strip_comments + start: 134780 + targetStatement: + chapter: 9 + file: temp2kappa.tex + label: zkbound + mtref: '9.68' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 134931 + sourceTransform: tex2mdx.strip_comments + start: 134780 + type: proof-containment + via: + - proof-contains +generator: tools/poincare_tex_extract.py +inference: structural +kind: containment +ref: +- ee314aacd207 +- f61f72020102 +rel: contains +reviewStatus: accepted +scope: proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof-contains +--- +Morgan--Tian proof containment. The source is the statement whose proof contains the target statement. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/08cac6bed200.md b/projects/poincare-conjecture/.astrolabe/edges/08cac6bed200.md new file mode 100644 index 00000000..0bcee988 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/08cac6bed200.md @@ -0,0 +1,65 @@ +--- +confidence: 0.45 +edgeClass: semantic +evidence: + endpointSemanticDegreeBeforeTermBackfill: + source: 0 + target: 2 + selectionReason: semantic-connectivity-backfill + sourceMatchSpan: + coordinateSpace: comment-stripped-tex + end: 54760 + file: surgery.tex + sourceTransform: tex2mdx.strip_comments + start: 54737 + sourceStatement: + chapter: 14 + file: surgery.tex + label: defnkappa + mtref: '14.15' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 55165 + sourceTransform: tex2mdx.strip_comments + start: 54665 + targetStatement: + chapter: 14 + file: surgery.tex + label: '' + mtref: '14.8' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 47114 + sourceTransform: tex2mdx.strip_comments + start: 46506 + targetTermSpan: + coordinateSpace: comment-stripped-tex + end: 46550 + file: surgery.tex + sourceTransform: tex2mdx.strip_comments + start: 46527 + term: ricci flow with surgery + termAmbiguity: unique-in-chapter + termSource: italic + termSpecificity: no-longer-defined-term + type: same-chapter-definition-term-match + via: + - definition-term:ricci flow with surgery +generator: tools/poincare_tex_extract.py +inference: weak +kind: definition-use +ref: +- 75f3f7263208 +- 842f5a7c856b +rel: uses +reviewStatus: candidate +scope: same-chapter-term-match +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- definition-term:ricci flow with surgery +--- +Morgan--Tian same-chapter definition term match. This is a low-confidence candidate edge for review. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/08f8403b9b19.md b/projects/poincare-conjecture/.astrolabe/edges/08f8403b9b19.md new file mode 100644 index 00000000..48b6d637 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/08f8403b9b19.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 16 + file: surgery.tex + label: '' + mtref: '16.25' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 167879 + sourceTransform: tex2mdx.strip_comments + start: 167677 + targetStatement: + chapter: 16 + file: surgery.tex + label: lvalue + mtref: '16.26' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 171497 + sourceTransform: tex2mdx.strip_comments + start: 171318 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- fc5c13ed2e67 +- 8cae73a6a02c +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/09490d2424b1.md b/projects/poincare-conjecture/.astrolabe/edges/09490d2424b1.md new file mode 100644 index 00000000..ba3ef6a6 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/09490d2424b1.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: THM + proofSpan: + bodyEnd: 20911 + bodyStart: 18068 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 20922 + file: noncoll.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 18055 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 20855 + file: noncoll.tex + sourceTransform: tex2mdx.strip_comments + start: 20846 + via: proof + sourceStatement: + chapter: 8 + file: noncoll.tex + label: '' + mtref: '8.10' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 18052 + sourceTransform: tex2mdx.strip_comments + start: 17502 + targetStatement: + chapter: 8 + file: noncoll.tex + label: THM + mtref: '8.1' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 1970 + sourceTransform: tex2mdx.strip_comments + start: 682 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 198e7475ad2f +- 3defa9b2209c +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/09755fc5dfd2.md b/projects/poincare-conjecture/.astrolabe/edges/09755fc5dfd2.md new file mode 100644 index 00000000..489cc89c --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/09755fc5dfd2.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 18 + file: energy1.tex + label: '' + mtref: '18.16' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 47275 + sourceTransform: tex2mdx.strip_comments + start: 47013 + targetStatement: + chapter: 18 + file: energy1.tex + label: Wdefn + mtref: '18.17' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 49255 + sourceTransform: tex2mdx.strip_comments + start: 48287 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 39199318fe64 +- d88ab31ec11f +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/09ad4eb992db.md b/projects/poincare-conjecture/.astrolabe/edges/09ad4eb992db.md new file mode 100644 index 00000000..39727ff0 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/09ad4eb992db.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 3 + file: flowbasics.tex + label: '' + mtref: '3.3' + sort: example + span: + coordinateSpace: comment-stripped-tex + end: 4618 + sourceTransform: tex2mdx.strip_comments + start: 4083 + targetStatement: + chapter: 3 + file: flowbasics.tex + label: '' + mtref: '3.4' + sort: example + span: + coordinateSpace: comment-stripped-tex + end: 4828 + sourceTransform: tex2mdx.strip_comments + start: 4620 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 43be10053c44 +- fbf53047d0c1 +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/09fd2166d50f.md b/projects/poincare-conjecture/.astrolabe/edges/09fd2166d50f.md new file mode 100644 index 00000000..ccc23897 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/09fd2166d50f.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: s2isotopic + proofSpan: + bodyEnd: 128391 + bodyStart: 125658 + coordinateSpace: comment-stripped-tex + depth: 2 + end: 128402 + file: surgery.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 125645 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 128045 + file: surgery.tex + sourceTransform: tex2mdx.strip_comments + start: 128029 + via: proof + sourceStatement: + chapter: 16 + file: surgery.tex + label: '' + mtref: '16.10' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 125643 + sourceTransform: tex2mdx.strip_comments + start: 125039 + targetStatement: + chapter: 19 + file: canonnbhd.tex + label: s2isotopic + mtref: '19.3' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 3924 + sourceTransform: tex2mdx.strip_comments + start: 3327 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 683bb06c1059 +- 98c6a95f8556 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/0a2d64c3ca00.md b/projects/poincare-conjecture/.astrolabe/edges/0a2d64c3ca00.md new file mode 100644 index 00000000..d4729356 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/0a2d64c3ca00.md @@ -0,0 +1,84 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: controlnbhd + proofSpan: + bodyEnd: 41515 + bodyStart: 39183 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 41526 + file: singlimit2.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 39170 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 39583 + file: singlimit2.tex + sourceTransform: tex2mdx.strip_comments + start: 39566 + via: proof + - context: proof + label: controlnbhd + proofSpan: + bodyEnd: 41515 + bodyStart: 39183 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 41526 + file: singlimit2.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 39170 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 41000 + file: singlimit2.tex + sourceTransform: tex2mdx.strip_comments + start: 40983 + via: proof + sourceStatement: + chapter: 11 + file: singlimit2.tex + label: Rproper + mtref: '11.21' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 39168 + sourceTransform: tex2mdx.strip_comments + start: 38496 + targetStatement: + chapter: 11 + file: singlimit2.tex + label: controlnbhd + mtref: '11.2' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 4360 + sourceTransform: tex2mdx.strip_comments + start: 3597 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- a99536ac8110 +- 0ddfe578961e +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/0a3d937a5dd1.md b/projects/poincare-conjecture/.astrolabe/edges/0a3d937a5dd1.md new file mode 100644 index 00000000..5aa01876 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/0a3d937a5dd1.md @@ -0,0 +1,63 @@ +--- +confidence: 0.8 +edgeClass: semantic +evidence: + proseTriggers: + - label: fordiffquot + refSpan: + coordinateSpace: comment-stripped-tex + end: 73314 + file: energy1.tex + sourceTransform: tex2mdx.strip_comments + start: 73297 + sentence: The following is immediate from the previous lemma and Lemma~\ref{fordiffquot}. + sentenceSpan: + coordinateSpace: comment-stripped-tex + end: 73315 + file: energy1.tex + sourceTransform: tex2mdx.strip_comments + start: 73235 + sourceStrategy: following-statement-after-inferential-prose + via: + - prose-dependency-following + sourceStatement: + chapter: 18 + file: energy1.tex + label: '' + mtref: '18.31' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 73544 + sourceTransform: tex2mdx.strip_comments + start: 73317 + targetStatement: + chapter: 2 + file: prelim.tex + label: fordiffquot + mtref: '2.22' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 82182 + sourceTransform: tex2mdx.strip_comments + start: 81704 + type: inferential-prose-reference + via: + - prose-dependency-following +generator: tools/poincare_tex_extract.py +inference: inferred +kind: dependency +ref: +- 9e694ee343bf +- 67c378f87457 +rel: depends +reviewStatus: accepted +scope: prose +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- prose-dependency-following +--- +Morgan--Tian high-confidence prose dependency. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/0a5bb617c9a7.md b/projects/poincare-conjecture/.astrolabe/edges/0a5bb617c9a7.md new file mode 100644 index 00000000..6d4b1100 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/0a5bb617c9a7.md @@ -0,0 +1,67 @@ +--- +confidence: 0.72 +edgeClass: semantic +evidence: + anaphoraTriggers: + - context: proof + direction: previous + phrase: previous theorem + proofSpan: + bodyEnd: 54015 + bodyStart: 53800 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 54026 + file: converge2.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 53787 + sort: theorem + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 54013 + file: converge2.tex + sourceTransform: tex2mdx.strip_comments + start: 53997 + via: local-anaphora-proof + sourceStatement: + chapter: 5 + file: converge2.tex + label: '' + mtref: '5.34' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 53784 + sourceTransform: tex2mdx.strip_comments + start: 53414 + targetStatement: + chapter: 5 + file: converge2.tex + label: blowupposcurv + mtref: '5.33' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 51450 + sourceTransform: tex2mdx.strip_comments + start: 50791 + type: local-anaphora-reference + via: + - local-anaphora-proof +generator: tools/poincare_tex_extract.py +inference: inferred +kind: dependency +ref: +- e67bd2c19008 +- 2ddfbc74b081 +rel: depends +reviewStatus: accepted +scope: local-context +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- local-anaphora-proof +--- +Morgan--Tian local anaphora dependency. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/0a706b69c728.md b/projects/poincare-conjecture/.astrolabe/edges/0a706b69c728.md new file mode 100644 index 00000000..2f8da760 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/0a706b69c728.md @@ -0,0 +1,63 @@ +--- +confidence: 0.35 +edgeClass: prose +evidence: + mentionTriggers: + - label: Uinfty + refSpan: + coordinateSpace: comment-stripped-tex + end: 34725 + file: bddcurvbdddist.tex + sourceTransform: tex2mdx.strip_comments + start: 34713 + sentence: The next step is to study the nature of the limit $U_\infty$ given in + Proposition~\ref{Uinfty} + sentenceSpan: + coordinateSpace: comment-stripped-tex + end: 34725 + file: bddcurvbdddist.tex + sourceTransform: tex2mdx.strip_comments + start: 34629 + sourceStrategy: nearest-previous-statement + via: prose + sourceStatement: + chapter: 10 + file: bddcurvbdddist.tex + label: '' + mtref: '10.11' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 33981 + sourceTransform: tex2mdx.strip_comments + start: 33869 + targetStatement: + chapter: 10 + file: bddcurvbdddist.tex + label: Uinfty + mtref: '10.7' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 26826 + sourceTransform: tex2mdx.strip_comments + start: 25406 + type: prose-mention-nearest-previous-statement + via: + - prose +generator: tools/poincare_tex_extract.py +inference: weak +kind: mention +ref: +- 2d6c9dcad762 +- e2e64cdb0b75 +rel: mentions +reviewStatus: candidate +scope: prose +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- prose +--- +Morgan--Tian prose mention. The source is the nearest preceding statement in the same chapter segment. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/0a758d419c91.md b/projects/poincare-conjecture/.astrolabe/edges/0a758d419c91.md new file mode 100644 index 00000000..cc2fc84c --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/0a758d419c91.md @@ -0,0 +1,56 @@ +--- +confidence: 0.85 +edgeClass: semantic +evidence: + containmentReason: target theorem-like statement occurs inside source proof + proofSpan: + bodyEnd: 23189 + bodyStart: 20006 + coordinateSpace: comment-stripped-tex + depth: 3 + end: 23200 + file: singlimit2.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 19993 + sourceStatement: + chapter: 11 + file: singlimit2.tex + label: compactbd + mtref: '11.11' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 19990 + sourceTransform: tex2mdx.strip_comments + start: 19475 + targetStatement: + chapter: 11 + file: singlimit2.tex + label: distch + mtref: '11.12' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 20248 + sourceTransform: tex2mdx.strip_comments + start: 20027 + type: proof-contained-theorem-like-statement + via: + - proof-contained-statement +generator: tools/poincare_tex_extract.py +inference: inferred +kind: dependency +ref: +- cc3a26ca3809 +- 002f3496c54c +rel: depends +reviewStatus: accepted +scope: proof-containment +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof-contained-statement +--- +Morgan--Tian proof-contained statement dependency. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/0a941b7d2871.md b/projects/poincare-conjecture/.astrolabe/edges/0a941b7d2871.md new file mode 100644 index 00000000..64c062c4 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/0a941b7d2871.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 19 + file: canonnbhd.tex + label: firstchain + mtref: '19.17' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 28487 + sourceTransform: tex2mdx.strip_comments + start: 27933 + targetStatement: + chapter: 19 + file: canonnbhd.tex + label: addaneck + mtref: '19.18' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 29774 + sourceTransform: tex2mdx.strip_comments + start: 29005 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 58804e22eff2 +- 6dc87ad116fc +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/0ac42de85f1b.md b/projects/poincare-conjecture/.astrolabe/edges/0ac42de85f1b.md new file mode 100644 index 00000000..c9c1c148 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/0ac42de85f1b.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: Hessformula + proofSpan: + bodyEnd: 37923 + bodyStart: 36740 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 37934 + file: prelim.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 36727 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 37146 + file: prelim.tex + sourceTransform: tex2mdx.strip_comments + start: 37129 + via: proof + sourceStatement: + chapter: 1 + file: prelim.tex + label: '' + mtref: '1.27' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 36725 + sourceTransform: tex2mdx.strip_comments + start: 36479 + targetStatement: + chapter: 1 + file: prelim.tex + label: Hessformula + mtref: '1.3' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 4780 + sourceTransform: tex2mdx.strip_comments + start: 4254 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 0cf778ecc2ec +- 98af9c99c3c2 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/0accb6b88faf.md b/projects/poincare-conjecture/.astrolabe/edges/0accb6b88faf.md new file mode 100644 index 00000000..3e963d4d --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/0accb6b88faf.md @@ -0,0 +1,84 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: defncanonnbhd + proofSpan: + bodyEnd: 72033 + bodyStart: 67204 + coordinateSpace: comment-stripped-tex + depth: 2 + end: 72044 + file: singlimit2.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 67191 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 71345 + file: singlimit2.tex + sourceTransform: tex2mdx.strip_comments + start: 71326 + via: proof + - context: proof + label: defncanonnbhd + proofSpan: + bodyEnd: 72033 + bodyStart: 67204 + coordinateSpace: comment-stripped-tex + depth: 2 + end: 72044 + file: singlimit2.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 67191 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 71657 + file: singlimit2.tex + sourceTransform: tex2mdx.strip_comments + start: 71638 + via: proof + sourceStatement: + chapter: 11 + file: singlimit2.tex + label: '' + mtref: '11.35' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 67188 + sourceTransform: tex2mdx.strip_comments + start: 66746 + targetStatement: + chapter: 9 + file: temp2kappa.tex + label: defncanonnbhd + mtref: '9.78' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 147589 + sourceTransform: tex2mdx.strip_comments + start: 145644 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 29f2350a588a +- 3aa5883ab629 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/0ad4cfa9bd28.md b/projects/poincare-conjecture/.astrolabe/edges/0ad4cfa9bd28.md new file mode 100644 index 00000000..78037ebc --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/0ad4cfa9bd28.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: smoothconv + proofSpan: + bodyEnd: 34517 + bodyStart: 26920 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 34528 + file: bddcurvbdddist.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 26907 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 27102 + file: bddcurvbdddist.tex + sourceTransform: tex2mdx.strip_comments + start: 27086 + via: proof + sourceStatement: + chapter: 10 + file: bddcurvbdddist.tex + label: Uinfty + mtref: '10.7' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 26826 + sourceTransform: tex2mdx.strip_comments + start: 25406 + targetStatement: + chapter: 5 + file: converge2.tex + label: smoothconv + mtref: '5.3' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 4757 + sourceTransform: tex2mdx.strip_comments + start: 2949 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- e2e64cdb0b75 +- 467f4b920031 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/0afc8ed3325f.md b/projects/poincare-conjecture/.astrolabe/edges/0afc8ed3325f.md new file mode 100644 index 00000000..26b4f476 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/0afc8ed3325f.md @@ -0,0 +1,56 @@ +--- +confidence: 0.85 +edgeClass: semantic +evidence: + containmentReason: target theorem-like statement occurs inside source proof + proofSpan: + bodyEnd: 47393 + bodyStart: 39592 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 47404 + file: maxprin.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 39579 + sourceStatement: + chapter: 4 + file: maxprin.tex + label: '' + mtref: '4.27' + sort: remark + span: + coordinateSpace: comment-stripped-tex + end: 39436 + sourceTransform: tex2mdx.strip_comments + start: 39187 + targetStatement: + chapter: 4 + file: maxprin.tex + label: '' + mtref: '4.29' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 42803 + sourceTransform: tex2mdx.strip_comments + start: 42730 + type: proof-contained-theorem-like-statement + via: + - proof-contained-statement +generator: tools/poincare_tex_extract.py +inference: inferred +kind: dependency +ref: +- 1160f53cc783 +- cfd7d6954385 +rel: depends +reviewStatus: accepted +scope: proof-containment +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof-contained-statement +--- +Morgan--Tian proof-contained statement dependency. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/0b04ab83342f.md b/projects/poincare-conjecture/.astrolabe/edges/0b04ab83342f.md new file mode 100644 index 00000000..295e2475 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/0b04ab83342f.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 11 + file: singlimit2.tex + label: '' + mtref: '11.9' + sort: remark + span: + coordinateSpace: comment-stripped-tex + end: 16852 + sourceTransform: tex2mdx.strip_comments + start: 16134 + targetStatement: + chapter: 11 + file: singlimit2.tex + label: maxT + mtref: '11.10' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 17643 + sourceTransform: tex2mdx.strip_comments + start: 17183 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 5e06d2593bfa +- d6919616448d +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/0b1eafb63756.md b/projects/poincare-conjecture/.astrolabe/edges/0b1eafb63756.md new file mode 100644 index 00000000..587307a9 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/0b1eafb63756.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: flowtoround + proofSpan: + bodyEnd: 37036 + bodyStart: 36484 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 37047 + file: maxprin.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 36471 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 36653 + file: maxprin.tex + sourceTransform: tex2mdx.strip_comments + start: 36636 + via: proof + sourceStatement: + chapter: 4 + file: maxprin.tex + label: sphsf + mtref: '4.24' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 36469 + sourceTransform: tex2mdx.strip_comments + start: 36157 + targetStatement: + chapter: 4 + file: maxprin.tex + label: flowtoround + mtref: '4.23' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 35932 + sourceTransform: tex2mdx.strip_comments + start: 34642 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 4f253de31e0c +- 86e5a5766bda +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/0b3f1cc1aafd.md b/projects/poincare-conjecture/.astrolabe/edges/0b3f1cc1aafd.md new file mode 100644 index 00000000..f12605a8 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/0b3f1cc1aafd.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: epschains + proofSpan: + bodyEnd: 53552 + bodyStart: 35551 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 53563 + file: canonnbhd.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 35538 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 50554 + file: canonnbhd.tex + sourceTransform: tex2mdx.strip_comments + start: 50539 + via: proof + sourceStatement: + chapter: 19 + file: canonnbhd.tex + label: Xcontainedin + mtref: '19.21' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 35375 + sourceTransform: tex2mdx.strip_comments + start: 34343 + targetStatement: + chapter: 19 + file: canonnbhd.tex + label: epschains + mtref: '19.19' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 31004 + sourceTransform: tex2mdx.strip_comments + start: 30360 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 0e9a1a167caa +- ba681e15e1a6 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/0b61b92d2978.md b/projects/poincare-conjecture/.astrolabe/edges/0b61b92d2978.md new file mode 100644 index 00000000..32e3ed85 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/0b61b92d2978.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 6 + file: newcompar.tex + label: minmax + mtref: '6.61' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 91839 + sourceTransform: tex2mdx.strip_comments + start: 91251 + targetStatement: + chapter: 6 + file: newcompar.tex + label: upperbd + mtref: '6.62' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 94207 + sourceTransform: tex2mdx.strip_comments + start: 92964 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- a7c2ebee07a3 +- 7d130fba9693 +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/0b78e3b042a4.md b/projects/poincare-conjecture/.astrolabe/edges/0b78e3b042a4.md new file mode 100644 index 00000000..d5d77125 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/0b78e3b042a4.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 3 + file: flowbasics.tex + label: 2ndI.8.3 + mtref: '3.25' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 36538 + sourceTransform: tex2mdx.strip_comments + start: 36082 + targetStatement: + chapter: 3 + file: flowbasics.tex + label: corI.8.3 + mtref: '3.26' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 37763 + sourceTransform: tex2mdx.strip_comments + start: 37390 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- d22d3cf47fbf +- 01a829e22cf6 +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/0b8019b26f38.md b/projects/poincare-conjecture/.astrolabe/edges/0b8019b26f38.md new file mode 100644 index 00000000..6ba7a88a --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/0b8019b26f38.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 9 + file: temp2kappa.tex + label: rho + mtref: '9.70' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 137332 + sourceTransform: tex2mdx.strip_comments + start: 137165 + targetStatement: + chapter: 9 + file: temp2kappa.tex + label: derivcor + mtref: '9.71' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 139489 + sourceTransform: tex2mdx.strip_comments + start: 139133 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 4bff7f1e7bc4 +- 8c090fbd21a0 +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/0b8782599d46.md b/projects/poincare-conjecture/.astrolabe/edges/0b8782599d46.md new file mode 100644 index 00000000..b8dda00c --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/0b8782599d46.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 10 + file: bddcurvbdddist.tex + label: morethanone + mtref: '10.18' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 42343 + sourceTransform: tex2mdx.strip_comments + start: 42251 + targetStatement: + chapter: 10 + file: bddcurvbdddist.tex + label: '' + mtref: '10.19' + sort: remark + span: + coordinateSpace: comment-stripped-tex + end: 42875 + sourceTransform: tex2mdx.strip_comments + start: 42673 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- dda8a7c7acb6 +- f640a0334396 +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/0bc2ce79539f.md b/projects/poincare-conjecture/.astrolabe/edges/0bc2ce79539f.md new file mode 100644 index 00000000..73799669 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/0bc2ce79539f.md @@ -0,0 +1,63 @@ +--- +confidence: 0.35 +edgeClass: prose +evidence: + mentionTriggers: + - label: star + refSpan: + coordinateSpace: comment-stripped-tex + end: 33721 + file: prelim.tex + sourceTransform: tex2mdx.strip_comments + start: 33711 + sentence: It follows from Corollary~\ref{star} that $U\subset T_pM$ is a star-shaped + open neighborhood of $0\in T_pM$. + sentenceSpan: + coordinateSpace: comment-stripped-tex + end: 33795 + file: prelim.tex + sourceTransform: tex2mdx.strip_comments + start: 33684 + sourceStrategy: nearest-previous-statement + via: prose + sourceStatement: + chapter: 1 + file: prelim.tex + label: '' + mtref: '1.24' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 33683 + sourceTransform: tex2mdx.strip_comments + start: 33241 + targetStatement: + chapter: 1 + file: prelim.tex + label: star + mtref: '1.23' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 32784 + sourceTransform: tex2mdx.strip_comments + start: 32478 + type: prose-mention-nearest-previous-statement + via: + - prose +generator: tools/poincare_tex_extract.py +inference: weak +kind: mention +ref: +- 19afab2e3115 +- 8141585de23a +rel: mentions +reviewStatus: candidate +scope: prose +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- prose +--- +Morgan--Tian prose mention. The source is the nearest preceding statement in the same chapter segment. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/0bddb3272303.md b/projects/poincare-conjecture/.astrolabe/edges/0bddb3272303.md new file mode 100644 index 00000000..ee81d408 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/0bddb3272303.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 17 + file: surgery.tex + label: extend + mtref: '17.1' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 185406 + sourceTransform: tex2mdx.strip_comments + start: 184285 + targetStatement: + chapter: 17 + file: surgery.tex + label: '16.2' + mtref: '17.2' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 186718 + sourceTransform: tex2mdx.strip_comments + start: 186200 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 254e8dae97e6 +- 778d690c4e7c +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/0bfbdab758c3.md b/projects/poincare-conjecture/.astrolabe/edges/0bfbdab758c3.md new file mode 100644 index 00000000..832bf4cc --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/0bfbdab758c3.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: conelimit + proofSpan: + bodyEnd: 113303 + bodyStart: 106682 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 113314 + file: temp2kappa.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 106669 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 109798 + file: temp2kappa.tex + sourceTransform: tex2mdx.strip_comments + start: 109783 + via: proof + sourceStatement: + chapter: 9 + file: temp2kappa.tex + label: asympscal + mtref: '9.56' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 106666 + sourceTransform: tex2mdx.strip_comments + start: 106402 + targetStatement: + chapter: 5 + file: converge2.tex + label: conelimit + mtref: '5.31' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 45881 + sourceTransform: tex2mdx.strip_comments + start: 45285 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 5d9f1b75170d +- 88a75d6dd09e +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/0c53c2237d75.md b/projects/poincare-conjecture/.astrolabe/edges/0c53c2237d75.md new file mode 100644 index 00000000..1cc66c19 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/0c53c2237d75.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: unianc + proofSpan: + bodyEnd: 45252 + bodyStart: 45022 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 45263 + file: temp2kappa.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 45009 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 45049 + file: temp2kappa.tex + sourceTransform: tex2mdx.strip_comments + start: 45037 + via: proof + sourceStatement: + chapter: 9 + file: temp2kappa.tex + label: absconv + mtref: '9.28' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 45006 + sourceTransform: tex2mdx.strip_comments + start: 44788 + targetStatement: + chapter: 9 + file: temp2kappa.tex + label: unianc + mtref: '9.10' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 10335 + sourceTransform: tex2mdx.strip_comments + start: 9624 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 0e9703c15036 +- 8b8e3a0f3e36 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/0c60f932c94e.md b/projects/poincare-conjecture/.astrolabe/edges/0c60f932c94e.md new file mode 100644 index 00000000..262c2418 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/0c60f932c94e.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: flatssplit + proofSpan: + bodyEnd: 91252 + bodyStart: 90912 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 91263 + file: temp2kappa.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 90899 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 90938 + file: temp2kappa.tex + sourceTransform: tex2mdx.strip_comments + start: 90922 + via: proof + sourceStatement: + chapter: 9 + file: temp2kappa.tex + label: compact2Dsol + mtref: '9.49' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 90897 + sourceTransform: tex2mdx.strip_comments + start: 90165 + targetStatement: + chapter: 9 + file: temp2kappa.tex + label: flatssplit + mtref: '9.45' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 81068 + sourceTransform: tex2mdx.strip_comments + start: 80573 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- fd11f1f8def3 +- afc86a351f0c +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/0c7628c7087d.md b/projects/poincare-conjecture/.astrolabe/edges/0c7628c7087d.md new file mode 100644 index 00000000..629d2588 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/0c7628c7087d.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 1 + file: prelim.tex + label: localdiffeo + mtref: '1.32' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 42376 + sourceTransform: tex2mdx.strip_comments + start: 42132 + targetStatement: + chapter: 1 + file: prelim.tex + label: riccurvcomp + mtref: '1.33' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 42870 + sourceTransform: tex2mdx.strip_comments + start: 42464 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 4a0173b99805 +- 569be40ea88e +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/0c78e8c40a18.md b/projects/poincare-conjecture/.astrolabe/edges/0c78e8c40a18.md new file mode 100644 index 00000000..0fc92565 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/0c78e8c40a18.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: n/2 + proofSpan: + bodyEnd: 20911 + bodyStart: 18068 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 20922 + file: noncoll.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 18055 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 20064 + file: noncoll.tex + sourceTransform: tex2mdx.strip_comments + start: 20055 + via: proof + sourceStatement: + chapter: 8 + file: noncoll.tex + label: '' + mtref: '8.10' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 18052 + sourceTransform: tex2mdx.strip_comments + start: 17502 + targetStatement: + chapter: 7 + file: newcomp2.tex + label: n/2 + mtref: '7.10' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 15803 + sourceTransform: tex2mdx.strip_comments + start: 15517 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 198e7475ad2f +- aebc3ff2bd3b +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/0c9c5ba18b76.md b/projects/poincare-conjecture/.astrolabe/edges/0c9c5ba18b76.md new file mode 100644 index 00000000..8fad354c --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/0c9c5ba18b76.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: Harnack + proofSpan: + bodyEnd: 9608 + bodyStart: 7590 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 9619 + file: temp2kappa.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 7577 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 7802 + file: temp2kappa.tex + sourceTransform: tex2mdx.strip_comments + start: 7789 + via: proof + sourceStatement: + chapter: 9 + file: temp2kappa.tex + label: uniformlip + mtref: '9.9' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 7575 + sourceTransform: tex2mdx.strip_comments + start: 6851 + targetStatement: + chapter: 4 + file: maxprin.tex + label: Harnack + mtref: '4.37' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 52804 + sourceTransform: tex2mdx.strip_comments + start: 52280 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 9387b1b04824 +- aa9d836895d8 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/0cc467240695.md b/projects/poincare-conjecture/.astrolabe/edges/0cc467240695.md new file mode 100644 index 00000000..f14aad56 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/0cc467240695.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: posderiv + proofSpan: + bodyEnd: 137790 + bodyStart: 127828 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 137801 + file: temp2kappa.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 127815 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 132986 + file: temp2kappa.tex + sourceTransform: tex2mdx.strip_comments + start: 132972 + via: proof + sourceStatement: + chapter: 9 + file: temp2kappa.tex + label: proposition + mtref: '9.65' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 127812 + sourceTransform: tex2mdx.strip_comments + start: 127549 + targetStatement: + chapter: 4 + file: maxprin.tex + label: posderiv + mtref: '4.39' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 53144 + sourceTransform: tex2mdx.strip_comments + start: 52904 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- ee314aacd207 +- 36dd2f6fa9c2 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/0ce0a079a85c.md b/projects/poincare-conjecture/.astrolabe/edges/0ce0a079a85c.md new file mode 100644 index 00000000..0dc3384d --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/0ce0a079a85c.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: sliceconv + proofSpan: + bodyEnd: 229182 + bodyStart: 185422 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 229193 + file: surgery.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 185409 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 224855 + file: surgery.tex + sourceTransform: tex2mdx.strip_comments + start: 224840 + via: proof + sourceStatement: + chapter: 17 + file: surgery.tex + label: extend + mtref: '17.1' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 185406 + sourceTransform: tex2mdx.strip_comments + start: 184285 + targetStatement: + chapter: 5 + file: converge2.tex + label: sliceconv + mtref: '5.11' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 21468 + sourceTransform: tex2mdx.strip_comments + start: 20456 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 254e8dae97e6 +- d9f3522dec12 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/0cea2bf396c8.md b/projects/poincare-conjecture/.astrolabe/edges/0cea2bf396c8.md new file mode 100644 index 00000000..3fa49690 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/0cea2bf396c8.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 7 + file: newcomp2.tex + label: lineineq + mtref: '7.19' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 33702 + sourceTransform: tex2mdx.strip_comments + start: 33163 + targetStatement: + chapter: 7 + file: newcomp2.tex + label: '' + mtref: '7.20' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 35127 + sourceTransform: tex2mdx.strip_comments + start: 33923 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 528f2ee171eb +- 675c76833e83 +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/0cfaeb472a7e.md b/projects/poincare-conjecture/.astrolabe/edges/0cfaeb472a7e.md new file mode 100644 index 00000000..bd418af3 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/0cfaeb472a7e.md @@ -0,0 +1,84 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: neckdistcomp + proofSpan: + bodyEnd: 23979 + bodyStart: 21546 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 23990 + file: bddcurvbdddist.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 21533 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 22635 + file: bddcurvbdddist.tex + sourceTransform: tex2mdx.strip_comments + start: 22617 + via: proof + - context: proof + label: neckdistcomp + proofSpan: + bodyEnd: 23979 + bodyStart: 21546 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 23990 + file: bddcurvbdddist.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 21533 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 23081 + file: bddcurvbdddist.tex + sourceTransform: tex2mdx.strip_comments + start: 23063 + via: proof + sourceStatement: + chapter: 10 + file: bddcurvbdddist.tex + label: '' + mtref: '10.6' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 21531 + sourceTransform: tex2mdx.strip_comments + start: 21390 + targetStatement: + chapter: 19 + file: canonnbhd.tex + label: neckdistcomp + mtref: '19.9' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 14663 + sourceTransform: tex2mdx.strip_comments + start: 13813 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 6a13c64a90e4 +- 6cf1cbc39ada +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/0d0dd7460740.md b/projects/poincare-conjecture/.astrolabe/edges/0d0dd7460740.md new file mode 100644 index 00000000..45c125e3 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/0d0dd7460740.md @@ -0,0 +1,63 @@ +--- +confidence: 0.35 +edgeClass: prose +evidence: + mentionTriggers: + - label: dtuniqtouniq + refSpan: + coordinateSpace: comment-stripped-tex + end: 82129 + file: stdsoln.tex + sourceTransform: tex2mdx.strip_comments + start: 82111 + sentence: Invoking Corollary~\ref{dtuniqtouniq} we conclude that $g_1(t)=g_2(t)$ + for all $t\in [0,T']$, contradicting the maximality of the interval $I$. + sentenceSpan: + coordinateSpace: comment-stripped-tex + end: 82235 + file: stdsoln.tex + sourceTransform: tex2mdx.strip_comments + start: 82091 + sourceStrategy: nearest-previous-statement + via: prose + sourceStatement: + chapter: 12 + file: stdsoln.tex + label: duniq + mtref: '12.27' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 80569 + sourceTransform: tex2mdx.strip_comments + start: 80181 + targetStatement: + chapter: 12 + file: stdsoln.tex + label: dtuniqtouniq + mtref: '12.16' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 40918 + sourceTransform: tex2mdx.strip_comments + start: 40307 + type: prose-mention-nearest-previous-statement + via: + - prose +generator: tools/poincare_tex_extract.py +inference: weak +kind: mention +ref: +- a501760fb0e6 +- 5b79e248c5cb +rel: mentions +reviewStatus: candidate +scope: prose +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- prose +--- +Morgan--Tian prose mention. The source is the nearest preceding statement in the same chapter segment. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/0d47063670b4.md b/projects/poincare-conjecture/.astrolabe/edges/0d47063670b4.md new file mode 100644 index 00000000..f8223f93 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/0d47063670b4.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: inffrontier + proofSpan: + bodyEnd: 32963 + bodyStart: 31019 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 32974 + file: canonnbhd.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 31006 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 32104 + file: canonnbhd.tex + sourceTransform: tex2mdx.strip_comments + start: 32087 + via: proof + sourceStatement: + chapter: 19 + file: canonnbhd.tex + label: epschains + mtref: '19.19' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 31004 + sourceTransform: tex2mdx.strip_comments + start: 30360 + targetStatement: + chapter: 19 + file: canonnbhd.tex + label: inffrontier + mtref: '19.15' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 24968 + sourceTransform: tex2mdx.strip_comments + start: 24756 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- ba681e15e1a6 +- f6496966d395 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/0d4833cf91b6.md b/projects/poincare-conjecture/.astrolabe/edges/0d4833cf91b6.md new file mode 100644 index 00000000..6cc45e1a --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/0d4833cf91b6.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: DL + proofSpan: + bodyEnd: 41607 + bodyStart: 36687 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 41618 + file: temp2kappa.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 36674 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 37951 + file: temp2kappa.tex + sourceTransform: tex2mdx.strip_comments + start: 37943 + via: proof + sourceStatement: + chapter: 9 + file: temp2kappa.tex + label: lowerbdd + mtref: '9.25' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 36672 + sourceTransform: tex2mdx.strip_comments + start: 36453 + targetStatement: + chapter: 6 + file: newcompar.tex + label: DL + mtref: '6.29' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 39591 + sourceTransform: tex2mdx.strip_comments + start: 39266 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- a6ee3b2ab615 +- 033786a89c6d +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/0d4f75f86f37.md b/projects/poincare-conjecture/.astrolabe/edges/0d4f75f86f37.md new file mode 100644 index 00000000..851d469a --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/0d4f75f86f37.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: DL + proofSpan: + bodyEnd: 58768 + bodyStart: 57518 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 58779 + file: newcompar.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 57505 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 58277 + file: newcompar.tex + sourceTransform: tex2mdx.strip_comments + start: 58269 + via: proof + sourceStatement: + chapter: 6 + file: newcompar.tex + label: Jacobi + mtref: '6.40' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 57503 + sourceTransform: tex2mdx.strip_comments + start: 57209 + targetStatement: + chapter: 6 + file: newcompar.tex + label: DL + mtref: '6.29' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 39591 + sourceTransform: tex2mdx.strip_comments + start: 39266 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 44abf07e60ec +- 033786a89c6d +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/0d501e17ed4d.md b/projects/poincare-conjecture/.astrolabe/edges/0d501e17ed4d.md new file mode 100644 index 00000000..94508ed1 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/0d501e17ed4d.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: vclaim + proofSpan: + bodyEnd: 127580 + bodyStart: 124899 + coordinateSpace: comment-stripped-tex + depth: 2 + end: 127591 + file: energy1.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 124886 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 126347 + file: energy1.tex + sourceTransform: tex2mdx.strip_comments + start: 126335 + via: proof + sourceStatement: + chapter: 18 + file: energy1.tex + label: C1C4delta + mtref: '18.54' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 124884 + sourceTransform: tex2mdx.strip_comments + start: 124119 + targetStatement: + chapter: 18 + file: energy1.tex + label: vclaim + mtref: '18.26' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 60140 + sourceTransform: tex2mdx.strip_comments + start: 59893 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 7e0b4b1362c0 +- eaf31795ca89 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/0e2bccdfe171.md b/projects/poincare-conjecture/.astrolabe/edges/0e2bccdfe171.md new file mode 100644 index 00000000..469f12dd --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/0e2bccdfe171.md @@ -0,0 +1,62 @@ +--- +confidence: 0.9 +edgeClass: structural +evidence: + containmentDepth: 1 + proofSpan: + bodyEnd: 36068 + bodyStart: 28129 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 36079 + file: flowbasics.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 28116 + sourceStatement: + chapter: 3 + file: flowbasics.tex + label: '' + mtref: '3.22' + sort: remark + span: + coordinateSpace: comment-stripped-tex + end: 28114 + sourceTransform: tex2mdx.strip_comments + start: 27844 + targetContainedSpan: + coordinateSpace: comment-stripped-tex + end: 28871 + file: flowbasics.tex + sourceTransform: tex2mdx.strip_comments + start: 28258 + targetStatement: + chapter: 3 + file: flowbasics.tex + label: geosuff + mtref: '3.23' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 28871 + sourceTransform: tex2mdx.strip_comments + start: 28258 + type: proof-containment + via: + - proof-contains +generator: tools/poincare_tex_extract.py +inference: structural +kind: containment +ref: +- e485edf8d992 +- 0f76a0750b40 +rel: contains +reviewStatus: accepted +scope: proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof-contains +--- +Morgan--Tian proof containment. The source is the statement whose proof contains the target statement. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/0e443f600a0d.md b/projects/poincare-conjecture/.astrolabe/edges/0e443f600a0d.md new file mode 100644 index 00000000..07165136 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/0e443f600a0d.md @@ -0,0 +1,65 @@ +--- +confidence: 0.35 +edgeClass: prose +evidence: + mentionTriggers: + - label: redvolconst + refSpan: + coordinateSpace: comment-stripped-tex + end: 48372 + file: temp2kappa.tex + sourceTransform: tex2mdx.strip_comments + start: 48355 + sentence: Lebesgue dominated convergence implies that the following limit exists + $${\rm lim}_{k\rightarrow\infty}\widetilde V_k(\tau)\equiv\widetilde V_\infty(\tau)=\int_{M_\infty\times\{-\tau\}}\tau^{-n/2}e^{- + l_\infty(q,\tau)}d{\rm vol}_{g_\infty(\tau)}.$$\index{$\widetilde V_x$} By Corollary~\ref{redvolconst}, + the function $\tau\rightarrow \widetilde V_\infty(\tau)$ is constant + sentenceSpan: + coordinateSpace: comment-stripped-tex + end: 48442 + file: temp2kappa.tex + sourceTransform: tex2mdx.strip_comments + start: 48068 + sourceStrategy: nearest-previous-statement + via: prose + sourceStatement: + chapter: 9 + file: temp2kappa.tex + label: weakneg + mtref: '9.30' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 46581 + sourceTransform: tex2mdx.strip_comments + start: 46159 + targetStatement: + chapter: 9 + file: temp2kappa.tex + label: redvolconst + mtref: '9.14' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 17598 + sourceTransform: tex2mdx.strip_comments + start: 17342 + type: prose-mention-nearest-previous-statement + via: + - prose +generator: tools/poincare_tex_extract.py +inference: weak +kind: mention +ref: +- 42d9f91e59f3 +- e08895f2b0d8 +rel: mentions +reviewStatus: candidate +scope: prose +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- prose +--- +Morgan--Tian prose mention. The source is the nearest preceding statement in the same chapter segment. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/0e532fbe2f08.md b/projects/poincare-conjecture/.astrolabe/edges/0e532fbe2f08.md new file mode 100644 index 00000000..e165c8f6 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/0e532fbe2f08.md @@ -0,0 +1,63 @@ +--- +confidence: 0.35 +edgeClass: prose +evidence: + mentionTriggers: + - label: lips + refSpan: + coordinateSpace: comment-stripped-tex + end: 85370 + file: newcompar.tex + sourceTransform: tex2mdx.strip_comments + start: 85360 + sentence: Proposition~\ref{lips} follows from a much more precise, though more + complicated to state, result + sentenceSpan: + coordinateSpace: comment-stripped-tex + end: 85445 + file: newcompar.tex + sourceTransform: tex2mdx.strip_comments + start: 85347 + sourceStrategy: nearest-previous-statement + via: prose + sourceStatement: + chapter: 6 + file: newcompar.tex + label: '' + mtref: '6.57' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 85285 + sourceTransform: tex2mdx.strip_comments + start: 85074 + targetStatement: + chapter: 6 + file: newcompar.tex + label: lips + mtref: '6.56' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 85066 + sourceTransform: tex2mdx.strip_comments + start: 84130 + type: prose-mention-nearest-previous-statement + via: + - prose +generator: tools/poincare_tex_extract.py +inference: weak +kind: mention +ref: +- a89a833cf7cb +- db4ced24b05f +rel: mentions +reviewStatus: candidate +scope: prose +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- prose +--- +Morgan--Tian prose mention. The source is the nearest preceding statement in the same chapter segment. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/0e5a0624e3db.md b/projects/poincare-conjecture/.astrolabe/edges/0e5a0624e3db.md new file mode 100644 index 00000000..db4010c5 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/0e5a0624e3db.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: epscap + proofSpan: + bodyEnd: 40983 + bodyStart: 36070 + coordinateSpace: comment-stripped-tex + depth: 2 + end: 40994 + file: canonnbhd.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 36057 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 37731 + file: canonnbhd.tex + sourceTransform: tex2mdx.strip_comments + start: 37719 + via: proof + sourceStatement: + chapter: 19 + file: canonnbhd.tex + label: '' + mtref: '19.22' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 36055 + sourceTransform: tex2mdx.strip_comments + start: 35803 + targetStatement: + chapter: 9 + file: temp2kappa.tex + label: epscap + mtref: '9.72' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 142452 + sourceTransform: tex2mdx.strip_comments + start: 140833 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 9737d037f97b +- 757e83881e62 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/0e6b13da9bf2.md b/projects/poincare-conjecture/.astrolabe/edges/0e6b13da9bf2.md new file mode 100644 index 00000000..149c1357 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/0e6b13da9bf2.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: corI.8.3 + proofSpan: + bodyEnd: 123000 + bodyStart: 121805 + coordinateSpace: comment-stripped-tex + depth: 2 + end: 123011 + file: temp2kappa.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 121792 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 122288 + file: temp2kappa.tex + sourceTransform: tex2mdx.strip_comments + start: 122274 + via: proof + sourceStatement: + chapter: 9 + file: temp2kappa.tex + label: '' + mtref: '9.61' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 121789 + sourceTransform: tex2mdx.strip_comments + start: 121292 + targetStatement: + chapter: 3 + file: flowbasics.tex + label: corI.8.3 + mtref: '3.26' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 37763 + sourceTransform: tex2mdx.strip_comments + start: 37390 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- d2259da4d67d +- 01a829e22cf6 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/0e6f602d25d3.md b/projects/poincare-conjecture/.astrolabe/edges/0e6f602d25d3.md new file mode 100644 index 00000000..5ba369d3 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/0e6f602d25d3.md @@ -0,0 +1,62 @@ +--- +confidence: 0.9 +edgeClass: structural +evidence: + containmentDepth: 2 + proofSpan: + bodyEnd: 71155 + bodyStart: 67995 + coordinateSpace: comment-stripped-tex + depth: 2 + end: 71166 + file: prelim.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 67982 + sourceStatement: + chapter: 2 + file: prelim.tex + label: line + mtref: '2.14' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 67980 + sourceTransform: tex2mdx.strip_comments + start: 67775 + targetContainedSpan: + coordinateSpace: comment-stripped-tex + end: 69160 + file: prelim.tex + sourceTransform: tex2mdx.strip_comments + start: 68871 + targetStatement: + chapter: 2 + file: prelim.tex + label: '' + mtref: '2.15' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 69160 + sourceTransform: tex2mdx.strip_comments + start: 68871 + type: proof-containment + via: + - proof-contains +generator: tools/poincare_tex_extract.py +inference: structural +kind: containment +ref: +- 006e42ef5bb5 +- 0d534e8417eb +rel: contains +reviewStatus: accepted +scope: proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof-contains +--- +Morgan--Tian proof containment. The source is the statement whose proof contains the target statement. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/0e81582d27a6.md b/projects/poincare-conjecture/.astrolabe/edges/0e81582d27a6.md new file mode 100644 index 00000000..3af29fee --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/0e81582d27a6.md @@ -0,0 +1,45 @@ +--- +confidence: 0.2 +edgeClass: structural +evidence: + sourceStatement: + chapter: 6 + file: newcompar.tex + label: lformula + mtref: '6.51' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 81082 + sourceTransform: tex2mdx.strip_comments + start: 80194 + targetStatement: + chapter: 6 + file: newcompar.tex + label: '' + mtref: '6.52' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 81660 + sourceTransform: tex2mdx.strip_comments + start: 81446 + type: chapter-reading-order-adjacency + via: + - chapter-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 967c0f8bb774 +- 77ed5048ae37 +rel: in-chapter +reviewStatus: accepted +scope: chapter +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- chapter-sequence +--- +Morgan--Tian reading-order adjacency across sections in a chapter. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/0ed48714f1e6.md b/projects/poincare-conjecture/.astrolabe/edges/0ed48714f1e6.md new file mode 100644 index 00000000..c4e7fd55 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/0ed48714f1e6.md @@ -0,0 +1,62 @@ +--- +confidence: 0.9 +edgeClass: structural +evidence: + containmentDepth: 3 + proofSpan: + bodyEnd: 29291 + bodyStart: 25300 + coordinateSpace: comment-stripped-tex + depth: 3 + end: 29302 + file: singlimit2.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 25287 + sourceStatement: + chapter: 11 + file: singlimit2.tex + label: outerbd + mtref: '11.14' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 25285 + sourceTransform: tex2mdx.strip_comments + start: 24900 + targetContainedSpan: + coordinateSpace: comment-stripped-tex + end: 26714 + file: singlimit2.tex + sourceTransform: tex2mdx.strip_comments + start: 26365 + targetStatement: + chapter: 11 + file: singlimit2.tex + label: '' + mtref: '11.15' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 26714 + sourceTransform: tex2mdx.strip_comments + start: 26365 + type: proof-containment + via: + - proof-contains +generator: tools/poincare_tex_extract.py +inference: structural +kind: containment +ref: +- 9a7e3fe2e5c4 +- b3fae89d3264 +rel: contains +reviewStatus: accepted +scope: proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof-contains +--- +Morgan--Tian proof containment. The source is the statement whose proof contains the target statement. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/0f0ee429b8b0.md b/projects/poincare-conjecture/.astrolabe/edges/0f0ee429b8b0.md new file mode 100644 index 00000000..16700ac6 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/0f0ee429b8b0.md @@ -0,0 +1,45 @@ +--- +confidence: 0.2 +edgeClass: structural +evidence: + sourceStatement: + chapter: 18 + file: energy1.tex + label: xicomp + mtref: '18.57' + sort: remark + span: + coordinateSpace: comment-stripped-tex + end: 133717 + sourceTransform: tex2mdx.strip_comments + start: 133485 + targetStatement: + chapter: 18 + file: energy1.tex + label: muarea + mtref: '18.58' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 138479 + sourceTransform: tex2mdx.strip_comments + start: 137860 + type: chapter-reading-order-adjacency + via: + - chapter-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 012eefbbc44a +- 10dace7fb165 +rel: in-chapter +reviewStatus: accepted +scope: chapter +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- chapter-sequence +--- +Morgan--Tian reading-order adjacency across sections in a chapter. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/0f28688bc616.md b/projects/poincare-conjecture/.astrolabe/edges/0f28688bc616.md new file mode 100644 index 00000000..93ed9dd7 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/0f28688bc616.md @@ -0,0 +1,64 @@ +--- +confidence: 0.35 +edgeClass: prose +evidence: + mentionTriggers: + - label: xistronger + refSpan: + coordinateSpace: comment-stripped-tex + end: 136675 + file: energy1.tex + sourceTransform: tex2mdx.strip_comments + start: 136659 + sentence: Now we wish to pass from Lemma~\ref{xistronger} which deals with an + individual $c\in S^2$ to a proof of Proposition~\ref{XI} which deals with the + entire family $\widetilde\Gamma$ + sentenceSpan: + coordinateSpace: comment-stripped-tex + end: 136806 + file: energy1.tex + sourceTransform: tex2mdx.strip_comments + start: 136626 + sourceStrategy: nearest-previous-statement + via: prose + sourceStatement: + chapter: 18 + file: energy1.tex + label: xicomp + mtref: '18.57' + sort: remark + span: + coordinateSpace: comment-stripped-tex + end: 133717 + sourceTransform: tex2mdx.strip_comments + start: 133485 + targetStatement: + chapter: 18 + file: energy1.tex + label: xistronger + mtref: '18.53' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 123454 + sourceTransform: tex2mdx.strip_comments + start: 122771 + type: prose-mention-nearest-previous-statement + via: + - prose +generator: tools/poincare_tex_extract.py +inference: weak +kind: mention +ref: +- 012eefbbc44a +- 9df99ae6f4cb +rel: mentions +reviewStatus: candidate +scope: prose +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- prose +--- +Morgan--Tian prose mention. The source is the nearest preceding statement in the same chapter segment. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/0f54b721bea1.md b/projects/poincare-conjecture/.astrolabe/edges/0f54b721bea1.md new file mode 100644 index 00000000..5546195b --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/0f54b721bea1.md @@ -0,0 +1,45 @@ +--- +confidence: 0.2 +edgeClass: structural +evidence: + sourceStatement: + chapter: 18 + file: energy1.tex + label: '' + mtref: '18.8' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 23392 + sourceTransform: tex2mdx.strip_comments + start: 23128 + targetStatement: + chapter: 18 + file: energy1.tex + label: pi2 + mtref: '18.9' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 30087 + sourceTransform: tex2mdx.strip_comments + start: 29657 + type: chapter-reading-order-adjacency + via: + - chapter-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 37efbeaa9eec +- 4415092c2079 +rel: in-chapter +reviewStatus: accepted +scope: chapter +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- chapter-sequence +--- +Morgan--Tian reading-order adjacency across sections in a chapter. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/0fc8f329f8c1.md b/projects/poincare-conjecture/.astrolabe/edges/0fc8f329f8c1.md new file mode 100644 index 00000000..dfb7b3fa --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/0fc8f329f8c1.md @@ -0,0 +1,55 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: statement + label: MAIN + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 233845 + file: surgery.tex + sourceTransform: tex2mdx.strip_comments + start: 233835 + via: statement + sourceStatement: + chapter: 17 + file: surgery.tex + label: bddsurgery + mtref: '17.12' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 234130 + sourceTransform: tex2mdx.strip_comments + start: 233569 + targetStatement: + chapter: 15 + file: surgery.tex + label: MAIN + mtref: '15.9' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 91186 + sourceTransform: tex2mdx.strip_comments + start: 90004 + type: tex-reference + via: + - statement +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 6efe562bcc06 +- 703e4cfeece7 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- statement +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/0fcf704591b0.md b/projects/poincare-conjecture/.astrolabe/edges/0fcf704591b0.md new file mode 100644 index 00000000..93bc011f --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/0fcf704591b0.md @@ -0,0 +1,84 @@ +--- +confidence: 0.35 +edgeClass: prose +evidence: + mentionTriggers: + - label: prodatinf + refSpan: + coordinateSpace: comment-stripped-tex + end: 89369 + file: temp2kappa.tex + sourceTransform: tex2mdx.strip_comments + start: 89354 + sentence: Thus, this flow satisfies all the hypotheses of Proposition~\ref{prodatinf} + sentenceSpan: + coordinateSpace: comment-stripped-tex + end: 89369 + file: temp2kappa.tex + sourceTransform: tex2mdx.strip_comments + start: 89293 + sourceStrategy: nearest-previous-statement + via: prose + - label: prodatinf + refSpan: + coordinateSpace: comment-stripped-tex + end: 89490 + file: temp2kappa.tex + sourceTransform: tex2mdx.strip_comments + start: 89475 + sentence: Hence, by Corollary~\ref{compact2D} we see that $n$ cannot be equal + to two. Furthermore, by Proposition~\ref{prodatinf}, when $n=3$ there is another + subsequence $q_i$ tending to infinity in $M$ such that there is a geometric + limit $(M_\infty,g_\infty(t),(q_\infty,-1)), \ -\inftyt_0$ for the Ricci flows $g_1(t)$ + and $g_2(t)$ + sentenceSpan: + coordinateSpace: comment-stripped-tex + end: 79483 + file: stdsoln.tex + sourceTransform: tex2mdx.strip_comments + start: 79139 + sourceStrategy: nearest-previous-statement + via: prose + - label: harmapflow + refSpan: + coordinateSpace: comment-stripped-tex + end: 79930 + file: stdsoln.tex + sourceTransform: tex2mdx.strip_comments + start: 79914 + sentence: It follows from Lemma~\ref{item5} and the decay estimate in Proposition~\ref{harmapflow} + that $\hat{g}_1(t)$ and $\hat{g}_2(t)$ are bounded solutions and that they have + same sequential asymptotic behavior at infinity + sentenceSpan: + coordinateSpace: comment-stripped-tex + end: 80058 + file: stdsoln.tex + sourceTransform: tex2mdx.strip_comments + start: 79841 + sourceStrategy: nearest-previous-statement + via: prose + sourceStatement: + chapter: 12 + file: stdsoln.tex + label: RDuniq + mtref: '12.26' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 75403 + sourceTransform: tex2mdx.strip_comments + start: 74267 + targetStatement: + chapter: 12 + file: stdsoln.tex + label: harmapflow + mtref: '12.25' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 74100 + sourceTransform: tex2mdx.strip_comments + start: 73601 + type: prose-mention-nearest-previous-statement + via: + - prose +generator: tools/poincare_tex_extract.py +inference: weak +kind: mention +ref: +- c5e225b067f1 +- 5fbeadbba61c +rel: mentions +reviewStatus: candidate +scope: prose +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- prose +--- +Morgan--Tian prose mention. The source is the nearest preceding statement in the same chapter segment. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/1c1489283465.md b/projects/poincare-conjecture/.astrolabe/edges/1c1489283465.md new file mode 100644 index 00000000..ac941a6d --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/1c1489283465.md @@ -0,0 +1,65 @@ +--- +confidence: 0.45 +edgeClass: semantic +evidence: + endpointSemanticDegreeBeforeTermBackfill: + source: 0 + target: 6 + selectionReason: semantic-connectivity-backfill + sourceMatchSpan: + coordinateSpace: comment-stripped-tex + end: 50121 + file: converge2.tex + sourceTransform: tex2mdx.strip_comments + start: 50106 + sourceStatement: + chapter: 5 + file: converge2.tex + label: '' + mtref: '5.32' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 50532 + sourceTransform: tex2mdx.strip_comments + start: 49697 + targetStatement: + chapter: 5 + file: converge2.tex + label: smoothconv + mtref: '5.3' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 4757 + sourceTransform: tex2mdx.strip_comments + start: 2949 + targetTermSpan: + coordinateSpace: comment-stripped-tex + end: 3078 + file: converge2.tex + sourceTransform: tex2mdx.strip_comments + start: 3063 + term: geometric limit + termAmbiguity: unique-in-chapter + termSource: italic + termSpecificity: not-covered-by-longer-defined-term + type: same-chapter-definition-term-match + via: + - definition-term:geometric limit +generator: tools/poincare_tex_extract.py +inference: weak +kind: definition-use +ref: +- 666538bf577e +- 467f4b920031 +rel: uses +reviewStatus: candidate +scope: same-chapter-term-match +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- definition-term:geometric limit +--- +Morgan--Tian same-chapter definition term match. This is a low-confidence candidate edge for review. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/1c1e10a42edc.md b/projects/poincare-conjecture/.astrolabe/edges/1c1e10a42edc.md new file mode 100644 index 00000000..6b963de2 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/1c1e10a42edc.md @@ -0,0 +1,62 @@ +--- +confidence: 0.9 +edgeClass: structural +evidence: + containmentDepth: 1 + proofSpan: + bodyEnd: 106622 + bodyStart: 101268 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 106633 + file: newcompar.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 101255 + sourceStatement: + chapter: 6 + file: newcompar.tex + label: fullmeasure + mtref: '6.67' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 101253 + sourceTransform: tex2mdx.strip_comments + start: 100568 + targetContainedSpan: + coordinateSpace: comment-stripped-tex + end: 105466 + file: newcompar.tex + sourceTransform: tex2mdx.strip_comments + start: 104872 + targetStatement: + chapter: 6 + file: newcompar.tex + label: '' + mtref: '6.69' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 105466 + sourceTransform: tex2mdx.strip_comments + start: 104872 + type: proof-containment + via: + - proof-contains +generator: tools/poincare_tex_extract.py +inference: structural +kind: containment +ref: +- 7aad50e49af2 +- c25742cc5546 +rel: contains +reviewStatus: accepted +scope: proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof-contains +--- +Morgan--Tian proof containment. The source is the statement whose proof contains the target statement. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/1c2de5b39605.md b/projects/poincare-conjecture/.astrolabe/edges/1c2de5b39605.md new file mode 100644 index 00000000..2afbd135 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/1c2de5b39605.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: topsplit + proofSpan: + bodyEnd: 161702 + bodyStart: 159636 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 161713 + file: temp2kappa.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 159623 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 160494 + file: temp2kappa.tex + sourceTransform: tex2mdx.strip_comments + start: 160480 + via: proof + sourceStatement: + chapter: 9 + file: temp2kappa.tex + label: '' + mtref: '9.84' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 159621 + sourceTransform: tex2mdx.strip_comments + start: 158978 + targetStatement: + chapter: 5 + file: converge2.tex + label: topsplit + mtref: '5.35' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 55387 + sourceTransform: tex2mdx.strip_comments + start: 54529 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 6bda3981dbaa +- e9a9172d16e2 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/1c42d436b45e.md b/projects/poincare-conjecture/.astrolabe/edges/1c42d436b45e.md new file mode 100644 index 00000000..235f5c65 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/1c42d436b45e.md @@ -0,0 +1,64 @@ +--- +confidence: 0.35 +edgeClass: prose +evidence: + mentionTriggers: + - label: L_0 + refSpan: + coordinateSpace: comment-stripped-tex + end: 150532 + file: surgery.tex + sourceTransform: tex2mdx.strip_comments + start: 150523 + sentence: Now using Lemmas~\ref{L_0} and~\ref{15.17} we show that it is possible + to take limits of certain types of sequences of paths parameterized by backward + time to create minimizing ${\mathcal L}$-geodesics. + sentenceSpan: + coordinateSpace: comment-stripped-tex + end: 150708 + file: surgery.tex + sourceTransform: tex2mdx.strip_comments + start: 150505 + sourceStrategy: nearest-previous-statement + via: prose + sourceStatement: + chapter: 16 + file: surgery.tex + label: '15.17' + mtref: '16.19' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 149253 + sourceTransform: tex2mdx.strip_comments + start: 148262 + targetStatement: + chapter: 16 + file: surgery.tex + label: L_0 + mtref: '16.15' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 138538 + sourceTransform: tex2mdx.strip_comments + start: 137276 + type: prose-mention-nearest-previous-statement + via: + - prose +generator: tools/poincare_tex_extract.py +inference: weak +kind: mention +ref: +- 5ed7c639bd97 +- 86043191a3c2 +rel: mentions +reviewStatus: candidate +scope: prose +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- prose +--- +Morgan--Tian prose mention. The source is the nearest preceding statement in the same chapter segment. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/1c5c4ab3687a.md b/projects/poincare-conjecture/.astrolabe/edges/1c5c4ab3687a.md new file mode 100644 index 00000000..68a0b4b2 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/1c5c4ab3687a.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 9 + file: temp2kappa.tex + label: 2DGSS + mtref: '9.50' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 91949 + sourceTransform: tex2mdx.strip_comments + start: 91266 + targetStatement: + chapter: 9 + file: temp2kappa.tex + label: '' + mtref: '9.51' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 94920 + sourceTransform: tex2mdx.strip_comments + start: 94840 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- da767c71996b +- 07605ec62528 +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/1c8767667246.md b/projects/poincare-conjecture/.astrolabe/edges/1c8767667246.md new file mode 100644 index 00000000..e30753db --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/1c8767667246.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: Qformula + proofSpan: + bodyEnd: 117788 + bodyStart: 114121 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 117799 + file: newcompar.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 114108 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 117430 + file: newcompar.tex + sourceTransform: tex2mdx.strip_comments + start: 117416 + via: proof + sourceStatement: + chapter: 6 + file: newcompar.tex + label: Jlimit + mtref: '6.78' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 114106 + sourceTransform: tex2mdx.strip_comments + start: 113458 + targetStatement: + chapter: 6 + file: newcompar.tex + label: Qformula + mtref: '6.75' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 111495 + sourceTransform: tex2mdx.strip_comments + start: 111142 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 4e6330b2aa46 +- 76c5f88ad2b4 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/1cb4ebfa423f.md b/projects/poincare-conjecture/.astrolabe/edges/1cb4ebfa423f.md new file mode 100644 index 00000000..484f9d11 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/1cb4ebfa423f.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 15 + file: surgery.tex + label: RFSexists + mtref: '15.10' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 92173 + sourceTransform: tex2mdx.strip_comments + start: 91337 + targetStatement: + chapter: 15 + file: surgery.tex + label: '14.26' + mtref: '15.11' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 93849 + sourceTransform: tex2mdx.strip_comments + start: 93211 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- c86aa999ce8a +- c8e7931dbf65 +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/1cfda0a3f193.md b/projects/poincare-conjecture/.astrolabe/edges/1cfda0a3f193.md new file mode 100644 index 00000000..32700275 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/1cfda0a3f193.md @@ -0,0 +1,62 @@ +--- +confidence: 0.9 +edgeClass: structural +evidence: + containmentDepth: 1 + proofSpan: + bodyEnd: 100555 + bodyStart: 97739 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 100566 + file: newcompar.tex + ownerLabel: lipaty + sourceTransform: tex2mdx.strip_comments + start: 97726 + sourceStatement: + chapter: 6 + file: newcompar.tex + label: lipaty + mtref: '6.59' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 87473 + sourceTransform: tex2mdx.strip_comments + start: 86282 + targetContainedSpan: + coordinateSpace: comment-stripped-tex + end: 98830 + file: newcompar.tex + sourceTransform: tex2mdx.strip_comments + start: 98666 + targetStatement: + chapter: 6 + file: newcompar.tex + label: '' + mtref: '6.65' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 98830 + sourceTransform: tex2mdx.strip_comments + start: 98666 + type: proof-containment + via: + - proof-contains +generator: tools/poincare_tex_extract.py +inference: structural +kind: containment +ref: +- 96ced8c484df +- e27e5f5d2bf7 +rel: contains +reviewStatus: accepted +scope: proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof-contains +--- +Morgan--Tian proof containment. The source is the statement whose proof contains the target statement. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/1d09980cedaf.md b/projects/poincare-conjecture/.astrolabe/edges/1d09980cedaf.md new file mode 100644 index 00000000..62621bb5 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/1d09980cedaf.md @@ -0,0 +1,64 @@ +--- +confidence: 0.35 +edgeClass: prose +evidence: + mentionTriggers: + - label: lformula + refSpan: + coordinateSpace: comment-stripped-tex + end: 51671 + file: temp2kappa.tex + sourceTransform: tex2mdx.strip_comments + start: 51657 + sentence: 'Now from the last two equations in Corollary~\ref{lformula} and the + convergence of the $l_{x_k}$ to $l_\infty$, we conclude that the following equation + also holds: \begin{equation}\label{wk=}2\triangle l_\infty-|\nabla l_\infty|^2+R+\frac{l_\infty-n}{\tau}=0.\end{equation}' + sentenceSpan: + coordinateSpace: comment-stripped-tex + end: 51885 + file: temp2kappa.tex + sourceTransform: tex2mdx.strip_comments + start: 51611 + sourceStrategy: nearest-previous-statement + via: prose + sourceStatement: + chapter: 9 + file: temp2kappa.tex + label: '' + mtref: '9.31' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 49297 + sourceTransform: tex2mdx.strip_comments + start: 48976 + targetStatement: + chapter: 6 + file: newcompar.tex + label: lformula + mtref: '6.51' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 81082 + sourceTransform: tex2mdx.strip_comments + start: 80194 + type: prose-mention-nearest-previous-statement + via: + - prose +generator: tools/poincare_tex_extract.py +inference: weak +kind: mention +ref: +- a8ab8a19cc61 +- 967c0f8bb774 +rel: mentions +reviewStatus: candidate +scope: prose +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- prose +--- +Morgan--Tian prose mention. The source is the nearest preceding statement in the same chapter segment. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/1d2287650fd7.md b/projects/poincare-conjecture/.astrolabe/edges/1d2287650fd7.md new file mode 100644 index 00000000..4c60425d --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/1d2287650fd7.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 6 + file: newcompar.tex + label: Amono + mtref: '6.80' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 121245 + sourceTransform: tex2mdx.strip_comments + start: 120839 + targetStatement: + chapter: 6 + file: newcompar.tex + label: fmonotone + mtref: '6.81' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 122525 + sourceTransform: tex2mdx.strip_comments + start: 122141 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 6a52537bef2f +- 4dc40f37cd6d +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/1d2d0b30af09.md b/projects/poincare-conjecture/.astrolabe/edges/1d2d0b30af09.md new file mode 100644 index 00000000..0674f3d8 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/1d2d0b30af09.md @@ -0,0 +1,62 @@ +--- +confidence: 0.35 +edgeClass: prose +evidence: + mentionTriggers: + - label: shi + refSpan: + coordinateSpace: comment-stripped-tex + end: 43170 + file: flowbasics.tex + sourceTransform: tex2mdx.strip_comments + start: 43161 + sentence: For a proof of Theorem~\ref{shi} see \cite{Shi1, Shi2} + sentenceSpan: + coordinateSpace: comment-stripped-tex + end: 43193 + file: flowbasics.tex + sourceTransform: tex2mdx.strip_comments + start: 43136 + sourceStrategy: nearest-previous-statement + via: prose + sourceStatement: + chapter: 3 + file: flowbasics.tex + label: Cinftybd + mtref: '3.31' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 43134 + sourceTransform: tex2mdx.strip_comments + start: 42648 + targetStatement: + chapter: 3 + file: flowbasics.tex + label: shi + mtref: '3.28' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 40541 + sourceTransform: tex2mdx.strip_comments + start: 39904 + type: prose-mention-nearest-previous-statement + via: + - prose +generator: tools/poincare_tex_extract.py +inference: weak +kind: mention +ref: +- c7a0e64aa85c +- ed4b72caf9f9 +rel: mentions +reviewStatus: candidate +scope: prose +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- prose +--- +Morgan--Tian prose mention. The source is the nearest preceding statement in the same chapter segment. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/1d3b14090010.md b/projects/poincare-conjecture/.astrolabe/edges/1d3b14090010.md new file mode 100644 index 00000000..a8bf16fa --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/1d3b14090010.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 6 + file: newcompar.tex + label: 2ndvari + mtref: '6.14' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 17350 + sourceTransform: tex2mdx.strip_comments + start: 16270 + targetStatement: + chapter: 6 + file: newcompar.tex + label: variremark + mtref: '6.15' + sort: remark + span: + coordinateSpace: comment-stripped-tex + end: 19144 + sourceTransform: tex2mdx.strip_comments + start: 18610 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- f48558c60ca2 +- 2fef81872a01 +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/1d48e78ee10d.md b/projects/poincare-conjecture/.astrolabe/edges/1d48e78ee10d.md new file mode 100644 index 00000000..4b76c0b6 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/1d48e78ee10d.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: Jlimit + proofSpan: + bodyEnd: 125200 + bodyStart: 122543 + coordinateSpace: comment-stripped-tex + depth: 2 + end: 125211 + file: newcompar.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 122530 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 125198 + file: newcompar.tex + sourceTransform: tex2mdx.strip_comments + start: 125186 + via: proof + sourceStatement: + chapter: 6 + file: newcompar.tex + label: fmonotone + mtref: '6.81' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 122525 + sourceTransform: tex2mdx.strip_comments + start: 122141 + targetStatement: + chapter: 6 + file: newcompar.tex + label: Jlimit + mtref: '6.78' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 114106 + sourceTransform: tex2mdx.strip_comments + start: 113458 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 4dc40f37cd6d +- 4e6330b2aa46 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/1d6f1bda60af.md b/projects/poincare-conjecture/.astrolabe/edges/1d6f1bda60af.md new file mode 100644 index 00000000..64bb9777 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/1d6f1bda60af.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 19 + file: canonnbhd.tex + label: neckdistcomp + mtref: '19.9' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 14663 + sourceTransform: tex2mdx.strip_comments + start: 13813 + targetStatement: + chapter: 19 + file: canonnbhd.tex + label: topiso + mtref: '19.10' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 15949 + sourceTransform: tex2mdx.strip_comments + start: 15078 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 6cf1cbc39ada +- 04dd3c3ea1c8 +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/1dbdecd2e512.md b/projects/poincare-conjecture/.astrolabe/edges/1dbdecd2e512.md new file mode 100644 index 00000000..ced75908 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/1dbdecd2e512.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 18 + file: energy1.tex + label: onec + mtref: '18.47' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 101619 + sourceTransform: tex2mdx.strip_comments + start: 101151 + targetStatement: + chapter: 18 + file: energy1.tex + label: uniformc + mtref: '18.48' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 102679 + sourceTransform: tex2mdx.strip_comments + start: 102426 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 981d63c19e50 +- 028d3b5182b5 +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/1dde7078b7cd.md b/projects/poincare-conjecture/.astrolabe/edges/1dde7078b7cd.md new file mode 100644 index 00000000..300f68e9 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/1dde7078b7cd.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: varyingZ + proofSpan: + bodyEnd: 47393 + bodyStart: 39592 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 47404 + file: maxprin.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 39579 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 46693 + file: maxprin.tex + sourceTransform: tex2mdx.strip_comments + start: 46679 + via: proof + sourceStatement: + chapter: 4 + file: maxprin.tex + label: '' + mtref: '4.27' + sort: remark + span: + coordinateSpace: comment-stripped-tex + end: 39436 + sourceTransform: tex2mdx.strip_comments + start: 39187 + targetStatement: + chapter: 4 + file: maxprin.tex + label: varyingZ + mtref: '4.8' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 10335 + sourceTransform: tex2mdx.strip_comments + start: 9140 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 1160f53cc783 +- 8f737f2b0f7d +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/1de67eab5ac6.md b/projects/poincare-conjecture/.astrolabe/edges/1de67eab5ac6.md new file mode 100644 index 00000000..63200adb --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/1de67eab5ac6.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 4 + file: maxprin.tex + label: nullspace + mtref: '4.20' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 31663 + sourceTransform: tex2mdx.strip_comments + start: 30958 + targetStatement: + chapter: 4 + file: maxprin.tex + label: '' + mtref: '4.21' + sort: remark + span: + coordinateSpace: comment-stripped-tex + end: 32120 + sourceTransform: tex2mdx.strip_comments + start: 31666 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- d21a675e4ea8 +- ce85fb7318bf +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/1e3ac42ca443.md b/projects/poincare-conjecture/.astrolabe/edges/1e3ac42ca443.md new file mode 100644 index 00000000..7acbbfd8 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/1e3ac42ca443.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: geosuff + proofSpan: + bodyEnd: 36068 + bodyStart: 28129 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 36079 + file: flowbasics.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 28116 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 36020 + file: flowbasics.tex + sourceTransform: tex2mdx.strip_comments + start: 36007 + via: proof + sourceStatement: + chapter: 3 + file: flowbasics.tex + label: '' + mtref: '3.22' + sort: remark + span: + coordinateSpace: comment-stripped-tex + end: 28114 + sourceTransform: tex2mdx.strip_comments + start: 27844 + targetStatement: + chapter: 3 + file: flowbasics.tex + label: geosuff + mtref: '3.23' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 28871 + sourceTransform: tex2mdx.strip_comments + start: 28258 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- e485edf8d992 +- 0f76a0750b40 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/1e66b07cdf03.md b/projects/poincare-conjecture/.astrolabe/edges/1e66b07cdf03.md new file mode 100644 index 00000000..69b9d90f --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/1e66b07cdf03.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: geoexist + proofSpan: + bodyEnd: 164255 + bodyStart: 154729 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 164266 + file: surgery.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 154716 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 159336 + file: surgery.tex + sourceTransform: tex2mdx.strip_comments + start: 159322 + via: proof + sourceStatement: + chapter: 16 + file: surgery.tex + label: Lgeoexist + mtref: '16.21' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 154713 + sourceTransform: tex2mdx.strip_comments + start: 152982 + targetStatement: + chapter: 7 + file: newcomp2.tex + label: geoexist + mtref: '7.2' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 2760 + sourceTransform: tex2mdx.strip_comments + start: 2498 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- d98ad25d9bc3 +- cd062526b91a +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/1e6addc7f755.md b/projects/poincare-conjecture/.astrolabe/edges/1e6addc7f755.md new file mode 100644 index 00000000..37c47054 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/1e6addc7f755.md @@ -0,0 +1,45 @@ +--- +confidence: 0.2 +edgeClass: structural +evidence: + sourceStatement: + chapter: 12 + file: stdsoln.tex + label: duniq + mtref: '12.27' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 80569 + sourceTransform: tex2mdx.strip_comments + start: 80181 + targetStatement: + chapter: 12 + file: stdsoln.tex + label: R_0cannbhd + mtref: '12.28' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 83517 + sourceTransform: tex2mdx.strip_comments + start: 82969 + type: chapter-reading-order-adjacency + via: + - chapter-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- a501760fb0e6 +- 85e3c925177e +rel: in-chapter +reviewStatus: accepted +scope: chapter +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- chapter-sequence +--- +Morgan--Tian reading-order adjacency across sections in a chapter. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/1e8db1a1567d.md b/projects/poincare-conjecture/.astrolabe/edges/1e8db1a1567d.md new file mode 100644 index 00000000..754e77e1 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/1e8db1a1567d.md @@ -0,0 +1,65 @@ +--- +confidence: 0.45 +edgeClass: semantic +evidence: + endpointSemanticDegreeBeforeTermBackfill: + source: 0 + target: 4 + selectionReason: semantic-connectivity-backfill + sourceMatchSpan: + coordinateSpace: comment-stripped-tex + end: 42654 + file: prelim.tex + sourceTransform: tex2mdx.strip_comments + start: 42646 + sourceStatement: + chapter: 1 + file: prelim.tex + label: riccurvcomp + mtref: '1.33' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 42870 + sourceTransform: tex2mdx.strip_comments + start: 42464 + targetStatement: + chapter: 1 + file: prelim.tex + label: '' + mtref: '1.17' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 19309 + sourceTransform: tex2mdx.strip_comments + start: 19124 + targetTermSpan: + coordinateSpace: comment-stripped-tex + end: 19240 + file: prelim.tex + sourceTransform: tex2mdx.strip_comments + start: 19232 + term: geodesic + termAmbiguity: unique-in-chapter + termSource: italic + termSpecificity: no-longer-defined-term + type: same-chapter-definition-term-match + via: + - definition-term:geodesic +generator: tools/poincare_tex_extract.py +inference: weak +kind: definition-use +ref: +- 569be40ea88e +- c4ccdd484637 +rel: uses +reviewStatus: candidate +scope: same-chapter-term-match +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- definition-term:geodesic +--- +Morgan--Tian same-chapter definition term match. This is a low-confidence candidate edge for review. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/1edd7f6ddde7.md b/projects/poincare-conjecture/.astrolabe/edges/1edd7f6ddde7.md new file mode 100644 index 00000000..c4228473 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/1edd7f6ddde7.md @@ -0,0 +1,45 @@ +--- +confidence: 0.2 +edgeClass: structural +evidence: + sourceStatement: + chapter: 18 + file: energy1.tex + label: focussummary + mtref: '18.84' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 189625 + sourceTransform: tex2mdx.strip_comments + start: 188975 + targetStatement: + chapter: 18 + file: energy1.tex + label: '' + mtref: '18.85' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 192706 + sourceTransform: tex2mdx.strip_comments + start: 192549 + type: chapter-reading-order-adjacency + via: + - chapter-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 8abfaea5aa9a +- 692ac8d2c602 +rel: in-chapter +reviewStatus: accepted +scope: chapter +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- chapter-sequence +--- +Morgan--Tian reading-order adjacency across sections in a chapter. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/1ee5b89f8787.md b/projects/poincare-conjecture/.astrolabe/edges/1ee5b89f8787.md new file mode 100644 index 00000000..a08209b0 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/1ee5b89f8787.md @@ -0,0 +1,56 @@ +--- +confidence: 0.85 +edgeClass: semantic +evidence: + containmentReason: target theorem-like statement occurs inside source proof + proofSpan: + bodyEnd: 33794 + bodyStart: 17659 + coordinateSpace: comment-stripped-tex + depth: 2 + end: 33805 + file: singlimit2.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 17646 + sourceStatement: + chapter: 11 + file: singlimit2.tex + label: maxT + mtref: '11.10' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 17643 + sourceTransform: tex2mdx.strip_comments + start: 17183 + targetStatement: + chapter: 11 + file: singlimit2.tex + label: '' + mtref: '11.13' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 23634 + sourceTransform: tex2mdx.strip_comments + start: 23355 + type: proof-contained-theorem-like-statement + via: + - proof-contained-statement +generator: tools/poincare_tex_extract.py +inference: inferred +kind: dependency +ref: +- d6919616448d +- 385a31c5d191 +rel: depends +reviewStatus: accepted +scope: proof-containment +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof-contained-statement +--- +Morgan--Tian proof-contained statement dependency. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/1eeb8a720227.md b/projects/poincare-conjecture/.astrolabe/edges/1eeb8a720227.md new file mode 100644 index 00000000..4b91dcfc --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/1eeb8a720227.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: localprod + proofSpan: + bodyEnd: 13332 + bodyStart: 11568 + coordinateSpace: comment-stripped-tex + depth: 2 + end: 13343 + file: singlimit2.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 11555 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 11837 + file: singlimit2.tex + sourceTransform: tex2mdx.strip_comments + start: 11822 + via: proof + sourceStatement: + chapter: 11 + file: singlimit2.tex + label: '' + mtref: '11.7' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 11552 + sourceTransform: tex2mdx.strip_comments + start: 11452 + targetStatement: + chapter: 4 + file: maxprin.tex + label: localprod + mtref: '4.19' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 25705 + sourceTransform: tex2mdx.strip_comments + start: 25065 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 620ff6c41ef4 +- 8ec855a2aab5 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/1ef8d2f33b67.md b/projects/poincare-conjecture/.astrolabe/edges/1ef8d2f33b67.md new file mode 100644 index 00000000..d2e43036 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/1ef8d2f33b67.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: defncanonnbhd + proofSpan: + bodyEnd: 204939 + bodyStart: 186733 + coordinateSpace: comment-stripped-tex + depth: 2 + end: 204950 + file: surgery.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 186720 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 189370 + file: surgery.tex + sourceTransform: tex2mdx.strip_comments + start: 189351 + via: proof + sourceStatement: + chapter: 17 + file: surgery.tex + label: '16.2' + mtref: '17.2' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 186718 + sourceTransform: tex2mdx.strip_comments + start: 186200 + targetStatement: + chapter: 9 + file: temp2kappa.tex + label: defncanonnbhd + mtref: '9.78' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 147589 + sourceTransform: tex2mdx.strip_comments + start: 145644 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 778d690c4e7c +- 3aa5883ab629 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/1f644114614e.md b/projects/poincare-conjecture/.astrolabe/edges/1f644114614e.md new file mode 100644 index 00000000..7378236f --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/1f644114614e.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 6 + file: newcompar.tex + label: 2ndvariation + mtref: '6.33' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 47655 + sourceTransform: tex2mdx.strip_comments + start: 46570 + targetStatement: + chapter: 6 + file: newcompar.tex + label: '' + mtref: '6.34' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 48328 + sourceTransform: tex2mdx.strip_comments + start: 47847 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 3add96057999 +- a75b1ef40db8 +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/1f75725eff67.md b/projects/poincare-conjecture/.astrolabe/edges/1f75725eff67.md new file mode 100644 index 00000000..ccd38cdb --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/1f75725eff67.md @@ -0,0 +1,62 @@ +--- +confidence: 0.35 +edgeClass: prose +evidence: + mentionTriggers: + - label: statement + refSpan: + coordinateSpace: comment-stripped-tex + end: 52462 + file: bddcurvbdddist.tex + sourceTransform: tex2mdx.strip_comments + start: 52447 + sentence: Now we are ready to prove Proposition~\ref{statement} + sentenceSpan: + coordinateSpace: comment-stripped-tex + end: 52462 + file: bddcurvbdddist.tex + sourceTransform: tex2mdx.strip_comments + start: 52408 + sourceStrategy: nearest-previous-statement + via: prose + sourceStatement: + chapter: 10 + file: bddcurvbdddist.tex + label: deltanet + mtref: '10.28' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 52334 + sourceTransform: tex2mdx.strip_comments + start: 51848 + targetStatement: + chapter: 10 + file: bddcurvbdddist.tex + label: statement + mtref: '10.12' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 36191 + sourceTransform: tex2mdx.strip_comments + start: 35406 + type: prose-mention-nearest-previous-statement + via: + - prose +generator: tools/poincare_tex_extract.py +inference: weak +kind: mention +ref: +- f7e1d840a49f +- 9cf6ad839dbb +rel: mentions +reviewStatus: candidate +scope: prose +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- prose +--- +Morgan--Tian prose mention. The source is the nearest preceding statement in the same chapter segment. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/1f8d69f8004a.md b/projects/poincare-conjecture/.astrolabe/edges/1f8d69f8004a.md new file mode 100644 index 00000000..f288a622 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/1f8d69f8004a.md @@ -0,0 +1,45 @@ +--- +confidence: 0.2 +edgeClass: structural +evidence: + sourceStatement: + chapter: 16 + file: surgery.tex + label: '' + mtref: '16.17' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 140633 + sourceTransform: tex2mdx.strip_comments + start: 140555 + targetStatement: + chapter: 16 + file: surgery.tex + label: J + mtref: '16.18' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 148100 + sourceTransform: tex2mdx.strip_comments + start: 146759 + type: chapter-reading-order-adjacency + via: + - chapter-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 9b7f421ab0b9 +- 0ea74aba0c14 +rel: in-chapter +reviewStatus: accepted +scope: chapter +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- chapter-sequence +--- +Morgan--Tian reading-order adjacency across sections in a chapter. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/1fa52d30cc98.md b/projects/poincare-conjecture/.astrolabe/edges/1fa52d30cc98.md new file mode 100644 index 00000000..258ae106 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/1fa52d30cc98.md @@ -0,0 +1,56 @@ +--- +confidence: 0.85 +edgeClass: semantic +evidence: + containmentReason: target theorem-like statement occurs inside source proof + proofSpan: + bodyEnd: 113303 + bodyStart: 106682 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 113314 + file: temp2kappa.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 106669 + sourceStatement: + chapter: 9 + file: temp2kappa.tex + label: asympscal + mtref: '9.56' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 106666 + sourceTransform: tex2mdx.strip_comments + start: 106402 + targetStatement: + chapter: 9 + file: temp2kappa.tex + label: '' + mtref: '9.57' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 111901 + sourceTransform: tex2mdx.strip_comments + start: 111822 + type: proof-contained-theorem-like-statement + via: + - proof-contained-statement +generator: tools/poincare_tex_extract.py +inference: inferred +kind: dependency +ref: +- 5d9f1b75170d +- 6448a516850a +rel: depends +reviewStatus: accepted +scope: proof-containment +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof-contained-statement +--- +Morgan--Tian proof-contained statement dependency. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/1fa94ba14929.md b/projects/poincare-conjecture/.astrolabe/edges/1fa94ba14929.md new file mode 100644 index 00000000..07ded900 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/1fa94ba14929.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: fordiffquot + proofSpan: + bodyEnd: 77920 + bodyStart: 74111 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 77931 + file: energy1.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 74098 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 77918 + file: energy1.tex + sourceTransform: tex2mdx.strip_comments + start: 77901 + via: proof + sourceStatement: + chapter: 18 + file: energy1.tex + label: nonembeasycase + mtref: '18.32' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 74095 + sourceTransform: tex2mdx.strip_comments + start: 73662 + targetStatement: + chapter: 2 + file: prelim.tex + label: fordiffquot + mtref: '2.22' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 82182 + sourceTransform: tex2mdx.strip_comments + start: 81704 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 27b452d6cb48 +- 67c378f87457 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/1fb862ab1cb2.md b/projects/poincare-conjecture/.astrolabe/edges/1fb862ab1cb2.md new file mode 100644 index 00000000..0f0884c2 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/1fb862ab1cb2.md @@ -0,0 +1,56 @@ +--- +confidence: 0.85 +edgeClass: semantic +evidence: + containmentReason: target theorem-like statement occurs inside source proof + proofSpan: + bodyEnd: 54967 + bodyStart: 37263 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 54978 + file: energy1.tex + ownerLabel: W_2 + sourceTransform: tex2mdx.strip_comments + start: 37250 + sourceStatement: + chapter: 18 + file: energy1.tex + label: W_2 + mtref: '18.11' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 36484 + sourceTransform: tex2mdx.strip_comments + start: 36186 + targetStatement: + chapter: 18 + file: energy1.tex + label: '' + mtref: '18.19' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 52551 + sourceTransform: tex2mdx.strip_comments + start: 52480 + type: proof-contained-theorem-like-statement + via: + - proof-contained-statement +generator: tools/poincare_tex_extract.py +inference: inferred +kind: dependency +ref: +- 7bb0fa0c9412 +- 6a1625220073 +rel: depends +reviewStatus: accepted +scope: proof-containment +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof-contained-statement +--- +Morgan--Tian proof-contained statement dependency. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/1ffa718a4fd9.md b/projects/poincare-conjecture/.astrolabe/edges/1ffa718a4fd9.md new file mode 100644 index 00000000..76cd34e7 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/1ffa718a4fd9.md @@ -0,0 +1,80 @@ +--- +confidence: 0.35 +edgeClass: prose +evidence: + mentionTriggers: + - label: W_3fordiff + refSpan: + coordinateSpace: comment-stripped-tex + end: 55090 + file: energy1.tex + sourceTransform: tex2mdx.strip_comments + start: 55074 + sentence: Thus, to complete the argument for Theorem~\ref{extinct} it remains + only to prove Proposition~\ref{W_3fordiff}. + sentenceSpan: + coordinateSpace: comment-stripped-tex + end: 55091 + file: energy1.tex + sourceTransform: tex2mdx.strip_comments + start: 54979 + sourceStrategy: nearest-previous-statement + via: prose + - label: W_3fordiff + refSpan: + coordinateSpace: comment-stripped-tex + end: 55248 + file: energy1.tex + sourceTransform: tex2mdx.strip_comments + start: 55232 + sentence: In this subsection we establish the two continuity conditions for $W_\xi(t)$ + stated in Proposition~\ref{W_3fordiff}. + sentenceSpan: + coordinateSpace: comment-stripped-tex + end: 55249 + file: energy1.tex + sourceTransform: tex2mdx.strip_comments + start: 55132 + sourceStrategy: nearest-previous-statement + via: prose + sourceStatement: + chapter: 18 + file: energy1.tex + label: '' + mtref: '18.20' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 53052 + sourceTransform: tex2mdx.strip_comments + start: 52883 + targetStatement: + chapter: 18 + file: energy1.tex + label: W_3fordiff + mtref: '18.18' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 51926 + sourceTransform: tex2mdx.strip_comments + start: 51222 + type: prose-mention-nearest-previous-statement + via: + - prose +generator: tools/poincare_tex_extract.py +inference: weak +kind: mention +ref: +- a04af7434ef0 +- b187c06d1432 +rel: mentions +reviewStatus: candidate +scope: prose +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- prose +--- +Morgan--Tian prose mention. The source is the nearest preceding statement in the same chapter segment. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/2024721e6a31.md b/projects/poincare-conjecture/.astrolabe/edges/2024721e6a31.md new file mode 100644 index 00000000..3d0e0e70 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/2024721e6a31.md @@ -0,0 +1,66 @@ +--- +confidence: 0.35 +edgeClass: prose +evidence: + mentionTriggers: + - label: MAIN + refSpan: + coordinateSpace: comment-stripped-tex + end: 27596 + file: energy1.tex + sourceTransform: tex2mdx.strip_comments + start: 27586 + sentence: We have just established that given any Ricci flow with surgery $({\mathcal + M},G)$ satisfying the conclusion of Theorem~\ref{MAIN} there is $T_0<\infty$, + depending on $({\mathcal M},G)$, such that all surgeries after time $T_0$ either + are along homotopically trivial $2$-spheres or remove entire components of the + manifold + sentenceSpan: + coordinateSpace: comment-stripped-tex + end: 27789 + file: energy1.tex + sourceTransform: tex2mdx.strip_comments + start: 27465 + sourceStrategy: nearest-previous-statement + via: prose + sourceStatement: + chapter: 18 + file: energy1.tex + label: '' + mtref: '18.8' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 23392 + sourceTransform: tex2mdx.strip_comments + start: 23128 + targetStatement: + chapter: 15 + file: surgery.tex + label: MAIN + mtref: '15.9' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 91186 + sourceTransform: tex2mdx.strip_comments + start: 90004 + type: prose-mention-nearest-previous-statement + via: + - prose +generator: tools/poincare_tex_extract.py +inference: weak +kind: mention +ref: +- 37efbeaa9eec +- 703e4cfeece7 +rel: mentions +reviewStatus: candidate +scope: prose +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- prose +--- +Morgan--Tian prose mention. The source is the nearest preceding statement in the same chapter segment. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/2034dccdc914.md b/projects/poincare-conjecture/.astrolabe/edges/2034dccdc914.md new file mode 100644 index 00000000..c7b30dac --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/2034dccdc914.md @@ -0,0 +1,67 @@ +--- +confidence: 0.72 +edgeClass: semantic +evidence: + anaphoraTriggers: + - context: proof + direction: previous + phrase: previous lemma + proofSpan: + bodyEnd: 172251 + bodyStart: 172003 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 172262 + file: energy1.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 171990 + sort: lemma + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 172060 + file: energy1.tex + sourceTransform: tex2mdx.strip_comments + start: 172046 + via: local-anaphora-proof + sourceStatement: + chapter: 18 + file: energy1.tex + label: muepsbd + mtref: '18.72' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 171988 + sourceTransform: tex2mdx.strip_comments + start: 171484 + targetStatement: + chapter: 18 + file: energy1.tex + label: laest + mtref: '18.66' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 161338 + sourceTransform: tex2mdx.strip_comments + start: 161080 + type: local-anaphora-reference + via: + - local-anaphora-proof +generator: tools/poincare_tex_extract.py +inference: inferred +kind: dependency +ref: +- 9c8d4163f700 +- 45c783d384c1 +rel: depends +reviewStatus: accepted +scope: local-context +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- local-anaphora-proof +--- +Morgan--Tian local anaphora dependency. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/20595aefcc3b.md b/projects/poincare-conjecture/.astrolabe/edges/20595aefcc3b.md new file mode 100644 index 00000000..d2a0f60f --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/20595aefcc3b.md @@ -0,0 +1,63 @@ +--- +confidence: 0.35 +edgeClass: prose +evidence: + mentionTriggers: + - label: Dxacross + refSpan: + coordinateSpace: comment-stripped-tex + end: 191135 + file: energy1.tex + sourceTransform: tex2mdx.strip_comments + start: 191121 + sentence: By Lemma~\ref{Dxacross}, the only possibility for the final endpoints + of all these $D_z$'s is that they lie in $c_1$. + sentenceSpan: + coordinateSpace: comment-stripped-tex + end: 191229 + file: energy1.tex + sourceTransform: tex2mdx.strip_comments + start: 191111 + sourceStrategy: nearest-previous-statement + via: prose + sourceStatement: + chapter: 18 + file: energy1.tex + label: focussummary + mtref: '18.84' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 189625 + sourceTransform: tex2mdx.strip_comments + start: 188975 + targetStatement: + chapter: 18 + file: energy1.tex + label: Dxacross + mtref: '18.74' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 173839 + sourceTransform: tex2mdx.strip_comments + start: 173679 + type: prose-mention-nearest-previous-statement + via: + - prose +generator: tools/poincare_tex_extract.py +inference: weak +kind: mention +ref: +- 8abfaea5aa9a +- 0d6f74e37efb +rel: mentions +reviewStatus: candidate +scope: prose +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- prose +--- +Morgan--Tian prose mention. The source is the nearest preceding statement in the same chapter segment. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/206021b52889.md b/projects/poincare-conjecture/.astrolabe/edges/206021b52889.md new file mode 100644 index 00000000..4a2f1769 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/206021b52889.md @@ -0,0 +1,56 @@ +--- +confidence: 0.85 +edgeClass: semantic +evidence: + containmentReason: target theorem-like statement occurs inside source proof + proofSpan: + bodyEnd: 229182 + bodyStart: 185422 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 229193 + file: surgery.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 185409 + sourceStatement: + chapter: 17 + file: surgery.tex + label: extend + mtref: '17.1' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 185406 + sourceTransform: tex2mdx.strip_comments + start: 184285 + targetStatement: + chapter: 17 + file: surgery.tex + label: '' + mtref: '17.6' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 208075 + sourceTransform: tex2mdx.strip_comments + start: 207857 + type: proof-contained-theorem-like-statement + via: + - proof-contained-statement +generator: tools/poincare_tex_extract.py +inference: inferred +kind: dependency +ref: +- 254e8dae97e6 +- 8de850960905 +rel: depends +reviewStatus: accepted +scope: proof-containment +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof-contained-statement +--- +Morgan--Tian proof-contained statement dependency. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/20615a6047be.md b/projects/poincare-conjecture/.astrolabe/edges/20615a6047be.md new file mode 100644 index 00000000..d2e39ff3 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/20615a6047be.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 11 + file: singlimit2.tex + label: '' + mtref: '11.35' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 67188 + sourceTransform: tex2mdx.strip_comments + start: 66746 + targetStatement: + chapter: 11 + file: singlimit2.tex + label: hmonotone + mtref: '11.36' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 75122 + sourceTransform: tex2mdx.strip_comments + start: 74827 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 29f2350a588a +- ea6d9419271a +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/2067e63ca3e9.md b/projects/poincare-conjecture/.astrolabe/edges/2067e63ca3e9.md new file mode 100644 index 00000000..7344d714 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/2067e63ca3e9.md @@ -0,0 +1,67 @@ +--- +confidence: 0.72 +edgeClass: semantic +evidence: + anaphoraTriggers: + - context: proof + direction: previous + phrase: previous lemma + proofSpan: + bodyEnd: 80069 + bodyStart: 80009 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 80080 + file: newcompar.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 79996 + sort: lemma + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 80067 + file: newcompar.tex + sourceTransform: tex2mdx.strip_comments + start: 80053 + via: local-anaphora-proof + sourceStatement: + chapter: 6 + file: newcompar.tex + label: Ueqn + mtref: '6.50' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 79994 + sourceTransform: tex2mdx.strip_comments + start: 79183 + targetStatement: + chapter: 6 + file: newcompar.tex + label: linequal + mtref: '6.49' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 78401 + sourceTransform: tex2mdx.strip_comments + start: 77391 + type: local-anaphora-reference + via: + - local-anaphora-proof +generator: tools/poincare_tex_extract.py +inference: inferred +kind: dependency +ref: +- 4917e81d7684 +- 2b8ab39364e7 +rel: depends +reviewStatus: accepted +scope: local-context +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- local-anaphora-proof +--- +Morgan--Tian local anaphora dependency. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/206bf24ed50b.md b/projects/poincare-conjecture/.astrolabe/edges/206bf24ed50b.md new file mode 100644 index 00000000..2605024c --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/206bf24ed50b.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 3 + file: flowbasics.tex + label: '' + mtref: '3.1' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 943 + sourceTransform: tex2mdx.strip_comments + start: 265 + targetStatement: + chapter: 3 + file: flowbasics.tex + label: defnRicci + mtref: '3.2' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 3015 + sourceTransform: tex2mdx.strip_comments + start: 1649 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 7e9caf7ba4da +- 95eb1bef8937 +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/207f2144a4af.md b/projects/poincare-conjecture/.astrolabe/edges/207f2144a4af.md new file mode 100644 index 00000000..f99e9189 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/207f2144a4af.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 15 + file: surgery.tex + label: '' + mtref: '15.6' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 79252 + sourceTransform: tex2mdx.strip_comments + start: 79163 + targetStatement: + chapter: 15 + file: surgery.tex + label: surgeryparams + mtref: '15.7' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 80864 + sourceTransform: tex2mdx.strip_comments + start: 79853 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 1f2e2e449b34 +- 22c40a5664ac +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/208f8f1e1056.md b/projects/poincare-conjecture/.astrolabe/edges/208f8f1e1056.md new file mode 100644 index 00000000..1f952d39 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/208f8f1e1056.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 5 + file: converge2.tex + label: partialflowlimit + mtref: '5.14' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 25227 + sourceTransform: tex2mdx.strip_comments + start: 23995 + targetStatement: + chapter: 5 + file: converge2.tex + label: flowlimit + mtref: '5.15' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 27817 + sourceTransform: tex2mdx.strip_comments + start: 26565 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 4673eb9f4424 +- a0212d0eefcf +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/20a6f0133561.md b/projects/poincare-conjecture/.astrolabe/edges/20a6f0133561.md new file mode 100644 index 00000000..30cdb605 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/20a6f0133561.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 1 + file: prelim.tex + label: '' + mtref: '1.12' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 15646 + sourceTransform: tex2mdx.strip_comments + start: 15459 + targetStatement: + chapter: 1 + file: prelim.tex + label: '' + mtref: '1.13' + sort: example + span: + coordinateSpace: comment-stripped-tex + end: 15924 + sourceTransform: tex2mdx.strip_comments + start: 15648 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 76958d4968e1 +- 35089ff7c79e +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/20a977f26cc2.md b/projects/poincare-conjecture/.astrolabe/edges/20a977f26cc2.md new file mode 100644 index 00000000..3f4fafe2 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/20a977f26cc2.md @@ -0,0 +1,97 @@ +--- +confidence: 0.35 +edgeClass: prose +evidence: + mentionTriggers: + - label: asympGSS + refSpan: + coordinateSpace: comment-stripped-tex + end: 62046 + file: temp2kappa.tex + sourceTransform: tex2mdx.strip_comments + start: 62032 + sentence: At this point, setting $f=l_\infty$, we have established all the results + claimed in Theorem~\ref{asympGSS} except for the fact that the limit is not + flat + sentenceSpan: + coordinateSpace: comment-stripped-tex + end: 62093 + file: temp2kappa.tex + sourceTransform: tex2mdx.strip_comments + start: 61939 + sourceStrategy: nearest-previous-statement + via: prose + - label: asympGSS + refSpan: + coordinateSpace: comment-stripped-tex + end: 62206 + file: temp2kappa.tex + sourceTransform: tex2mdx.strip_comments + start: 62192 + sentence: \subsection{Completion of the proof of Theorem~\protect{\ref{asympGSS}}} + sentenceSpan: + coordinateSpace: comment-stripped-tex + end: 62208 + file: temp2kappa.tex + sourceTransform: tex2mdx.strip_comments + start: 62135 + sourceStrategy: nearest-previous-statement + via: prose + - label: asympGSS + refSpan: + coordinateSpace: comment-stripped-tex + end: 62258 + file: temp2kappa.tex + sourceTransform: tex2mdx.strip_comments + start: 62244 + sentence: To complete the proof of Theorem~\ref{asympGSS} it remains to show that + for no $t\in (-\infty,0)$ is $(M_\infty,g_\infty(t))$ flat. + sentenceSpan: + coordinateSpace: comment-stripped-tex + end: 62342 + file: temp2kappa.tex + sourceTransform: tex2mdx.strip_comments + start: 62209 + sourceStrategy: nearest-previous-statement + via: prose + sourceStatement: + chapter: 9 + file: temp2kappa.tex + label: '' + mtref: '9.35' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 60315 + sourceTransform: tex2mdx.strip_comments + start: 60102 + targetStatement: + chapter: 9 + file: temp2kappa.tex + label: asympGSS + mtref: '9.11' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 13767 + sourceTransform: tex2mdx.strip_comments + start: 12464 + type: prose-mention-nearest-previous-statement + via: + - prose +generator: tools/poincare_tex_extract.py +inference: weak +kind: mention +ref: +- e8f087a9cbc6 +- 671402eb454b +rel: mentions +reviewStatus: candidate +scope: prose +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- prose +--- +Morgan--Tian prose mention. The source is the nearest preceding statement in the same chapter segment. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/20e544d295ab.md b/projects/poincare-conjecture/.astrolabe/edges/20e544d295ab.md new file mode 100644 index 00000000..b88b39be --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/20e544d295ab.md @@ -0,0 +1,65 @@ +--- +confidence: 0.45 +edgeClass: semantic +evidence: + endpointSemanticDegreeBeforeTermBackfill: + source: 0 + target: 3 + selectionReason: semantic-connectivity-backfill + sourceMatchSpan: + coordinateSpace: comment-stripped-tex + end: 157231 + file: temp2kappa.tex + sourceTransform: tex2mdx.strip_comments + start: 157209 + sourceStatement: + chapter: 9 + file: temp2kappa.tex + label: '' + mtref: '9.82' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 157267 + sourceTransform: tex2mdx.strip_comments + start: 157017 + targetStatement: + chapter: 9 + file: temp2kappa.tex + label: defncanonnbhd + mtref: '9.78' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 147589 + sourceTransform: tex2mdx.strip_comments + start: 145644 + targetTermSpan: + coordinateSpace: comment-stripped-tex + end: 146790 + file: temp2kappa.tex + sourceTransform: tex2mdx.strip_comments + start: 146768 + term: strong neck + termAmbiguity: unique-in-chapter + termSource: italic + termSpecificity: no-longer-defined-term + type: same-chapter-definition-term-match + via: + - definition-term:strong neck +generator: tools/poincare_tex_extract.py +inference: weak +kind: definition-use +ref: +- 1138b028cc69 +- 3aa5883ab629 +rel: uses +reviewStatus: candidate +scope: same-chapter-term-match +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- definition-term:strong neck +--- +Morgan--Tian same-chapter definition term match. This is a low-confidence candidate edge for review. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/210979986632.md b/projects/poincare-conjecture/.astrolabe/edges/210979986632.md new file mode 100644 index 00000000..3a8de93d --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/210979986632.md @@ -0,0 +1,45 @@ +--- +confidence: 0.2 +edgeClass: structural +evidence: + sourceStatement: + chapter: 9 + file: temp2kappa.tex + label: weakeqn + mtref: '9.20' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 25064 + sourceTransform: tex2mdx.strip_comments + start: 24639 + targetStatement: + chapter: 9 + file: temp2kappa.tex + label: nablaconv + mtref: '9.21' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 25890 + sourceTransform: tex2mdx.strip_comments + start: 25631 + type: chapter-reading-order-adjacency + via: + - chapter-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 2e9e6c89f40c +- 366e3fdadf6f +rel: in-chapter +reviewStatus: accepted +scope: chapter +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- chapter-sequence +--- +Morgan--Tian reading-order adjacency across sections in a chapter. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/212609e88fde.md b/projects/poincare-conjecture/.astrolabe/edges/212609e88fde.md new file mode 100644 index 00000000..32b0401c --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/212609e88fde.md @@ -0,0 +1,65 @@ +--- +confidence: 0.45 +edgeClass: semantic +evidence: + endpointSemanticDegreeBeforeTermBackfill: + source: 0 + target: 2 + selectionReason: semantic-connectivity-backfill + sourceMatchSpan: + coordinateSpace: comment-stripped-tex + end: 71749 + file: flowbasics.tex + sourceTransform: tex2mdx.strip_comments + start: 71727 + sourceStatement: + chapter: 3 + file: flowbasics.tex + label: '' + mtref: '3.41' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 72509 + sourceTransform: tex2mdx.strip_comments + start: 71686 + targetStatement: + chapter: 3 + file: flowbasics.tex + label: '' + mtref: '3.36' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 68154 + sourceTransform: tex2mdx.strip_comments + start: 67865 + targetTermSpan: + coordinateSpace: comment-stripped-tex + end: 67924 + file: flowbasics.tex + sourceTransform: tex2mdx.strip_comments + start: 67902 + term: generalized ricci flow + termAmbiguity: unique-in-chapter + termSource: italic + termSpecificity: no-longer-defined-term + type: same-chapter-definition-term-match + via: + - definition-term:generalized ricci flow +generator: tools/poincare_tex_extract.py +inference: weak +kind: definition-use +ref: +- 0f8fc616f97c +- 7048ed283c84 +rel: uses +reviewStatus: candidate +scope: same-chapter-term-match +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- definition-term:generalized ricci flow +--- +Morgan--Tian same-chapter definition term match. This is a low-confidence candidate edge for review. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/2142d2cb999f.md b/projects/poincare-conjecture/.astrolabe/edges/2142d2cb999f.md new file mode 100644 index 00000000..7bf665e6 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/2142d2cb999f.md @@ -0,0 +1,65 @@ +--- +confidence: 0.45 +edgeClass: semantic +evidence: + endpointSemanticDegreeBeforeTermBackfill: + source: 0 + target: 1 + selectionReason: semantic-connectivity-backfill + sourceMatchSpan: + coordinateSpace: comment-stripped-tex + end: 14378 + file: prelim.tex + sourceTransform: tex2mdx.strip_comments + start: 14359 + sourceStatement: + chapter: 1 + file: prelim.tex + label: '' + mtref: '1.11' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 14976 + sourceTransform: tex2mdx.strip_comments + start: 14241 + targetStatement: + chapter: 1 + file: prelim.tex + label: '' + mtref: '1.1' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 1437 + sourceTransform: tex2mdx.strip_comments + start: 797 + targetTermSpan: + coordinateSpace: comment-stripped-tex + end: 1362 + file: prelim.tex + sourceTransform: tex2mdx.strip_comments + start: 1343 + term: riemannian manifold + termAmbiguity: unique-in-chapter + termSource: italic + termSpecificity: no-longer-defined-term + type: same-chapter-definition-term-match + via: + - definition-term:riemannian manifold +generator: tools/poincare_tex_extract.py +inference: weak +kind: definition-use +ref: +- 5982e4029637 +- 0c6b379e7dca +rel: uses +reviewStatus: candidate +scope: same-chapter-term-match +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- definition-term:riemannian manifold +--- +Morgan--Tian same-chapter definition term match. This is a low-confidence candidate edge for review. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/2173784df0f3.md b/projects/poincare-conjecture/.astrolabe/edges/2173784df0f3.md new file mode 100644 index 00000000..ce9a0aa4 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/2173784df0f3.md @@ -0,0 +1,45 @@ +--- +confidence: 0.2 +edgeClass: structural +evidence: + sourceStatement: + chapter: 9 + file: temp2kappa.tex + label: compact2D + mtref: '9.40' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 68764 + sourceTransform: tex2mdx.strip_comments + start: 68635 + targetStatement: + chapter: 9 + file: temp2kappa.tex + label: '' + mtref: '9.41' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 74254 + sourceTransform: tex2mdx.strip_comments + start: 73978 + type: chapter-reading-order-adjacency + via: + - chapter-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 18ee66757f32 +- 2104e6f56e28 +rel: in-chapter +reviewStatus: accepted +scope: chapter +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- chapter-sequence +--- +Morgan--Tian reading-order adjacency across sections in a chapter. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/2181b95860b0.md b/projects/poincare-conjecture/.astrolabe/edges/2181b95860b0.md new file mode 100644 index 00000000..b5f47f59 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/2181b95860b0.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: 1stcurvshr + proofSpan: + bodyEnd: 89507 + bodyStart: 88828 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 89518 + file: energy1.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 88815 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 88985 + file: energy1.tex + sourceTransform: tex2mdx.strip_comments + start: 88969 + via: proof + sourceStatement: + chapter: 18 + file: energy1.tex + label: rampcomp + mtref: '18.39' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 88812 + sourceTransform: tex2mdx.strip_comments + start: 88619 + targetStatement: + chapter: 18 + file: energy1.tex + label: 1stcurvshr + mtref: '18.34' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 78816 + sourceTransform: tex2mdx.strip_comments + start: 78146 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 2f115406780a +- 4fc24fa1d31e +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/21a4eee62146.md b/projects/poincare-conjecture/.astrolabe/edges/21a4eee62146.md new file mode 100644 index 00000000..5e6777b8 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/21a4eee62146.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 0 + file: intro.tex + label: corequiv + mtref: '0.5' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 14939 + sourceTransform: tex2mdx.strip_comments + start: 13757 + targetStatement: + chapter: 0 + file: intro.tex + label: '' + mtref: '0.6' + sort: remark + span: + coordinateSpace: comment-stripped-tex + end: 19243 + sourceTransform: tex2mdx.strip_comments + start: 16436 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 7078952cb9e4 +- 48f3d2b5dea3 +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/21b1ca2035bf.md b/projects/poincare-conjecture/.astrolabe/edges/21b1ca2035bf.md new file mode 100644 index 00000000..60a49b38 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/21b1ca2035bf.md @@ -0,0 +1,56 @@ +--- +confidence: 0.85 +edgeClass: semantic +evidence: + containmentReason: target theorem-like statement occurs inside source proof + proofSpan: + bodyEnd: 129463 + bodyStart: 113176 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 129474 + file: surgery.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 113163 + sourceStatement: + chapter: 16 + file: surgery.tex + label: altern + mtref: '16.5' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 112999 + sourceTransform: tex2mdx.strip_comments + start: 110961 + targetStatement: + chapter: 16 + file: surgery.tex + label: '' + mtref: '16.6' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 114511 + sourceTransform: tex2mdx.strip_comments + start: 113887 + type: proof-contained-theorem-like-statement + via: + - proof-contained-statement +generator: tools/poincare_tex_extract.py +inference: inferred +kind: dependency +ref: +- ad4f67ea75bc +- ef94451b5785 +rel: depends +reviewStatus: accepted +scope: proof-containment +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof-contained-statement +--- +Morgan--Tian proof-contained statement dependency. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/21b1f9e58fc1.md b/projects/poincare-conjecture/.astrolabe/edges/21b1f9e58fc1.md new file mode 100644 index 00000000..bf09aa52 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/21b1f9e58fc1.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 3 + file: flowbasics.tex + label: '' + mtref: '3.38' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 70309 + sourceTransform: tex2mdx.strip_comments + start: 68864 + targetStatement: + chapter: 3 + file: flowbasics.tex + label: '' + mtref: '3.39' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 70990 + sourceTransform: tex2mdx.strip_comments + start: 70312 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- faf16f1a175d +- b5e6e97be37d +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/22160b2faba2.md b/projects/poincare-conjecture/.astrolabe/edges/22160b2faba2.md new file mode 100644 index 00000000..7ef16e1f --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/22160b2faba2.md @@ -0,0 +1,55 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: statement + label: newkappa0r0t0 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 80213 + file: surgery.tex + sourceTransform: tex2mdx.strip_comments + start: 80194 + via: statement + sourceStatement: + chapter: 15 + file: surgery.tex + label: surgeryparams + mtref: '15.7' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 80864 + sourceTransform: tex2mdx.strip_comments + start: 79853 + targetStatement: + chapter: 15 + file: surgery.tex + label: newkappa0r0t0 + mtref: '15.1' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 60765 + sourceTransform: tex2mdx.strip_comments + start: 60324 + type: tex-reference + via: + - statement +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 22c40a5664ac +- f5e180ecfd18 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- statement +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/222f368b6d01.md b/projects/poincare-conjecture/.astrolabe/edges/222f368b6d01.md new file mode 100644 index 00000000..46922185 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/222f368b6d01.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 9 + file: temp2kappa.tex + label: epscap + mtref: '9.72' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 142452 + sourceTransform: tex2mdx.strip_comments + start: 140833 + targetStatement: + chapter: 9 + file: temp2kappa.tex + label: '' + mtref: '9.73' + sort: remark + span: + coordinateSpace: comment-stripped-tex + end: 143057 + sourceTransform: tex2mdx.strip_comments + start: 142454 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 757e83881e62 +- 5ab50584f917 +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/223ad038fc74.md b/projects/poincare-conjecture/.astrolabe/edges/223ad038fc74.md new file mode 100644 index 00000000..07a9c57c --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/223ad038fc74.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: '15.17' + proofSpan: + bodyEnd: 158767 + bodyStart: 158110 + coordinateSpace: comment-stripped-tex + depth: 2 + end: 158778 + file: surgery.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 158097 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 158765 + file: surgery.tex + sourceTransform: tex2mdx.strip_comments + start: 158754 + via: proof + sourceStatement: + chapter: 16 + file: surgery.tex + label: '' + mtref: '16.23' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 158095 + sourceTransform: tex2mdx.strip_comments + start: 157903 + targetStatement: + chapter: 16 + file: surgery.tex + label: '15.17' + mtref: '16.19' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 149253 + sourceTransform: tex2mdx.strip_comments + start: 148262 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 5ba0ec2c68db +- 5ed7c639bd97 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/2261ba764a20.md b/projects/poincare-conjecture/.astrolabe/edges/2261ba764a20.md new file mode 100644 index 00000000..5549057e --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/2261ba764a20.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: corI.8.3 + proofSpan: + bodyEnd: 21238 + bodyStart: 20263 + coordinateSpace: comment-stripped-tex + depth: 4 + end: 21249 + file: singlimit2.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 20250 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 21066 + file: singlimit2.tex + sourceTransform: tex2mdx.strip_comments + start: 21052 + via: proof + sourceStatement: + chapter: 11 + file: singlimit2.tex + label: distch + mtref: '11.12' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 20248 + sourceTransform: tex2mdx.strip_comments + start: 20027 + targetStatement: + chapter: 3 + file: flowbasics.tex + label: corI.8.3 + mtref: '3.26' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 37763 + sourceTransform: tex2mdx.strip_comments + start: 37390 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 002f3496c54c +- 01a829e22cf6 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/2269f7c6606a.md b/projects/poincare-conjecture/.astrolabe/edges/2269f7c6606a.md new file mode 100644 index 00000000..f93b850a --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/2269f7c6606a.md @@ -0,0 +1,45 @@ +--- +confidence: 0.2 +edgeClass: structural +evidence: + sourceStatement: + chapter: 16 + file: surgery.tex + label: bardelta1 + mtref: '16.24' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 164841 + sourceTransform: tex2mdx.strip_comments + start: 164269 + targetStatement: + chapter: 16 + file: surgery.tex + label: '' + mtref: '16.25' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 167879 + sourceTransform: tex2mdx.strip_comments + start: 167677 + type: chapter-reading-order-adjacency + via: + - chapter-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- ae8837c505cc +- fc5c13ed2e67 +rel: in-chapter +reviewStatus: accepted +scope: chapter +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- chapter-sequence +--- +Morgan--Tian reading-order adjacency across sections in a chapter. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/227c86401495.md b/projects/poincare-conjecture/.astrolabe/edges/227c86401495.md new file mode 100644 index 00000000..be595828 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/227c86401495.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: 4times + proofSpan: + bodyEnd: 126183 + bodyStart: 124278 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 126194 + file: temp2kappa.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 124265 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 124751 + file: temp2kappa.tex + sourceTransform: tex2mdx.strip_comments + start: 124739 + via: proof + sourceStatement: + chapter: 9 + file: temp2kappa.tex + label: bdvolbdcurv + mtref: '9.62' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 124263 + sourceTransform: tex2mdx.strip_comments + start: 123826 + targetStatement: + chapter: 9 + file: temp2kappa.tex + label: 4times + mtref: '9.37' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 65333 + sourceTransform: tex2mdx.strip_comments + start: 64727 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 3227c39e1892 +- 351728d82639 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/2290d013e2d3.md b/projects/poincare-conjecture/.astrolabe/edges/2290d013e2d3.md new file mode 100644 index 00000000..7c26a7ba --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/2290d013e2d3.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: nullspace + proofSpan: + bodyEnd: 82074 + bodyStart: 81084 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 82085 + file: temp2kappa.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 81071 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 81160 + file: temp2kappa.tex + sourceTransform: tex2mdx.strip_comments + start: 81145 + via: proof + sourceStatement: + chapter: 9 + file: temp2kappa.tex + label: flatssplit + mtref: '9.45' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 81068 + sourceTransform: tex2mdx.strip_comments + start: 80573 + targetStatement: + chapter: 4 + file: maxprin.tex + label: nullspace + mtref: '4.20' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 31663 + sourceTransform: tex2mdx.strip_comments + start: 30958 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- afc86a351f0c +- d21a675e4ea8 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/22afa20d99f7.md b/projects/poincare-conjecture/.astrolabe/edges/22afa20d99f7.md new file mode 100644 index 00000000..198c9408 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/22afa20d99f7.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: uniformc + proofSpan: + bodyEnd: 104957 + bodyStart: 104054 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 104968 + file: energy1.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 104041 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 104955 + file: energy1.tex + sourceTransform: tex2mdx.strip_comments + start: 104941 + via: proof + sourceStatement: + chapter: 18 + file: energy1.tex + label: Gammaapprox + mtref: '18.49' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 104039 + sourceTransform: tex2mdx.strip_comments + start: 103290 + targetStatement: + chapter: 18 + file: energy1.tex + label: uniformc + mtref: '18.48' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 102679 + sourceTransform: tex2mdx.strip_comments + start: 102426 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 7a8f55be175e +- 028d3b5182b5 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/22cb09fcca84.md b/projects/poincare-conjecture/.astrolabe/edges/22cb09fcca84.md new file mode 100644 index 00000000..a594677c --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/22cb09fcca84.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: pinch + proofSpan: + bodyEnd: 54015 + bodyStart: 53800 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 54026 + file: converge2.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 53787 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 53833 + file: converge2.tex + sourceTransform: tex2mdx.strip_comments + start: 53822 + via: proof + sourceStatement: + chapter: 5 + file: converge2.tex + label: '' + mtref: '5.34' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 53784 + sourceTransform: tex2mdx.strip_comments + start: 53414 + targetStatement: + chapter: 4 + file: maxprin.tex + label: pinch + mtref: '4.26' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 38776 + sourceTransform: tex2mdx.strip_comments + start: 38259 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- e67bd2c19008 +- 92827a66b5dd +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/22d1249264cc.md b/projects/poincare-conjecture/.astrolabe/edges/22d1249264cc.md new file mode 100644 index 00000000..1b1b49af --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/22d1249264cc.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 10 + file: bddcurvbdddist.tex + label: '' + mtref: '10.13' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 37224 + sourceTransform: tex2mdx.strip_comments + start: 37094 + targetStatement: + chapter: 10 + file: bddcurvbdddist.tex + label: '' + mtref: '10.14' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 38669 + sourceTransform: tex2mdx.strip_comments + start: 38296 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- ffb86c646793 +- 8d3abe3790d7 +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/23585a80610b.md b/projects/poincare-conjecture/.astrolabe/edges/23585a80610b.md new file mode 100644 index 00000000..d5845faa --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/23585a80610b.md @@ -0,0 +1,63 @@ +--- +confidence: 0.35 +edgeClass: prose +evidence: + mentionTriggers: + - label: lncurvgrowth + refSpan: + coordinateSpace: comment-stripped-tex + end: 112638 + file: energy1.tex + sourceTransform: tex2mdx.strip_comments + start: 112620 + sentence: Hence, by Corollary~\ref{lncurvgrowth} the same is true for $\widetilde\Gamma_c^\lambda(t)$ + for all $c\in S^2$ and all $t\in [t_0,t_1]$. + sentenceSpan: + coordinateSpace: comment-stripped-tex + end: 112736 + file: energy1.tex + sourceTransform: tex2mdx.strip_comments + start: 112599 + sourceStrategy: nearest-previous-statement + via: prose + sourceStatement: + chapter: 18 + file: energy1.tex + label: '' + mtref: '18.50' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 107575 + sourceTransform: tex2mdx.strip_comments + start: 107348 + targetStatement: + chapter: 18 + file: energy1.tex + label: lncurvgrowth + mtref: '18.38' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 87232 + sourceTransform: tex2mdx.strip_comments + start: 86861 + type: prose-mention-nearest-previous-statement + via: + - prose +generator: tools/poincare_tex_extract.py +inference: weak +kind: mention +ref: +- cb69804a2166 +- 156a3da68fbd +rel: mentions +reviewStatus: candidate +scope: prose +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- prose +--- +Morgan--Tian prose mention. The source is the nearest preceding statement in the same chapter segment. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/2364337e7d41.md b/projects/poincare-conjecture/.astrolabe/edges/2364337e7d41.md new file mode 100644 index 00000000..3b3bc36a --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/2364337e7d41.md @@ -0,0 +1,45 @@ +--- +confidence: 0.2 +edgeClass: structural +evidence: + sourceStatement: + chapter: 5 + file: converge2.tex + label: 2ndmfdconv + mtref: '5.10' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 18602 + sourceTransform: tex2mdx.strip_comments + start: 18149 + targetStatement: + chapter: 5 + file: converge2.tex + label: sliceconv + mtref: '5.11' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 21468 + sourceTransform: tex2mdx.strip_comments + start: 20456 + type: chapter-reading-order-adjacency + via: + - chapter-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- b45765a3070e +- d9f3522dec12 +rel: in-chapter +reviewStatus: accepted +scope: chapter +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- chapter-sequence +--- +Morgan--Tian reading-order adjacency across sections in a chapter. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/238d085855d7.md b/projects/poincare-conjecture/.astrolabe/edges/238d085855d7.md new file mode 100644 index 00000000..4b31356e --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/238d085855d7.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: epstopology + proofSpan: + bodyEnd: 52184 + bodyStart: 51256 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 52195 + file: singlimit2.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 51243 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 52182 + file: singlimit2.tex + sourceTransform: tex2mdx.strip_comments + start: 52165 + via: proof + sourceStatement: + chapter: 11 + file: singlimit2.tex + label: '' + mtref: '11.28' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 51239 + sourceTransform: tex2mdx.strip_comments + start: 50230 + targetStatement: + chapter: 19 + file: canonnbhd.tex + label: epstopology + mtref: '19.25' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 54460 + sourceTransform: tex2mdx.strip_comments + start: 53603 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 2a16041cf547 +- fa8d9a43f34a +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/2411aab2ca6a.md b/projects/poincare-conjecture/.astrolabe/edges/2411aab2ca6a.md new file mode 100644 index 00000000..5a77559d --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/2411aab2ca6a.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 12 + file: stdsoln.tex + label: defn:stinmetr + mtref: '12.1' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 1454 + sourceTransform: tex2mdx.strip_comments + start: 657 + targetStatement: + chapter: 12 + file: stdsoln.tex + label: stdinit + mtref: '12.2' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 2328 + sourceTransform: tex2mdx.strip_comments + start: 2256 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- c5e007547ff3 +- f1d3021d32b8 +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/2433b54351ad.md b/projects/poincare-conjecture/.astrolabe/edges/2433b54351ad.md new file mode 100644 index 00000000..68a9f628 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/2433b54351ad.md @@ -0,0 +1,62 @@ +--- +confidence: 0.9 +edgeClass: structural +evidence: + containmentDepth: 1 + proofSpan: + bodyEnd: 54967 + bodyStart: 37263 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 54978 + file: energy1.tex + ownerLabel: W_2 + sourceTransform: tex2mdx.strip_comments + start: 37250 + sourceStatement: + chapter: 18 + file: energy1.tex + label: W_2 + mtref: '18.11' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 36484 + sourceTransform: tex2mdx.strip_comments + start: 36186 + targetContainedSpan: + coordinateSpace: comment-stripped-tex + end: 46028 + file: energy1.tex + sourceTransform: tex2mdx.strip_comments + start: 45155 + targetStatement: + chapter: 18 + file: energy1.tex + label: '' + mtref: '18.15' + sort: remark + span: + coordinateSpace: comment-stripped-tex + end: 46028 + sourceTransform: tex2mdx.strip_comments + start: 45155 + type: proof-containment + via: + - proof-contains +generator: tools/poincare_tex_extract.py +inference: structural +kind: containment +ref: +- 7bb0fa0c9412 +- e800ffebd2ce +rel: contains +reviewStatus: accepted +scope: proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof-contains +--- +Morgan--Tian proof containment. The source is the statement whose proof contains the target statement. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/24374e8f52cd.md b/projects/poincare-conjecture/.astrolabe/edges/24374e8f52cd.md new file mode 100644 index 00000000..4cf0d090 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/24374e8f52cd.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 11 + file: singlimit2.tex + label: assumptions + mtref: '11.18' + sort: assumption + span: + coordinateSpace: comment-stripped-tex + end: 36515 + sourceTransform: tex2mdx.strip_comments + start: 35753 + targetStatement: + chapter: 11 + file: singlimit2.tex + label: omega + mtref: '11.19' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 38153 + sourceTransform: tex2mdx.strip_comments + start: 36599 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 3009be0a37fc +- af00774dbaa9 +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/24409a4d6ad7.md b/projects/poincare-conjecture/.astrolabe/edges/24409a4d6ad7.md new file mode 100644 index 00000000..bc7727ae --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/24409a4d6ad7.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: sliceconv + proofSpan: + bodyEnd: 28420 + bodyStart: 27833 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 28431 + file: converge2.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 27820 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 27860 + file: converge2.tex + sourceTransform: tex2mdx.strip_comments + start: 27845 + via: proof + sourceStatement: + chapter: 5 + file: converge2.tex + label: flowlimit + mtref: '5.15' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 27817 + sourceTransform: tex2mdx.strip_comments + start: 26565 + targetStatement: + chapter: 5 + file: converge2.tex + label: sliceconv + mtref: '5.11' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 21468 + sourceTransform: tex2mdx.strip_comments + start: 20456 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- a0212d0eefcf +- d9f3522dec12 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/24588848297a.md b/projects/poincare-conjecture/.astrolabe/edges/24588848297a.md new file mode 100644 index 00000000..b688f0c0 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/24588848297a.md @@ -0,0 +1,82 @@ +--- +confidence: 0.72 +edgeClass: semantic +evidence: + anaphoraTriggers: + - context: statement + direction: previous + phrase: previous lemma + sort: lemma + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 119774 + file: surgery.tex + sourceTransform: tex2mdx.strip_comments + start: 119760 + via: local-anaphora-statement + - context: proof + direction: previous + phrase: 'previous + + lemma' + proofSpan: + bodyEnd: 122171 + bodyStart: 120127 + coordinateSpace: comment-stripped-tex + depth: 2 + end: 122182 + file: surgery.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 120114 + sort: lemma + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 120257 + file: surgery.tex + sourceTransform: tex2mdx.strip_comments + start: 120243 + via: local-anaphora-proof + sourceStatement: + chapter: 16 + file: surgery.tex + label: goodtotheta + mtref: '16.9' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 120111 + sourceTransform: tex2mdx.strip_comments + start: 119688 + targetStatement: + chapter: 16 + file: surgery.tex + label: '' + mtref: '16.8' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 117471 + sourceTransform: tex2mdx.strip_comments + start: 116211 + type: local-anaphora-reference + via: + - local-anaphora-proof + - local-anaphora-statement +generator: tools/poincare_tex_extract.py +inference: inferred +kind: dependency +ref: +- 712124d7ea02 +- 4b6ee3b7a329 +rel: depends +reviewStatus: accepted +scope: local-context +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- local-anaphora-proof +- local-anaphora-statement +--- +Morgan--Tian local anaphora dependency. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/2473a5ef1b0a.md b/projects/poincare-conjecture/.astrolabe/edges/2473a5ef1b0a.md new file mode 100644 index 00000000..42edefa5 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/2473a5ef1b0a.md @@ -0,0 +1,45 @@ +--- +confidence: 0.2 +edgeClass: structural +evidence: + sourceStatement: + chapter: 18 + file: energy1.tex + label: psiclaim + mtref: '18.94' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 216766 + sourceTransform: tex2mdx.strip_comments + start: 216425 + targetStatement: + chapter: 18 + file: energy1.tex + label: '' + mtref: '18.95' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 227168 + sourceTransform: tex2mdx.strip_comments + start: 227049 + type: chapter-reading-order-adjacency + via: + - chapter-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- d3f828726e2b +- 25dda5cb31e0 +rel: in-chapter +reviewStatus: accepted +scope: chapter +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- chapter-sequence +--- +Morgan--Tian reading-order adjacency across sections in a chapter. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/24dc5a555c62.md b/projects/poincare-conjecture/.astrolabe/edges/24dc5a555c62.md new file mode 100644 index 00000000..03f01394 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/24dc5a555c62.md @@ -0,0 +1,64 @@ +--- +confidence: 0.35 +edgeClass: prose +evidence: + mentionTriggers: + - label: item5 + refSpan: + coordinateSpace: comment-stripped-tex + end: 79875 + file: stdsoln.tex + sourceTransform: tex2mdx.strip_comments + start: 79864 + sentence: It follows from Lemma~\ref{item5} and the decay estimate in Proposition~\ref{harmapflow} + that $\hat{g}_1(t)$ and $\hat{g}_2(t)$ are bounded solutions and that they have + same sequential asymptotic behavior at infinity + sentenceSpan: + coordinateSpace: comment-stripped-tex + end: 80058 + file: stdsoln.tex + sourceTransform: tex2mdx.strip_comments + start: 79841 + sourceStrategy: nearest-previous-statement + via: prose + sourceStatement: + chapter: 12 + file: stdsoln.tex + label: RDuniq + mtref: '12.26' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 75403 + sourceTransform: tex2mdx.strip_comments + start: 74267 + targetStatement: + chapter: 12 + file: stdsoln.tex + label: item5 + mtref: '12.7' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 18988 + sourceTransform: tex2mdx.strip_comments + start: 18657 + type: prose-mention-nearest-previous-statement + via: + - prose +generator: tools/poincare_tex_extract.py +inference: weak +kind: mention +ref: +- c5e225b067f1 +- 77ca4669ffb3 +rel: mentions +reviewStatus: candidate +scope: prose +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- prose +--- +Morgan--Tian prose mention. The source is the nearest preceding statement in the same chapter segment. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/24e1dc4316a5.md b/projects/poincare-conjecture/.astrolabe/edges/24e1dc4316a5.md new file mode 100644 index 00000000..6208eb57 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/24e1dc4316a5.md @@ -0,0 +1,62 @@ +--- +confidence: 0.9 +edgeClass: structural +evidence: + containmentDepth: 1 + proofSpan: + bodyEnd: 53194 + bodyStart: 43311 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 53205 + file: flowbasics.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 43298 + sourceStatement: + chapter: 3 + file: flowbasics.tex + label: Cinftybd + mtref: '3.31' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 43134 + sourceTransform: tex2mdx.strip_comments + start: 42648 + targetContainedSpan: + coordinateSpace: comment-stripped-tex + end: 47994 + file: flowbasics.tex + sourceTransform: tex2mdx.strip_comments + start: 47039 + targetStatement: + chapter: 3 + file: flowbasics.tex + label: etaprop + mtref: '3.33' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 47994 + sourceTransform: tex2mdx.strip_comments + start: 47039 + type: proof-containment + via: + - proof-contains +generator: tools/poincare_tex_extract.py +inference: structural +kind: containment +ref: +- c7a0e64aa85c +- 81f8de5102de +rel: contains +reviewStatus: accepted +scope: proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof-contains +--- +Morgan--Tian proof containment. The source is the statement whose proof contains the target statement. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/24f4f2e0e0dc.md b/projects/poincare-conjecture/.astrolabe/edges/24f4f2e0e0dc.md new file mode 100644 index 00000000..e3d0dee5 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/24f4f2e0e0dc.md @@ -0,0 +1,70 @@ +--- +confidence: 0.35 +edgeClass: prose +evidence: + mentionTriggers: + - label: fordiffquot + refSpan: + coordinateSpace: comment-stripped-tex + end: 10304 + file: stdsoln.tex + sourceTransform: tex2mdx.strip_comments + start: 10287 + sentence: For each $n$, let $(S^3_{R_n},g_{R_n}(t)),\ 0\le t0$ sufficiently small such that for any $0<\epsilon\le + \epsilon_0$ all the results of the Appendix hold for $2\epsilon$ and $\alpha=10^{-2}$, + and Proposition~\ref{narrows} holds for $2\epsilon$. + sentenceSpan: + coordinateSpace: comment-stripped-tex + end: 1974 + file: bddcurvbdddist.tex + sourceTransform: tex2mdx.strip_comments + start: 1755 + sourceStrategy: nearest-previous-statement + via: prose + sourceStatement: + chapter: 10 + file: bddcurvbdddist.tex + label: pinchdef + mtref: '10.1' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 1390 + sourceTransform: tex2mdx.strip_comments + start: 647 + targetStatement: + chapter: 2 + file: prelim.tex + label: narrows + mtref: '2.19' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 76622 + sourceTransform: tex2mdx.strip_comments + start: 76377 + type: prose-mention-nearest-previous-statement + via: + - prose +generator: tools/poincare_tex_extract.py +inference: weak +kind: mention +ref: +- 7464980c4085 +- cb9ca6d6db9f +rel: mentions +reviewStatus: candidate +scope: prose +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- prose +--- +Morgan--Tian prose mention. The source is the nearest preceding statement in the same chapter segment. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/38dd17721cc2.md b/projects/poincare-conjecture/.astrolabe/edges/38dd17721cc2.md new file mode 100644 index 00000000..a561e17c --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/38dd17721cc2.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: topsplit + proofSpan: + bodyEnd: 169692 + bodyStart: 163171 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 169703 + file: temp2kappa.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 163158 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 168167 + file: temp2kappa.tex + sourceTransform: tex2mdx.strip_comments + start: 168153 + via: proof + sourceStatement: + chapter: 9 + file: temp2kappa.tex + label: '' + mtref: '9.86' + sort: remark + span: + coordinateSpace: comment-stripped-tex + end: 163155 + sourceTransform: tex2mdx.strip_comments + start: 163012 + targetStatement: + chapter: 5 + file: converge2.tex + label: topsplit + mtref: '5.35' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 55387 + sourceTransform: tex2mdx.strip_comments + start: 54529 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 51da06702a9f +- e9a9172d16e2 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/3909753f121f.md b/projects/poincare-conjecture/.astrolabe/edges/3909753f121f.md new file mode 100644 index 00000000..34aa0b19 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/3909753f121f.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 2 + file: prelim.tex + label: soul + mtref: '2.7' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 57159 + sourceTransform: tex2mdx.strip_comments + start: 56544 + targetStatement: + chapter: 2 + file: prelim.tex + label: '' + mtref: '2.8' + sort: remark + span: + coordinateSpace: comment-stripped-tex + end: 57370 + sourceTransform: tex2mdx.strip_comments + start: 57162 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 2b5f18292e62 +- 8e408e4f9091 +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/3927ecd341e7.md b/projects/poincare-conjecture/.astrolabe/edges/3927ecd341e7.md new file mode 100644 index 00000000..76810c2e --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/3927ecd341e7.md @@ -0,0 +1,84 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: canonvary + proofSpan: + bodyEnd: 182446 + bodyStart: 180935 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 182457 + file: temp2kappa.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 180922 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 181945 + file: temp2kappa.tex + sourceTransform: tex2mdx.strip_comments + start: 181930 + via: proof + - context: proof + label: canonvary + proofSpan: + bodyEnd: 182446 + bodyStart: 180935 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 182457 + file: temp2kappa.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 180922 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 182332 + file: temp2kappa.tex + sourceTransform: tex2mdx.strip_comments + start: 182317 + via: proof + sourceStatement: + chapter: 9 + file: temp2kappa.tex + label: limitcannbhd + mtref: '9.95' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 180919 + sourceTransform: tex2mdx.strip_comments + start: 180281 + targetStatement: + chapter: 9 + file: temp2kappa.tex + label: canonvary + mtref: '9.79' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 149236 + sourceTransform: tex2mdx.strip_comments + start: 147728 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- e724298e5bc3 +- 7c2758f0e2b3 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/393597df93d0.md b/projects/poincare-conjecture/.astrolabe/edges/393597df93d0.md new file mode 100644 index 00000000..ac47d51a --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/393597df93d0.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: Ric>0 + proofSpan: + bodyEnd: 30823 + bodyStart: 25720 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 30834 + file: maxprin.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 25707 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 26325 + file: maxprin.tex + sourceTransform: tex2mdx.strip_comments + start: 26314 + via: proof + sourceStatement: + chapter: 4 + file: maxprin.tex + label: localprod + mtref: '4.19' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 25705 + sourceTransform: tex2mdx.strip_comments + start: 25065 + targetStatement: + chapter: 4 + file: maxprin.tex + label: Ric>0 + mtref: '4.15' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 18295 + sourceTransform: tex2mdx.strip_comments + start: 18076 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 8ec855a2aab5 +- f8afc1c61977 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/393753609467.md b/projects/poincare-conjecture/.astrolabe/edges/393753609467.md new file mode 100644 index 00000000..29486f91 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/393753609467.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 10 + file: bddcurvbdddist.tex + label: secondstatment + mtref: '10.29' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 53510 + sourceTransform: tex2mdx.strip_comments + start: 52640 + targetStatement: + chapter: 10 + file: bddcurvbdddist.tex + label: '' + mtref: '10.30' + sort: remark + span: + coordinateSpace: comment-stripped-tex + end: 58650 + sourceTransform: tex2mdx.strip_comments + start: 58353 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- b34d555cd7b5 +- c397ced906ac +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/393b400013fc.md b/projects/poincare-conjecture/.astrolabe/edges/393b400013fc.md new file mode 100644 index 00000000..ef5b4dbb --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/393b400013fc.md @@ -0,0 +1,84 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: areachange + proofSpan: + bodyEnd: 136538 + bodyStart: 123471 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 136549 + file: energy1.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 123458 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 129355 + file: energy1.tex + sourceTransform: tex2mdx.strip_comments + start: 129339 + via: proof + - context: proof + label: areachange + proofSpan: + bodyEnd: 136538 + bodyStart: 123471 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 136549 + file: energy1.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 123458 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 135477 + file: energy1.tex + sourceTransform: tex2mdx.strip_comments + start: 135461 + via: proof + sourceStatement: + chapter: 18 + file: energy1.tex + label: xistronger + mtref: '18.53' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 123454 + sourceTransform: tex2mdx.strip_comments + start: 122771 + targetStatement: + chapter: 18 + file: energy1.tex + label: areachange + mtref: '18.51' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 113141 + sourceTransform: tex2mdx.strip_comments + start: 112738 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 9df99ae6f4cb +- 52123e3c5a27 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/395b857e02f9.md b/projects/poincare-conjecture/.astrolabe/edges/395b857e02f9.md new file mode 100644 index 00000000..da92f658 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/395b857e02f9.md @@ -0,0 +1,56 @@ +--- +confidence: 0.85 +edgeClass: semantic +evidence: + containmentReason: target theorem-like statement occurs inside source proof + proofSpan: + bodyEnd: 129463 + bodyStart: 113176 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 129474 + file: surgery.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 113163 + sourceStatement: + chapter: 16 + file: surgery.tex + label: altern + mtref: '16.5' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 112999 + sourceTransform: tex2mdx.strip_comments + start: 110961 + targetStatement: + chapter: 16 + file: surgery.tex + label: '' + mtref: '16.8' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 117471 + sourceTransform: tex2mdx.strip_comments + start: 116211 + type: proof-contained-theorem-like-statement + via: + - proof-contained-statement +generator: tools/poincare_tex_extract.py +inference: inferred +kind: dependency +ref: +- ad4f67ea75bc +- 4b6ee3b7a329 +rel: depends +reviewStatus: accepted +scope: proof-containment +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof-contained-statement +--- +Morgan--Tian proof-contained statement dependency. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/3977286834a3.md b/projects/poincare-conjecture/.astrolabe/edges/3977286834a3.md new file mode 100644 index 00000000..56d705ec --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/3977286834a3.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: EL + proofSpan: + bodyEnd: 13637 + bodyStart: 12589 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 13648 + file: newcompar.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 12576 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 12675 + file: newcompar.tex + sourceTransform: tex2mdx.strip_comments + start: 12667 + via: proof + sourceStatement: + chapter: 6 + file: newcompar.tex + label: '' + mtref: '6.10' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 12573 + sourceTransform: tex2mdx.strip_comments + start: 11870 + targetStatement: + chapter: 6 + file: newcompar.tex + label: EL + mtref: '6.4' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 6659 + sourceTransform: tex2mdx.strip_comments + start: 6364 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- deb9b83d83f2 +- 75695d1b1b54 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/399fac434235.md b/projects/poincare-conjecture/.astrolabe/edges/399fac434235.md new file mode 100644 index 00000000..8be157a1 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/399fac434235.md @@ -0,0 +1,65 @@ +--- +confidence: 0.45 +edgeClass: semantic +evidence: + endpointSemanticDegreeBeforeTermBackfill: + source: 0 + target: 6 + selectionReason: semantic-connectivity-backfill + sourceMatchSpan: + coordinateSpace: comment-stripped-tex + end: 83921 + file: newcompar.tex + sourceTransform: tex2mdx.strip_comments + start: 83913 + sourceStatement: + chapter: 6 + file: newcompar.tex + label: '' + mtref: '6.55' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 84083 + sourceTransform: tex2mdx.strip_comments + start: 83631 + targetStatement: + chapter: 6 + file: newcompar.tex + label: '' + mtref: '6.7' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 10481 + sourceTransform: tex2mdx.strip_comments + start: 10161 + targetTermSpan: + coordinateSpace: comment-stripped-tex + end: 10318 + file: newcompar.tex + sourceTransform: tex2mdx.strip_comments + start: 10310 + term: geodesic + termAmbiguity: unique-in-chapter + termSource: italic + termSpecificity: not-covered-by-longer-defined-term + type: same-chapter-definition-term-match + via: + - definition-term:geodesic +generator: tools/poincare_tex_extract.py +inference: weak +kind: definition-use +ref: +- 27c072f30ccf +- d23dc29f2aae +rel: uses +reviewStatus: candidate +scope: same-chapter-term-match +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- definition-term:geodesic +--- +Morgan--Tian same-chapter definition term match. This is a low-confidence candidate edge for review. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/39a56dc5d6b2.md b/projects/poincare-conjecture/.astrolabe/edges/39a56dc5d6b2.md new file mode 100644 index 00000000..cadafe0d --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/39a56dc5d6b2.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 2 + file: prelim.tex + label: homeo + mtref: '2.10' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 60913 + sourceTransform: tex2mdx.strip_comments + start: 60407 + targetStatement: + chapter: 2 + file: prelim.tex + label: htpytype + mtref: '2.11' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 63443 + sourceTransform: tex2mdx.strip_comments + start: 63143 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 64a0f82323d3 +- 68624c3d26bd +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/39ca3a081e3a.md b/projects/poincare-conjecture/.astrolabe/edges/39ca3a081e3a.md new file mode 100644 index 00000000..6903b7ba --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/39ca3a081e3a.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 3 + file: flowbasics.tex + label: geosuff + mtref: '3.23' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 28871 + sourceTransform: tex2mdx.strip_comments + start: 28258 + targetStatement: + chapter: 3 + file: flowbasics.tex + label: '3.23' + mtref: '3.24' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 29260 + sourceTransform: tex2mdx.strip_comments + start: 29003 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 0f76a0750b40 +- a43d37f014e8 +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/39cbb505e1d6.md b/projects/poincare-conjecture/.astrolabe/edges/39cbb505e1d6.md new file mode 100644 index 00000000..a51d1d33 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/39cbb505e1d6.md @@ -0,0 +1,84 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: S2intersection + proofSpan: + bodyEnd: 49745 + bodyStart: 44181 + coordinateSpace: comment-stripped-tex + depth: 2 + end: 49756 + file: canonnbhd.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 44168 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 47135 + file: canonnbhd.tex + sourceTransform: tex2mdx.strip_comments + start: 47115 + via: proof + - context: proof + label: S2intersection + proofSpan: + bodyEnd: 49745 + bodyStart: 44181 + coordinateSpace: comment-stripped-tex + depth: 2 + end: 49756 + file: canonnbhd.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 44168 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 48357 + file: canonnbhd.tex + sourceTransform: tex2mdx.strip_comments + start: 48337 + via: proof + sourceStatement: + chapter: 19 + file: canonnbhd.tex + label: cccomponent + mtref: '19.23' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 44166 + sourceTransform: tex2mdx.strip_comments + start: 43962 + targetStatement: + chapter: 19 + file: canonnbhd.tex + label: S2intersection + mtref: '19.11' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 19482 + sourceTransform: tex2mdx.strip_comments + start: 17256 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 327067f8535b +- a46d72b87e07 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/3a54be7beaf7.md b/projects/poincare-conjecture/.astrolabe/edges/3a54be7beaf7.md new file mode 100644 index 00000000..577a9e80 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/3a54be7beaf7.md @@ -0,0 +1,57 @@ +--- +confidence: 0.72 +edgeClass: semantic +evidence: + anaphoraTriggers: + - context: statement + direction: previous + phrase: previous definition + sort: definition + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 43506 + file: bddcurvbdddist.tex + sourceTransform: tex2mdx.strip_comments + start: 43487 + via: local-anaphora-statement + sourceStatement: + chapter: 10 + file: bddcurvbdddist.tex + label: thetaconv + mtref: '10.21' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 44375 + sourceTransform: tex2mdx.strip_comments + start: 43422 + targetStatement: + chapter: 10 + file: bddcurvbdddist.tex + label: defntheta + mtref: '10.20' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 43420 + sourceTransform: tex2mdx.strip_comments + start: 42950 + type: local-anaphora-reference + via: + - local-anaphora-statement +generator: tools/poincare_tex_extract.py +inference: inferred +kind: dependency +ref: +- 5b34d0c84cbb +- f4ee075394ef +rel: depends +reviewStatus: accepted +scope: local-context +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- local-anaphora-statement +--- +Morgan--Tian local anaphora dependency. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/3a713a70380c.md b/projects/poincare-conjecture/.astrolabe/edges/3a713a70380c.md new file mode 100644 index 00000000..85a53a2e --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/3a713a70380c.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: 1stcurvshr + proofSpan: + bodyEnd: 202245 + bodyStart: 200903 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 202256 + file: energy1.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 200890 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 201412 + file: energy1.tex + sourceTransform: tex2mdx.strip_comments + start: 201396 + via: proof + sourceStatement: + chapter: 18 + file: energy1.tex + label: '1887' + mtref: '18.88' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 200887 + sourceTransform: tex2mdx.strip_comments + start: 200520 + targetStatement: + chapter: 18 + file: energy1.tex + label: 1stcurvshr + mtref: '18.34' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 78816 + sourceTransform: tex2mdx.strip_comments + start: 78146 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 80e5fc6759b2 +- 4fc24fa1d31e +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/3a9373c94dc2.md b/projects/poincare-conjecture/.astrolabe/edges/3a9373c94dc2.md new file mode 100644 index 00000000..2d078dc4 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/3a9373c94dc2.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 10 + file: bddcurvbdddist.tex + label: '' + mtref: '10.9' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 31429 + sourceTransform: tex2mdx.strip_comments + start: 31297 + targetStatement: + chapter: 10 + file: bddcurvbdddist.tex + label: '' + mtref: '10.10' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 32796 + sourceTransform: tex2mdx.strip_comments + start: 32551 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- d9ac883eac42 +- 1418d3c286c7 +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/3aad648dd462.md b/projects/poincare-conjecture/.astrolabe/edges/3aad648dd462.md new file mode 100644 index 00000000..cc79cad9 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/3aad648dd462.md @@ -0,0 +1,85 @@ +--- +confidence: 0.35 +edgeClass: prose +evidence: + mentionTriggers: + - label: narrows + refSpan: + coordinateSpace: comment-stripped-tex + end: 63918 + file: surgery.tex + sourceTransform: tex2mdx.strip_comments + start: 63905 + sentence: We fix $0<\epsilon\le {\rm min}(1/200,\left(\sqrt{D}(A_0+5)\right)^{-1},\bar\epsilon_1/2,\bar + \epsilon'/2,\epsilon_0)$ where $\bar\epsilon_1$ is the constant from Proposition~\ref{narrows}, + $\bar\epsilon'$ is the constant from Theorem~\ref{kappasummary}, $\epsilon_0$ + is the constant from Section~\ref{10.1}, and $A_0$ and $D$ are the constants + from Lemma~\ref{A_0}. We fix $\beta<1/2$, the constant from Proposition~\ref{neckglue} + sentenceSpan: + coordinateSpace: comment-stripped-tex + end: 64162 + file: surgery.tex + sourceTransform: tex2mdx.strip_comments + start: 63728 + sourceStrategy: nearest-previous-statement + via: prose + - label: narrows + refSpan: + coordinateSpace: comment-stripped-tex + end: 64420 + file: surgery.tex + sourceTransform: tex2mdx.strip_comments + start: 64407 + sentence: For all such $\epsilon$, Theorem~\ref{bcbd} holds for $\epsilon$ and + Proposition~\ref{narrows}, Proposition~\ref{canonvary} and Corollaries~\ref{kappacannbhd} + and~\ref{limitcannbhd} and Theorems~\ref{smlmtflow} and~\ref{kaplimit} hold + for $2\epsilon$ + sentenceSpan: + coordinateSpace: comment-stripped-tex + end: 64577 + file: surgery.tex + sourceTransform: tex2mdx.strip_comments + start: 64325 + sourceStrategy: nearest-previous-statement + via: prose + sourceStatement: + chapter: 15 + file: surgery.tex + label: neckglue + mtref: '15.2' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 61465 + sourceTransform: tex2mdx.strip_comments + start: 60812 + targetStatement: + chapter: 2 + file: prelim.tex + label: narrows + mtref: '2.19' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 76622 + sourceTransform: tex2mdx.strip_comments + start: 76377 + type: prose-mention-nearest-previous-statement + via: + - prose +generator: tools/poincare_tex_extract.py +inference: weak +kind: mention +ref: +- a992d6a89479 +- cb9ca6d6db9f +rel: mentions +reviewStatus: candidate +scope: prose +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- prose +--- +Morgan--Tian prose mention. The source is the nearest preceding statement in the same chapter segment. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/3ad40ad9079b.md b/projects/poincare-conjecture/.astrolabe/edges/3ad40ad9079b.md new file mode 100644 index 00000000..d3fac24a --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/3ad40ad9079b.md @@ -0,0 +1,56 @@ +--- +confidence: 0.85 +edgeClass: semantic +evidence: + containmentReason: target theorem-like statement occurs inside source proof + proofSpan: + bodyEnd: 164255 + bodyStart: 154729 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 164266 + file: surgery.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 154716 + sourceStatement: + chapter: 16 + file: surgery.tex + label: Lgeoexist + mtref: '16.21' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 154713 + sourceTransform: tex2mdx.strip_comments + start: 152982 + targetStatement: + chapter: 16 + file: surgery.tex + label: '' + mtref: '16.22' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 156009 + sourceTransform: tex2mdx.strip_comments + start: 155858 + type: proof-contained-theorem-like-statement + via: + - proof-contained-statement +generator: tools/poincare_tex_extract.py +inference: inferred +kind: dependency +ref: +- d98ad25d9bc3 +- cfb464e3cf35 +rel: depends +reviewStatus: accepted +scope: proof-containment +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof-contained-statement +--- +Morgan--Tian proof-contained statement dependency. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/3ae9be7f5975.md b/projects/poincare-conjecture/.astrolabe/edges/3ae9be7f5975.md new file mode 100644 index 00000000..efdb7aa3 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/3ae9be7f5975.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 3 + file: flowbasics.tex + label: '' + mtref: '3.5' + sort: example + span: + coordinateSpace: comment-stripped-tex + end: 5230 + sourceTransform: tex2mdx.strip_comments + start: 4831 + targetStatement: + chapter: 3 + file: flowbasics.tex + label: strongepsneck + mtref: '3.6' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 6733 + sourceTransform: tex2mdx.strip_comments + start: 5816 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 78d9cbe9be13 +- 6d6227668950 +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/3b2753255bf8.md b/projects/poincare-conjecture/.astrolabe/edges/3b2753255bf8.md new file mode 100644 index 00000000..3227c942 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/3b2753255bf8.md @@ -0,0 +1,45 @@ +--- +confidence: 0.2 +edgeClass: structural +evidence: + sourceStatement: + chapter: 19 + file: canonnbhd.tex + label: S2times01 + mtref: '19.20' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 33333 + sourceTransform: tex2mdx.strip_comments + start: 32976 + targetStatement: + chapter: 19 + file: canonnbhd.tex + label: Xcontainedin + mtref: '19.21' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 35375 + sourceTransform: tex2mdx.strip_comments + start: 34343 + type: chapter-reading-order-adjacency + via: + - chapter-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- aa8a682b4513 +- 0e9a1a167caa +rel: in-chapter +reviewStatus: accepted +scope: chapter +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- chapter-sequence +--- +Morgan--Tian reading-order adjacency across sections in a chapter. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/3b3e212002f6.md b/projects/poincare-conjecture/.astrolabe/edges/3b3e212002f6.md new file mode 100644 index 00000000..dd7ac2a3 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/3b3e212002f6.md @@ -0,0 +1,65 @@ +--- +confidence: 0.45 +edgeClass: semantic +evidence: + endpointSemanticDegreeBeforeTermBackfill: + source: 0 + target: 3 + selectionReason: semantic-connectivity-backfill + sourceMatchSpan: + coordinateSpace: comment-stripped-tex + end: 33396 + file: prelim.tex + sourceTransform: tex2mdx.strip_comments + start: 33388 + sourceStatement: + chapter: 1 + file: prelim.tex + label: '' + mtref: '1.24' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 33683 + sourceTransform: tex2mdx.strip_comments + start: 33241 + targetStatement: + chapter: 1 + file: prelim.tex + label: '' + mtref: '1.17' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 19309 + sourceTransform: tex2mdx.strip_comments + start: 19124 + targetTermSpan: + coordinateSpace: comment-stripped-tex + end: 19240 + file: prelim.tex + sourceTransform: tex2mdx.strip_comments + start: 19232 + term: geodesic + termAmbiguity: unique-in-chapter + termSource: italic + termSpecificity: no-longer-defined-term + type: same-chapter-definition-term-match + via: + - definition-term:geodesic +generator: tools/poincare_tex_extract.py +inference: weak +kind: definition-use +ref: +- 19afab2e3115 +- c4ccdd484637 +rel: uses +reviewStatus: candidate +scope: same-chapter-term-match +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- definition-term:geodesic +--- +Morgan--Tian same-chapter definition term match. This is a low-confidence candidate edge for review. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/3b47e88e204c.md b/projects/poincare-conjecture/.astrolabe/edges/3b47e88e204c.md new file mode 100644 index 00000000..60364c54 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/3b47e88e204c.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: bcbd + proofSpan: + bodyEnd: 222542 + bodyStart: 220786 + coordinateSpace: comment-stripped-tex + depth: 2 + end: 222553 + file: surgery.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 220773 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 221690 + file: surgery.tex + sourceTransform: tex2mdx.strip_comments + start: 221680 + via: proof + sourceStatement: + chapter: 17 + file: surgery.tex + label: D(A)delta(A) + mtref: '17.9' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 220771 + sourceTransform: tex2mdx.strip_comments + start: 220446 + targetStatement: + chapter: 10 + file: bddcurvbdddist.tex + label: bcbd + mtref: '10.2' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 2942 + sourceTransform: tex2mdx.strip_comments + start: 2168 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 7549bace9302 +- ecd62fc2ea2b +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/3b492c309e67.md b/projects/poincare-conjecture/.astrolabe/edges/3b492c309e67.md new file mode 100644 index 00000000..af88be67 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/3b492c309e67.md @@ -0,0 +1,45 @@ +--- +confidence: 0.2 +edgeClass: structural +evidence: + sourceStatement: + chapter: 0 + file: intro.tex + label: '' + mtref: '0.7' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 48115 + sourceTransform: tex2mdx.strip_comments + start: 47473 + targetStatement: + chapter: 0 + file: intro.tex + label: '' + mtref: '0.8' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 60281 + sourceTransform: tex2mdx.strip_comments + start: 60167 + type: chapter-reading-order-adjacency + via: + - chapter-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 56d435df5c6d +- 42089b0a1d9e +rel: in-chapter +reviewStatus: accepted +scope: chapter +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- chapter-sequence +--- +Morgan--Tian reading-order adjacency across sections in a chapter. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/3b62b9d4cf63.md b/projects/poincare-conjecture/.astrolabe/edges/3b62b9d4cf63.md new file mode 100644 index 00000000..d1e90484 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/3b62b9d4cf63.md @@ -0,0 +1,103 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: unianc + proofSpan: + bodyEnd: 41607 + bodyStart: 36687 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 41618 + file: temp2kappa.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 36674 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 38420 + file: temp2kappa.tex + sourceTransform: tex2mdx.strip_comments + start: 38408 + via: proof + - context: proof + label: unianc + proofSpan: + bodyEnd: 41607 + bodyStart: 36687 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 41618 + file: temp2kappa.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 36674 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 39107 + file: temp2kappa.tex + sourceTransform: tex2mdx.strip_comments + start: 39095 + via: proof + - context: proof + label: unianc + proofSpan: + bodyEnd: 41607 + bodyStart: 36687 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 41618 + file: temp2kappa.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 36674 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 39635 + file: temp2kappa.tex + sourceTransform: tex2mdx.strip_comments + start: 39623 + via: proof + sourceStatement: + chapter: 9 + file: temp2kappa.tex + label: lowerbdd + mtref: '9.25' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 36672 + sourceTransform: tex2mdx.strip_comments + start: 36453 + targetStatement: + chapter: 9 + file: temp2kappa.tex + label: unianc + mtref: '9.10' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 10335 + sourceTransform: tex2mdx.strip_comments + start: 9624 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- a6ee3b2ab615 +- 8b8e3a0f3e36 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/3b64e09e5886.md b/projects/poincare-conjecture/.astrolabe/edges/3b64e09e5886.md new file mode 100644 index 00000000..e6d95335 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/3b64e09e5886.md @@ -0,0 +1,66 @@ +--- +confidence: 0.35 +edgeClass: prose +evidence: + mentionTriggers: + - label: 4pi + refSpan: + coordinateSpace: comment-stripped-tex + end: 16673 + file: temp2kappa.tex + sourceTransform: tex2mdx.strip_comments + start: 16664 + sentence: By Theorem~\ref{4pi} the reduced volume function $\widetilde V_x(\tau)$ + is a non-increasing function of $\tau$ with ${\rm lim}_{\tau\rightarrow 0}\widetilde + V_x(\tau)=(4\pi)^{\frac{n}{2}}$. Since the integrand for $\widetilde V_x(\tau)$ + is everywhere positive, it is clear that $\widetilde V_x(\tau)>0$ for all $\tau$. + Hence, ${\rm lim}_{\tau\rightarrow\infty}\widetilde V_x(\tau)$ exists + sentenceSpan: + coordinateSpace: comment-stripped-tex + end: 17041 + file: temp2kappa.tex + sourceTransform: tex2mdx.strip_comments + start: 16652 + sourceStrategy: nearest-previous-statement + via: prose + sourceStatement: + chapter: 9 + file: temp2kappa.tex + label: scaleinv2 + mtref: '9.13' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 16522 + sourceTransform: tex2mdx.strip_comments + start: 16053 + targetStatement: + chapter: 7 + file: newcomp2.tex + label: 4pi + mtref: '7.26' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 45413 + sourceTransform: tex2mdx.strip_comments + start: 44820 + type: prose-mention-nearest-previous-statement + via: + - prose +generator: tools/poincare_tex_extract.py +inference: weak +kind: mention +ref: +- dbf6a3c0bae7 +- ae6f1c6d00d9 +rel: mentions +reviewStatus: candidate +scope: prose +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- prose +--- +Morgan--Tian prose mention. The source is the nearest preceding statement in the same chapter segment. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/3b715125b1f8.md b/projects/poincare-conjecture/.astrolabe/edges/3b715125b1f8.md new file mode 100644 index 00000000..02ee5b1e --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/3b715125b1f8.md @@ -0,0 +1,45 @@ +--- +confidence: 0.2 +edgeClass: structural +evidence: + sourceStatement: + chapter: 1 + file: prelim.tex + label: '' + mtref: '1.26' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 34361 + sourceTransform: tex2mdx.strip_comments + start: 33952 + targetStatement: + chapter: 1 + file: prelim.tex + label: '' + mtref: '1.27' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 36725 + sourceTransform: tex2mdx.strip_comments + start: 36479 + type: chapter-reading-order-adjacency + via: + - chapter-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 8ebd93835797 +- 0cf778ecc2ec +rel: in-chapter +reviewStatus: accepted +scope: chapter +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- chapter-sequence +--- +Morgan--Tian reading-order adjacency across sections in a chapter. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/3b944f27fa5d.md b/projects/poincare-conjecture/.astrolabe/edges/3b944f27fa5d.md new file mode 100644 index 00000000..9175942e --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/3b944f27fa5d.md @@ -0,0 +1,45 @@ +--- +confidence: 0.2 +edgeClass: structural +evidence: + sourceStatement: + chapter: 18 + file: energy1.tex + label: muepsbd + mtref: '18.72' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 171988 + sourceTransform: tex2mdx.strip_comments + start: 171484 + targetStatement: + chapter: 18 + file: energy1.tex + label: noloops + mtref: '18.73' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 172812 + sourceTransform: tex2mdx.strip_comments + start: 172656 + type: chapter-reading-order-adjacency + via: + - chapter-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 9c8d4163f700 +- b6fb2b9c364b +rel: in-chapter +reviewStatus: accepted +scope: chapter +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- chapter-sequence +--- +Morgan--Tian reading-order adjacency across sections in a chapter. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/3b9b4825fe13.md b/projects/poincare-conjecture/.astrolabe/edges/3b9b4825fe13.md new file mode 100644 index 00000000..030f9580 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/3b9b4825fe13.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 9 + file: temp2kappa.tex + label: uniformlip + mtref: '9.9' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 7575 + sourceTransform: tex2mdx.strip_comments + start: 6851 + targetStatement: + chapter: 9 + file: temp2kappa.tex + label: unianc + mtref: '9.10' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 10335 + sourceTransform: tex2mdx.strip_comments + start: 9624 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 9387b1b04824 +- 8b8e3a0f3e36 +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/3bb3eeabf60e.md b/projects/poincare-conjecture/.astrolabe/edges/3bb3eeabf60e.md new file mode 100644 index 00000000..57d75726 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/3bb3eeabf60e.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 6 + file: newcompar.tex + label: '' + mtref: '6.5' + sort: remark + span: + coordinateSpace: comment-stripped-tex + end: 6833 + sourceTransform: tex2mdx.strip_comments + start: 6662 + targetStatement: + chapter: 6 + file: newcompar.tex + label: '' + mtref: '6.6' + sort: remark + span: + coordinateSpace: comment-stripped-tex + end: 10159 + sourceTransform: tex2mdx.strip_comments + start: 9991 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 91d88796541d +- 388e599c4bee +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/3bc9448f7363.md b/projects/poincare-conjecture/.astrolabe/edges/3bc9448f7363.md new file mode 100644 index 00000000..987e0146 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/3bc9448f7363.md @@ -0,0 +1,84 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: altern + proofSpan: + bodyEnd: 214277 + bodyStart: 212637 + coordinateSpace: comment-stripped-tex + depth: 3 + end: 214288 + file: surgery.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 212624 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 213574 + file: surgery.tex + sourceTransform: tex2mdx.strip_comments + start: 213562 + via: proof + - context: proof + label: altern + proofSpan: + bodyEnd: 214277 + bodyStart: 212637 + coordinateSpace: comment-stripped-tex + depth: 3 + end: 214288 + file: surgery.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 212624 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 213957 + file: surgery.tex + sourceTransform: tex2mdx.strip_comments + start: 213945 + via: proof + sourceStatement: + chapter: 17 + file: surgery.tex + label: '' + mtref: '17.8' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 212622 + sourceTransform: tex2mdx.strip_comments + start: 212469 + targetStatement: + chapter: 16 + file: surgery.tex + label: altern + mtref: '16.5' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 112999 + sourceTransform: tex2mdx.strip_comments + start: 110961 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- a01b63baabde +- ad4f67ea75bc +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/3be6224b3bd0.md b/projects/poincare-conjecture/.astrolabe/edges/3be6224b3bd0.md new file mode 100644 index 00000000..28fe0db8 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/3be6224b3bd0.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: GSSclass + proofSpan: + bodyEnd: 176513 + bodyStart: 172329 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 176524 + file: temp2kappa.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 172316 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 176186 + file: temp2kappa.tex + sourceTransform: tex2mdx.strip_comments + start: 176172 + via: proof + sourceStatement: + chapter: 9 + file: temp2kappa.tex + label: thmep3 + mtref: '9.89' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 172313 + sourceTransform: tex2mdx.strip_comments + start: 171695 + targetStatement: + chapter: 9 + file: temp2kappa.tex + label: GSSclass + mtref: '9.42' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 75321 + sourceTransform: tex2mdx.strip_comments + start: 74299 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- a4ea411d0765 +- 2d68880771ff +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/3c15ca368e7f.md b/projects/poincare-conjecture/.astrolabe/edges/3c15ca368e7f.md new file mode 100644 index 00000000..5dedf0c5 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/3c15ca368e7f.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: flowtoround + proofSpan: + bodyEnd: 80505 + bodyStart: 78984 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 80516 + file: temp2kappa.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 78971 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 80040 + file: temp2kappa.tex + sourceTransform: tex2mdx.strip_comments + start: 80023 + via: proof + sourceStatement: + chapter: 9 + file: temp2kappa.tex + label: compactcase + mtref: '9.44' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 78969 + sourceTransform: tex2mdx.strip_comments + start: 78694 + targetStatement: + chapter: 4 + file: maxprin.tex + label: flowtoround + mtref: '4.23' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 35932 + sourceTransform: tex2mdx.strip_comments + start: 34642 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 01028c93347d +- 86e5a5766bda +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/3c4fa6fcdb14.md b/projects/poincare-conjecture/.astrolabe/edges/3c4fa6fcdb14.md new file mode 100644 index 00000000..46b43c68 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/3c4fa6fcdb14.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: LOCALSURGERY + proofSpan: + bodyEnd: 96236 + bodyStart: 93864 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 96247 + file: surgery.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 93851 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 95716 + file: surgery.tex + sourceTransform: tex2mdx.strip_comments + start: 95698 + via: proof + sourceStatement: + chapter: 15 + file: surgery.tex + label: '14.26' + mtref: '15.11' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 93849 + sourceTransform: tex2mdx.strip_comments + start: 93211 + targetStatement: + chapter: 13 + file: surgery.tex + label: LOCALSURGERY + mtref: '13.2' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 6442 + sourceTransform: tex2mdx.strip_comments + start: 4875 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- c8e7931dbf65 +- cf61b731c6f2 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/3c83a171d59f.md b/projects/poincare-conjecture/.astrolabe/edges/3c83a171d59f.md new file mode 100644 index 00000000..64f59633 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/3c83a171d59f.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: compactcase + proofSpan: + bodyEnd: 176513 + bodyStart: 172329 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 176524 + file: temp2kappa.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 172316 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 176115 + file: temp2kappa.tex + sourceTransform: tex2mdx.strip_comments + start: 176098 + via: proof + sourceStatement: + chapter: 9 + file: temp2kappa.tex + label: thmep3 + mtref: '9.89' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 172313 + sourceTransform: tex2mdx.strip_comments + start: 171695 + targetStatement: + chapter: 9 + file: temp2kappa.tex + label: compactcase + mtref: '9.44' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 78969 + sourceTransform: tex2mdx.strip_comments + start: 78694 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- a4ea411d0765 +- 01028c93347d +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/3ca6d7ca03da.md b/projects/poincare-conjecture/.astrolabe/edges/3ca6d7ca03da.md new file mode 100644 index 00000000..474f6658 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/3ca6d7ca03da.md @@ -0,0 +1,65 @@ +--- +confidence: 0.45 +edgeClass: semantic +evidence: + endpointSemanticDegreeBeforeTermBackfill: + source: 3 + target: 0 + selectionReason: semantic-connectivity-backfill + sourceMatchSpan: + coordinateSpace: comment-stripped-tex + end: 77169 + file: prelim.tex + sourceTransform: tex2mdx.strip_comments + start: 77151 + sourceStatement: + chapter: 2 + file: prelim.tex + label: neckseparate + mtref: '2.20' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 77469 + sourceTransform: tex2mdx.strip_comments + start: 76866 + targetStatement: + chapter: 2 + file: prelim.tex + label: epsneck + mtref: '2.18' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 76303 + sourceTransform: tex2mdx.strip_comments + start: 73778 + targetTermSpan: + coordinateSpace: comment-stripped-tex + end: 74612 + file: prelim.tex + sourceTransform: tex2mdx.strip_comments + start: 74594 + term: central sphere + termAmbiguity: unique-in-chapter + termSource: italic + termSpecificity: no-longer-defined-term + type: same-chapter-definition-term-match + via: + - definition-term:central sphere +generator: tools/poincare_tex_extract.py +inference: weak +kind: definition-use +ref: +- 818772530e5a +- d91d54ddbd1c +rel: uses +reviewStatus: candidate +scope: same-chapter-term-match +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- definition-term:central sphere +--- +Morgan--Tian same-chapter definition term match. This is a low-confidence candidate edge for review. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/3caaf6105589.md b/projects/poincare-conjecture/.astrolabe/edges/3caaf6105589.md new file mode 100644 index 00000000..8b970cb0 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/3caaf6105589.md @@ -0,0 +1,45 @@ +--- +confidence: 0.2 +edgeClass: structural +evidence: + sourceStatement: + chapter: 9 + file: temp2kappa.tex + label: '' + mtref: '9.51' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 94920 + sourceTransform: tex2mdx.strip_comments + start: 94840 + targetStatement: + chapter: 9 + file: temp2kappa.tex + label: 3Dinfty + mtref: '9.52' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 100781 + sourceTransform: tex2mdx.strip_comments + start: 100280 + type: chapter-reading-order-adjacency + via: + - chapter-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 07605ec62528 +- f8ff249efb81 +rel: in-chapter +reviewStatus: accepted +scope: chapter +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- chapter-sequence +--- +Morgan--Tian reading-order adjacency across sections in a chapter. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/3cc1e2733d83.md b/projects/poincare-conjecture/.astrolabe/edges/3cc1e2733d83.md new file mode 100644 index 00000000..e06dc037 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/3cc1e2733d83.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 13 + file: surgery.tex + label: newbasis + mtref: '13.7' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 14007 + sourceTransform: tex2mdx.strip_comments + start: 13167 + targetStatement: + chapter: 13 + file: surgery.tex + label: 2ndnewbasis + mtref: '13.8' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 15714 + sourceTransform: tex2mdx.strip_comments + start: 15256 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 52d3cb562bdc +- ad2042196c95 +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/3cc41d359e36.md b/projects/poincare-conjecture/.astrolabe/edges/3cc41d359e36.md new file mode 100644 index 00000000..4d5267ce --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/3cc41d359e36.md @@ -0,0 +1,55 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: statement + label: lformula + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 22381 + file: newcomp2.tex + sourceTransform: tex2mdx.strip_comments + start: 22367 + via: statement + sourceStatement: + chapter: 7 + file: newcomp2.tex + label: weak + mtref: '7.13' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 23409 + sourceTransform: tex2mdx.strip_comments + start: 22180 + targetStatement: + chapter: 6 + file: newcompar.tex + label: lformula + mtref: '6.51' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 81082 + sourceTransform: tex2mdx.strip_comments + start: 80194 + type: tex-reference + via: + - statement +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 7d5d94efe4e9 +- 967c0f8bb774 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- statement +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/3ce80d1f9b0d.md b/projects/poincare-conjecture/.astrolabe/edges/3ce80d1f9b0d.md new file mode 100644 index 00000000..a8588770 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/3ce80d1f9b0d.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 18 + file: energy1.tex + label: laest + mtref: '18.66' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 161338 + sourceTransform: tex2mdx.strip_comments + start: 161080 + targetStatement: + chapter: 18 + file: energy1.tex + label: theta/Kbd + mtref: '18.67' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 162289 + sourceTransform: tex2mdx.strip_comments + start: 162047 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 45c783d384c1 +- 512c74b9ad1f +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/3cf029d6ec89.md b/projects/poincare-conjecture/.astrolabe/edges/3cf029d6ec89.md new file mode 100644 index 00000000..c307fd30 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/3cf029d6ec89.md @@ -0,0 +1,162 @@ +--- +confidence: 0.35 +edgeClass: prose +evidence: + mentionTriggers: + - label: XI + refSpan: + coordinateSpace: comment-stripped-tex + end: 62285 + file: energy1.tex + sourceTransform: tex2mdx.strip_comments + start: 62277 + sentence: Now we return to the proof that Proposition~\ref{XI} implies Proposition~\ref{W_3fordiff} + sentenceSpan: + coordinateSpace: comment-stripped-tex + end: 62322 + file: energy1.tex + sourceTransform: tex2mdx.strip_comments + start: 62232 + sourceStrategy: nearest-previous-statement + via: prose + - label: XI + refSpan: + coordinateSpace: comment-stripped-tex + end: 63416 + file: energy1.tex + sourceTransform: tex2mdx.strip_comments + start: 63408 + sentence: According to Proposition~\ref{XI} there is a one-parameter family $\widetilde\Gamma(t),\ + t_0\le t\le t_1$, of maps $S^2\to \Lambda{\mathcal X}(t)$, whose images consist + of homotopically trivial loops, with $[\widetilde\Gamma(t_0)]=[\Gamma]=\xi$ + such that for every $c\in S^2$ we have $A(\widetilde\Gamma(t_0)(c))0$ and is equal to $(4\pi)^{n/2}$ + sentenceSpan: + coordinateSpace: comment-stripped-tex + end: 47573 + file: newcomp2.tex + sourceTransform: tex2mdx.strip_comments + start: 47339 + sourceStrategy: nearest-previous-statement + via: prose + - label: flat + refSpan: + coordinateSpace: comment-stripped-tex + end: 47694 + file: newcomp2.tex + sourceTransform: tex2mdx.strip_comments + start: 47684 + sentence: In this subsection we use the monotonicity results of the last subsection + to establish the converse to Lemma~\ref{flat}, namely to show that if $(M,g(t)),0\le + t\le T$, is a Ricci flow complete with bounded curvature and if $\widetilde + V_x(M,\bar\tau)=(4\pi)^{n/2}$ for some $\bar\tau>0$ and some $x\in M\times\{T\}$, + then the flow on the interval $[T-\bar\tau,T]$ is the trivial flow on flat Euclidean + $n$-space. + sentenceSpan: + coordinateSpace: comment-stripped-tex + end: 47987 + file: newcomp2.tex + sourceTransform: tex2mdx.strip_comments + start: 47574 + sourceStrategy: nearest-previous-statement + via: prose + sourceStatement: + chapter: 7 + file: newcomp2.tex + label: 4pi + mtref: '7.26' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 45413 + sourceTransform: tex2mdx.strip_comments + start: 44820 + targetStatement: + chapter: 7 + file: newcomp2.tex + label: flat + mtref: '7.25' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 43846 + sourceTransform: tex2mdx.strip_comments + start: 43634 + type: prose-mention-nearest-previous-statement + via: + - prose +generator: tools/poincare_tex_extract.py +inference: weak +kind: mention +ref: +- ae6f1c6d00d9 +- 4a4c8aae3a8d +rel: mentions +reviewStatus: candidate +scope: prose +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- prose +--- +Morgan--Tian prose mention. The source is the nearest preceding statement in the same chapter segment. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/4f7d1cc868ce.md b/projects/poincare-conjecture/.astrolabe/edges/4f7d1cc868ce.md new file mode 100644 index 00000000..a8243a77 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/4f7d1cc868ce.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: flat + proofSpan: + bodyEnd: 51188 + bodyStart: 48388 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 51199 + file: newcomp2.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 48375 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 48459 + file: newcomp2.tex + sourceTransform: tex2mdx.strip_comments + start: 48449 + via: proof + sourceStatement: + chapter: 7 + file: newcomp2.tex + label: 4picase + mtref: '7.27' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 48373 + sourceTransform: tex2mdx.strip_comments + start: 47991 + targetStatement: + chapter: 7 + file: newcomp2.tex + label: flat + mtref: '7.25' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 43846 + sourceTransform: tex2mdx.strip_comments + start: 43634 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 613fb3994956 +- 4a4c8aae3a8d +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/4f964d76fdb1.md b/projects/poincare-conjecture/.astrolabe/edges/4f964d76fdb1.md new file mode 100644 index 00000000..79e71f84 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/4f964d76fdb1.md @@ -0,0 +1,62 @@ +--- +confidence: 0.35 +edgeClass: prose +evidence: + mentionTriggers: + - label: stdsoln + refSpan: + coordinateSpace: comment-stripped-tex + end: 90048 + file: stdsoln.tex + sourceTransform: tex2mdx.strip_comments + start: 90035 + sentence: This completes the proof of Theorem~\ref{stdsoln}. + sentenceSpan: + coordinateSpace: comment-stripped-tex + end: 90049 + file: stdsoln.tex + sourceTransform: tex2mdx.strip_comments + start: 89998 + sourceStrategy: nearest-previous-statement + via: prose + sourceStatement: + chapter: 12 + file: stdsoln.tex + label: '' + mtref: '12.30' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 88901 + sourceTransform: tex2mdx.strip_comments + start: 88738 + targetStatement: + chapter: 12 + file: stdsoln.tex + label: stdsoln + mtref: '12.5' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 8086 + sourceTransform: tex2mdx.strip_comments + start: 6763 + type: prose-mention-nearest-previous-statement + via: + - prose +generator: tools/poincare_tex_extract.py +inference: weak +kind: mention +ref: +- cf6eed7b1da4 +- f44d182ef27e +rel: mentions +reviewStatus: candidate +scope: prose +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- prose +--- +Morgan--Tian prose mention. The source is the nearest preceding statement in the same chapter segment. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/4fc058d21cf3.md b/projects/poincare-conjecture/.astrolabe/edges/4fc058d21cf3.md new file mode 100644 index 00000000..6eb19edd --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/4fc058d21cf3.md @@ -0,0 +1,63 @@ +--- +confidence: 0.35 +edgeClass: prose +evidence: + mentionTriggers: + - label: tau0cl + refSpan: + coordinateSpace: comment-stripped-tex + end: 177334 + file: surgery.tex + sourceTransform: tex2mdx.strip_comments + start: 177322 + sentence: Now fix $\tau_0$ satisfying Claim~\ref{tau0cl}. Let $\gamma_1$ be the + restriction of $\gamma$ to the subinterval $[0,\tau_0]$, and let $y=\gamma_1(\tau_0)$ + sentenceSpan: + coordinateSpace: comment-stripped-tex + end: 177444 + file: surgery.tex + sourceTransform: tex2mdx.strip_comments + start: 177287 + sourceStrategy: nearest-previous-statement + via: prose + sourceStatement: + chapter: 16 + file: surgery.tex + label: '' + mtref: '16.28' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 176062 + sourceTransform: tex2mdx.strip_comments + start: 175778 + targetStatement: + chapter: 16 + file: surgery.tex + label: tau0cl + mtref: '16.27' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 175063 + sourceTransform: tex2mdx.strip_comments + start: 174895 + type: prose-mention-nearest-previous-statement + via: + - prose +generator: tools/poincare_tex_extract.py +inference: weak +kind: mention +ref: +- 3e9580da0ff5 +- 8aac9c08c058 +rel: mentions +reviewStatus: candidate +scope: prose +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- prose +--- +Morgan--Tian prose mention. The source is the nearest preceding statement in the same chapter segment. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/50112929eda2.md b/projects/poincare-conjecture/.astrolabe/edges/50112929eda2.md new file mode 100644 index 00000000..1f20249f --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/50112929eda2.md @@ -0,0 +1,45 @@ +--- +confidence: 0.2 +edgeClass: structural +evidence: + sourceStatement: + chapter: 9 + file: temp2kappa.tex + label: '' + mtref: '9.54' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 103864 + sourceTransform: tex2mdx.strip_comments + start: 103562 + targetStatement: + chapter: 9 + file: temp2kappa.tex + label: '' + mtref: '9.55' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 106400 + sourceTransform: tex2mdx.strip_comments + start: 106063 + type: chapter-reading-order-adjacency + via: + - chapter-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 92ee3ee82b58 +- 1998da8dab3d +rel: in-chapter +reviewStatus: accepted +scope: chapter +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- chapter-sequence +--- +Morgan--Tian reading-order adjacency across sections in a chapter. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/502aa12f3433.md b/projects/poincare-conjecture/.astrolabe/edges/502aa12f3433.md new file mode 100644 index 00000000..3aea4951 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/502aa12f3433.md @@ -0,0 +1,45 @@ +--- +confidence: 0.2 +edgeClass: structural +evidence: + sourceStatement: + chapter: 6 + file: newcompar.tex + label: redvoltilde + mtref: '6.71' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 107671 + sourceTransform: tex2mdx.strip_comments + start: 107167 + targetStatement: + chapter: 6 + file: newcompar.tex + label: Lrescale + mtref: '6.72' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 109575 + sourceTransform: tex2mdx.strip_comments + start: 108645 + type: chapter-reading-order-adjacency + via: + - chapter-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- c3d4eaed00fd +- ad48b42bb0a0 +rel: in-chapter +reviewStatus: accepted +scope: chapter +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- chapter-sequence +--- +Morgan--Tian reading-order adjacency across sections in a chapter. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/5062252ef769.md b/projects/poincare-conjecture/.astrolabe/edges/5062252ef769.md new file mode 100644 index 00000000..2d445959 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/5062252ef769.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 12 + file: stdsoln.tex + label: Restim + mtref: '12.31' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 90335 + sourceTransform: tex2mdx.strip_comments + start: 90184 + targetStatement: + chapter: 12 + file: stdsoln.tex + label: stdsolncannbhd + mtref: '12.32' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 94479 + sourceTransform: tex2mdx.strip_comments + start: 93886 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- ffae4504318a +- 6ff28dba4e56 +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/50d28f4b6d49.md b/projects/poincare-conjecture/.astrolabe/edges/50d28f4b6d49.md new file mode 100644 index 00000000..8ef5e95d --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/50d28f4b6d49.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 6 + file: newcompar.tex + label: variremark + mtref: '6.15' + sort: remark + span: + coordinateSpace: comment-stripped-tex + end: 19144 + sourceTransform: tex2mdx.strip_comments + start: 18610 + targetStatement: + chapter: 6 + file: newcompar.tex + label: symmY + mtref: '6.16' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 19484 + sourceTransform: tex2mdx.strip_comments + start: 19147 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 2fef81872a01 +- abd40a465b53 +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/50fbeeefbe1f.md b/projects/poincare-conjecture/.astrolabe/edges/50fbeeefbe1f.md new file mode 100644 index 00000000..032c8bef --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/50fbeeefbe1f.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 9 + file: temp2kappa.tex + label: redvolconst + mtref: '9.14' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 17598 + sourceTransform: tex2mdx.strip_comments + start: 17342 + targetStatement: + chapter: 9 + file: temp2kappa.tex + label: tausqtau + mtref: '9.15' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 17779 + sourceTransform: tex2mdx.strip_comments + start: 17653 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- e08895f2b0d8 +- 88571a22a2bc +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/5125cf598078.md b/projects/poincare-conjecture/.astrolabe/edges/5125cf598078.md new file mode 100644 index 00000000..2aff8e09 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/5125cf598078.md @@ -0,0 +1,45 @@ +--- +confidence: 0.2 +edgeClass: structural +evidence: + sourceStatement: + chapter: 18 + file: energy1.tex + label: '' + mtref: '18.81' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 185425 + sourceTransform: tex2mdx.strip_comments + start: 185320 + targetStatement: + chapter: 18 + file: energy1.tex + label: '' + mtref: '18.82' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 187562 + sourceTransform: tex2mdx.strip_comments + start: 187208 + type: chapter-reading-order-adjacency + via: + - chapter-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- fb6be46de671 +- 9e3a849a6a4c +rel: in-chapter +reviewStatus: accepted +scope: chapter +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- chapter-sequence +--- +Morgan--Tian reading-order adjacency across sections in a chapter. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/512ecdf6724f.md b/projects/poincare-conjecture/.astrolabe/edges/512ecdf6724f.md new file mode 100644 index 00000000..9bb49b42 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/512ecdf6724f.md @@ -0,0 +1,65 @@ +--- +confidence: 0.45 +edgeClass: semantic +evidence: + endpointSemanticDegreeBeforeTermBackfill: + source: 0 + target: 1 + selectionReason: semantic-connectivity-backfill + sourceMatchSpan: + coordinateSpace: comment-stripped-tex + end: 47374 + file: surgery.tex + sourceTransform: tex2mdx.strip_comments + start: 47351 + sourceStatement: + chapter: 14 + file: surgery.tex + label: '' + mtref: '14.10' + sort: example + span: + coordinateSpace: comment-stripped-tex + end: 47779 + sourceTransform: tex2mdx.strip_comments + start: 47321 + targetStatement: + chapter: 14 + file: surgery.tex + label: '' + mtref: '14.8' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 47114 + sourceTransform: tex2mdx.strip_comments + start: 46506 + targetTermSpan: + coordinateSpace: comment-stripped-tex + end: 46550 + file: surgery.tex + sourceTransform: tex2mdx.strip_comments + start: 46527 + term: ricci flow with surgery + termAmbiguity: unique-in-chapter + termSource: italic + termSpecificity: no-longer-defined-term + type: same-chapter-definition-term-match + via: + - definition-term:ricci flow with surgery +generator: tools/poincare_tex_extract.py +inference: weak +kind: definition-use +ref: +- 7e213cc8aad9 +- 842f5a7c856b +rel: uses +reviewStatus: candidate +scope: same-chapter-term-match +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- definition-term:ricci flow with surgery +--- +Morgan--Tian same-chapter definition term match. This is a low-confidence candidate edge for review. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/5152d7393c46.md b/projects/poincare-conjecture/.astrolabe/edges/5152d7393c46.md new file mode 100644 index 00000000..0f003fa6 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/5152d7393c46.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: Revol + proofSpan: + bodyEnd: 143530 + bodyStart: 143482 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 143541 + file: temp2kappa.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 143469 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 143527 + file: temp2kappa.tex + sourceTransform: tex2mdx.strip_comments + start: 143516 + via: proof-aux-label-proxy + sourceStatement: + chapter: 9 + file: temp2kappa.tex + label: partialtR + mtref: '9.74' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 143467 + sourceTransform: tex2mdx.strip_comments + start: 143149 + targetStatement: + chapter: 3 + file: flowbasics.tex + label: thmRmevol + mtref: '3.13' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 16962 + sourceTransform: tex2mdx.strip_comments + start: 16120 + type: tex-reference + via: + - proof-aux-label-proxy +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 1280a1901e2e +- 1515151790b2 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof-aux-label-proxy +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/5194108bb177.md b/projects/poincare-conjecture/.astrolabe/edges/5194108bb177.md new file mode 100644 index 00000000..920f5570 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/5194108bb177.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: anglemono + proofSpan: + bodyEnd: 24886 + bodyStart: 23650 + coordinateSpace: comment-stripped-tex + depth: 3 + end: 24897 + file: singlimit2.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 23637 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 24342 + file: singlimit2.tex + sourceTransform: tex2mdx.strip_comments + start: 24327 + via: proof + sourceStatement: + chapter: 11 + file: singlimit2.tex + label: '' + mtref: '11.13' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 23634 + sourceTransform: tex2mdx.strip_comments + start: 23355 + targetStatement: + chapter: 2 + file: prelim.tex + label: anglemono + mtref: '2.5' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 54639 + sourceTransform: tex2mdx.strip_comments + start: 54438 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 385a31c5d191 +- c81962d31706 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/51964c757383.md b/projects/poincare-conjecture/.astrolabe/edges/51964c757383.md new file mode 100644 index 00000000..66bae58d --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/51964c757383.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 11 + file: singlimit2.tex + label: omega + mtref: '11.19' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 38153 + sourceTransform: tex2mdx.strip_comments + start: 36599 + targetStatement: + chapter: 11 + file: singlimit2.tex + label: '' + mtref: '11.20' + sort: remark + span: + coordinateSpace: comment-stripped-tex + end: 38287 + sourceTransform: tex2mdx.strip_comments + start: 38155 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- af00774dbaa9 +- d6585ba392cf +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/51ba26ac2592.md b/projects/poincare-conjecture/.astrolabe/edges/51ba26ac2592.md new file mode 100644 index 00000000..23ec25f2 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/51ba26ac2592.md @@ -0,0 +1,84 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: S2intersection + proofSpan: + bodyEnd: 19521 + bodyStart: 13193 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 19532 + file: bddcurvbdddist.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 13180 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 18917 + file: bddcurvbdddist.tex + sourceTransform: tex2mdx.strip_comments + start: 18897 + via: proof + - context: proof + label: S2intersection + proofSpan: + bodyEnd: 19521 + bodyStart: 13193 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 19532 + file: bddcurvbdddist.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 13180 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 19151 + file: bddcurvbdddist.tex + sourceTransform: tex2mdx.strip_comments + start: 19131 + via: proof + sourceStatement: + chapter: 10 + file: bddcurvbdddist.tex + label: '' + mtref: '10.4' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 13178 + sourceTransform: tex2mdx.strip_comments + start: 12275 + targetStatement: + chapter: 19 + file: canonnbhd.tex + label: S2intersection + mtref: '19.11' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 19482 + sourceTransform: tex2mdx.strip_comments + start: 17256 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 96519f84d54e +- a46d72b87e07 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/51cc47d9b202.md b/projects/poincare-conjecture/.astrolabe/edges/51cc47d9b202.md new file mode 100644 index 00000000..d1dc956e --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/51cc47d9b202.md @@ -0,0 +1,45 @@ +--- +confidence: 0.2 +edgeClass: structural +evidence: + sourceStatement: + chapter: 11 + file: singlimit2.tex + label: '' + mtref: '11.7' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 11552 + sourceTransform: tex2mdx.strip_comments + start: 11452 + targetStatement: + chapter: 11 + file: singlimit2.tex + label: kaplimit + mtref: '11.8' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 16131 + sourceTransform: tex2mdx.strip_comments + start: 15040 + type: chapter-reading-order-adjacency + via: + - chapter-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 620ff6c41ef4 +- a9e023d082dc +rel: in-chapter +reviewStatus: accepted +scope: chapter +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- chapter-sequence +--- +Morgan--Tian reading-order adjacency across sections in a chapter. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/51ce2a0352c2.md b/projects/poincare-conjecture/.astrolabe/edges/51ce2a0352c2.md new file mode 100644 index 00000000..cd610813 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/51ce2a0352c2.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: altern + proofSpan: + bodyEnd: 164255 + bodyStart: 154729 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 164266 + file: surgery.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 154716 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 155162 + file: surgery.tex + sourceTransform: tex2mdx.strip_comments + start: 155150 + via: proof + sourceStatement: + chapter: 16 + file: surgery.tex + label: Lgeoexist + mtref: '16.21' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 154713 + sourceTransform: tex2mdx.strip_comments + start: 152982 + targetStatement: + chapter: 16 + file: surgery.tex + label: altern + mtref: '16.5' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 112999 + sourceTransform: tex2mdx.strip_comments + start: 110961 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- d98ad25d9bc3 +- ad4f67ea75bc +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/51dae6c369da.md b/projects/poincare-conjecture/.astrolabe/edges/51dae6c369da.md new file mode 100644 index 00000000..6646aebb --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/51dae6c369da.md @@ -0,0 +1,56 @@ +--- +confidence: 0.85 +edgeClass: semantic +evidence: + containmentReason: target theorem-like statement occurs inside source proof + proofSpan: + bodyEnd: 53552 + bodyStart: 35551 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 53563 + file: canonnbhd.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 35538 + sourceStatement: + chapter: 19 + file: canonnbhd.tex + label: Xcontainedin + mtref: '19.21' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 35375 + sourceTransform: tex2mdx.strip_comments + start: 34343 + targetStatement: + chapter: 19 + file: canonnbhd.tex + label: '' + mtref: '19.22' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 36055 + sourceTransform: tex2mdx.strip_comments + start: 35803 + type: proof-contained-theorem-like-statement + via: + - proof-contained-statement +generator: tools/poincare_tex_extract.py +inference: inferred +kind: dependency +ref: +- 0e9a1a167caa +- 9737d037f97b +rel: depends +reviewStatus: accepted +scope: proof-containment +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof-contained-statement +--- +Morgan--Tian proof-contained statement dependency. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/51f57ac657b9.md b/projects/poincare-conjecture/.astrolabe/edges/51f57ac657b9.md new file mode 100644 index 00000000..2c0c053b --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/51f57ac657b9.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 7 + file: newcomp2.tex + label: weaksense + mtref: '7.8' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 12878 + sourceTransform: tex2mdx.strip_comments + start: 12168 + targetStatement: + chapter: 7 + file: newcomp2.tex + label: '' + mtref: '7.9' + sort: remark + span: + coordinateSpace: comment-stripped-tex + end: 13133 + sourceTransform: tex2mdx.strip_comments + start: 12880 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 1ef3a78b1911 +- 6262ca70063e +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/5200d6c3b1e6.md b/projects/poincare-conjecture/.astrolabe/edges/5200d6c3b1e6.md new file mode 100644 index 00000000..9f194afa --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/5200d6c3b1e6.md @@ -0,0 +1,45 @@ +--- +confidence: 0.2 +edgeClass: structural +evidence: + sourceStatement: + chapter: 0 + file: intro.tex + label: '' + mtref: '0.2' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 3307 + sourceTransform: tex2mdx.strip_comments + start: 3098 + targetStatement: + chapter: 0 + file: intro.tex + label: surgery + mtref: '0.3' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 11419 + sourceTransform: tex2mdx.strip_comments + start: 10567 + type: chapter-reading-order-adjacency + via: + - chapter-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- d08303132397 +- 847af3ecd4a8 +rel: in-chapter +reviewStatus: accepted +scope: chapter +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- chapter-sequence +--- +Morgan--Tian reading-order adjacency across sections in a chapter. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/5205db52d098.md b/projects/poincare-conjecture/.astrolabe/edges/5205db52d098.md new file mode 100644 index 00000000..d8203620 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/5205db52d098.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 7 + file: newcomp2.tex + label: '' + mtref: '7.7' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 12044 + sourceTransform: tex2mdx.strip_comments + start: 11683 + targetStatement: + chapter: 7 + file: newcomp2.tex + label: weaksense + mtref: '7.8' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 12878 + sourceTransform: tex2mdx.strip_comments + start: 12168 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- d50ccf232266 +- 1ef3a78b1911 +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/52606ba40a5a.md b/projects/poincare-conjecture/.astrolabe/edges/52606ba40a5a.md new file mode 100644 index 00000000..b3c4ca84 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/52606ba40a5a.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 2 + file: prelim.tex + label: '' + mtref: '2.2' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 49959 + sourceTransform: tex2mdx.strip_comments + start: 49128 + targetStatement: + chapter: 2 + file: prelim.tex + label: Blambda + mtref: '2.3' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 50562 + sourceTransform: tex2mdx.strip_comments + start: 50322 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 4d6e47f3ec09 +- 7c06c4b22b84 +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/52ab26b0d683.md b/projects/poincare-conjecture/.astrolabe/edges/52ab26b0d683.md new file mode 100644 index 00000000..ebf6b174 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/52ab26b0d683.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 5 + file: converge2.tex + label: '' + mtref: '5.27' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 40739 + sourceTransform: tex2mdx.strip_comments + start: 40490 + targetStatement: + chapter: 5 + file: converge2.tex + label: GHcompact + mtref: '5.28' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 41436 + sourceTransform: tex2mdx.strip_comments + start: 40928 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 9b970f13e747 +- 17352ed43755 +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/535e200178c1.md b/projects/poincare-conjecture/.astrolabe/edges/535e200178c1.md new file mode 100644 index 00000000..51850e7e --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/535e200178c1.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: 2DGSS + proofSpan: + bodyEnd: 103549 + bodyStart: 101913 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 103560 + file: temp2kappa.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 101900 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 102349 + file: temp2kappa.tex + sourceTransform: tex2mdx.strip_comments + start: 102338 + via: proof + sourceStatement: + chapter: 9 + file: temp2kappa.tex + label: 3DGSSkappa + mtref: '9.53' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 101898 + sourceTransform: tex2mdx.strip_comments + start: 101535 + targetStatement: + chapter: 9 + file: temp2kappa.tex + label: 2DGSS + mtref: '9.50' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 91949 + sourceTransform: tex2mdx.strip_comments + start: 91266 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 351e4ac14e4e +- da767c71996b +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/53707f681570.md b/projects/poincare-conjecture/.astrolabe/edges/53707f681570.md new file mode 100644 index 00000000..ddf93a3c --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/53707f681570.md @@ -0,0 +1,55 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: statement + label: distdecrease + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 5984 + file: surgery.tex + sourceTransform: tex2mdx.strip_comments + start: 5966 + via: statement + sourceStatement: + chapter: 13 + file: surgery.tex + label: LOCALSURGERY + mtref: '13.2' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 6442 + sourceTransform: tex2mdx.strip_comments + start: 4875 + targetStatement: + chapter: 13 + file: surgery.tex + label: distdecrease + mtref: '13.1' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 3514 + sourceTransform: tex2mdx.strip_comments + start: 3099 + type: tex-reference + via: + - statement +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- cf61b731c6f2 +- 6168ffcb3bed +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- statement +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/537437974b31.md b/projects/poincare-conjecture/.astrolabe/edges/537437974b31.md new file mode 100644 index 00000000..2ec80a59 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/537437974b31.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: bigell + proofSpan: + bodyEnd: 204939 + bodyStart: 186733 + coordinateSpace: comment-stripped-tex + depth: 2 + end: 204950 + file: surgery.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 186720 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 198772 + file: surgery.tex + sourceTransform: tex2mdx.strip_comments + start: 198760 + via: proof + sourceStatement: + chapter: 17 + file: surgery.tex + label: '16.2' + mtref: '17.2' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 186718 + sourceTransform: tex2mdx.strip_comments + start: 186200 + targetStatement: + chapter: 16 + file: surgery.tex + label: bigell + mtref: '16.13' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 131955 + sourceTransform: tex2mdx.strip_comments + start: 130235 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 778d690c4e7c +- 0a4d6f17810d +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/538670d5f4b8.md b/projects/poincare-conjecture/.astrolabe/edges/538670d5f4b8.md new file mode 100644 index 00000000..24bce241 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/538670d5f4b8.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 6 + file: newcompar.tex + label: '' + mtref: '6.55' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 84083 + sourceTransform: tex2mdx.strip_comments + start: 83631 + targetStatement: + chapter: 6 + file: newcompar.tex + label: lips + mtref: '6.56' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 85066 + sourceTransform: tex2mdx.strip_comments + start: 84130 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 27c072f30ccf +- db4ced24b05f +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/53fe66958758.md b/projects/poincare-conjecture/.astrolabe/edges/53fe66958758.md new file mode 100644 index 00000000..472dfb42 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/53fe66958758.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 18 + file: energy1.tex + label: vclaim + mtref: '18.26' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 60140 + sourceTransform: tex2mdx.strip_comments + start: 59893 + targetStatement: + chapter: 18 + file: energy1.tex + label: shortpi2triv + mtref: '18.27' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 60569 + sourceTransform: tex2mdx.strip_comments + start: 60196 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- eaf31795ca89 +- ed0e92ce8f57 +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/5429043af5e7.md b/projects/poincare-conjecture/.astrolabe/edges/5429043af5e7.md new file mode 100644 index 00000000..530aa03d --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/5429043af5e7.md @@ -0,0 +1,67 @@ +--- +confidence: 0.72 +edgeClass: semantic +evidence: + anaphoraTriggers: + - context: proof + direction: previous + phrase: previous lemma + proofSpan: + bodyEnd: 5394 + bodyStart: 3941 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 5405 + file: canonnbhd.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 3928 + sort: lemma + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 3963 + file: canonnbhd.tex + sourceTransform: tex2mdx.strip_comments + start: 3949 + via: local-anaphora-proof + sourceStatement: + chapter: 19 + file: canonnbhd.tex + label: s2isotopic + mtref: '19.3' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 3924 + sourceTransform: tex2mdx.strip_comments + start: 3327 + targetStatement: + chapter: 19 + file: canonnbhd.tex + label: neckcurv + mtref: '19.2' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 3124 + sourceTransform: tex2mdx.strip_comments + start: 2315 + type: local-anaphora-reference + via: + - local-anaphora-proof +generator: tools/poincare_tex_extract.py +inference: inferred +kind: dependency +ref: +- 98c6a95f8556 +- dc5709757703 +rel: depends +reviewStatus: accepted +scope: local-context +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- local-anaphora-proof +--- +Morgan--Tian local anaphora dependency. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/5461bb5f77f0.md b/projects/poincare-conjecture/.astrolabe/edges/5461bb5f77f0.md new file mode 100644 index 00000000..1ae032ff --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/5461bb5f77f0.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 6 + file: newcompar.tex + label: EL + mtref: '6.4' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 6659 + sourceTransform: tex2mdx.strip_comments + start: 6364 + targetStatement: + chapter: 6 + file: newcompar.tex + label: '' + mtref: '6.5' + sort: remark + span: + coordinateSpace: comment-stripped-tex + end: 6833 + sourceTransform: tex2mdx.strip_comments + start: 6662 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 75695d1b1b54 +- 91d88796541d +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/547b59d2b516.md b/projects/poincare-conjecture/.astrolabe/edges/547b59d2b516.md new file mode 100644 index 00000000..023bdacd --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/547b59d2b516.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 16 + file: surgery.tex + label: J + mtref: '16.18' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 148100 + sourceTransform: tex2mdx.strip_comments + start: 146759 + targetStatement: + chapter: 16 + file: surgery.tex + label: '15.17' + mtref: '16.19' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 149253 + sourceTransform: tex2mdx.strip_comments + start: 148262 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 0ea74aba0c14 +- 5ed7c639bd97 +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/54a401770e98.md b/projects/poincare-conjecture/.astrolabe/edges/54a401770e98.md new file mode 100644 index 00000000..e51e1c74 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/54a401770e98.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: BishopGromov + proofSpan: + bodyEnd: 20196 + bodyStart: 18617 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 20207 + file: converge2.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 18604 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 19168 + file: converge2.tex + sourceTransform: tex2mdx.strip_comments + start: 19150 + via: proof + sourceStatement: + chapter: 5 + file: converge2.tex + label: 2ndmfdconv + mtref: '5.10' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 18602 + sourceTransform: tex2mdx.strip_comments + start: 18149 + targetStatement: + chapter: 1 + file: prelim.tex + label: BishopGromov + mtref: '1.34' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 43978 + sourceTransform: tex2mdx.strip_comments + start: 43265 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- b45765a3070e +- 1a1b2c4fdc9b +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/54ae976e83da.md b/projects/poincare-conjecture/.astrolabe/edges/54ae976e83da.md new file mode 100644 index 00000000..87bad19b --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/54ae976e83da.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 9 + file: temp2kappa.tex + label: '' + mtref: '9.91' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 177332 + sourceTransform: tex2mdx.strip_comments + start: 176526 + targetStatement: + chapter: 9 + file: temp2kappa.tex + label: '' + mtref: '9.92' + sort: remark + span: + coordinateSpace: comment-stripped-tex + end: 177845 + sourceTransform: tex2mdx.strip_comments + start: 177446 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 7744a698d8da +- 744167bbcef7 +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/54cce3189224.md b/projects/poincare-conjecture/.astrolabe/edges/54cce3189224.md new file mode 100644 index 00000000..965faf42 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/54cce3189224.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: bcbd + proofSpan: + bodyEnd: 14787 + bodyStart: 6954 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 14798 + file: singlimit2.tex + ownerLabel: smlmtflow + sourceTransform: tex2mdx.strip_comments + start: 6941 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 7132 + file: singlimit2.tex + sourceTransform: tex2mdx.strip_comments + start: 7122 + via: proof + sourceStatement: + chapter: 11 + file: singlimit2.tex + label: smlmtflow + mtref: '11.1' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 3459 + sourceTransform: tex2mdx.strip_comments + start: 1498 + targetStatement: + chapter: 10 + file: bddcurvbdddist.tex + label: bcbd + mtref: '10.2' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 2942 + sourceTransform: tex2mdx.strip_comments + start: 2168 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- d51aab8a8065 +- ecd62fc2ea2b +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/553b91de1d64.md b/projects/poincare-conjecture/.astrolabe/edges/553b91de1d64.md new file mode 100644 index 00000000..1bc1a882 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/553b91de1d64.md @@ -0,0 +1,56 @@ +--- +confidence: 0.85 +edgeClass: semantic +evidence: + containmentReason: target theorem-like statement occurs inside source proof + proofSpan: + bodyEnd: 34517 + bodyStart: 26920 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 34528 + file: bddcurvbdddist.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 26907 + sourceStatement: + chapter: 10 + file: bddcurvbdddist.tex + label: Uinfty + mtref: '10.7' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 26826 + sourceTransform: tex2mdx.strip_comments + start: 25406 + targetStatement: + chapter: 10 + file: bddcurvbdddist.tex + label: '' + mtref: '10.11' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 33981 + sourceTransform: tex2mdx.strip_comments + start: 33869 + type: proof-contained-theorem-like-statement + via: + - proof-contained-statement +generator: tools/poincare_tex_extract.py +inference: inferred +kind: dependency +ref: +- e2e64cdb0b75 +- 2d6c9dcad762 +rel: depends +reviewStatus: accepted +scope: proof-containment +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof-contained-statement +--- +Morgan--Tian proof-contained statement dependency. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/555e1e75c9ad.md b/projects/poincare-conjecture/.astrolabe/edges/555e1e75c9ad.md new file mode 100644 index 00000000..5dcbc3b1 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/555e1e75c9ad.md @@ -0,0 +1,80 @@ +--- +confidence: 0.72 +edgeClass: semantic +evidence: + anaphoraTriggers: + - context: statement + direction: previous + phrase: previous claim + sort: claim + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 115715 + file: surgery.tex + sourceTransform: tex2mdx.strip_comments + start: 115701 + via: local-anaphora-statement + - context: proof + direction: previous + phrase: previous claim + proofSpan: + bodyEnd: 116198 + bodyStart: 116009 + coordinateSpace: comment-stripped-tex + depth: 2 + end: 116209 + file: surgery.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 115996 + sort: claim + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 116092 + file: surgery.tex + sourceTransform: tex2mdx.strip_comments + start: 116078 + via: local-anaphora-proof + sourceStatement: + chapter: 16 + file: surgery.tex + label: '' + mtref: '16.7' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 115994 + sourceTransform: tex2mdx.strip_comments + start: 115513 + targetStatement: + chapter: 16 + file: surgery.tex + label: '' + mtref: '16.6' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 114511 + sourceTransform: tex2mdx.strip_comments + start: 113887 + type: local-anaphora-reference + via: + - local-anaphora-proof + - local-anaphora-statement +generator: tools/poincare_tex_extract.py +inference: inferred +kind: dependency +ref: +- 6cc16821de1e +- ef94451b5785 +rel: depends +reviewStatus: accepted +scope: local-context +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- local-anaphora-proof +- local-anaphora-statement +--- +Morgan--Tian local anaphora dependency. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/55b7a4440925.md b/projects/poincare-conjecture/.astrolabe/edges/55b7a4440925.md new file mode 100644 index 00000000..21402443 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/55b7a4440925.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 4 + file: maxprin.tex + label: sphsf + mtref: '4.24' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 36469 + sourceTransform: tex2mdx.strip_comments + start: 36157 + targetStatement: + chapter: 4 + file: maxprin.tex + label: '' + mtref: '4.25' + sort: remark + span: + coordinateSpace: comment-stripped-tex + end: 37294 + sourceTransform: tex2mdx.strip_comments + start: 37050 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 4f253de31e0c +- 142b52ba2d68 +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/55cac3e384f0.md b/projects/poincare-conjecture/.astrolabe/edges/55cac3e384f0.md new file mode 100644 index 00000000..ea42dea7 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/55cac3e384f0.md @@ -0,0 +1,62 @@ +--- +confidence: 0.9 +edgeClass: structural +evidence: + containmentDepth: 1 + proofSpan: + bodyEnd: 20703 + bodyStart: 15818 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 20714 + file: newcomp2.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 15805 + sourceStatement: + chapter: 7 + file: newcomp2.tex + label: n/2 + mtref: '7.10' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 15803 + sourceTransform: tex2mdx.strip_comments + start: 15517 + targetContainedSpan: + coordinateSpace: comment-stripped-tex + end: 16332 + file: newcomp2.tex + sourceTransform: tex2mdx.strip_comments + start: 16045 + targetStatement: + chapter: 7 + file: newcomp2.tex + label: '8.1' + mtref: '7.11' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 16332 + sourceTransform: tex2mdx.strip_comments + start: 16045 + type: proof-containment + via: + - proof-contains +generator: tools/poincare_tex_extract.py +inference: structural +kind: containment +ref: +- aebc3ff2bd3b +- db2397f9dacd +rel: contains +reviewStatus: accepted +scope: proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof-contains +--- +Morgan--Tian proof containment. The source is the statement whose proof contains the target statement. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/55cb9c5dfdcc.md b/projects/poincare-conjecture/.astrolabe/edges/55cb9c5dfdcc.md new file mode 100644 index 00000000..c25c661c --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/55cb9c5dfdcc.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 6 + file: newcompar.tex + label: '' + mtref: '6.11' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 14389 + sourceTransform: tex2mdx.strip_comments + start: 13955 + targetStatement: + chapter: 6 + file: newcompar.tex + label: '' + mtref: '6.12' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 14734 + sourceTransform: tex2mdx.strip_comments + start: 14448 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 200f280a8e3b +- 6b205e5b51a2 +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/55dcb6233eff.md b/projects/poincare-conjecture/.astrolabe/edges/55dcb6233eff.md new file mode 100644 index 00000000..24f1800c --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/55dcb6233eff.md @@ -0,0 +1,45 @@ +--- +confidence: 0.2 +edgeClass: structural +evidence: + sourceStatement: + chapter: 6 + file: newcompar.tex + label: '' + mtref: '6.9' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 11539 + sourceTransform: tex2mdx.strip_comments + start: 11300 + targetStatement: + chapter: 6 + file: newcompar.tex + label: '' + mtref: '6.10' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 12573 + sourceTransform: tex2mdx.strip_comments + start: 11870 + type: chapter-reading-order-adjacency + via: + - chapter-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- f09cbffcbdce +- deb9b83d83f2 +rel: in-chapter +reviewStatus: accepted +scope: chapter +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- chapter-sequence +--- +Morgan--Tian reading-order adjacency across sections in a chapter. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/5612790f3554.md b/projects/poincare-conjecture/.astrolabe/edges/5612790f3554.md new file mode 100644 index 00000000..db7c06e0 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/5612790f3554.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 9 + file: temp2kappa.tex + label: bdvolbdcurv + mtref: '9.62' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 124263 + sourceTransform: tex2mdx.strip_comments + start: 123826 + targetStatement: + chapter: 9 + file: temp2kappa.tex + label: global + mtref: '9.63' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 126734 + sourceTransform: tex2mdx.strip_comments + start: 126308 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 3227c39e1892 +- 329d0175aed8 +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/5635fc5529b3.md b/projects/poincare-conjecture/.astrolabe/edges/5635fc5529b3.md new file mode 100644 index 00000000..b224ac9c --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/5635fc5529b3.md @@ -0,0 +1,141 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: canonvary + proofSpan: + bodyEnd: 169692 + bodyStart: 163171 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 169703 + file: temp2kappa.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 163158 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 164035 + file: temp2kappa.tex + sourceTransform: tex2mdx.strip_comments + start: 164020 + via: proof + - context: proof + label: canonvary + proofSpan: + bodyEnd: 169692 + bodyStart: 163171 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 169703 + file: temp2kappa.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 163158 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 165268 + file: temp2kappa.tex + sourceTransform: tex2mdx.strip_comments + start: 165253 + via: proof + - context: proof + label: canonvary + proofSpan: + bodyEnd: 169692 + bodyStart: 163171 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 169703 + file: temp2kappa.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 163158 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 166397 + file: temp2kappa.tex + sourceTransform: tex2mdx.strip_comments + start: 166382 + via: proof + - context: proof + label: canonvary + proofSpan: + bodyEnd: 169692 + bodyStart: 163171 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 169703 + file: temp2kappa.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 163158 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 167195 + file: temp2kappa.tex + sourceTransform: tex2mdx.strip_comments + start: 167180 + via: proof + - context: proof + label: canonvary + proofSpan: + bodyEnd: 169692 + bodyStart: 163171 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 169703 + file: temp2kappa.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 163158 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 168912 + file: temp2kappa.tex + sourceTransform: tex2mdx.strip_comments + start: 168897 + via: proof + sourceStatement: + chapter: 9 + file: temp2kappa.tex + label: '' + mtref: '9.86' + sort: remark + span: + coordinateSpace: comment-stripped-tex + end: 163155 + sourceTransform: tex2mdx.strip_comments + start: 163012 + targetStatement: + chapter: 9 + file: temp2kappa.tex + label: canonvary + mtref: '9.79' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 149236 + sourceTransform: tex2mdx.strip_comments + start: 147728 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 51da06702a9f +- 7c2758f0e2b3 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/5637b5eebdbb.md b/projects/poincare-conjecture/.astrolabe/edges/5637b5eebdbb.md new file mode 100644 index 00000000..f4fce715 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/5637b5eebdbb.md @@ -0,0 +1,55 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: statement + label: locstr + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 34141 + file: newcompar.tex + sourceTransform: tex2mdx.strip_comments + start: 34129 + via: statement + sourceStatement: + chapter: 6 + file: newcompar.tex + label: backwards + mtref: '6.24' + sort: remark + span: + coordinateSpace: comment-stripped-tex + end: 34221 + sourceTransform: tex2mdx.strip_comments + start: 33508 + targetStatement: + chapter: 6 + file: newcompar.tex + label: locstr + mtref: '6.23' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 32019 + sourceTransform: tex2mdx.strip_comments + start: 31685 + type: tex-reference + via: + - statement +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- fa3234930898 +- 4180719ab2c8 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- statement +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/566c474481e1.md b/projects/poincare-conjecture/.astrolabe/edges/566c474481e1.md new file mode 100644 index 00000000..a9e996fd --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/566c474481e1.md @@ -0,0 +1,45 @@ +--- +confidence: 0.2 +edgeClass: structural +evidence: + sourceStatement: + chapter: 18 + file: energy1.tex + label: '' + mtref: '18.80' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 181286 + sourceTransform: tex2mdx.strip_comments + start: 181184 + targetStatement: + chapter: 18 + file: energy1.tex + label: '' + mtref: '18.81' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 185425 + sourceTransform: tex2mdx.strip_comments + start: 185320 + type: chapter-reading-order-adjacency + via: + - chapter-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 5f77e346e12a +- fb6be46de671 +rel: in-chapter +reviewStatus: accepted +scope: chapter +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- chapter-sequence +--- +Morgan--Tian reading-order adjacency across sections in a chapter. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/56b86fe373b1.md b/projects/poincare-conjecture/.astrolabe/edges/56b86fe373b1.md new file mode 100644 index 00000000..142b7f2c --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/56b86fe373b1.md @@ -0,0 +1,45 @@ +--- +confidence: 0.2 +edgeClass: structural +evidence: + sourceStatement: + chapter: 12 + file: stdsoln.tex + label: '' + mtref: '12.14' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 32654 + sourceTransform: tex2mdx.strip_comments + start: 32551 + targetStatement: + chapter: 12 + file: stdsoln.tex + label: '12.7' + mtref: '12.15' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 39561 + sourceTransform: tex2mdx.strip_comments + start: 39191 + type: chapter-reading-order-adjacency + via: + - chapter-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- f26ee81ded19 +- a8675ea40ba2 +rel: in-chapter +reviewStatus: accepted +scope: chapter +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- chapter-sequence +--- +Morgan--Tian reading-order adjacency across sections in a chapter. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/56d62aaf9796.md b/projects/poincare-conjecture/.astrolabe/edges/56d62aaf9796.md new file mode 100644 index 00000000..d928fde4 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/56d62aaf9796.md @@ -0,0 +1,62 @@ +--- +confidence: 0.9 +edgeClass: structural +evidence: + containmentDepth: 1 + proofSpan: + bodyEnd: 28148 + bodyStart: 26997 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 28159 + file: newcompar.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 26984 + sourceStatement: + chapter: 6 + file: newcompar.tex + label: DLJacobi + mtref: '6.19' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 26981 + sourceTransform: tex2mdx.strip_comments + start: 26317 + targetContainedSpan: + coordinateSpace: comment-stripped-tex + end: 27985 + file: newcompar.tex + sourceTransform: tex2mdx.strip_comments + start: 27774 + targetStatement: + chapter: 6 + file: newcompar.tex + label: '' + mtref: '6.20' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 27985 + sourceTransform: tex2mdx.strip_comments + start: 27774 + type: proof-containment + via: + - proof-contains +generator: tools/poincare_tex_extract.py +inference: structural +kind: containment +ref: +- d2e61ac52277 +- da7306fc733d +rel: contains +reviewStatus: accepted +scope: proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof-contains +--- +Morgan--Tian proof containment. The source is the statement whose proof contains the target statement. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/570858b96ad8.md b/projects/poincare-conjecture/.astrolabe/edges/570858b96ad8.md new file mode 100644 index 00000000..a0587485 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/570858b96ad8.md @@ -0,0 +1,62 @@ +--- +confidence: 0.35 +edgeClass: prose +evidence: + mentionTriggers: + - label: noholes + refSpan: + coordinateSpace: comment-stripped-tex + end: 190502 + file: energy1.tex + sourceTransform: tex2mdx.strip_comments + start: 190489 + sentence: This is the content of Corollary~\ref{noholes} + sentenceSpan: + coordinateSpace: comment-stripped-tex + end: 190502 + file: energy1.tex + sourceTransform: tex2mdx.strip_comments + start: 190455 + sourceStrategy: nearest-previous-statement + via: prose + sourceStatement: + chapter: 18 + file: energy1.tex + label: focussummary + mtref: '18.84' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 189625 + sourceTransform: tex2mdx.strip_comments + start: 188975 + targetStatement: + chapter: 18 + file: energy1.tex + label: noholes + mtref: '18.83' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 188550 + sourceTransform: tex2mdx.strip_comments + start: 188167 + type: prose-mention-nearest-previous-statement + via: + - prose +generator: tools/poincare_tex_extract.py +inference: weak +kind: mention +ref: +- 8abfaea5aa9a +- ad36aef968b6 +rel: mentions +reviewStatus: candidate +scope: prose +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- prose +--- +Morgan--Tian prose mention. The source is the nearest preceding statement in the same chapter segment. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/575cf4d340a5.md b/projects/poincare-conjecture/.astrolabe/edges/575cf4d340a5.md new file mode 100644 index 00000000..2d402b27 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/575cf4d340a5.md @@ -0,0 +1,45 @@ +--- +confidence: 0.2 +edgeClass: structural +evidence: + sourceStatement: + chapter: 18 + file: energy1.tex + label: W_3fordiff + mtref: '18.18' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 51926 + sourceTransform: tex2mdx.strip_comments + start: 51222 + targetStatement: + chapter: 18 + file: energy1.tex + label: '' + mtref: '18.19' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 52551 + sourceTransform: tex2mdx.strip_comments + start: 52480 + type: chapter-reading-order-adjacency + via: + - chapter-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- b187c06d1432 +- 6a1625220073 +rel: in-chapter +reviewStatus: accepted +scope: chapter +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- chapter-sequence +--- +Morgan--Tian reading-order adjacency across sections in a chapter. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/5792c2f45021.md b/projects/poincare-conjecture/.astrolabe/edges/5792c2f45021.md new file mode 100644 index 00000000..84548922 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/5792c2f45021.md @@ -0,0 +1,84 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: conecurv + proofSpan: + bodyEnd: 34465 + bodyStart: 32850 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 34476 + file: maxprin.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 32837 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 33144 + file: maxprin.tex + sourceTransform: tex2mdx.strip_comments + start: 33130 + via: proof + - context: proof + label: conecurv + proofSpan: + bodyEnd: 34465 + bodyStart: 32850 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 34476 + file: maxprin.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 32837 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 33551 + file: maxprin.tex + sourceTransform: tex2mdx.strip_comments + start: 33537 + via: proof + sourceStatement: + chapter: 4 + file: maxprin.tex + label: nocones + mtref: '4.22' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 32834 + sourceTransform: tex2mdx.strip_comments + start: 32471 + targetStatement: + chapter: 1 + file: prelim.tex + label: conecurv + mtref: '1.15' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 18132 + sourceTransform: tex2mdx.strip_comments + start: 17714 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 92b5c664399e +- 1b517dfd4bd7 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/579ee7afdd0f.md b/projects/poincare-conjecture/.astrolabe/edges/579ee7afdd0f.md new file mode 100644 index 00000000..f5a4e776 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/579ee7afdd0f.md @@ -0,0 +1,122 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: bigell + proofSpan: + bodyEnd: 146058 + bodyStart: 138906 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 146069 + file: surgery.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 138893 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 139102 + file: surgery.tex + sourceTransform: tex2mdx.strip_comments + start: 139090 + via: proof + - context: proof + label: bigell + proofSpan: + bodyEnd: 146058 + bodyStart: 138906 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 146069 + file: surgery.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 138893 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 139189 + file: surgery.tex + sourceTransform: tex2mdx.strip_comments + start: 139177 + via: proof + - context: proof + label: bigell + proofSpan: + bodyEnd: 146058 + bodyStart: 138906 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 146069 + file: surgery.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 138893 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 144848 + file: surgery.tex + sourceTransform: tex2mdx.strip_comments + start: 144836 + via: proof + - context: proof + label: bigell + proofSpan: + bodyEnd: 146058 + bodyStart: 138906 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 146069 + file: surgery.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 138893 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 145484 + file: surgery.tex + sourceTransform: tex2mdx.strip_comments + start: 145472 + via: proof + sourceStatement: + chapter: 16 + file: surgery.tex + label: '' + mtref: '16.16' + sort: remark + span: + coordinateSpace: comment-stripped-tex + end: 138750 + sourceTransform: tex2mdx.strip_comments + start: 138540 + targetStatement: + chapter: 16 + file: surgery.tex + label: bigell + mtref: '16.13' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 131955 + sourceTransform: tex2mdx.strip_comments + start: 130235 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 4ee1bc9acc48 +- 0a4d6f17810d +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/57a835310a73.md b/projects/poincare-conjecture/.astrolabe/edges/57a835310a73.md new file mode 100644 index 00000000..1029c87f --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/57a835310a73.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 7 + file: newcomp2.tex + label: flat + mtref: '7.25' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 43846 + sourceTransform: tex2mdx.strip_comments + start: 43634 + targetStatement: + chapter: 7 + file: newcomp2.tex + label: 4pi + mtref: '7.26' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 45413 + sourceTransform: tex2mdx.strip_comments + start: 44820 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 4a4c8aae3a8d +- ae6f1c6d00d9 +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/582a099e760f.md b/projects/poincare-conjecture/.astrolabe/edges/582a099e760f.md new file mode 100644 index 00000000..a9a716ae --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/582a099e760f.md @@ -0,0 +1,45 @@ +--- +confidence: 0.2 +edgeClass: structural +evidence: + sourceStatement: + chapter: 4 + file: maxprin.tex + label: Rminprime + mtref: '4.3' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 4069 + sourceTransform: tex2mdx.strip_comments + start: 3817 + targetStatement: + chapter: 4 + file: maxprin.tex + label: '' + mtref: '4.4' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 6365 + sourceTransform: tex2mdx.strip_comments + start: 6101 + type: chapter-reading-order-adjacency + via: + - chapter-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 84e573c0f2b2 +- d12fecfdfae3 +rel: in-chapter +reviewStatus: accepted +scope: chapter +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- chapter-sequence +--- +Morgan--Tian reading-order adjacency across sections in a chapter. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/5853fb0cdb0f.md b/projects/poincare-conjecture/.astrolabe/edges/5853fb0cdb0f.md new file mode 100644 index 00000000..2aac9c3e --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/5853fb0cdb0f.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 9 + file: temp2kappa.tex + label: canonvary + mtref: '9.79' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 149236 + sourceTransform: tex2mdx.strip_comments + start: 147728 + targetStatement: + chapter: 9 + file: temp2kappa.tex + label: strepsopen + mtref: '9.80' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 153741 + sourceTransform: tex2mdx.strip_comments + start: 153588 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 7c2758f0e2b3 +- 75b4c67c9141 +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/58b22405d1ea.md b/projects/poincare-conjecture/.astrolabe/edges/58b22405d1ea.md new file mode 100644 index 00000000..073e8318 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/58b22405d1ea.md @@ -0,0 +1,65 @@ +--- +confidence: 0.45 +edgeClass: semantic +evidence: + endpointSemanticDegreeBeforeTermBackfill: + source: 1 + target: 0 + selectionReason: semantic-connectivity-backfill + sourceMatchSpan: + coordinateSpace: comment-stripped-tex + end: 13892 + file: prelim.tex + sourceTransform: tex2mdx.strip_comments + start: 13873 + sourceStatement: + chapter: 1 + file: prelim.tex + label: lapformula + mtref: '1.10' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 14031 + sourceTransform: tex2mdx.strip_comments + start: 13808 + targetStatement: + chapter: 1 + file: prelim.tex + label: '' + mtref: '1.1' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 1437 + sourceTransform: tex2mdx.strip_comments + start: 797 + targetTermSpan: + coordinateSpace: comment-stripped-tex + end: 1362 + file: prelim.tex + sourceTransform: tex2mdx.strip_comments + start: 1343 + term: riemannian manifold + termAmbiguity: unique-in-chapter + termSource: italic + termSpecificity: no-longer-defined-term + type: same-chapter-definition-term-match + via: + - definition-term:riemannian manifold +generator: tools/poincare_tex_extract.py +inference: weak +kind: definition-use +ref: +- 8d03f3ca0aea +- 0c6b379e7dca +rel: uses +reviewStatus: candidate +scope: same-chapter-term-match +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- definition-term:riemannian manifold +--- +Morgan--Tian same-chapter definition term match. This is a low-confidence candidate edge for review. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/58d73786c564.md b/projects/poincare-conjecture/.astrolabe/edges/58d73786c564.md new file mode 100644 index 00000000..fca91606 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/58d73786c564.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: smlmtflow + proofSpan: + bodyEnd: 33794 + bodyStart: 17659 + coordinateSpace: comment-stripped-tex + depth: 2 + end: 33805 + file: singlimit2.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 17646 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 18377 + file: singlimit2.tex + sourceTransform: tex2mdx.strip_comments + start: 18362 + via: proof + sourceStatement: + chapter: 11 + file: singlimit2.tex + label: maxT + mtref: '11.10' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 17643 + sourceTransform: tex2mdx.strip_comments + start: 17183 + targetStatement: + chapter: 11 + file: singlimit2.tex + label: smlmtflow + mtref: '11.1' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 3459 + sourceTransform: tex2mdx.strip_comments + start: 1498 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- d6919616448d +- d51aab8a8065 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/5936211f78a0.md b/projects/poincare-conjecture/.astrolabe/edges/5936211f78a0.md new file mode 100644 index 00000000..09e84f60 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/5936211f78a0.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: 2DGSS + proofSpan: + bodyEnd: 101521 + bodyStart: 100796 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 101532 + file: temp2kappa.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 100783 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 101454 + file: temp2kappa.tex + sourceTransform: tex2mdx.strip_comments + start: 101443 + via: proof + sourceStatement: + chapter: 9 + file: temp2kappa.tex + label: 3Dinfty + mtref: '9.52' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 100781 + sourceTransform: tex2mdx.strip_comments + start: 100280 + targetStatement: + chapter: 9 + file: temp2kappa.tex + label: 2DGSS + mtref: '9.50' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 91949 + sourceTransform: tex2mdx.strip_comments + start: 91266 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- f8ff249efb81 +- da767c71996b +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/594115680fc6.md b/projects/poincare-conjecture/.astrolabe/edges/594115680fc6.md new file mode 100644 index 00000000..4b9bb0d9 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/594115680fc6.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 1 + file: prelim.tex + label: '' + mtref: '1.19' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 26794 + sourceTransform: tex2mdx.strip_comments + start: 26539 + targetStatement: + chapter: 1 + file: prelim.tex + label: jacmin + mtref: '1.20' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 27147 + sourceTransform: tex2mdx.strip_comments + start: 26797 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- b209b247be34 +- 50646105e5e0 +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/5993747531ba.md b/projects/poincare-conjecture/.astrolabe/edges/5993747531ba.md new file mode 100644 index 00000000..c0014a82 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/5993747531ba.md @@ -0,0 +1,56 @@ +--- +confidence: 0.85 +edgeClass: semantic +evidence: + containmentReason: target theorem-like statement occurs inside source proof + proofSpan: + bodyEnd: 35466 + bodyStart: 32231 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 35477 + file: stdsoln.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 32218 + sourceStatement: + chapter: 12 + file: stdsoln.tex + label: stdkappa + mtref: '12.13' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 32215 + sourceTransform: tex2mdx.strip_comments + start: 32043 + targetStatement: + chapter: 12 + file: stdsoln.tex + label: '' + mtref: '12.14' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 32654 + sourceTransform: tex2mdx.strip_comments + start: 32551 + type: proof-contained-theorem-like-statement + via: + - proof-contained-statement +generator: tools/poincare_tex_extract.py +inference: inferred +kind: dependency +ref: +- ed069175cdf9 +- f26ee81ded19 +rel: depends +reviewStatus: accepted +scope: proof-containment +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof-contained-statement +--- +Morgan--Tian proof-contained statement dependency. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/59a82e418bf5.md b/projects/poincare-conjecture/.astrolabe/edges/59a82e418bf5.md new file mode 100644 index 00000000..37bbcd2c --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/59a82e418bf5.md @@ -0,0 +1,62 @@ +--- +confidence: 0.9 +edgeClass: structural +evidence: + containmentDepth: 1 + proofSpan: + bodyEnd: 129463 + bodyStart: 113176 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 129474 + file: surgery.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 113163 + sourceStatement: + chapter: 16 + file: surgery.tex + label: altern + mtref: '16.5' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 112999 + sourceTransform: tex2mdx.strip_comments + start: 110961 + targetContainedSpan: + coordinateSpace: comment-stripped-tex + end: 115994 + file: surgery.tex + sourceTransform: tex2mdx.strip_comments + start: 115513 + targetStatement: + chapter: 16 + file: surgery.tex + label: '' + mtref: '16.7' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 115994 + sourceTransform: tex2mdx.strip_comments + start: 115513 + type: proof-containment + via: + - proof-contains +generator: tools/poincare_tex_extract.py +inference: structural +kind: containment +ref: +- ad4f67ea75bc +- 6cc16821de1e +rel: contains +reviewStatus: accepted +scope: proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof-contains +--- +Morgan--Tian proof containment. The source is the statement whose proof contains the target statement. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/59c0a7a05024.md b/projects/poincare-conjecture/.astrolabe/edges/59c0a7a05024.md new file mode 100644 index 00000000..02e3ff34 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/59c0a7a05024.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 6 + file: newcompar.tex + label: injdefn + mtref: '6.25' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 35883 + sourceTransform: tex2mdx.strip_comments + start: 34564 + targetStatement: + chapter: 6 + file: newcompar.tex + label: L_x + mtref: '6.26' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 36613 + sourceTransform: tex2mdx.strip_comments + start: 36222 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 90fe97584bdd +- 440e7d27aee8 +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/59e9891de3be.md b/projects/poincare-conjecture/.astrolabe/edges/59e9891de3be.md new file mode 100644 index 00000000..8253eb52 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/59e9891de3be.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: scalarevol + proofSpan: + bodyEnd: 31467 + bodyStart: 29842 + coordinateSpace: comment-stripped-tex + depth: 3 + end: 31478 + file: singlimit2.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 29829 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 30208 + file: singlimit2.tex + sourceTransform: tex2mdx.strip_comments + start: 30192 + via: proof + sourceStatement: + chapter: 11 + file: singlimit2.tex + label: ancientlimit + mtref: '11.16' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 29827 + sourceTransform: tex2mdx.strip_comments + start: 29411 + targetStatement: + chapter: 4 + file: maxprin.tex + label: scalarevol + mtref: '4.1' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 2998 + sourceTransform: tex2mdx.strip_comments + start: 2387 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- a66fa0ea8720 +- 95cc838cfc6d +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/59ec7e956f19.md b/projects/poincare-conjecture/.astrolabe/edges/59ec7e956f19.md new file mode 100644 index 00000000..cadb0ae4 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/59ec7e956f19.md @@ -0,0 +1,55 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: statement + label: absconv1 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 45447 + file: temp2kappa.tex + sourceTransform: tex2mdx.strip_comments + start: 45433 + via: statement + sourceStatement: + chapter: 9 + file: temp2kappa.tex + label: absconv2 + mtref: '9.29' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 45666 + sourceTransform: tex2mdx.strip_comments + start: 45351 + targetStatement: + chapter: 9 + file: temp2kappa.tex + label: absconv1 + mtref: '9.27' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 44147 + sourceTransform: tex2mdx.strip_comments + start: 43269 + type: tex-reference + via: + - statement +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 5b4c94102f41 +- 1766f37a13cf +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- statement +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/5a655ebcb758.md b/projects/poincare-conjecture/.astrolabe/edges/5a655ebcb758.md new file mode 100644 index 00000000..139d9e34 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/5a655ebcb758.md @@ -0,0 +1,65 @@ +--- +confidence: 0.45 +edgeClass: semantic +evidence: + endpointSemanticDegreeBeforeTermBackfill: + source: 0 + target: 1 + selectionReason: semantic-connectivity-backfill + sourceMatchSpan: + coordinateSpace: comment-stripped-tex + end: 10208 + file: flowbasics.tex + sourceTransform: tex2mdx.strip_comments + start: 10182 + sourceStatement: + chapter: 3 + file: flowbasics.tex + label: '' + mtref: '3.10' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 10220 + sourceTransform: tex2mdx.strip_comments + start: 10093 + targetStatement: + chapter: 3 + file: flowbasics.tex + label: '' + mtref: '3.8' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 9455 + sourceTransform: tex2mdx.strip_comments + start: 9202 + targetTermSpan: + coordinateSpace: comment-stripped-tex + end: 9305 + file: flowbasics.tex + sourceTransform: tex2mdx.strip_comments + start: 9279 + term: gradient shrinking soliton + termAmbiguity: unique-in-chapter + termSource: italic + termSpecificity: no-longer-defined-term + type: same-chapter-definition-term-match + via: + - definition-term:gradient shrinking soliton +generator: tools/poincare_tex_extract.py +inference: weak +kind: definition-use +ref: +- a560479acf35 +- 6ace86bb0b70 +rel: uses +reviewStatus: candidate +scope: same-chapter-term-match +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- definition-term:gradient shrinking soliton +--- +Morgan--Tian same-chapter definition term match. This is a low-confidence candidate edge for review. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/5aa0140daf29.md b/projects/poincare-conjecture/.astrolabe/edges/5aa0140daf29.md new file mode 100644 index 00000000..4a81b702 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/5aa0140daf29.md @@ -0,0 +1,63 @@ +--- +confidence: 0.35 +edgeClass: prose +evidence: + mentionTriggers: + - label: omega + refSpan: + coordinateSpace: comment-stripped-tex + end: 52755 + file: singlimit2.tex + sourceTransform: tex2mdx.strip_comments + start: 52744 + sentence: To complete the proof of Theorem~\ref{omega}, it remains only to establish + the same result for the components of $\Omega$ that meet $\Omega_\rho$ + sentenceSpan: + coordinateSpace: comment-stripped-tex + end: 52856 + file: singlimit2.tex + sourceTransform: tex2mdx.strip_comments + start: 52710 + sourceStrategy: nearest-previous-statement + via: prose + sourceStatement: + chapter: 11 + file: singlimit2.tex + label: '' + mtref: '11.29' + sort: remark + span: + coordinateSpace: comment-stripped-tex + end: 52523 + sourceTransform: tex2mdx.strip_comments + start: 52352 + targetStatement: + chapter: 11 + file: singlimit2.tex + label: omega + mtref: '11.19' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 38153 + sourceTransform: tex2mdx.strip_comments + start: 36599 + type: prose-mention-nearest-previous-statement + via: + - prose +generator: tools/poincare_tex_extract.py +inference: weak +kind: mention +ref: +- 1c2d9d005b0e +- af00774dbaa9 +rel: mentions +reviewStatus: candidate +scope: prose +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- prose +--- +Morgan--Tian prose mention. The source is the nearest preceding statement in the same chapter segment. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/5ae48e28736c.md b/projects/poincare-conjecture/.astrolabe/edges/5ae48e28736c.md new file mode 100644 index 00000000..d7a818b5 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/5ae48e28736c.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 12 + file: stdsoln.tex + label: '12.9' + mtref: '12.9' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 21288 + sourceTransform: tex2mdx.strip_comments + start: 20809 + targetStatement: + chapter: 12 + file: stdsoln.tex + label: '' + mtref: '12.10' + sort: remark + span: + coordinateSpace: comment-stripped-tex + end: 22639 + sourceTransform: tex2mdx.strip_comments + start: 21290 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 78cab9a6159b +- f9aa8c556cab +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/5ae53aea242e.md b/projects/poincare-conjecture/.astrolabe/edges/5ae53aea242e.md new file mode 100644 index 00000000..45fde138 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/5ae53aea242e.md @@ -0,0 +1,55 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: statement + label: surgery + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 12494 + file: intro.tex + sourceTransform: tex2mdx.strip_comments + start: 12481 + via: statement + sourceStatement: + chapter: 0 + file: intro.tex + label: finiteext + mtref: '0.4' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 12644 + sourceTransform: tex2mdx.strip_comments + start: 11822 + targetStatement: + chapter: 0 + file: intro.tex + label: surgery + mtref: '0.3' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 11419 + sourceTransform: tex2mdx.strip_comments + start: 10567 + type: tex-reference + via: + - statement +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 127db1dc9315 +- 847af3ecd4a8 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- statement +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/5af187f1425f.md b/projects/poincare-conjecture/.astrolabe/edges/5af187f1425f.md new file mode 100644 index 00000000..74fe743c --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/5af187f1425f.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 9 + file: temp2kappa.tex + label: lxkest + mtref: '9.16' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 18673 + sourceTransform: tex2mdx.strip_comments + start: 18334 + targetStatement: + chapter: 9 + file: temp2kappa.tex + label: '' + mtref: '9.17' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 20870 + sourceTransform: tex2mdx.strip_comments + start: 20456 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 611a73900deb +- 42b05b37c3ad +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/5b6b8e0deee8.md b/projects/poincare-conjecture/.astrolabe/edges/5b6b8e0deee8.md new file mode 100644 index 00000000..2a9f10dc --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/5b6b8e0deee8.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 14 + file: surgery.tex + label: '' + mtref: '14.4' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 42753 + sourceTransform: tex2mdx.strip_comments + start: 42165 + targetStatement: + chapter: 14 + file: surgery.tex + label: '' + mtref: '14.5' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 43469 + sourceTransform: tex2mdx.strip_comments + start: 42757 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- bdc4fc3c0d3e +- 6564ce56e2fd +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/5bc872371efe.md b/projects/poincare-conjecture/.astrolabe/edges/5bc872371efe.md new file mode 100644 index 00000000..3b96008b --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/5bc872371efe.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 9 + file: temp2kappa.tex + label: noncompkappa + mtref: '9.88' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 170123 + sourceTransform: tex2mdx.strip_comments + start: 169708 + targetStatement: + chapter: 9 + file: temp2kappa.tex + label: thmep3 + mtref: '9.89' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 172313 + sourceTransform: tex2mdx.strip_comments + start: 171695 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 274cda9be677 +- a4ea411d0765 +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/5be6347e6d20.md b/projects/poincare-conjecture/.astrolabe/edges/5be6347e6d20.md new file mode 100644 index 00000000..2c28f4cd --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/5be6347e6d20.md @@ -0,0 +1,56 @@ +--- +confidence: 0.85 +edgeClass: semantic +evidence: + containmentReason: target theorem-like statement occurs inside source proof + proofSpan: + bodyEnd: 229182 + bodyStart: 185422 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 229193 + file: surgery.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 185409 + sourceStatement: + chapter: 17 + file: surgery.tex + label: extend + mtref: '17.1' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 185406 + sourceTransform: tex2mdx.strip_comments + start: 184285 + targetStatement: + chapter: 17 + file: surgery.tex + label: '' + mtref: '17.11' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 226210 + sourceTransform: tex2mdx.strip_comments + start: 226024 + type: proof-contained-theorem-like-statement + via: + - proof-contained-statement +generator: tools/poincare_tex_extract.py +inference: inferred +kind: dependency +ref: +- 254e8dae97e6 +- 8dafe9c95b7f +rel: depends +reviewStatus: accepted +scope: proof-containment +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof-contained-statement +--- +Morgan--Tian proof-contained statement dependency. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/5c04c761a8c0.md b/projects/poincare-conjecture/.astrolabe/edges/5c04c761a8c0.md new file mode 100644 index 00000000..3fd3de67 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/5c04c761a8c0.md @@ -0,0 +1,80 @@ +--- +confidence: 0.8 +edgeClass: semantic +evidence: + proseTriggers: + - label: finiteext + refSpan: + coordinateSpace: comment-stripped-tex + end: 12878 + file: intro.tex + sourceTransform: tex2mdx.strip_comments + start: 12863 + sentence: 'We immediately deduce Theorem~\ref{Theorem1} from Theorems~\ref{surgery} + and~\ref{finiteext} as follows: Let $M$ be a $3$-manifold satisfying the hypothesis + of Theorem~\ref{Theorem1}. Then there is a finite sequence $M=M_0,M_1,\ldots,M_k=\emptyset$ + such that for each $i,\ 1\le i\le k$, $M_i$ is obtained from $M_{i-1}$ by a + connected sum decomposition or $M_i$ is obtained from $M_{i-1}$ by removing + a component diffeomorphic to one of $S^2\times S^1$, $\Ar P^3\#\Ar P^3$, a non-orientable + $2$-sphere bundle over $S^1$, or a $3$-dimensional spherical space-form. Clearly, + it follows by downward induction on $i$ that each connected component of $M_i$ + is diffeomorphic to a connected sum of $3$-dimensional spherical space-forms, + copies of $S^2\times S^1$, and copies of the non-orientable $2$-sphere bundle + over $S^1$' + sentenceSpan: + coordinateSpace: comment-stripped-tex + end: 13608 + file: intro.tex + sourceTransform: tex2mdx.strip_comments + start: 12785 + sourceLabel: Theorem1 + sourceRefSpan: + coordinateSpace: comment-stripped-tex + end: 12830 + file: intro.tex + sourceTransform: tex2mdx.strip_comments + start: 12816 + sourceStrategy: refs-in-inferential-sentence + via: + - prose-deduce-from + sourceStatement: + chapter: 0 + file: intro.tex + label: Theorem1 + mtref: '0.1' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 2955 + sourceTransform: tex2mdx.strip_comments + start: 2574 + targetStatement: + chapter: 0 + file: intro.tex + label: finiteext + mtref: '0.4' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 12644 + sourceTransform: tex2mdx.strip_comments + start: 11822 + type: inferential-prose-reference + via: + - prose-deduce-from +generator: tools/poincare_tex_extract.py +inference: inferred +kind: dependency +ref: +- 9999efa2be2c +- 127db1dc9315 +rel: depends +reviewStatus: accepted +scope: prose +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- prose-deduce-from +--- +Morgan--Tian high-confidence prose dependency. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/5c374bd9a76a.md b/projects/poincare-conjecture/.astrolabe/edges/5c374bd9a76a.md new file mode 100644 index 00000000..d808bb2e --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/5c374bd9a76a.md @@ -0,0 +1,56 @@ +--- +confidence: 0.85 +edgeClass: semantic +evidence: + containmentReason: target theorem-like statement occurs inside source proof + proofSpan: + bodyEnd: 100555 + bodyStart: 97739 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 100566 + file: newcompar.tex + ownerLabel: lipaty + sourceTransform: tex2mdx.strip_comments + start: 97726 + sourceStatement: + chapter: 6 + file: newcompar.tex + label: lipaty + mtref: '6.59' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 87473 + sourceTransform: tex2mdx.strip_comments + start: 86282 + targetStatement: + chapter: 6 + file: newcompar.tex + label: '' + mtref: '6.65' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 98830 + sourceTransform: tex2mdx.strip_comments + start: 98666 + type: proof-contained-theorem-like-statement + via: + - proof-contained-statement +generator: tools/poincare_tex_extract.py +inference: inferred +kind: dependency +ref: +- 96ced8c484df +- e27e5f5d2bf7 +rel: depends +reviewStatus: accepted +scope: proof-containment +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof-contained-statement +--- +Morgan--Tian proof-contained statement dependency. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/5c38d9a242d2.md b/projects/poincare-conjecture/.astrolabe/edges/5c38d9a242d2.md new file mode 100644 index 00000000..52f753cc --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/5c38d9a242d2.md @@ -0,0 +1,45 @@ +--- +confidence: 0.2 +edgeClass: structural +evidence: + sourceStatement: + chapter: 18 + file: energy1.tex + label: '' + mtref: '18.2' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 14028 + sourceTransform: tex2mdx.strip_comments + start: 13563 + targetStatement: + chapter: 18 + file: energy1.tex + label: grouptheory + mtref: '18.3' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 19372 + sourceTransform: tex2mdx.strip_comments + start: 19084 + type: chapter-reading-order-adjacency + via: + - chapter-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 5b649fe97568 +- 858af0fef413 +rel: in-chapter +reviewStatus: accepted +scope: chapter +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- chapter-sequence +--- +Morgan--Tian reading-order adjacency across sections in a chapter. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/5c3f816ab60d.md b/projects/poincare-conjecture/.astrolabe/edges/5c3f816ab60d.md new file mode 100644 index 00000000..0878f29b --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/5c3f816ab60d.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 7 + file: newcomp2.tex + label: '' + mtref: '7.16' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 30129 + sourceTransform: tex2mdx.strip_comments + start: 29149 + targetStatement: + chapter: 7 + file: newcomp2.tex + label: '' + mtref: '7.17' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 31480 + sourceTransform: tex2mdx.strip_comments + start: 31161 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- d427ad10ceee +- b58a798d5eea +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/5ca18bbafba8.md b/projects/poincare-conjecture/.astrolabe/edges/5ca18bbafba8.md new file mode 100644 index 00000000..0832a315 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/5ca18bbafba8.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 10 + file: bddcurvbdddist.tex + label: '' + mtref: '10.17' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 42249 + sourceTransform: tex2mdx.strip_comments + start: 41852 + targetStatement: + chapter: 10 + file: bddcurvbdddist.tex + label: morethanone + mtref: '10.18' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 42343 + sourceTransform: tex2mdx.strip_comments + start: 42251 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- e98dee3d7485 +- dda8a7c7acb6 +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/5caef3da399d.md b/projects/poincare-conjecture/.astrolabe/edges/5caef3da399d.md new file mode 100644 index 00000000..733c1457 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/5caef3da399d.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 18 + file: energy1.tex + label: csforramps + mtref: '18.42' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 91212 + sourceTransform: tex2mdx.strip_comments + start: 90872 + targetStatement: + chapter: 18 + file: energy1.tex + label: mugrowth + mtref: '18.43' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 93643 + sourceTransform: tex2mdx.strip_comments + start: 93070 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 11ec367eae93 +- 6590af0738c7 +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/5cdb7e143488.md b/projects/poincare-conjecture/.astrolabe/edges/5cdb7e143488.md new file mode 100644 index 00000000..eac5a2ed --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/5cdb7e143488.md @@ -0,0 +1,55 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: statement + label: MAIN + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 51338 + file: energy1.tex + sourceTransform: tex2mdx.strip_comments + start: 51328 + via: statement + sourceStatement: + chapter: 18 + file: energy1.tex + label: W_3fordiff + mtref: '18.18' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 51926 + sourceTransform: tex2mdx.strip_comments + start: 51222 + targetStatement: + chapter: 15 + file: surgery.tex + label: MAIN + mtref: '15.9' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 91186 + sourceTransform: tex2mdx.strip_comments + start: 90004 + type: tex-reference + via: + - statement +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- b187c06d1432 +- 703e4cfeece7 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- statement +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/5d16cafa2cbc.md b/projects/poincare-conjecture/.astrolabe/edges/5d16cafa2cbc.md new file mode 100644 index 00000000..b15c88a6 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/5d16cafa2cbc.md @@ -0,0 +1,45 @@ +--- +confidence: 0.2 +edgeClass: structural +evidence: + sourceStatement: + chapter: 18 + file: energy1.tex + label: mulength + mtref: '18.60' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 141140 + sourceTransform: tex2mdx.strip_comments + start: 140651 + targetStatement: + chapter: 18 + file: energy1.tex + label: '' + mtref: '18.61' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 149799 + sourceTransform: tex2mdx.strip_comments + start: 149158 + type: chapter-reading-order-adjacency + via: + - chapter-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 52f36e456355 +- d91a6d9473a7 +rel: in-chapter +reviewStatus: accepted +scope: chapter +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- chapter-sequence +--- +Morgan--Tian reading-order adjacency across sections in a chapter. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/5d1df58f9d3a.md b/projects/poincare-conjecture/.astrolabe/edges/5d1df58f9d3a.md new file mode 100644 index 00000000..1d6d97b9 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/5d1df58f9d3a.md @@ -0,0 +1,55 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: statement + label: MAIN + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 29780 + file: energy1.tex + sourceTransform: tex2mdx.strip_comments + start: 29770 + via: statement + sourceStatement: + chapter: 18 + file: energy1.tex + label: pi2 + mtref: '18.9' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 30087 + sourceTransform: tex2mdx.strip_comments + start: 29657 + targetStatement: + chapter: 15 + file: surgery.tex + label: MAIN + mtref: '15.9' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 91186 + sourceTransform: tex2mdx.strip_comments + start: 90004 + type: tex-reference + via: + - statement +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 4415092c2079 +- 703e4cfeece7 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- statement +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/5d2904f9129b.md b/projects/poincare-conjecture/.astrolabe/edges/5d2904f9129b.md new file mode 100644 index 00000000..73971922 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/5d2904f9129b.md @@ -0,0 +1,45 @@ +--- +confidence: 0.2 +edgeClass: structural +evidence: + sourceStatement: + chapter: 18 + file: energy1.tex + label: '' + mtref: '18.85' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 192706 + sourceTransform: tex2mdx.strip_comments + start: 192549 + targetStatement: + chapter: 18 + file: energy1.tex + label: csshi1 + mtref: '18.86' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 195723 + sourceTransform: tex2mdx.strip_comments + start: 194943 + type: chapter-reading-order-adjacency + via: + - chapter-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 692ac8d2c602 +- ff5d51a483d9 +rel: in-chapter +reviewStatus: accepted +scope: chapter +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- chapter-sequence +--- +Morgan--Tian reading-order adjacency across sections in a chapter. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/5d54ee7e3807.md b/projects/poincare-conjecture/.astrolabe/edges/5d54ee7e3807.md new file mode 100644 index 00000000..cfb782ea --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/5d54ee7e3807.md @@ -0,0 +1,84 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: derivcor + proofSpan: + bodyEnd: 171628 + bodyStart: 170139 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 171639 + file: temp2kappa.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 170126 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 170365 + file: temp2kappa.tex + sourceTransform: tex2mdx.strip_comments + start: 170351 + via: proof + - context: proof + label: derivcor + proofSpan: + bodyEnd: 171628 + bodyStart: 170139 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 171639 + file: temp2kappa.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 170126 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 170591 + file: temp2kappa.tex + sourceTransform: tex2mdx.strip_comments + start: 170577 + via: proof + sourceStatement: + chapter: 9 + file: temp2kappa.tex + label: noncompkappa + mtref: '9.88' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 170123 + sourceTransform: tex2mdx.strip_comments + start: 169708 + targetStatement: + chapter: 9 + file: temp2kappa.tex + label: derivcor + mtref: '9.71' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 139489 + sourceTransform: tex2mdx.strip_comments + start: 139133 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 274cda9be677 +- 8c090fbd21a0 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/5da685a5265b.md b/projects/poincare-conjecture/.astrolabe/edges/5da685a5265b.md new file mode 100644 index 00000000..ab173e78 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/5da685a5265b.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: deltanet + proofSpan: + bodyEnd: 58340 + bodyStart: 53526 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 58351 + file: bddcurvbdddist.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 53513 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 55354 + file: bddcurvbdddist.tex + sourceTransform: tex2mdx.strip_comments + start: 55340 + via: proof + sourceStatement: + chapter: 10 + file: bddcurvbdddist.tex + label: secondstatment + mtref: '10.29' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 53510 + sourceTransform: tex2mdx.strip_comments + start: 52640 + targetStatement: + chapter: 10 + file: bddcurvbdddist.tex + label: deltanet + mtref: '10.28' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 52334 + sourceTransform: tex2mdx.strip_comments + start: 51848 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- b34d555cd7b5 +- f7e1d840a49f +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/5dee5a60c75f.md b/projects/poincare-conjecture/.astrolabe/edges/5dee5a60c75f.md new file mode 100644 index 00000000..447c1a5d --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/5dee5a60c75f.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: shi + proofSpan: + bodyEnd: 21736 + bodyStart: 21484 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 21747 + file: converge2.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 21471 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 21614 + file: converge2.tex + sourceTransform: tex2mdx.strip_comments + start: 21605 + via: proof + sourceStatement: + chapter: 5 + file: converge2.tex + label: sliceconv + mtref: '5.11' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 21468 + sourceTransform: tex2mdx.strip_comments + start: 20456 + targetStatement: + chapter: 3 + file: flowbasics.tex + label: shi + mtref: '3.28' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 40541 + sourceTransform: tex2mdx.strip_comments + start: 39904 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- d9f3522dec12 +- ed4b72caf9f9 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/5df474278f45.md b/projects/poincare-conjecture/.astrolabe/edges/5df474278f45.md new file mode 100644 index 00000000..0656ceed --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/5df474278f45.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 18 + file: energy1.tex + label: '' + mtref: '18.19' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 52551 + sourceTransform: tex2mdx.strip_comments + start: 52480 + targetStatement: + chapter: 18 + file: energy1.tex + label: '' + mtref: '18.20' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 53052 + sourceTransform: tex2mdx.strip_comments + start: 52883 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 6a1625220073 +- a04af7434ef0 +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/5df67ffa18e8.md b/projects/poincare-conjecture/.astrolabe/edges/5df67ffa18e8.md new file mode 100644 index 00000000..683bae1a --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/5df67ffa18e8.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: soul + proofSpan: + bodyEnd: 80533 + bodyStart: 76639 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 80544 + file: prelim.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 76626 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 76769 + file: prelim.tex + sourceTransform: tex2mdx.strip_comments + start: 76759 + via: proof + sourceStatement: + chapter: 2 + file: prelim.tex + label: narrows + mtref: '2.19' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 76622 + sourceTransform: tex2mdx.strip_comments + start: 76377 + targetStatement: + chapter: 2 + file: prelim.tex + label: soul + mtref: '2.7' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 57159 + sourceTransform: tex2mdx.strip_comments + start: 56544 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- cb9ca6d6db9f +- 2b5f18292e62 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/5e191aa84f04.md b/projects/poincare-conjecture/.astrolabe/edges/5e191aa84f04.md new file mode 100644 index 00000000..67238e5f --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/5e191aa84f04.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 12 + file: stdsoln.tex + label: '' + mtref: '12.10' + sort: remark + span: + coordinateSpace: comment-stripped-tex + end: 22639 + sourceTransform: tex2mdx.strip_comments + start: 21290 + targetStatement: + chapter: 12 + file: stdsoln.tex + label: '' + mtref: '12.11' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 23953 + sourceTransform: tex2mdx.strip_comments + start: 23764 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- f9aa8c556cab +- a5e421ef8adf +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/5e2dc8d0e51d.md b/projects/poincare-conjecture/.astrolabe/edges/5e2dc8d0e51d.md new file mode 100644 index 00000000..b65d4aea --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/5e2dc8d0e51d.md @@ -0,0 +1,45 @@ +--- +confidence: 0.2 +edgeClass: structural +evidence: + sourceStatement: + chapter: 3 + file: flowbasics.tex + label: etaprop + mtref: '3.33' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 47994 + sourceTransform: tex2mdx.strip_comments + start: 47039 + targetStatement: + chapter: 3 + file: flowbasics.tex + label: '' + mtref: '3.34' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 65890 + sourceTransform: tex2mdx.strip_comments + start: 64937 + type: chapter-reading-order-adjacency + via: + - chapter-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 81f8de5102de +- 6aa2c1fdf94e +rel: in-chapter +reviewStatus: accepted +scope: chapter +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- chapter-sequence +--- +Morgan--Tian reading-order adjacency across sections in a chapter. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/5e6790deba5d.md b/projects/poincare-conjecture/.astrolabe/edges/5e6790deba5d.md new file mode 100644 index 00000000..38ddfc30 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/5e6790deba5d.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 18 + file: energy1.tex + label: claim1885 + mtref: '18.89' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 202725 + sourceTransform: tex2mdx.strip_comments + start: 202260 + targetStatement: + chapter: 18 + file: energy1.tex + label: '' + mtref: '18.90' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 204623 + sourceTransform: tex2mdx.strip_comments + start: 204360 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 1f7d2e438fef +- f7b6896c9e7d +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/5e7280041c29.md b/projects/poincare-conjecture/.astrolabe/edges/5e7280041c29.md new file mode 100644 index 00000000..53e4572c --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/5e7280041c29.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: nullspace + proofSpan: + bodyEnd: 158965 + bodyStart: 157813 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 158976 + file: temp2kappa.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 157800 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 157976 + file: temp2kappa.tex + sourceTransform: tex2mdx.strip_comments + start: 157961 + via: proof + sourceStatement: + chapter: 9 + file: temp2kappa.tex + label: infprod2 + mtref: '9.83' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 157798 + sourceTransform: tex2mdx.strip_comments + start: 157331 + targetStatement: + chapter: 4 + file: maxprin.tex + label: nullspace + mtref: '4.20' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 31663 + sourceTransform: tex2mdx.strip_comments + start: 30958 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 25e0946e7222 +- d21a675e4ea8 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/5e7909e3260b.md b/projects/poincare-conjecture/.astrolabe/edges/5e7909e3260b.md new file mode 100644 index 00000000..856320ac --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/5e7909e3260b.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 2 + file: prelim.tex + label: '' + mtref: '2.9' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 58094 + sourceTransform: tex2mdx.strip_comments + start: 57706 + targetStatement: + chapter: 2 + file: prelim.tex + label: homeo + mtref: '2.10' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 60913 + sourceTransform: tex2mdx.strip_comments + start: 60407 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- dc7704c76958 +- 64a0f82323d3 +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/5ea7b626ad9f.md b/projects/poincare-conjecture/.astrolabe/edges/5ea7b626ad9f.md new file mode 100644 index 00000000..e91f95d5 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/5ea7b626ad9f.md @@ -0,0 +1,62 @@ +--- +confidence: 0.9 +edgeClass: structural +evidence: + containmentDepth: 1 + proofSpan: + bodyEnd: 5289 + bodyStart: 3014 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 5300 + file: maxprin.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 3001 + sourceStatement: + chapter: 4 + file: maxprin.tex + label: scalarevol + mtref: '4.1' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 2998 + sourceTransform: tex2mdx.strip_comments + start: 2387 + targetContainedSpan: + coordinateSpace: comment-stripped-tex + end: 4069 + file: maxprin.tex + sourceTransform: tex2mdx.strip_comments + start: 3817 + targetStatement: + chapter: 4 + file: maxprin.tex + label: Rminprime + mtref: '4.3' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 4069 + sourceTransform: tex2mdx.strip_comments + start: 3817 + type: proof-containment + via: + - proof-contains +generator: tools/poincare_tex_extract.py +inference: structural +kind: containment +ref: +- 95cc838cfc6d +- 84e573c0f2b2 +rel: contains +reviewStatus: accepted +scope: proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof-contains +--- +Morgan--Tian proof containment. The source is the statement whose proof contains the target statement. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/5ec6b1a8e710.md b/projects/poincare-conjecture/.astrolabe/edges/5ec6b1a8e710.md new file mode 100644 index 00000000..616f4cfa --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/5ec6b1a8e710.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: blowupposcurv + proofSpan: + bodyEnd: 35374 + bodyStart: 16867 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 35385 + file: singlimit2.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 16854 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 34373 + file: singlimit2.tex + sourceTransform: tex2mdx.strip_comments + start: 34354 + via: proof + sourceStatement: + chapter: 11 + file: singlimit2.tex + label: '' + mtref: '11.9' + sort: remark + span: + coordinateSpace: comment-stripped-tex + end: 16852 + sourceTransform: tex2mdx.strip_comments + start: 16134 + targetStatement: + chapter: 5 + file: converge2.tex + label: blowupposcurv + mtref: '5.33' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 51450 + sourceTransform: tex2mdx.strip_comments + start: 50791 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 5e06d2593bfa +- 2ddfbc74b081 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/5ecf1d4c58af.md b/projects/poincare-conjecture/.astrolabe/edges/5ecf1d4c58af.md new file mode 100644 index 00000000..ae852695 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/5ecf1d4c58af.md @@ -0,0 +1,45 @@ +--- +confidence: 0.2 +edgeClass: structural +evidence: + sourceStatement: + chapter: 9 + file: temp2kappa.tex + label: '' + mtref: '9.17' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 20870 + sourceTransform: tex2mdx.strip_comments + start: 20456 + targetStatement: + chapter: 9 + file: temp2kappa.tex + label: linfest + mtref: '9.18' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 24152 + sourceTransform: tex2mdx.strip_comments + start: 23894 + type: chapter-reading-order-adjacency + via: + - chapter-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 42b05b37c3ad +- ff5eee52ee0e +rel: in-chapter +reviewStatus: accepted +scope: chapter +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- chapter-sequence +--- +Morgan--Tian reading-order adjacency across sections in a chapter. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/5ed095335ca8.md b/projects/poincare-conjecture/.astrolabe/edges/5ed095335ca8.md new file mode 100644 index 00000000..58ca3632 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/5ed095335ca8.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: nullspace + proofSpan: + bodyEnd: 105941 + bodyStart: 103880 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 105952 + file: temp2kappa.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 103867 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 105582 + file: temp2kappa.tex + sourceTransform: tex2mdx.strip_comments + start: 105567 + via: proof + sourceStatement: + chapter: 9 + file: temp2kappa.tex + label: '' + mtref: '9.54' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 103864 + sourceTransform: tex2mdx.strip_comments + start: 103562 + targetStatement: + chapter: 4 + file: maxprin.tex + label: nullspace + mtref: '4.20' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 31663 + sourceTransform: tex2mdx.strip_comments + start: 30958 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 92ee3ee82b58 +- d21a675e4ea8 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/5f3192161398.md b/projects/poincare-conjecture/.astrolabe/edges/5f3192161398.md new file mode 100644 index 00000000..ee144a69 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/5f3192161398.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: 3rdcurvshr + proofSpan: + bodyEnd: 92953 + bodyStart: 91228 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 92964 + file: energy1.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 91215 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 91835 + file: energy1.tex + sourceTransform: tex2mdx.strip_comments + start: 91819 + via: proof + sourceStatement: + chapter: 18 + file: energy1.tex + label: csforramps + mtref: '18.42' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 91212 + sourceTransform: tex2mdx.strip_comments + start: 90872 + targetStatement: + chapter: 18 + file: energy1.tex + label: 3rdcurvshr + mtref: '18.36' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 84184 + sourceTransform: tex2mdx.strip_comments + start: 83899 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 11ec367eae93 +- 67b855d0d6b5 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/5f3f3a3ca01d.md b/projects/poincare-conjecture/.astrolabe/edges/5f3f3a3ca01d.md new file mode 100644 index 00000000..6440e889 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/5f3f3a3ca01d.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: smeigen + proofSpan: + bodyEnd: 24647 + bodyStart: 23636 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 24658 + file: surgery.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 23623 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 23673 + file: surgery.tex + sourceTransform: tex2mdx.strip_comments + start: 23660 + via: proof + sourceStatement: + chapter: 13 + file: surgery.tex + label: '' + mtref: '13.14' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 23621 + sourceTransform: tex2mdx.strip_comments + start: 23086 + targetStatement: + chapter: 13 + file: surgery.tex + label: smeigen + mtref: '13.12' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 20688 + sourceTransform: tex2mdx.strip_comments + start: 20353 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- c9d8cb9d34c3 +- e70bacf8aab0 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/5f413b255a45.md b/projects/poincare-conjecture/.astrolabe/edges/5f413b255a45.md new file mode 100644 index 00000000..ab112ea1 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/5f413b255a45.md @@ -0,0 +1,62 @@ +--- +confidence: 0.9 +edgeClass: structural +evidence: + containmentDepth: 1 + proofSpan: + bodyEnd: 71454 + bodyStart: 67635 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 71465 + file: prelim.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 67622 + sourceStatement: + chapter: 2 + file: prelim.tex + label: splitting + mtref: '2.13' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 67619 + sourceTransform: tex2mdx.strip_comments + start: 67365 + targetContainedSpan: + coordinateSpace: comment-stripped-tex + end: 67980 + file: prelim.tex + sourceTransform: tex2mdx.strip_comments + start: 67775 + targetStatement: + chapter: 2 + file: prelim.tex + label: line + mtref: '2.14' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 67980 + sourceTransform: tex2mdx.strip_comments + start: 67775 + type: proof-containment + via: + - proof-contains +generator: tools/poincare_tex_extract.py +inference: structural +kind: containment +ref: +- e5b196773169 +- 006e42ef5bb5 +rel: contains +reviewStatus: accepted +scope: proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof-contains +--- +Morgan--Tian proof containment. The source is the statement whose proof contains the target statement. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/5f4826cb83ee.md b/projects/poincare-conjecture/.astrolabe/edges/5f4826cb83ee.md new file mode 100644 index 00000000..9a70d42d --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/5f4826cb83ee.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 5 + file: converge2.tex + label: '' + mtref: '5.5' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 5439 + sourceTransform: tex2mdx.strip_comments + start: 5336 + targetStatement: + chapter: 5 + file: converge2.tex + label: basicconv + mtref: '5.6' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 9928 + sourceTransform: tex2mdx.strip_comments + start: 8668 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 1a98d21b3855 +- 74c7682a68ed +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/5f7a9889b2f8.md b/projects/poincare-conjecture/.astrolabe/edges/5f7a9889b2f8.md new file mode 100644 index 00000000..87ea68ec --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/5f7a9889b2f8.md @@ -0,0 +1,62 @@ +--- +confidence: 0.9 +edgeClass: structural +evidence: + containmentDepth: 1 + proofSpan: + bodyEnd: 229182 + bodyStart: 185422 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 229193 + file: surgery.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 185409 + sourceStatement: + chapter: 17 + file: surgery.tex + label: extend + mtref: '17.1' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 185406 + sourceTransform: tex2mdx.strip_comments + start: 184285 + targetContainedSpan: + coordinateSpace: comment-stripped-tex + end: 211471 + file: surgery.tex + sourceTransform: tex2mdx.strip_comments + start: 210838 + targetStatement: + chapter: 17 + file: surgery.tex + label: lemstdcan + mtref: '17.7' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 211471 + sourceTransform: tex2mdx.strip_comments + start: 210838 + type: proof-containment + via: + - proof-contains +generator: tools/poincare_tex_extract.py +inference: structural +kind: containment +ref: +- 254e8dae97e6 +- 111551d7831b +rel: contains +reviewStatus: accepted +scope: proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof-contains +--- +Morgan--Tian proof containment. The source is the statement whose proof contains the target statement. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/5fc342cac4d8.md b/projects/poincare-conjecture/.astrolabe/edges/5fc342cac4d8.md new file mode 100644 index 00000000..4aa8c8dc --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/5fc342cac4d8.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: shiw/deriv + proofSpan: + bodyEnd: 119675 + bodyStart: 117487 + coordinateSpace: comment-stripped-tex + depth: 2 + end: 119686 + file: surgery.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 117474 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 118912 + file: surgery.tex + sourceTransform: tex2mdx.strip_comments + start: 118896 + via: proof + sourceStatement: + chapter: 16 + file: surgery.tex + label: '' + mtref: '16.8' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 117471 + sourceTransform: tex2mdx.strip_comments + start: 116211 + targetStatement: + chapter: 3 + file: flowbasics.tex + label: shiw/deriv + mtref: '3.29' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 42464 + sourceTransform: tex2mdx.strip_comments + start: 41312 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 4b6ee3b7a329 +- e0b6a1f646a7 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/5ff05d6a6785.md b/projects/poincare-conjecture/.astrolabe/edges/5ff05d6a6785.md new file mode 100644 index 00000000..98e7318a --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/5ff05d6a6785.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: blowupposcurv + proofSpan: + bodyEnd: 34311 + bodyStart: 33996 + coordinateSpace: comment-stripped-tex + depth: 2 + end: 34322 + file: bddcurvbdddist.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 33983 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 34308 + file: bddcurvbdddist.tex + sourceTransform: tex2mdx.strip_comments + start: 34289 + via: proof + sourceStatement: + chapter: 10 + file: bddcurvbdddist.tex + label: '' + mtref: '10.11' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 33981 + sourceTransform: tex2mdx.strip_comments + start: 33869 + targetStatement: + chapter: 5 + file: converge2.tex + label: blowupposcurv + mtref: '5.33' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 51450 + sourceTransform: tex2mdx.strip_comments + start: 50791 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 2d6c9dcad762 +- 2ddfbc74b081 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/602bf578476b.md b/projects/poincare-conjecture/.astrolabe/edges/602bf578476b.md new file mode 100644 index 00000000..54eb6a00 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/602bf578476b.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: canonvary + proofSpan: + bodyEnd: 220240 + bodyStart: 211487 + coordinateSpace: comment-stripped-tex + depth: 2 + end: 220251 + file: surgery.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 211474 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 218174 + file: surgery.tex + sourceTransform: tex2mdx.strip_comments + start: 218159 + via: proof + sourceStatement: + chapter: 17 + file: surgery.tex + label: lemstdcan + mtref: '17.7' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 211471 + sourceTransform: tex2mdx.strip_comments + start: 210838 + targetStatement: + chapter: 9 + file: temp2kappa.tex + label: canonvary + mtref: '9.79' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 149236 + sourceTransform: tex2mdx.strip_comments + start: 147728 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 111551d7831b +- 7c2758f0e2b3 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/6046c73233a2.md b/projects/poincare-conjecture/.astrolabe/edges/6046c73233a2.md new file mode 100644 index 00000000..f39ab84f --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/6046c73233a2.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 7 + file: newcomp2.tex + label: lipcomplete + mtref: '7.5' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 9437 + sourceTransform: tex2mdx.strip_comments + start: 8692 + targetStatement: + chapter: 7 + file: newcomp2.tex + label: '' + mtref: '7.6' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 11044 + sourceTransform: tex2mdx.strip_comments + start: 10270 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 16fbe263a7c8 +- 2e14dd57a334 +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/6096b597aff6.md b/projects/poincare-conjecture/.astrolabe/edges/6096b597aff6.md new file mode 100644 index 00000000..92438f55 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/6096b597aff6.md @@ -0,0 +1,45 @@ +--- +confidence: 0.2 +edgeClass: structural +evidence: + sourceStatement: + chapter: 1 + file: prelim.tex + label: '' + mtref: '1.8' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 11793 + sourceTransform: tex2mdx.strip_comments + start: 10898 + targetStatement: + chapter: 1 + file: prelim.tex + label: divRic + mtref: '1.9' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 12596 + sourceTransform: tex2mdx.strip_comments + start: 12512 + type: chapter-reading-order-adjacency + via: + - chapter-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- c5b8a4d0d061 +- 9c83686198e8 +rel: in-chapter +reviewStatus: accepted +scope: chapter +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- chapter-sequence +--- +Morgan--Tian reading-order adjacency across sections in a chapter. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/6099b06baac8.md b/projects/poincare-conjecture/.astrolabe/edges/6099b06baac8.md new file mode 100644 index 00000000..a0d4ee57 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/6099b06baac8.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: MP + proofSpan: + bodyEnd: 19584 + bodyStart: 18311 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 19595 + file: maxprin.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 18298 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 19565 + file: maxprin.tex + sourceTransform: tex2mdx.strip_comments + start: 19557 + via: proof + sourceStatement: + chapter: 4 + file: maxprin.tex + label: Ric>0 + mtref: '4.15' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 18295 + sourceTransform: tex2mdx.strip_comments + start: 18076 + targetStatement: + chapter: 4 + file: maxprin.tex + label: MP + mtref: '4.7' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 8808 + sourceTransform: tex2mdx.strip_comments + start: 7939 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- f8afc1c61977 +- c954e95b5d89 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/61154c9e8f40.md b/projects/poincare-conjecture/.astrolabe/edges/61154c9e8f40.md new file mode 100644 index 00000000..c09a3ac8 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/61154c9e8f40.md @@ -0,0 +1,56 @@ +--- +confidence: 0.85 +edgeClass: semantic +evidence: + containmentReason: target theorem-like statement occurs inside source proof + proofSpan: + bodyEnd: 71454 + bodyStart: 67635 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 71465 + file: prelim.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 67622 + sourceStatement: + chapter: 2 + file: prelim.tex + label: splitting + mtref: '2.13' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 67619 + sourceTransform: tex2mdx.strip_comments + start: 67365 + targetStatement: + chapter: 2 + file: prelim.tex + label: line + mtref: '2.14' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 67980 + sourceTransform: tex2mdx.strip_comments + start: 67775 + type: proof-contained-theorem-like-statement + via: + - proof-contained-statement +generator: tools/poincare_tex_extract.py +inference: inferred +kind: dependency +ref: +- e5b196773169 +- 006e42ef5bb5 +rel: depends +reviewStatus: accepted +scope: proof-containment +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof-contained-statement +--- +Morgan--Tian proof-contained statement dependency. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/613005954541.md b/projects/poincare-conjecture/.astrolabe/edges/613005954541.md new file mode 100644 index 00000000..0a1ed10e --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/613005954541.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: omega + proofSpan: + bodyEnd: 93873 + bodyStart: 90350 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 93884 + file: stdsoln.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 90337 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 91591 + file: stdsoln.tex + sourceTransform: tex2mdx.strip_comments + start: 91580 + via: proof + sourceStatement: + chapter: 12 + file: stdsoln.tex + label: Restim + mtref: '12.31' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 90335 + sourceTransform: tex2mdx.strip_comments + start: 90184 + targetStatement: + chapter: 11 + file: singlimit2.tex + label: omega + mtref: '11.19' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 38153 + sourceTransform: tex2mdx.strip_comments + start: 36599 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- ffae4504318a +- af00774dbaa9 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/61ab0b7a48f6.md b/projects/poincare-conjecture/.astrolabe/edges/61ab0b7a48f6.md new file mode 100644 index 00000000..f9c88d35 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/61ab0b7a48f6.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 8 + file: noncoll.tex + label: small + mtref: '8.3' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 5471 + sourceTransform: tex2mdx.strip_comments + start: 5161 + targetStatement: + chapter: 8 + file: noncoll.tex + label: '' + mtref: '8.4' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 5966 + sourceTransform: tex2mdx.strip_comments + start: 5756 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 26e1a6292fef +- b49cbb9f66c6 +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/61b3fd82e784.md b/projects/poincare-conjecture/.astrolabe/edges/61b3fd82e784.md new file mode 100644 index 00000000..38a9bf97 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/61b3fd82e784.md @@ -0,0 +1,56 @@ +--- +confidence: 0.85 +edgeClass: semantic +evidence: + containmentReason: target theorem-like statement occurs inside source proof + proofSpan: + bodyEnd: 74813 + bodyStart: 59556 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 74824 + file: singlimit2.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 59543 + sourceStatement: + chapter: 11 + file: singlimit2.tex + label: hexist + mtref: '11.31' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 59267 + sourceTransform: tex2mdx.strip_comments + start: 58375 + targetStatement: + chapter: 11 + file: singlimit2.tex + label: '' + mtref: '11.34' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 63736 + sourceTransform: tex2mdx.strip_comments + start: 63514 + type: proof-contained-theorem-like-statement + via: + - proof-contained-statement +generator: tools/poincare_tex_extract.py +inference: inferred +kind: dependency +ref: +- 074e02b067b8 +- ee7e22fb33f6 +rel: depends +reviewStatus: accepted +scope: proof-containment +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof-contained-statement +--- +Morgan--Tian proof-contained statement dependency. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/61c7dd3960fb.md b/projects/poincare-conjecture/.astrolabe/edges/61c7dd3960fb.md new file mode 100644 index 00000000..e8747c29 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/61c7dd3960fb.md @@ -0,0 +1,103 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: compactkappa + proofSpan: + bodyEnd: 169692 + bodyStart: 163171 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 169703 + file: temp2kappa.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 163158 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 163675 + file: temp2kappa.tex + sourceTransform: tex2mdx.strip_comments + start: 163657 + via: proof + - context: proof + label: compactkappa + proofSpan: + bodyEnd: 169692 + bodyStart: 163171 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 169703 + file: temp2kappa.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 163158 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 166965 + file: temp2kappa.tex + sourceTransform: tex2mdx.strip_comments + start: 166947 + via: proof + - context: proof + label: compactkappa + proofSpan: + bodyEnd: 169692 + bodyStart: 163171 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 169703 + file: temp2kappa.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 163158 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 169434 + file: temp2kappa.tex + sourceTransform: tex2mdx.strip_comments + start: 169416 + via: proof + sourceStatement: + chapter: 9 + file: temp2kappa.tex + label: '' + mtref: '9.86' + sort: remark + span: + coordinateSpace: comment-stripped-tex + end: 163155 + sourceTransform: tex2mdx.strip_comments + start: 163012 + targetStatement: + chapter: 9 + file: temp2kappa.tex + label: compactkappa + mtref: '9.64' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 127434 + sourceTransform: tex2mdx.strip_comments + start: 127205 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 51da06702a9f +- 49e865228b19 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/61cf017436c6.md b/projects/poincare-conjecture/.astrolabe/edges/61cf017436c6.md new file mode 100644 index 00000000..5b8221bb --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/61cf017436c6.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: minmax + proofSpan: + bodyEnd: 20606 + bodyStart: 18514 + coordinateSpace: comment-stripped-tex + depth: 2 + end: 20617 + file: newcomp2.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 18501 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 19001 + file: newcomp2.tex + sourceTransform: tex2mdx.strip_comments + start: 18989 + via: proof + sourceStatement: + chapter: 7 + file: newcomp2.tex + label: '8.1' + mtref: '7.11' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 16332 + sourceTransform: tex2mdx.strip_comments + start: 16045 + targetStatement: + chapter: 6 + file: newcompar.tex + label: minmax + mtref: '6.61' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 91839 + sourceTransform: tex2mdx.strip_comments + start: 91251 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- db2397f9dacd +- a7c2ebee07a3 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/61df93811329.md b/projects/poincare-conjecture/.astrolabe/edges/61df93811329.md new file mode 100644 index 00000000..99820bc1 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/61df93811329.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 2 + file: prelim.tex + label: ends + mtref: '2.1' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 49124 + sourceTransform: tex2mdx.strip_comments + start: 48865 + targetStatement: + chapter: 2 + file: prelim.tex + label: '' + mtref: '2.2' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 49959 + sourceTransform: tex2mdx.strip_comments + start: 49128 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 89a6041c42b9 +- 4d6e47f3ec09 +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/61eef75f7fb2.md b/projects/poincare-conjecture/.astrolabe/edges/61eef75f7fb2.md new file mode 100644 index 00000000..a3ea1f86 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/61eef75f7fb2.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 13 + file: surgery.tex + label: '13.10' + mtref: '13.10' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 18442 + sourceTransform: tex2mdx.strip_comments + start: 17697 + targetStatement: + chapter: 13 + file: surgery.tex + label: improve + mtref: '13.11' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 19510 + sourceTransform: tex2mdx.strip_comments + start: 18654 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 347896f0385b +- 41a5f172b2dc +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/6257691dee44.md b/projects/poincare-conjecture/.astrolabe/edges/6257691dee44.md new file mode 100644 index 00000000..2ed18ee8 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/6257691dee44.md @@ -0,0 +1,62 @@ +--- +confidence: 0.9 +edgeClass: structural +evidence: + containmentDepth: 1 + proofSpan: + bodyEnd: 125334 + bodyStart: 121264 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 125345 + file: newcompar.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 121251 + sourceStatement: + chapter: 6 + file: newcompar.tex + label: Amono + mtref: '6.80' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 121245 + sourceTransform: tex2mdx.strip_comments + start: 120839 + targetContainedSpan: + coordinateSpace: comment-stripped-tex + end: 122525 + file: newcompar.tex + sourceTransform: tex2mdx.strip_comments + start: 122141 + targetStatement: + chapter: 6 + file: newcompar.tex + label: fmonotone + mtref: '6.81' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 122525 + sourceTransform: tex2mdx.strip_comments + start: 122141 + type: proof-containment + via: + - proof-contains +generator: tools/poincare_tex_extract.py +inference: structural +kind: containment +ref: +- 6a52537bef2f +- 4dc40f37cd6d +rel: contains +reviewStatus: accepted +scope: proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof-contains +--- +Morgan--Tian proof containment. The source is the statement whose proof contains the target statement. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/6280188f009c.md b/projects/poincare-conjecture/.astrolabe/edges/6280188f009c.md new file mode 100644 index 00000000..24ff212f --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/6280188f009c.md @@ -0,0 +1,55 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: statement + label: assumptions + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 36773 + file: singlimit2.tex + sourceTransform: tex2mdx.strip_comments + start: 36756 + via: statement + sourceStatement: + chapter: 11 + file: singlimit2.tex + label: omega + mtref: '11.19' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 38153 + sourceTransform: tex2mdx.strip_comments + start: 36599 + targetStatement: + chapter: 11 + file: singlimit2.tex + label: assumptions + mtref: '11.18' + sort: assumption + span: + coordinateSpace: comment-stripped-tex + end: 36515 + sourceTransform: tex2mdx.strip_comments + start: 35753 + type: tex-reference + via: + - statement +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- af00774dbaa9 +- 3009be0a37fc +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- statement +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/628d798783a7.md b/projects/poincare-conjecture/.astrolabe/edges/628d798783a7.md new file mode 100644 index 00000000..76736fed --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/628d798783a7.md @@ -0,0 +1,55 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: statement + label: KAPPALIMIT + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 108063 + file: surgery.tex + sourceTransform: tex2mdx.strip_comments + start: 108047 + via: statement + sourceStatement: + chapter: 16 + file: surgery.tex + label: delta0ri+1 + mtref: '16.4' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 109162 + sourceTransform: tex2mdx.strip_comments + start: 107682 + targetStatement: + chapter: 16 + file: surgery.tex + label: KAPPALIMIT + mtref: '16.1' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 103576 + sourceTransform: tex2mdx.strip_comments + start: 102336 + type: tex-reference + via: + - statement +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 89a60eb69c3f +- 4ce8b5da9c58 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- statement +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/62a282014f25.md b/projects/poincare-conjecture/.astrolabe/edges/62a282014f25.md new file mode 100644 index 00000000..59b22b1f --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/62a282014f25.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 9 + file: temp2kappa.tex + label: '' + mtref: '9.73' + sort: remark + span: + coordinateSpace: comment-stripped-tex + end: 143057 + sourceTransform: tex2mdx.strip_comments + start: 142454 + targetStatement: + chapter: 9 + file: temp2kappa.tex + label: partialtR + mtref: '9.74' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 143467 + sourceTransform: tex2mdx.strip_comments + start: 143149 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 5ab50584f917 +- 1280a1901e2e +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/62e3b633e220.md b/projects/poincare-conjecture/.astrolabe/edges/62e3b633e220.md new file mode 100644 index 00000000..9da730c7 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/62e3b633e220.md @@ -0,0 +1,66 @@ +--- +confidence: 0.35 +edgeClass: prose +evidence: + mentionTriggers: + - label: annlengthest + refSpan: + coordinateSpace: comment-stripped-tex + end: 145717 + file: energy1.tex + sourceTransform: tex2mdx.strip_comments + start: 145699 + sentence: Except for the brief comments that follow, our proof involves geometric + analysis that takes place on an abstract annulus with bounds on its area, upper + bounds on its Gaussian curvature, and on integrals of the geodesic curvature + on the boundary. Proposition~\ref{annlengthest} below gives the precise result + along these lines + sentenceSpan: + coordinateSpace: comment-stripped-tex + end: 145766 + file: energy1.tex + sourceTransform: tex2mdx.strip_comments + start: 145440 + sourceStrategy: nearest-previous-statement + via: prose + sourceStatement: + chapter: 18 + file: energy1.tex + label: mulength + mtref: '18.60' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 141140 + sourceTransform: tex2mdx.strip_comments + start: 140651 + targetStatement: + chapter: 18 + file: energy1.tex + label: annlengthest + mtref: '18.63' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 153504 + sourceTransform: tex2mdx.strip_comments + start: 152870 + type: prose-mention-nearest-previous-statement + via: + - prose +generator: tools/poincare_tex_extract.py +inference: weak +kind: mention +ref: +- 52f36e456355 +- 6e9739b0c59b +rel: mentions +reviewStatus: candidate +scope: prose +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- prose +--- +Morgan--Tian prose mention. The source is the nearest preceding statement in the same chapter segment. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/62f6ba97e898.md b/projects/poincare-conjecture/.astrolabe/edges/62f6ba97e898.md new file mode 100644 index 00000000..877ff5c4 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/62f6ba97e898.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 6 + file: newcompar.tex + label: fmonotone + mtref: '6.81' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 122525 + sourceTransform: tex2mdx.strip_comments + start: 122141 + targetStatement: + chapter: 6 + file: newcompar.tex + label: finiteredvol + mtref: '6.82' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 125562 + sourceTransform: tex2mdx.strip_comments + start: 125396 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 4dc40f37cd6d +- e49b6b45c4d0 +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/6316f79e8539.md b/projects/poincare-conjecture/.astrolabe/edges/6316f79e8539.md new file mode 100644 index 00000000..a2b5985b --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/6316f79e8539.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 5 + file: converge2.tex + label: '' + mtref: '5.25' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 39043 + sourceTransform: tex2mdx.strip_comments + start: 38763 + targetStatement: + chapter: 5 + file: converge2.tex + label: '' + mtref: '5.26' + sort: example + span: + coordinateSpace: comment-stripped-tex + end: 40373 + sourceTransform: tex2mdx.strip_comments + start: 40144 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- beb6a354ee6a +- 4f373966523b +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/63375d14dcf9.md b/projects/poincare-conjecture/.astrolabe/edges/63375d14dcf9.md new file mode 100644 index 00000000..86114eee --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/63375d14dcf9.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: Amono + proofSpan: + bodyEnd: 47272 + bodyStart: 45428 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 47283 + file: newcomp2.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 45415 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 46445 + file: newcomp2.tex + sourceTransform: tex2mdx.strip_comments + start: 46434 + via: proof + sourceStatement: + chapter: 7 + file: newcomp2.tex + label: 4pi + mtref: '7.26' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 45413 + sourceTransform: tex2mdx.strip_comments + start: 44820 + targetStatement: + chapter: 6 + file: newcompar.tex + label: Amono + mtref: '6.80' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 121245 + sourceTransform: tex2mdx.strip_comments + start: 120839 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- ae6f1c6d00d9 +- 6a52537bef2f +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/63495d38d645.md b/projects/poincare-conjecture/.astrolabe/edges/63495d38d645.md new file mode 100644 index 00000000..4e6d400b --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/63495d38d645.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: firststrongmax + proofSpan: + bodyEnd: 132635 + bodyStart: 128771 + coordinateSpace: comment-stripped-tex + depth: 2 + end: 132646 + file: temp2kappa.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 128758 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 131362 + file: temp2kappa.tex + sourceTransform: tex2mdx.strip_comments + start: 131342 + via: proof + sourceStatement: + chapter: 9 + file: temp2kappa.tex + label: 2Rbound + mtref: '9.66' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 128756 + sourceTransform: tex2mdx.strip_comments + start: 128570 + targetStatement: + chapter: 4 + file: maxprin.tex + label: firststrongmax + mtref: '4.18' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 23916 + sourceTransform: tex2mdx.strip_comments + start: 23628 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- ea9b285e40cb +- 016cf36686b2 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/634dbe832a2e.md b/projects/poincare-conjecture/.astrolabe/edges/634dbe832a2e.md new file mode 100644 index 00000000..16136510 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/634dbe832a2e.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 18 + file: energy1.tex + label: ubd + mtref: '18.41' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 90315 + sourceTransform: tex2mdx.strip_comments + start: 90090 + targetStatement: + chapter: 18 + file: energy1.tex + label: csforramps + mtref: '18.42' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 91212 + sourceTransform: tex2mdx.strip_comments + start: 90872 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 0936bbe2abb8 +- 11ec367eae93 +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/6367594d0789.md b/projects/poincare-conjecture/.astrolabe/edges/6367594d0789.md new file mode 100644 index 00000000..fb5868f2 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/6367594d0789.md @@ -0,0 +1,55 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: statement + label: epsclose + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 146688 + file: temp2kappa.tex + sourceTransform: tex2mdx.strip_comments + start: 146674 + via: statement + sourceStatement: + chapter: 9 + file: temp2kappa.tex + label: defncanonnbhd + mtref: '9.78' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 147589 + sourceTransform: tex2mdx.strip_comments + start: 145644 + targetStatement: + chapter: 2 + file: prelim.tex + label: epsclose + mtref: '2.16' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 72976 + sourceTransform: tex2mdx.strip_comments + start: 71892 + type: tex-reference + via: + - statement +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 3aa5883ab629 +- 755d47f17b43 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- statement +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/63ab54bfe973.md b/projects/poincare-conjecture/.astrolabe/edges/63ab54bfe973.md new file mode 100644 index 00000000..1d59ef72 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/63ab54bfe973.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 16 + file: surgery.tex + label: L_0 + mtref: '16.15' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 138538 + sourceTransform: tex2mdx.strip_comments + start: 137276 + targetStatement: + chapter: 16 + file: surgery.tex + label: '' + mtref: '16.16' + sort: remark + span: + coordinateSpace: comment-stripped-tex + end: 138750 + sourceTransform: tex2mdx.strip_comments + start: 138540 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 86043191a3c2 +- 4ee1bc9acc48 +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/63abb45019fd.md b/projects/poincare-conjecture/.astrolabe/edges/63abb45019fd.md new file mode 100644 index 00000000..4068502b --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/63abb45019fd.md @@ -0,0 +1,45 @@ +--- +confidence: 0.2 +edgeClass: structural +evidence: + sourceStatement: + chapter: 5 + file: converge2.tex + label: '' + mtref: '5.16' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 28735 + sourceTransform: tex2mdx.strip_comments + start: 28434 + targetStatement: + chapter: 5 + file: converge2.tex + label: epsnet + mtref: '5.17' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 31206 + sourceTransform: tex2mdx.strip_comments + start: 30954 + type: chapter-reading-order-adjacency + via: + - chapter-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 00e5410e6aa1 +- 9baa1c0fea07 +rel: in-chapter +reviewStatus: accepted +scope: chapter +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- chapter-sequence +--- +Morgan--Tian reading-order adjacency across sections in a chapter. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/63b5390f0f64.md b/projects/poincare-conjecture/.astrolabe/edges/63b5390f0f64.md new file mode 100644 index 00000000..17eb7da7 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/63b5390f0f64.md @@ -0,0 +1,45 @@ +--- +confidence: 0.2 +edgeClass: structural +evidence: + sourceStatement: + chapter: 6 + file: newcompar.tex + label: COR + mtref: '6.74' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 110602 + sourceTransform: tex2mdx.strip_comments + start: 110260 + targetStatement: + chapter: 6 + file: newcompar.tex + label: Qformula + mtref: '6.75' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 111495 + sourceTransform: tex2mdx.strip_comments + start: 111142 + type: chapter-reading-order-adjacency + via: + - chapter-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 541f71387f84 +- 76c5f88ad2b4 +rel: in-chapter +reviewStatus: accepted +scope: chapter +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- chapter-sequence +--- +Morgan--Tian reading-order adjacency across sections in a chapter. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/644a9ec01c65.md b/projects/poincare-conjecture/.astrolabe/edges/644a9ec01c65.md new file mode 100644 index 00000000..3ba12f9b --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/644a9ec01c65.md @@ -0,0 +1,63 @@ +--- +confidence: 0.35 +edgeClass: prose +evidence: + mentionTriggers: + - label: gluingsoln + refSpan: + coordinateSpace: comment-stripped-tex + end: 88367 + file: surgery.tex + sourceTransform: tex2mdx.strip_comments + start: 88351 + sentence: According to Lemma~\ref{gluingsoln} the horizontal metrics glue together + to make a smooth metric on space-time satisfying the Ricci flow equation. + sentenceSpan: + coordinateSpace: comment-stripped-tex + end: 88478 + file: surgery.tex + sourceTransform: tex2mdx.strip_comments + start: 88331 + sourceStrategy: nearest-previous-statement + via: prose + sourceStatement: + chapter: 15 + file: surgery.tex + label: surgeryparams + mtref: '15.7' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 80864 + sourceTransform: tex2mdx.strip_comments + start: 79853 + targetStatement: + chapter: 14 + file: surgery.tex + label: gluingsoln + mtref: '14.11' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 49299 + sourceTransform: tex2mdx.strip_comments + start: 47892 + type: prose-mention-nearest-previous-statement + via: + - prose +generator: tools/poincare_tex_extract.py +inference: weak +kind: mention +ref: +- 22c40a5664ac +- b186908cc3b4 +rel: mentions +reviewStatus: candidate +scope: prose +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- prose +--- +Morgan--Tian prose mention. The source is the nearest preceding statement in the same chapter segment. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/644d616144a2.md b/projects/poincare-conjecture/.astrolabe/edges/644d616144a2.md new file mode 100644 index 00000000..2cf4beee --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/644d616144a2.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: kappa0r0t0 + proofSpan: + bodyEnd: 92972 + bodyStart: 92188 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 92983 + file: surgery.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 92175 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 92460 + file: surgery.tex + sourceTransform: tex2mdx.strip_comments + start: 92444 + via: proof + sourceStatement: + chapter: 15 + file: surgery.tex + label: RFSexists + mtref: '15.10' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 92173 + sourceTransform: tex2mdx.strip_comments + start: 91337 + targetStatement: + chapter: 4 + file: maxprin.tex + label: kappa0r0t0 + mtref: '4.11' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 13944 + sourceTransform: tex2mdx.strip_comments + start: 13445 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- c86aa999ce8a +- c1330c5a8be3 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/64c040a322eb.md b/projects/poincare-conjecture/.astrolabe/edges/64c040a322eb.md new file mode 100644 index 00000000..10ce2445 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/64c040a322eb.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 4 + file: maxprin.tex + label: MP + mtref: '4.7' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 8808 + sourceTransform: tex2mdx.strip_comments + start: 7939 + targetStatement: + chapter: 4 + file: maxprin.tex + label: varyingZ + mtref: '4.8' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 10335 + sourceTransform: tex2mdx.strip_comments + start: 9140 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- c954e95b5d89 +- 8f737f2b0f7d +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/64d94cfa422b.md b/projects/poincare-conjecture/.astrolabe/edges/64d94cfa422b.md new file mode 100644 index 00000000..88ab01eb --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/64d94cfa422b.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 19 + file: canonnbhd.tex + label: neckgeo + mtref: '19.5' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 8075 + sourceTransform: tex2mdx.strip_comments + start: 7413 + targetStatement: + chapter: 19 + file: canonnbhd.tex + label: compint + mtref: '19.6' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 8416 + sourceTransform: tex2mdx.strip_comments + start: 8079 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- b74cd360f72f +- 82f43e137897 +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/653b1f9778c8.md b/projects/poincare-conjecture/.astrolabe/edges/653b1f9778c8.md new file mode 100644 index 00000000..ed8a4667 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/653b1f9778c8.md @@ -0,0 +1,62 @@ +--- +confidence: 0.9 +edgeClass: structural +evidence: + containmentDepth: 1 + proofSpan: + bodyEnd: 129463 + bodyStart: 113176 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 129474 + file: surgery.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 113163 + sourceStatement: + chapter: 16 + file: surgery.tex + label: altern + mtref: '16.5' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 112999 + sourceTransform: tex2mdx.strip_comments + start: 110961 + targetContainedSpan: + coordinateSpace: comment-stripped-tex + end: 120111 + file: surgery.tex + sourceTransform: tex2mdx.strip_comments + start: 119688 + targetStatement: + chapter: 16 + file: surgery.tex + label: goodtotheta + mtref: '16.9' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 120111 + sourceTransform: tex2mdx.strip_comments + start: 119688 + type: proof-containment + via: + - proof-contains +generator: tools/poincare_tex_extract.py +inference: structural +kind: containment +ref: +- ad4f67ea75bc +- 712124d7ea02 +rel: contains +reviewStatus: accepted +scope: proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof-contains +--- +Morgan--Tian proof containment. The source is the statement whose proof contains the target statement. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/65b3ce0444c5.md b/projects/poincare-conjecture/.astrolabe/edges/65b3ce0444c5.md new file mode 100644 index 00000000..204dd6e9 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/65b3ce0444c5.md @@ -0,0 +1,55 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: statement + label: A_0 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 80615 + file: surgery.tex + sourceTransform: tex2mdx.strip_comments + start: 80606 + via: statement + sourceStatement: + chapter: 15 + file: surgery.tex + label: surgeryparams + mtref: '15.7' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 80864 + sourceTransform: tex2mdx.strip_comments + start: 79853 + targetStatement: + chapter: 12 + file: stdsoln.tex + label: A_0 + mtref: '12.3' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 6040 + sourceTransform: tex2mdx.strip_comments + start: 5568 + type: tex-reference + via: + - statement +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 22c40a5664ac +- 0b07beec84c0 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- statement +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/65b404e8b5be.md b/projects/poincare-conjecture/.astrolabe/edges/65b404e8b5be.md new file mode 100644 index 00000000..262c9690 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/65b404e8b5be.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 9 + file: temp2kappa.tex + label: nablaconv + mtref: '9.21' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 25890 + sourceTransform: tex2mdx.strip_comments + start: 25631 + targetStatement: + chapter: 9 + file: temp2kappa.tex + label: '' + mtref: '9.22' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 30396 + sourceTransform: tex2mdx.strip_comments + start: 30212 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 366e3fdadf6f +- 598f6059280c +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/65e7e1e9560e.md b/projects/poincare-conjecture/.astrolabe/edges/65e7e1e9560e.md new file mode 100644 index 00000000..9ce06971 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/65e7e1e9560e.md @@ -0,0 +1,62 @@ +--- +confidence: 0.9 +edgeClass: structural +evidence: + containmentDepth: 1 + proofSpan: + bodyEnd: 184898 + bodyStart: 173854 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 184909 + file: energy1.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 173841 + sourceStatement: + chapter: 18 + file: energy1.tex + label: Dxacross + mtref: '18.74' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 173839 + sourceTransform: tex2mdx.strip_comments + start: 173679 + targetContainedSpan: + coordinateSpace: comment-stripped-tex + end: 180809 + file: energy1.tex + sourceTransform: tex2mdx.strip_comments + start: 180734 + targetStatement: + chapter: 18 + file: energy1.tex + label: '' + mtref: '18.79' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 180809 + sourceTransform: tex2mdx.strip_comments + start: 180734 + type: proof-containment + via: + - proof-contains +generator: tools/poincare_tex_extract.py +inference: structural +kind: containment +ref: +- 0d6f74e37efb +- e967f00df0a8 +rel: contains +reviewStatus: accepted +scope: proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof-contains +--- +Morgan--Tian proof containment. The source is the statement whose proof contains the target statement. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/661f675ffcbd.md b/projects/poincare-conjecture/.astrolabe/edges/661f675ffcbd.md new file mode 100644 index 00000000..e69d2eed --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/661f675ffcbd.md @@ -0,0 +1,65 @@ +--- +confidence: 0.35 +edgeClass: prose +evidence: + mentionTriggers: + - label: kaplimit + refSpan: + coordinateSpace: comment-stripped-tex + end: 64555 + file: surgery.tex + sourceTransform: tex2mdx.strip_comments + start: 64541 + sentence: For all such $\epsilon$, Theorem~\ref{bcbd} holds for $\epsilon$ and + Proposition~\ref{narrows}, Proposition~\ref{canonvary} and Corollaries~\ref{kappacannbhd} + and~\ref{limitcannbhd} and Theorems~\ref{smlmtflow} and~\ref{kaplimit} hold + for $2\epsilon$ + sentenceSpan: + coordinateSpace: comment-stripped-tex + end: 64577 + file: surgery.tex + sourceTransform: tex2mdx.strip_comments + start: 64325 + sourceStrategy: nearest-previous-statement + via: prose + sourceStatement: + chapter: 15 + file: surgery.tex + label: neckglue + mtref: '15.2' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 61465 + sourceTransform: tex2mdx.strip_comments + start: 60812 + targetStatement: + chapter: 11 + file: singlimit2.tex + label: kaplimit + mtref: '11.8' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 16131 + sourceTransform: tex2mdx.strip_comments + start: 15040 + type: prose-mention-nearest-previous-statement + via: + - prose +generator: tools/poincare_tex_extract.py +inference: weak +kind: mention +ref: +- a992d6a89479 +- a9e023d082dc +rel: mentions +reviewStatus: candidate +scope: prose +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- prose +--- +Morgan--Tian prose mention. The source is the nearest preceding statement in the same chapter segment. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/6636a45a4a7b.md b/projects/poincare-conjecture/.astrolabe/edges/6636a45a4a7b.md new file mode 100644 index 00000000..95390b25 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/6636a45a4a7b.md @@ -0,0 +1,57 @@ +--- +confidence: 0.72 +edgeClass: semantic +evidence: + anaphoraTriggers: + - context: statement + direction: previous + phrase: previous example + sort: example + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 4730 + file: temp2kappa.tex + sourceTransform: tex2mdx.strip_comments + start: 4714 + via: local-anaphora-statement + sourceStatement: + chapter: 9 + file: temp2kappa.tex + label: '' + mtref: '9.4' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 4791 + sourceTransform: tex2mdx.strip_comments + start: 4628 + targetStatement: + chapter: 9 + file: temp2kappa.tex + label: '' + mtref: '9.3' + sort: example + span: + coordinateSpace: comment-stripped-tex + end: 4539 + sourceTransform: tex2mdx.strip_comments + start: 4142 + type: local-anaphora-reference + via: + - local-anaphora-statement +generator: tools/poincare_tex_extract.py +inference: inferred +kind: dependency +ref: +- 1b78c6bffb78 +- 41b9e2369bf0 +rel: depends +reviewStatus: accepted +scope: local-context +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- local-anaphora-statement +--- +Morgan--Tian local anaphora dependency. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/667fb1ddca1d.md b/projects/poincare-conjecture/.astrolabe/edges/667fb1ddca1d.md new file mode 100644 index 00000000..101dd922 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/667fb1ddca1d.md @@ -0,0 +1,65 @@ +--- +confidence: 0.45 +edgeClass: semantic +evidence: + endpointSemanticDegreeBeforeTermBackfill: + source: 0 + target: 1 + selectionReason: semantic-connectivity-backfill + sourceMatchSpan: + coordinateSpace: comment-stripped-tex + end: 4190 + file: flowbasics.tex + sourceTransform: tex2mdx.strip_comments + start: 4180 + sourceStatement: + chapter: 3 + file: flowbasics.tex + label: '' + mtref: '3.3' + sort: example + span: + coordinateSpace: comment-stripped-tex + end: 4618 + sourceTransform: tex2mdx.strip_comments + start: 4083 + targetStatement: + chapter: 3 + file: flowbasics.tex + label: '' + mtref: '3.1' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 943 + sourceTransform: tex2mdx.strip_comments + start: 265 + targetTermSpan: + coordinateSpace: comment-stripped-tex + end: 297 + file: flowbasics.tex + sourceTransform: tex2mdx.strip_comments + start: 287 + term: ricci flow + termAmbiguity: unique-in-chapter + termSource: italic + termSpecificity: not-covered-by-longer-defined-term + type: same-chapter-definition-term-match + via: + - definition-term:ricci flow +generator: tools/poincare_tex_extract.py +inference: weak +kind: definition-use +ref: +- 43be10053c44 +- 7e9caf7ba4da +rel: uses +reviewStatus: candidate +scope: same-chapter-term-match +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- definition-term:ricci flow +--- +Morgan--Tian same-chapter definition term match. This is a low-confidence candidate edge for review. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/66a81c4811f4.md b/projects/poincare-conjecture/.astrolabe/edges/66a81c4811f4.md new file mode 100644 index 00000000..be365ed3 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/66a81c4811f4.md @@ -0,0 +1,103 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: surgerydist + proofSpan: + bodyEnd: 54967 + bodyStart: 37263 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 54978 + file: energy1.tex + ownerLabel: W_2 + sourceTransform: tex2mdx.strip_comments + start: 37250 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 42351 + file: energy1.tex + sourceTransform: tex2mdx.strip_comments + start: 42334 + via: proof + - context: proof + label: surgerydist + proofSpan: + bodyEnd: 54967 + bodyStart: 37263 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 54978 + file: energy1.tex + ownerLabel: W_2 + sourceTransform: tex2mdx.strip_comments + start: 37250 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 50369 + file: energy1.tex + sourceTransform: tex2mdx.strip_comments + start: 50352 + via: proof + - context: proof + label: surgerydist + proofSpan: + bodyEnd: 54967 + bodyStart: 37263 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 54978 + file: energy1.tex + ownerLabel: W_2 + sourceTransform: tex2mdx.strip_comments + start: 37250 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 50857 + file: energy1.tex + sourceTransform: tex2mdx.strip_comments + start: 50840 + via: proof + sourceStatement: + chapter: 18 + file: energy1.tex + label: W_2 + mtref: '18.11' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 36484 + sourceTransform: tex2mdx.strip_comments + start: 36186 + targetStatement: + chapter: 15 + file: surgery.tex + label: surgerydist + mtref: '15.12' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 97780 + sourceTransform: tex2mdx.strip_comments + start: 96249 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 7bb0fa0c9412 +- 9021a4eca03f +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/66c76c1f2fbe.md b/projects/poincare-conjecture/.astrolabe/edges/66c76c1f2fbe.md new file mode 100644 index 00000000..3fb65946 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/66c76c1f2fbe.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: S2timesI + proofSpan: + bodyEnd: 32963 + bodyStart: 31019 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 32974 + file: canonnbhd.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 31006 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 32854 + file: canonnbhd.tex + sourceTransform: tex2mdx.strip_comments + start: 32840 + via: proof + sourceStatement: + chapter: 19 + file: canonnbhd.tex + label: epschains + mtref: '19.19' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 31004 + sourceTransform: tex2mdx.strip_comments + start: 30360 + targetStatement: + chapter: 19 + file: canonnbhd.tex + label: S2timesI + mtref: '19.13' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 23164 + sourceTransform: tex2mdx.strip_comments + start: 22970 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- ba681e15e1a6 +- 035b5e6ca144 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/66e0f775f1d0.md b/projects/poincare-conjecture/.astrolabe/edges/66e0f775f1d0.md new file mode 100644 index 00000000..63a61b95 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/66e0f775f1d0.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: grouptheory + proofSpan: + bodyEnd: 22144 + bodyStart: 20827 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 22155 + file: energy1.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 20814 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 22029 + file: energy1.tex + sourceTransform: tex2mdx.strip_comments + start: 22012 + via: proof + sourceStatement: + chapter: 18 + file: energy1.tex + label: '' + mtref: '18.5' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 20812 + sourceTransform: tex2mdx.strip_comments + start: 20598 + targetStatement: + chapter: 18 + file: energy1.tex + label: grouptheory + mtref: '18.3' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 19372 + sourceTransform: tex2mdx.strip_comments + start: 19084 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- a86000e8dd35 +- 858af0fef413 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/66fb9ff0c844.md b/projects/poincare-conjecture/.astrolabe/edges/66fb9ff0c844.md new file mode 100644 index 00000000..087fdd4a --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/66fb9ff0c844.md @@ -0,0 +1,65 @@ +--- +confidence: 0.45 +edgeClass: semantic +evidence: + endpointSemanticDegreeBeforeTermBackfill: + source: 2 + target: 0 + selectionReason: semantic-connectivity-backfill + sourceMatchSpan: + coordinateSpace: comment-stripped-tex + end: 20759 + file: energy1.tex + sourceTransform: tex2mdx.strip_comments + start: 20736 + sourceStatement: + chapter: 18 + file: energy1.tex + label: '' + mtref: '18.5' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 20812 + sourceTransform: tex2mdx.strip_comments + start: 20598 + targetStatement: + chapter: 18 + file: energy1.tex + label: '' + mtref: '18.4' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 20414 + sourceTransform: tex2mdx.strip_comments + start: 19743 + targetTermSpan: + coordinateSpace: comment-stripped-tex + end: 19887 + file: energy1.tex + sourceTransform: tex2mdx.strip_comments + start: 19864 + term: homotopically essential + termAmbiguity: unique-in-chapter + termSource: italic + termSpecificity: not-covered-by-longer-defined-term + type: same-chapter-definition-term-match + via: + - definition-term:homotopically essential +generator: tools/poincare_tex_extract.py +inference: weak +kind: definition-use +ref: +- a86000e8dd35 +- 5727e4c714af +rel: uses +reviewStatus: candidate +scope: same-chapter-term-match +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- definition-term:homotopically essential +--- +Morgan--Tian same-chapter definition term match. This is a low-confidence candidate edge for review. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/67083f375e8c.md b/projects/poincare-conjecture/.astrolabe/edges/67083f375e8c.md new file mode 100644 index 00000000..671cebfd --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/67083f375e8c.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 3 + file: flowbasics.tex + label: I.8.3 + mtref: '3.21' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 27841 + sourceTransform: tex2mdx.strip_comments + start: 27149 + targetStatement: + chapter: 3 + file: flowbasics.tex + label: '' + mtref: '3.22' + sort: remark + span: + coordinateSpace: comment-stripped-tex + end: 28114 + sourceTransform: tex2mdx.strip_comments + start: 27844 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 5d895f9c3918 +- e485edf8d992 +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/670f367a1f9b.md b/projects/poincare-conjecture/.astrolabe/edges/670f367a1f9b.md new file mode 100644 index 00000000..7bfa9ce7 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/670f367a1f9b.md @@ -0,0 +1,62 @@ +--- +confidence: 0.9 +edgeClass: structural +evidence: + containmentDepth: 1 + proofSpan: + bodyEnd: 14787 + bodyStart: 6954 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 14798 + file: singlimit2.tex + ownerLabel: smlmtflow + sourceTransform: tex2mdx.strip_comments + start: 6941 + sourceStatement: + chapter: 11 + file: singlimit2.tex + label: smlmtflow + mtref: '11.1' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 3459 + sourceTransform: tex2mdx.strip_comments + start: 1498 + targetContainedSpan: + coordinateSpace: comment-stripped-tex + end: 9079 + file: singlimit2.tex + sourceTransform: tex2mdx.strip_comments + start: 8940 + targetStatement: + chapter: 11 + file: singlimit2.tex + label: '' + mtref: '11.5' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 9079 + sourceTransform: tex2mdx.strip_comments + start: 8940 + type: proof-containment + via: + - proof-contains +generator: tools/poincare_tex_extract.py +inference: structural +kind: containment +ref: +- d51aab8a8065 +- 61143f134fcf +rel: contains +reviewStatus: accepted +scope: proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof-contains +--- +Morgan--Tian proof containment. The source is the statement whose proof contains the target statement. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/6727fc090815.md b/projects/poincare-conjecture/.astrolabe/edges/6727fc090815.md new file mode 100644 index 00000000..065ee7f3 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/6727fc090815.md @@ -0,0 +1,63 @@ +--- +confidence: 0.35 +edgeClass: prose +evidence: + mentionTriggers: + - label: fordiffquot + refSpan: + coordinateSpace: comment-stripped-tex + end: 36803 + file: energy1.tex + sourceTransform: tex2mdx.strip_comments + start: 36786 + sentence: Then by Lemma~\ref{fordiffquot} and Lemma~\ref{W_2} we have $W_2(t)\le + w_2(t)$ for all $t\ge T_0$ + sentenceSpan: + coordinateSpace: comment-stripped-tex + end: 36869 + file: energy1.tex + sourceTransform: tex2mdx.strip_comments + start: 36771 + sourceStrategy: nearest-previous-statement + via: prose + sourceStatement: + chapter: 18 + file: energy1.tex + label: W_2 + mtref: '18.11' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 36484 + sourceTransform: tex2mdx.strip_comments + start: 36186 + targetStatement: + chapter: 2 + file: prelim.tex + label: fordiffquot + mtref: '2.22' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 82182 + sourceTransform: tex2mdx.strip_comments + start: 81704 + type: prose-mention-nearest-previous-statement + via: + - prose +generator: tools/poincare_tex_extract.py +inference: weak +kind: mention +ref: +- 7bb0fa0c9412 +- 67c378f87457 +rel: mentions +reviewStatus: candidate +scope: prose +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- prose +--- +Morgan--Tian prose mention. The source is the nearest preceding statement in the same chapter segment. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/672b5aec4ef8.md b/projects/poincare-conjecture/.astrolabe/edges/672b5aec4ef8.md new file mode 100644 index 00000000..8fd04db5 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/672b5aec4ef8.md @@ -0,0 +1,65 @@ +--- +confidence: 0.45 +edgeClass: semantic +evidence: + endpointSemanticDegreeBeforeTermBackfill: + source: 0 + target: 0 + selectionReason: semantic-connectivity-backfill + sourceMatchSpan: + coordinateSpace: comment-stripped-tex + end: 2317 + file: stdsoln.tex + sourceTransform: tex2mdx.strip_comments + start: 2294 + sourceStatement: + chapter: 12 + file: stdsoln.tex + label: stdinit + mtref: '12.2' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 2328 + sourceTransform: tex2mdx.strip_comments + start: 2256 + targetStatement: + chapter: 12 + file: stdsoln.tex + label: defn:stinmetr + mtref: '12.1' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 1454 + sourceTransform: tex2mdx.strip_comments + start: 657 + targetTermSpan: + coordinateSpace: comment-stripped-tex + end: 721 + file: stdsoln.tex + sourceTransform: tex2mdx.strip_comments + start: 698 + term: standard initial metric + termAmbiguity: unique-in-chapter + termSource: italic + termSpecificity: no-longer-defined-term + type: same-chapter-definition-term-match + via: + - definition-term:standard initial metric +generator: tools/poincare_tex_extract.py +inference: weak +kind: definition-use +ref: +- f1d3021d32b8 +- c5e007547ff3 +rel: uses +reviewStatus: candidate +scope: same-chapter-term-match +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- definition-term:standard initial metric +--- +Morgan--Tian same-chapter definition term match. This is a low-confidence candidate edge for review. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/673aa12960fa.md b/projects/poincare-conjecture/.astrolabe/edges/673aa12960fa.md new file mode 100644 index 00000000..47b4b283 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/673aa12960fa.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 6 + file: newcompar.tex + label: tausmooth + mtref: '6.31' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 44056 + sourceTransform: tex2mdx.strip_comments + start: 43590 + targetStatement: + chapter: 6 + file: newcompar.tex + label: Ltau + mtref: '6.32' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 45443 + sourceTransform: tex2mdx.strip_comments + start: 45026 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 4b9af3660539 +- b3bf2f526cfc +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/67a0d930ebbc.md b/projects/poincare-conjecture/.astrolabe/edges/67a0d930ebbc.md new file mode 100644 index 00000000..c070445c --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/67a0d930ebbc.md @@ -0,0 +1,64 @@ +--- +confidence: 0.35 +edgeClass: prose +evidence: + mentionTriggers: + - label: lformula + refSpan: + coordinateSpace: comment-stripped-tex + end: 11345 + file: newcomp2.tex + sourceTransform: tex2mdx.strip_comments + start: 11331 + sentence: Our purpose here is to extend the first differential inequality given + in Corollary~\ref{lformula} to a differential inequality in the weak or distributional + sense for $l_{x}$ valid on all of $M\times (0,T)$ + sentenceSpan: + coordinateSpace: comment-stripped-tex + end: 11454 + file: newcomp2.tex + sourceTransform: tex2mdx.strip_comments + start: 11246 + sourceStrategy: nearest-previous-statement + via: prose + sourceStatement: + chapter: 7 + file: newcomp2.tex + label: '' + mtref: '7.6' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 11044 + sourceTransform: tex2mdx.strip_comments + start: 10270 + targetStatement: + chapter: 6 + file: newcompar.tex + label: lformula + mtref: '6.51' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 81082 + sourceTransform: tex2mdx.strip_comments + start: 80194 + type: prose-mention-nearest-previous-statement + via: + - prose +generator: tools/poincare_tex_extract.py +inference: weak +kind: mention +ref: +- 2e14dd57a334 +- 967c0f8bb774 +rel: mentions +reviewStatus: candidate +scope: prose +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- prose +--- +Morgan--Tian prose mention. The source is the nearest preceding statement in the same chapter segment. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/67c26b27498e.md b/projects/poincare-conjecture/.astrolabe/edges/67c26b27498e.md new file mode 100644 index 00000000..ec7eaed5 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/67c26b27498e.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: dvolevol + proofSpan: + bodyEnd: 18850 + bodyStart: 18573 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 18861 + file: flowbasics.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 18560 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 18847 + file: flowbasics.tex + sourceTransform: tex2mdx.strip_comments + start: 18833 + via: proof-aux-label-proxy + sourceStatement: + chapter: 3 + file: flowbasics.tex + label: '' + mtref: '3.15' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 18558 + sourceTransform: tex2mdx.strip_comments + start: 18005 + targetStatement: + chapter: 3 + file: flowbasics.tex + label: thmRmevol + mtref: '3.13' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 16962 + sourceTransform: tex2mdx.strip_comments + start: 16120 + type: tex-reference + via: + - proof-aux-label-proxy +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 41667bcb0cab +- 1515151790b2 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof-aux-label-proxy +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/67e0c4cf5a18.md b/projects/poincare-conjecture/.astrolabe/edges/67e0c4cf5a18.md new file mode 100644 index 00000000..caa65685 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/67e0c4cf5a18.md @@ -0,0 +1,99 @@ +--- +confidence: 0.35 +edgeClass: prose +evidence: + mentionTriggers: + - label: finiteext + refSpan: + coordinateSpace: comment-stripped-tex + end: 84273 + file: intro.tex + sourceTransform: tex2mdx.strip_comments + start: 84258 + sentence: The last topic we discuss is the proof of the finite-time extinction\index{finite-time + extinction} for Ricci flows with initial metrics satisfying the hypothesis of + Theorem~\ref{finiteext}. + sentenceSpan: + coordinateSpace: comment-stripped-tex + end: 84274 + file: intro.tex + sourceTransform: tex2mdx.strip_comments + start: 84082 + sourceStrategy: nearest-previous-statement + via: prose + - label: finiteext + refSpan: + coordinateSpace: comment-stripped-tex + end: 87244 + file: intro.tex + sourceTransform: tex2mdx.strip_comments + start: 87229 + sentence: This argument showing that eventually every component of the time-slice + $t$ has trivial $\pi_2$ is not necessary for the topological application (Theorem~\ref{finiteext}), + or indeed, for any other topological application + sentenceSpan: + coordinateSpace: comment-stripped-tex + end: 87295 + file: intro.tex + sourceTransform: tex2mdx.strip_comments + start: 87074 + sourceStrategy: nearest-previous-statement + via: prose + - label: finiteext + refSpan: + coordinateSpace: comment-stripped-tex + end: 92084 + file: intro.tex + sourceTransform: tex2mdx.strip_comments + start: 92069 + sentence: Hence, under the hypothesis on the fundamental group in Theorem~\ref{finiteext} + the entire manifold must disappear at finite time. + sentenceSpan: + coordinateSpace: comment-stripped-tex + end: 92135 + file: intro.tex + sourceTransform: tex2mdx.strip_comments + start: 92004 + sourceStrategy: nearest-previous-statement + via: prose + sourceStatement: + chapter: 0 + file: intro.tex + label: '' + mtref: '0.8' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 60281 + sourceTransform: tex2mdx.strip_comments + start: 60167 + targetStatement: + chapter: 0 + file: intro.tex + label: finiteext + mtref: '0.4' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 12644 + sourceTransform: tex2mdx.strip_comments + start: 11822 + type: prose-mention-nearest-previous-statement + via: + - prose +generator: tools/poincare_tex_extract.py +inference: weak +kind: mention +ref: +- 42089b0a1d9e +- 127db1dc9315 +rel: mentions +reviewStatus: candidate +scope: prose +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- prose +--- +Morgan--Tian prose mention. The source is the nearest preceding statement in the same chapter segment. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/67e8b3bc0cfb.md b/projects/poincare-conjecture/.astrolabe/edges/67e8b3bc0cfb.md new file mode 100644 index 00000000..9967dcf7 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/67e8b3bc0cfb.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 5 + file: converge2.tex + label: basicconv + mtref: '5.6' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 9928 + sourceTransform: tex2mdx.strip_comments + start: 8668 + targetStatement: + chapter: 5 + file: converge2.tex + label: '' + mtref: '5.7' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 10866 + sourceTransform: tex2mdx.strip_comments + start: 10009 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 74c7682a68ed +- e5006fafe096 +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/67ea47047424.md b/projects/poincare-conjecture/.astrolabe/edges/67ea47047424.md new file mode 100644 index 00000000..bc3ccb1a --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/67ea47047424.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 4 + file: maxprin.tex + label: posderiv + mtref: '4.39' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 53144 + sourceTransform: tex2mdx.strip_comments + start: 52904 + targetStatement: + chapter: 4 + file: maxprin.tex + label: Harnack2 + mtref: '4.40' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 53826 + sourceTransform: tex2mdx.strip_comments + start: 53456 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 36dd2f6fa9c2 +- 6b21448ff214 +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/680297f1f009.md b/projects/poincare-conjecture/.astrolabe/edges/680297f1f009.md new file mode 100644 index 00000000..55f6ee5d --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/680297f1f009.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: noncompkappa + proofSpan: + bodyEnd: 175215 + bodyStart: 173916 + coordinateSpace: comment-stripped-tex + depth: 2 + end: 175226 + file: temp2kappa.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 173903 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 174761 + file: temp2kappa.tex + sourceTransform: tex2mdx.strip_comments + start: 174743 + via: proof + sourceStatement: + chapter: 9 + file: temp2kappa.tex + label: '' + mtref: '9.90' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 173901 + sourceTransform: tex2mdx.strip_comments + start: 173501 + targetStatement: + chapter: 9 + file: temp2kappa.tex + label: noncompkappa + mtref: '9.88' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 170123 + sourceTransform: tex2mdx.strip_comments + start: 169708 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 494bc110f8ab +- 274cda9be677 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/6826b32a3fd0.md b/projects/poincare-conjecture/.astrolabe/edges/6826b32a3fd0.md new file mode 100644 index 00000000..f861aaeb --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/6826b32a3fd0.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: controlnbhd + proofSpan: + bodyEnd: 93873 + bodyStart: 90350 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 93884 + file: stdsoln.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 90337 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 90754 + file: stdsoln.tex + sourceTransform: tex2mdx.strip_comments + start: 90737 + via: proof + sourceStatement: + chapter: 12 + file: stdsoln.tex + label: Restim + mtref: '12.31' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 90335 + sourceTransform: tex2mdx.strip_comments + start: 90184 + targetStatement: + chapter: 11 + file: singlimit2.tex + label: controlnbhd + mtref: '11.2' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 4360 + sourceTransform: tex2mdx.strip_comments + start: 3597 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- ffae4504318a +- 0ddfe578961e +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/682e338dda9c.md b/projects/poincare-conjecture/.astrolabe/edges/682e338dda9c.md new file mode 100644 index 00000000..49772aba --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/682e338dda9c.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: neckcurv + proofSpan: + bodyEnd: 27479 + bodyStart: 27169 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 27490 + file: canonnbhd.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 27156 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 27209 + file: canonnbhd.tex + sourceTransform: tex2mdx.strip_comments + start: 27195 + via: proof + sourceStatement: + chapter: 19 + file: canonnbhd.tex + label: '' + mtref: '19.16' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 27153 + sourceTransform: tex2mdx.strip_comments + start: 26638 + targetStatement: + chapter: 19 + file: canonnbhd.tex + label: neckcurv + mtref: '19.2' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 3124 + sourceTransform: tex2mdx.strip_comments + start: 2315 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- d5a1d88909aa +- dc5709757703 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/68402dfa0649.md b/projects/poincare-conjecture/.astrolabe/edges/68402dfa0649.md new file mode 100644 index 00000000..0c8c6f9b --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/68402dfa0649.md @@ -0,0 +1,65 @@ +--- +confidence: 0.45 +edgeClass: semantic +evidence: + endpointSemanticDegreeBeforeTermBackfill: + source: 0 + target: 5 + selectionReason: semantic-connectivity-backfill + sourceMatchSpan: + coordinateSpace: comment-stripped-tex + end: 180205 + file: temp2kappa.tex + sourceTransform: tex2mdx.strip_comments + start: 180161 + sourceStatement: + chapter: 9 + file: temp2kappa.tex + label: kappacannbhd + mtref: '9.94' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 180278 + sourceTransform: tex2mdx.strip_comments + start: 180007 + targetStatement: + chapter: 9 + file: temp2kappa.tex + label: defncanonnbhd + mtref: '9.78' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 147589 + sourceTransform: tex2mdx.strip_comments + start: 145644 + targetTermSpan: + coordinateSpace: comment-stripped-tex + end: 147206 + file: temp2kappa.tex + sourceTransform: tex2mdx.strip_comments + start: 147162 + term: strong canonical neighborhood + termAmbiguity: unique-in-chapter + termSource: italic + termSpecificity: no-longer-defined-term + type: same-chapter-definition-term-match + via: + - definition-term:strong canonical neighborhood +generator: tools/poincare_tex_extract.py +inference: weak +kind: definition-use +ref: +- fc2d2036a96c +- 3aa5883ab629 +rel: uses +reviewStatus: candidate +scope: same-chapter-term-match +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- definition-term:strong canonical neighborhood +--- +Morgan--Tian same-chapter definition term match. This is a low-confidence candidate edge for review. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/68437980bb3d.md b/projects/poincare-conjecture/.astrolabe/edges/68437980bb3d.md new file mode 100644 index 00000000..11450b12 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/68437980bb3d.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 18 + file: energy1.tex + label: '' + mtref: '18.13' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 40079 + sourceTransform: tex2mdx.strip_comments + start: 39789 + targetStatement: + chapter: 18 + file: energy1.tex + label: spherethm + mtref: '18.14' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 43696 + sourceTransform: tex2mdx.strip_comments + start: 43231 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 1554c9624fa7 +- a679e0086677 +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/684e7375d351.md b/projects/poincare-conjecture/.astrolabe/edges/684e7375d351.md new file mode 100644 index 00000000..0fcab5ec --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/684e7375d351.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: controlnbhd + proofSpan: + bodyEnd: 227397 + bodyStart: 226225 + coordinateSpace: comment-stripped-tex + depth: 2 + end: 227408 + file: surgery.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 226212 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 226338 + file: surgery.tex + sourceTransform: tex2mdx.strip_comments + start: 226321 + via: proof + sourceStatement: + chapter: 17 + file: surgery.tex + label: '' + mtref: '17.11' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 226210 + sourceTransform: tex2mdx.strip_comments + start: 226024 + targetStatement: + chapter: 11 + file: singlimit2.tex + label: controlnbhd + mtref: '11.2' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 4360 + sourceTransform: tex2mdx.strip_comments + start: 3597 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 8dafe9c95b7f +- 0ddfe578961e +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/685acc43aed6.md b/projects/poincare-conjecture/.astrolabe/edges/685acc43aed6.md new file mode 100644 index 00000000..1bc18939 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/685acc43aed6.md @@ -0,0 +1,62 @@ +--- +confidence: 0.9 +edgeClass: structural +evidence: + containmentDepth: 1 + proofSpan: + bodyEnd: 53552 + bodyStart: 35551 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 53563 + file: canonnbhd.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 35538 + sourceStatement: + chapter: 19 + file: canonnbhd.tex + label: Xcontainedin + mtref: '19.21' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 35375 + sourceTransform: tex2mdx.strip_comments + start: 34343 + targetContainedSpan: + coordinateSpace: comment-stripped-tex + end: 36055 + file: canonnbhd.tex + sourceTransform: tex2mdx.strip_comments + start: 35803 + targetStatement: + chapter: 19 + file: canonnbhd.tex + label: '' + mtref: '19.22' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 36055 + sourceTransform: tex2mdx.strip_comments + start: 35803 + type: proof-containment + via: + - proof-contains +generator: tools/poincare_tex_extract.py +inference: structural +kind: containment +ref: +- 0e9a1a167caa +- 9737d037f97b +rel: contains +reviewStatus: accepted +scope: proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof-contains +--- +Morgan--Tian proof containment. The source is the statement whose proof contains the target statement. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/686209d3eb95.md b/projects/poincare-conjecture/.astrolabe/edges/686209d3eb95.md new file mode 100644 index 00000000..5d3c38f3 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/686209d3eb95.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 5 + file: converge2.tex + label: seqconv + mtref: '5.21' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 34707 + sourceTransform: tex2mdx.strip_comments + start: 34006 + targetStatement: + chapter: 5 + file: converge2.tex + label: '' + mtref: '5.22' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 35882 + sourceTransform: tex2mdx.strip_comments + start: 35265 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 7744e53809c5 +- 5be0c15e901a +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/68b0bad9b6af.md b/projects/poincare-conjecture/.astrolabe/edges/68b0bad9b6af.md new file mode 100644 index 00000000..40cbd07f --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/68b0bad9b6af.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 18 + file: energy1.tex + label: '1887' + mtref: '18.88' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 200887 + sourceTransform: tex2mdx.strip_comments + start: 200520 + targetStatement: + chapter: 18 + file: energy1.tex + label: claim1885 + mtref: '18.89' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 202725 + sourceTransform: tex2mdx.strip_comments + start: 202260 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 80e5fc6759b2 +- 1f7d2e438fef +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/68bfb105f61d.md b/projects/poincare-conjecture/.astrolabe/edges/68bfb105f61d.md new file mode 100644 index 00000000..8047dde6 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/68bfb105f61d.md @@ -0,0 +1,45 @@ +--- +confidence: 0.2 +edgeClass: structural +evidence: + sourceStatement: + chapter: 18 + file: energy1.tex + label: curvshrexist + mtref: '18.29' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 68623 + sourceTransform: tex2mdx.strip_comments + start: 68167 + targetStatement: + chapter: 18 + file: energy1.tex + label: embcase + mtref: '18.30' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 69934 + sourceTransform: tex2mdx.strip_comments + start: 69415 + type: chapter-reading-order-adjacency + via: + - chapter-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 656c8fbce129 +- f98ccf7ea70d +rel: in-chapter +reviewStatus: accepted +scope: chapter +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- chapter-sequence +--- +Morgan--Tian reading-order adjacency across sections in a chapter. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/68c52cb39152.md b/projects/poincare-conjecture/.astrolabe/edges/68c52cb39152.md new file mode 100644 index 00000000..a8397d59 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/68c52cb39152.md @@ -0,0 +1,76 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: statement + label: defnoverlinedelta0 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 198985 + file: surgery.tex + sourceTransform: tex2mdx.strip_comments + start: 198961 + via: statement + - context: proof + label: defnoverlinedelta0 + proofSpan: + bodyEnd: 200569 + bodyStart: 199052 + coordinateSpace: comment-stripped-tex + depth: 3 + end: 200580 + file: surgery.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 199039 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 200079 + file: surgery.tex + sourceTransform: tex2mdx.strip_comments + start: 200055 + via: proof + sourceStatement: + chapter: 17 + file: surgery.tex + label: '' + mtref: '17.3' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 199036 + sourceTransform: tex2mdx.strip_comments + start: 198777 + targetStatement: + chapter: 16 + file: surgery.tex + label: defnoverlinedelta0 + mtref: '16.12' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 130201 + sourceTransform: tex2mdx.strip_comments + start: 129888 + type: tex-reference + via: + - proof + - statement +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- d130b5adaed6 +- b92e1a990021 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +- statement +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/6906e4c7caea.md b/projects/poincare-conjecture/.astrolabe/edges/6906e4c7caea.md new file mode 100644 index 00000000..1a0941d1 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/6906e4c7caea.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: compactkappa + proofSpan: + bodyEnd: 139120 + bodyStart: 137924 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 139131 + file: temp2kappa.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 137911 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 139118 + file: temp2kappa.tex + sourceTransform: tex2mdx.strip_comments + start: 139100 + via: proof + sourceStatement: + chapter: 9 + file: temp2kappa.tex + label: rho + mtref: '9.70' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 137332 + sourceTransform: tex2mdx.strip_comments + start: 137165 + targetStatement: + chapter: 9 + file: temp2kappa.tex + label: compactkappa + mtref: '9.64' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 127434 + sourceTransform: tex2mdx.strip_comments + start: 127205 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 4bff7f1e7bc4 +- 49e865228b19 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/690c5f0b036d.md b/projects/poincare-conjecture/.astrolabe/edges/690c5f0b036d.md new file mode 100644 index 00000000..a0fb2cce --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/690c5f0b036d.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: Rminprime + proofSpan: + bodyEnd: 5289 + bodyStart: 3014 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 5300 + file: maxprin.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 3001 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 4228 + file: maxprin.tex + sourceTransform: tex2mdx.strip_comments + start: 4213 + via: proof + sourceStatement: + chapter: 4 + file: maxprin.tex + label: scalarevol + mtref: '4.1' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 2998 + sourceTransform: tex2mdx.strip_comments + start: 2387 + targetStatement: + chapter: 4 + file: maxprin.tex + label: Rminprime + mtref: '4.3' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 4069 + sourceTransform: tex2mdx.strip_comments + start: 3817 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 95cc838cfc6d +- 84e573c0f2b2 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/692f0818afee.md b/projects/poincare-conjecture/.astrolabe/edges/692f0818afee.md new file mode 100644 index 00000000..ca4b42e2 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/692f0818afee.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: controlnbhd + proofSpan: + bodyEnd: 33794 + bodyStart: 17659 + coordinateSpace: comment-stripped-tex + depth: 2 + end: 33805 + file: singlimit2.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 17646 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 18521 + file: singlimit2.tex + sourceTransform: tex2mdx.strip_comments + start: 18504 + via: proof + sourceStatement: + chapter: 11 + file: singlimit2.tex + label: maxT + mtref: '11.10' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 17643 + sourceTransform: tex2mdx.strip_comments + start: 17183 + targetStatement: + chapter: 11 + file: singlimit2.tex + label: controlnbhd + mtref: '11.2' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 4360 + sourceTransform: tex2mdx.strip_comments + start: 3597 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- d6919616448d +- 0ddfe578961e +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/697a6e2cc60d.md b/projects/poincare-conjecture/.astrolabe/edges/697a6e2cc60d.md new file mode 100644 index 00000000..c268d993 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/697a6e2cc60d.md @@ -0,0 +1,45 @@ +--- +confidence: 0.2 +edgeClass: structural +evidence: + sourceStatement: + chapter: 15 + file: surgery.tex + label: surgeryparams + mtref: '15.7' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 80864 + sourceTransform: tex2mdx.strip_comments + start: 79853 + targetStatement: + chapter: 15 + file: surgery.tex + label: '' + mtref: '15.8' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 89238 + sourceTransform: tex2mdx.strip_comments + start: 89057 + type: chapter-reading-order-adjacency + via: + - chapter-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 22c40a5664ac +- 6c167a310932 +rel: in-chapter +reviewStatus: accepted +scope: chapter +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- chapter-sequence +--- +Morgan--Tian reading-order adjacency across sections in a chapter. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/69a48723658a.md b/projects/poincare-conjecture/.astrolabe/edges/69a48723658a.md new file mode 100644 index 00000000..162c4e7e --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/69a48723658a.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 9 + file: temp2kappa.tex + label: '' + mtref: '9.86' + sort: remark + span: + coordinateSpace: comment-stripped-tex + end: 163155 + sourceTransform: tex2mdx.strip_comments + start: 163012 + targetStatement: + chapter: 9 + file: temp2kappa.tex + label: '' + mtref: '9.87' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 167340 + sourceTransform: tex2mdx.strip_comments + start: 167251 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 51da06702a9f +- b9f671d590f3 +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/69bd2931732f.md b/projects/poincare-conjecture/.astrolabe/edges/69bd2931732f.md new file mode 100644 index 00000000..a3d1f881 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/69bd2931732f.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: triL + proofSpan: + bodyEnd: 125200 + bodyStart: 122543 + coordinateSpace: comment-stripped-tex + depth: 2 + end: 125211 + file: newcompar.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 122530 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 124050 + file: newcompar.tex + sourceTransform: tex2mdx.strip_comments + start: 124040 + via: proof + sourceStatement: + chapter: 6 + file: newcompar.tex + label: fmonotone + mtref: '6.81' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 122525 + sourceTransform: tex2mdx.strip_comments + start: 122141 + targetStatement: + chapter: 6 + file: newcompar.tex + label: triL + mtref: '6.43' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 70273 + sourceTransform: tex2mdx.strip_comments + start: 68899 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 4dc40f37cd6d +- 882677eecdc1 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/6a3a9c49140b.md b/projects/poincare-conjecture/.astrolabe/edges/6a3a9c49140b.md new file mode 100644 index 00000000..76fcaf9f --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/6a3a9c49140b.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 5 + file: converge2.tex + label: '' + mtref: '5.32' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 50532 + sourceTransform: tex2mdx.strip_comments + start: 49697 + targetStatement: + chapter: 5 + file: converge2.tex + label: blowupposcurv + mtref: '5.33' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 51450 + sourceTransform: tex2mdx.strip_comments + start: 50791 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 666538bf577e +- 2ddfbc74b081 +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/6a3d0a94e3e1.md b/projects/poincare-conjecture/.astrolabe/edges/6a3d0a94e3e1.md new file mode 100644 index 00000000..0f12eb5e --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/6a3d0a94e3e1.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 9 + file: temp2kappa.tex + label: defnepsround + mtref: '9.76' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 144692 + sourceTransform: tex2mdx.strip_comments + start: 144236 + targetStatement: + chapter: 9 + file: temp2kappa.tex + label: '' + mtref: '9.77' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 145405 + sourceTransform: tex2mdx.strip_comments + start: 144759 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 14ff8384814f +- cdb7d427d07e +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/6b0711a3c4d1.md b/projects/poincare-conjecture/.astrolabe/edges/6b0711a3c4d1.md new file mode 100644 index 00000000..61f78373 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/6b0711a3c4d1.md @@ -0,0 +1,65 @@ +--- +confidence: 0.35 +edgeClass: prose +evidence: + mentionTriggers: + - label: xistronger + refSpan: + coordinateSpace: comment-stripped-tex + end: 111675 + file: energy1.tex + sourceTransform: tex2mdx.strip_comments + start: 111659 + sentence: (Actually, we establish a slightly stronger result, see Lemma~\ref{xistronger}.) + The next step in the argument is to take a finite subset ${\mathcal S}\subset + S^2$ so that for every $c\in S^2$ there is $\hat c\in {\mathcal S}$ such that + $\widetilde\Gamma(c)$ and $\widetilde\Gamma(\hat c)$ are sufficiently close + sentenceSpan: + coordinateSpace: comment-stripped-tex + end: 111909 + file: energy1.tex + sourceTransform: tex2mdx.strip_comments + start: 111596 + sourceStrategy: nearest-previous-statement + via: prose + sourceStatement: + chapter: 18 + file: energy1.tex + label: '' + mtref: '18.50' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 107575 + sourceTransform: tex2mdx.strip_comments + start: 107348 + targetStatement: + chapter: 18 + file: energy1.tex + label: xistronger + mtref: '18.53' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 123454 + sourceTransform: tex2mdx.strip_comments + start: 122771 + type: prose-mention-nearest-previous-statement + via: + - prose +generator: tools/poincare_tex_extract.py +inference: weak +kind: mention +ref: +- cb69804a2166 +- 9df99ae6f4cb +rel: mentions +reviewStatus: candidate +scope: prose +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- prose +--- +Morgan--Tian prose mention. The source is the nearest preceding statement in the same chapter segment. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/6b22884f9ce1.md b/projects/poincare-conjecture/.astrolabe/edges/6b22884f9ce1.md new file mode 100644 index 00000000..b99ac35d --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/6b22884f9ce1.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: THM + proofSpan: + bodyEnd: 117032 + bodyStart: 113723 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 117043 + file: temp2kappa.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 113710 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 116081 + file: temp2kappa.tex + sourceTransform: tex2mdx.strip_comments + start: 116072 + via: proof + sourceStatement: + chapter: 9 + file: temp2kappa.tex + label: kappa0 + mtref: '9.58' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 113708 + sourceTransform: tex2mdx.strip_comments + start: 113563 + targetStatement: + chapter: 8 + file: noncoll.tex + label: THM + mtref: '8.1' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 1970 + sourceTransform: tex2mdx.strip_comments + start: 682 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 9c54e0d0c7d8 +- 3defa9b2209c +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/6b3adc7d4041.md b/projects/poincare-conjecture/.astrolabe/edges/6b3adc7d4041.md new file mode 100644 index 00000000..b832887c --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/6b3adc7d4041.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: L_0 + proofSpan: + bodyEnd: 173338 + bodyStart: 165469 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 173349 + file: surgery.tex + ownerLabel: delta0ri+1 + sourceTransform: tex2mdx.strip_comments + start: 165456 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 165826 + file: surgery.tex + sourceTransform: tex2mdx.strip_comments + start: 165817 + via: proof + sourceStatement: + chapter: 16 + file: surgery.tex + label: delta0ri+1 + mtref: '16.4' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 109162 + sourceTransform: tex2mdx.strip_comments + start: 107682 + targetStatement: + chapter: 16 + file: surgery.tex + label: L_0 + mtref: '16.15' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 138538 + sourceTransform: tex2mdx.strip_comments + start: 137276 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 89a60eb69c3f +- 86043191a3c2 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/6b71ed7b949e.md b/projects/poincare-conjecture/.astrolabe/edges/6b71ed7b949e.md new file mode 100644 index 00000000..bb9ecc7d --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/6b71ed7b949e.md @@ -0,0 +1,62 @@ +--- +confidence: 0.35 +edgeClass: prose +evidence: + mentionTriggers: + - label: Hessineq + refSpan: + coordinateSpace: comment-stripped-tex + end: 60073 + file: newcompar.tex + sourceTransform: tex2mdx.strip_comments + start: 60059 + sentence: Now we are ready to begin the proof of Proposition~\ref{Hessineq}. + sentenceSpan: + coordinateSpace: comment-stripped-tex + end: 60074 + file: newcompar.tex + sourceTransform: tex2mdx.strip_comments + start: 60007 + sourceStrategy: nearest-previous-statement + via: prose + sourceStatement: + chapter: 6 + file: newcompar.tex + label: adapjab + mtref: '6.41' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 59462 + sourceTransform: tex2mdx.strip_comments + start: 58890 + targetStatement: + chapter: 6 + file: newcompar.tex + label: Hessineq + mtref: '6.37' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 56065 + sourceTransform: tex2mdx.strip_comments + start: 55015 + type: prose-mention-nearest-previous-statement + via: + - prose +generator: tools/poincare_tex_extract.py +inference: weak +kind: mention +ref: +- 72949e6350d4 +- 88d4ef67440f +rel: mentions +reviewStatus: candidate +scope: prose +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- prose +--- +Morgan--Tian prose mention. The source is the nearest preceding statement in the same chapter segment. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/6b7b71067fff.md b/projects/poincare-conjecture/.astrolabe/edges/6b7b71067fff.md new file mode 100644 index 00000000..b10fc008 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/6b7b71067fff.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 3 + file: flowbasics.tex + label: shi + mtref: '3.28' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 40541 + sourceTransform: tex2mdx.strip_comments + start: 39904 + targetStatement: + chapter: 3 + file: flowbasics.tex + label: shiw/deriv + mtref: '3.29' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 42464 + sourceTransform: tex2mdx.strip_comments + start: 41312 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- ed4b72caf9f9 +- e0b6a1f646a7 +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/6b85f10876b6.md b/projects/poincare-conjecture/.astrolabe/edges/6b85f10876b6.md new file mode 100644 index 00000000..534ffe2a --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/6b85f10876b6.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: flowtoround + proofSpan: + bodyEnd: 51714 + bodyStart: 50457 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 51725 + file: maxprin.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 50444 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 51472 + file: maxprin.tex + sourceTransform: tex2mdx.strip_comments + start: 51455 + via: proof + sourceStatement: + chapter: 4 + file: maxprin.tex + label: '' + mtref: '4.35' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 50442 + sourceTransform: tex2mdx.strip_comments + start: 50109 + targetStatement: + chapter: 4 + file: maxprin.tex + label: flowtoround + mtref: '4.23' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 35932 + sourceTransform: tex2mdx.strip_comments + start: 34642 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- c7dbaa8c2ca5 +- 86e5a5766bda +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/6bd7ba97b419.md b/projects/poincare-conjecture/.astrolabe/edges/6bd7ba97b419.md new file mode 100644 index 00000000..8853d823 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/6bd7ba97b419.md @@ -0,0 +1,64 @@ +--- +confidence: 0.35 +edgeClass: prose +evidence: + mentionTriggers: + - label: KAPPALIMIT + refSpan: + coordinateSpace: comment-stripped-tex + end: 104584 + file: surgery.tex + sourceTransform: tex2mdx.strip_comments + start: 104568 + sentence: We suppose that we have a Ricci flow with surgery $({\mathcal M},G)$ + defined for $0\le t0$ there is a point $q(\tau)\in M$ at which + the function $l_x(\cdot,^\tau)$ achieves its minimum, and furthermore, we have + $$l_x(q(\tau),\tau) \leq \frac{n}{2}.$$ + sentenceSpan: + coordinateSpace: comment-stripped-tex + end: 12202 + file: temp2kappa.tex + sourceTransform: tex2mdx.strip_comments + start: 11928 + sourceStrategy: nearest-previous-statement + via: prose + sourceStatement: + chapter: 9 + file: temp2kappa.tex + label: unianc + mtref: '9.10' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 10335 + sourceTransform: tex2mdx.strip_comments + start: 9624 + targetStatement: + chapter: 7 + file: newcomp2.tex + label: n/2 + mtref: '7.10' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 15803 + sourceTransform: tex2mdx.strip_comments + start: 15517 + type: prose-mention-nearest-previous-statement + via: + - prose +generator: tools/poincare_tex_extract.py +inference: weak +kind: mention +ref: +- 8b8e3a0f3e36 +- aebc3ff2bd3b +rel: mentions +reviewStatus: candidate +scope: prose +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- prose +--- +Morgan--Tian prose mention. The source is the nearest preceding statement in the same chapter segment. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/91bd125abd8b.md b/projects/poincare-conjecture/.astrolabe/edges/91bd125abd8b.md new file mode 100644 index 00000000..c63308a0 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/91bd125abd8b.md @@ -0,0 +1,84 @@ +--- +confidence: 0.35 +edgeClass: prose +evidence: + mentionTriggers: + - label: bcbd + refSpan: + coordinateSpace: comment-stripped-tex + end: 65252 + file: bddcurvbdddist.tex + sourceTransform: tex2mdx.strip_comments + start: 65242 + sentence: The contradiction shows that the limit $(U_\infty,g_\infty,x_\infty)$ + cannot exist. The only assumption that we made in order to construct this limit + was that Theorem~\ref{bcbd} did not hold for some $A_0<\infty$. Thus, we have + established Theorem~\ref{bcbd} by contradiction. + sentenceSpan: + coordinateSpace: comment-stripped-tex + end: 65353 + file: bddcurvbdddist.tex + sourceTransform: tex2mdx.strip_comments + start: 65074 + sourceStrategy: nearest-previous-statement + via: prose + - label: bcbd + refSpan: + coordinateSpace: comment-stripped-tex + end: 65333 + file: bddcurvbdddist.tex + sourceTransform: tex2mdx.strip_comments + start: 65323 + sentence: The contradiction shows that the limit $(U_\infty,g_\infty,x_\infty)$ + cannot exist. The only assumption that we made in order to construct this limit + was that Theorem~\ref{bcbd} did not hold for some $A_0<\infty$. Thus, we have + established Theorem~\ref{bcbd} by contradiction. + sentenceSpan: + coordinateSpace: comment-stripped-tex + end: 65353 + file: bddcurvbdddist.tex + sourceTransform: tex2mdx.strip_comments + start: 65074 + sourceStrategy: nearest-previous-statement + via: prose + sourceStatement: + chapter: 10 + file: bddcurvbdddist.tex + label: '' + mtref: '10.31' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 61603 + sourceTransform: tex2mdx.strip_comments + start: 61488 + targetStatement: + chapter: 10 + file: bddcurvbdddist.tex + label: bcbd + mtref: '10.2' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 2942 + sourceTransform: tex2mdx.strip_comments + start: 2168 + type: prose-mention-nearest-previous-statement + via: + - prose +generator: tools/poincare_tex_extract.py +inference: weak +kind: mention +ref: +- 6d6b9e050054 +- ecd62fc2ea2b +rel: mentions +reviewStatus: candidate +scope: prose +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- prose +--- +Morgan--Tian prose mention. The source is the nearest preceding statement in the same chapter segment. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/9249a06ee285.md b/projects/poincare-conjecture/.astrolabe/edges/9249a06ee285.md new file mode 100644 index 00000000..b477a0f7 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/9249a06ee285.md @@ -0,0 +1,56 @@ +--- +confidence: 0.85 +edgeClass: semantic +evidence: + containmentReason: target theorem-like statement occurs inside source proof + proofSpan: + bodyEnd: 146058 + bodyStart: 138906 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 146069 + file: surgery.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 138893 + sourceStatement: + chapter: 16 + file: surgery.tex + label: '' + mtref: '16.16' + sort: remark + span: + coordinateSpace: comment-stripped-tex + end: 138750 + sourceTransform: tex2mdx.strip_comments + start: 138540 + targetStatement: + chapter: 16 + file: surgery.tex + label: '' + mtref: '16.17' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 140633 + sourceTransform: tex2mdx.strip_comments + start: 140555 + type: proof-contained-theorem-like-statement + via: + - proof-contained-statement +generator: tools/poincare_tex_extract.py +inference: inferred +kind: dependency +ref: +- 4ee1bc9acc48 +- 9b7f421ab0b9 +rel: depends +reviewStatus: accepted +scope: proof-containment +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof-contained-statement +--- +Morgan--Tian proof-contained statement dependency. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/924eab23facd.md b/projects/poincare-conjecture/.astrolabe/edges/924eab23facd.md new file mode 100644 index 00000000..19455e32 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/924eab23facd.md @@ -0,0 +1,62 @@ +--- +confidence: 0.9 +edgeClass: structural +evidence: + containmentDepth: 2 + proofSpan: + bodyEnd: 33794 + bodyStart: 17659 + coordinateSpace: comment-stripped-tex + depth: 2 + end: 33805 + file: singlimit2.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 17646 + sourceStatement: + chapter: 11 + file: singlimit2.tex + label: maxT + mtref: '11.10' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 17643 + sourceTransform: tex2mdx.strip_comments + start: 17183 + targetContainedSpan: + coordinateSpace: comment-stripped-tex + end: 29827 + file: singlimit2.tex + sourceTransform: tex2mdx.strip_comments + start: 29411 + targetStatement: + chapter: 11 + file: singlimit2.tex + label: ancientlimit + mtref: '11.16' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 29827 + sourceTransform: tex2mdx.strip_comments + start: 29411 + type: proof-containment + via: + - proof-contains +generator: tools/poincare_tex_extract.py +inference: structural +kind: containment +ref: +- d6919616448d +- a66fa0ea8720 +rel: contains +reviewStatus: accepted +scope: proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof-contains +--- +Morgan--Tian proof containment. The source is the statement whose proof contains the target statement. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/928dc8f6e77e.md b/projects/poincare-conjecture/.astrolabe/edges/928dc8f6e77e.md new file mode 100644 index 00000000..897cb012 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/928dc8f6e77e.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: directions + proofSpan: + bodyEnd: 40692 + bodyStart: 38685 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 40703 + file: bddcurvbdddist.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 38672 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 39736 + file: bddcurvbdddist.tex + sourceTransform: tex2mdx.strip_comments + start: 39720 + via: proof + sourceStatement: + chapter: 10 + file: bddcurvbdddist.tex + label: '' + mtref: '10.14' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 38669 + sourceTransform: tex2mdx.strip_comments + start: 38296 + targetStatement: + chapter: 19 + file: canonnbhd.tex + label: directions + mtref: '19.4' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 6165 + sourceTransform: tex2mdx.strip_comments + start: 5408 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 8d3abe3790d7 +- 88a5a2233418 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/929ec25a0b4d.md b/projects/poincare-conjecture/.astrolabe/edges/929ec25a0b4d.md new file mode 100644 index 00000000..75e26d5d --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/929ec25a0b4d.md @@ -0,0 +1,62 @@ +--- +confidence: 0.35 +edgeClass: prose +evidence: + mentionTriggers: + - label: annlengthest + refSpan: + coordinateSpace: comment-stripped-tex + end: 8084 + file: energy1.tex + sourceTransform: tex2mdx.strip_comments + start: 8066 + sentence: see Proposition~\ref{annlengthest} for a precise statement + sentenceSpan: + coordinateSpace: comment-stripped-tex + end: 8108 + file: energy1.tex + sourceTransform: tex2mdx.strip_comments + start: 8049 + sourceStrategy: nearest-previous-statement + via: prose + sourceStatement: + chapter: 18 + file: energy1.tex + label: extinct + mtref: '18.1' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 1107 + sourceTransform: tex2mdx.strip_comments + start: 511 + targetStatement: + chapter: 18 + file: energy1.tex + label: annlengthest + mtref: '18.63' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 153504 + sourceTransform: tex2mdx.strip_comments + start: 152870 + type: prose-mention-nearest-previous-statement + via: + - prose +generator: tools/poincare_tex_extract.py +inference: weak +kind: mention +ref: +- 4b8a44aa316f +- 6e9739b0c59b +rel: mentions +reviewStatus: candidate +scope: prose +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- prose +--- +Morgan--Tian prose mention. The source is the nearest preceding statement in the same chapter segment. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/929ef8140f08.md b/projects/poincare-conjecture/.astrolabe/edges/929ef8140f08.md new file mode 100644 index 00000000..b940703f --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/929ef8140f08.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: symmY + proofSpan: + bodyEnd: 22715 + bodyStart: 21316 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 22726 + file: newcompar.tex + ownerLabel: 2ndvarj + sourceTransform: tex2mdx.strip_comments + start: 21303 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 22583 + file: newcompar.tex + sourceTransform: tex2mdx.strip_comments + start: 22572 + via: proof + sourceStatement: + chapter: 6 + file: newcompar.tex + label: 2ndvarj + mtref: '6.13' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 16190 + sourceTransform: tex2mdx.strip_comments + start: 15621 + targetStatement: + chapter: 6 + file: newcompar.tex + label: symmY + mtref: '6.16' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 19484 + sourceTransform: tex2mdx.strip_comments + start: 19147 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 26b7f69e803e +- abd40a465b53 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/92e1a1da89e2.md b/projects/poincare-conjecture/.astrolabe/edges/92e1a1da89e2.md new file mode 100644 index 00000000..51a2ba73 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/92e1a1da89e2.md @@ -0,0 +1,66 @@ +--- +confidence: 0.35 +edgeClass: prose +evidence: + mentionTriggers: + - label: GSSclass + refSpan: + coordinateSpace: comment-stripped-tex + end: 88815 + file: temp2kappa.tex + sourceTransform: tex2mdx.strip_comments + start: 88801 + sentence: Since the original Riemannian manifold $(M,g)$ given in the statement + of Theorem~\ref{GSSclass} is $\kappa$-non-collapsed, it follows that, for every + $t<0$, the Riemannian manifold $(M,G(t))$ is $\kappa$-non-collapsed. Consequently, + the Ricci flow $(M,G(t))$ is $\kappa$-non-collapsed. It clearly has bounded + non-negative curvature on each time-slice and is non-flat + sentenceSpan: + coordinateSpace: comment-stripped-tex + end: 89086 + file: temp2kappa.tex + sourceTransform: tex2mdx.strip_comments + start: 88719 + sourceStrategy: nearest-previous-statement + via: prose + sourceStatement: + chapter: 9 + file: temp2kappa.tex + label: '' + mtref: '9.48' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 86082 + sourceTransform: tex2mdx.strip_comments + start: 86005 + targetStatement: + chapter: 9 + file: temp2kappa.tex + label: GSSclass + mtref: '9.42' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 75321 + sourceTransform: tex2mdx.strip_comments + start: 74299 + type: prose-mention-nearest-previous-statement + via: + - prose +generator: tools/poincare_tex_extract.py +inference: weak +kind: mention +ref: +- 6060c7910950 +- 2d68880771ff +rel: mentions +reviewStatus: candidate +scope: prose +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- prose +--- +Morgan--Tian prose mention. The source is the nearest preceding statement in the same chapter segment. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/92f1f1c74b2b.md b/projects/poincare-conjecture/.astrolabe/edges/92f1f1c74b2b.md new file mode 100644 index 00000000..0253c098 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/92f1f1c74b2b.md @@ -0,0 +1,45 @@ +--- +confidence: 0.2 +edgeClass: structural +evidence: + sourceStatement: + chapter: 7 + file: newcomp2.tex + label: compactcoro + mtref: '7.12' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 21856 + sourceTransform: tex2mdx.strip_comments + start: 20802 + targetStatement: + chapter: 7 + file: newcomp2.tex + label: weak + mtref: '7.13' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 23409 + sourceTransform: tex2mdx.strip_comments + start: 22180 + type: chapter-reading-order-adjacency + via: + - chapter-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 71669e7fd5ec +- 7d5d94efe4e9 +rel: in-chapter +reviewStatus: accepted +scope: chapter +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- chapter-sequence +--- +Morgan--Tian reading-order adjacency across sections in a chapter. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/9319a71bb875.md b/projects/poincare-conjecture/.astrolabe/edges/9319a71bb875.md new file mode 100644 index 00000000..788dc2d0 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/9319a71bb875.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 9 + file: temp2kappa.tex + label: '' + mtref: '9.5' + sort: example + span: + coordinateSpace: comment-stripped-tex + end: 5373 + sourceTransform: tex2mdx.strip_comments + start: 4794 + targetStatement: + chapter: 9 + file: temp2kappa.tex + label: '' + mtref: '9.6' + sort: example + span: + coordinateSpace: comment-stripped-tex + end: 5640 + sourceTransform: tex2mdx.strip_comments + start: 5387 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- d506ee1eeb7e +- 6345ee49174f +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/93691264b4ab.md b/projects/poincare-conjecture/.astrolabe/edges/93691264b4ab.md new file mode 100644 index 00000000..573688c2 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/93691264b4ab.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 18 + file: energy1.tex + label: theta/Kbd + mtref: '18.67' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 162289 + sourceTransform: tex2mdx.strip_comments + start: 162047 + targetStatement: + chapter: 18 + file: energy1.tex + label: avgeo + mtref: '18.68' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 163185 + sourceTransform: tex2mdx.strip_comments + start: 162959 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 512c74b9ad1f +- ea7c58d04fc7 +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/938585e4b70b.md b/projects/poincare-conjecture/.astrolabe/edges/938585e4b70b.md new file mode 100644 index 00000000..8447b962 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/938585e4b70b.md @@ -0,0 +1,45 @@ +--- +confidence: 0.2 +edgeClass: structural +evidence: + sourceStatement: + chapter: 14 + file: surgery.tex + label: '' + mtref: '14.1' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 28555 + sourceTransform: tex2mdx.strip_comments + start: 27727 + targetStatement: + chapter: 14 + file: surgery.tex + label: spacetime + mtref: '14.2' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 40726 + sourceTransform: tex2mdx.strip_comments + start: 38087 + type: chapter-reading-order-adjacency + via: + - chapter-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 41da6871e864 +- 95fbdd1a8a2d +rel: in-chapter +reviewStatus: accepted +scope: chapter +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- chapter-sequence +--- +Morgan--Tian reading-order adjacency across sections in a chapter. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/9391c5892c6f.md b/projects/poincare-conjecture/.astrolabe/edges/9391c5892c6f.md new file mode 100644 index 00000000..f3b6cd4c --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/9391c5892c6f.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: fordiffquot + proofSpan: + bodyEnd: 38050 + bodyStart: 37779 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 38061 + file: flowbasics.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 37766 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 38048 + file: flowbasics.tex + sourceTransform: tex2mdx.strip_comments + start: 38031 + via: proof + sourceStatement: + chapter: 3 + file: flowbasics.tex + label: corI.8.3 + mtref: '3.26' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 37763 + sourceTransform: tex2mdx.strip_comments + start: 37390 + targetStatement: + chapter: 2 + file: prelim.tex + label: fordiffquot + mtref: '2.22' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 82182 + sourceTransform: tex2mdx.strip_comments + start: 81704 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 01a829e22cf6 +- 67c378f87457 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/939e36689d0e.md b/projects/poincare-conjecture/.astrolabe/edges/939e36689d0e.md new file mode 100644 index 00000000..afdcaf0b --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/939e36689d0e.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 4 + file: maxprin.tex + label: firststrongmax + mtref: '4.18' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 23916 + sourceTransform: tex2mdx.strip_comments + start: 23628 + targetStatement: + chapter: 4 + file: maxprin.tex + label: localprod + mtref: '4.19' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 25705 + sourceTransform: tex2mdx.strip_comments + start: 25065 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 016cf36686b2 +- 8ec855a2aab5 +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/93f562af1f5c.md b/projects/poincare-conjecture/.astrolabe/edges/93f562af1f5c.md new file mode 100644 index 00000000..1f57449f --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/93f562af1f5c.md @@ -0,0 +1,55 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: statement + label: assumptions + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 53073 + file: singlimit2.tex + sourceTransform: tex2mdx.strip_comments + start: 53056 + via: statement + sourceStatement: + chapter: 11 + file: singlimit2.tex + label: 2epshorns + mtref: '11.30' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 53428 + sourceTransform: tex2mdx.strip_comments + start: 52907 + targetStatement: + chapter: 11 + file: singlimit2.tex + label: assumptions + mtref: '11.18' + sort: assumption + span: + coordinateSpace: comment-stripped-tex + end: 36515 + sourceTransform: tex2mdx.strip_comments + start: 35753 + type: tex-reference + via: + - statement +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 01f0d3baa28f +- 3009be0a37fc +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- statement +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/94028129d46e.md b/projects/poincare-conjecture/.astrolabe/edges/94028129d46e.md new file mode 100644 index 00000000..eef99b4f --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/94028129d46e.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: global + proofSpan: + bodyEnd: 137790 + bodyStart: 127828 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 137801 + file: temp2kappa.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 127815 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 137566 + file: temp2kappa.tex + sourceTransform: tex2mdx.strip_comments + start: 137554 + via: proof + sourceStatement: + chapter: 9 + file: temp2kappa.tex + label: proposition + mtref: '9.65' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 127812 + sourceTransform: tex2mdx.strip_comments + start: 127549 + targetStatement: + chapter: 9 + file: temp2kappa.tex + label: global + mtref: '9.63' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 126734 + sourceTransform: tex2mdx.strip_comments + start: 126308 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- ee314aacd207 +- 329d0175aed8 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/9430356ec05b.md b/projects/poincare-conjecture/.astrolabe/edges/9430356ec05b.md new file mode 100644 index 00000000..d6bbdab0 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/9430356ec05b.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: neckgeo + proofSpan: + bodyEnd: 28989 + bodyStart: 28502 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 29000 + file: canonnbhd.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 28489 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 28529 + file: canonnbhd.tex + sourceTransform: tex2mdx.strip_comments + start: 28516 + via: proof + sourceStatement: + chapter: 19 + file: canonnbhd.tex + label: firstchain + mtref: '19.17' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 28487 + sourceTransform: tex2mdx.strip_comments + start: 27933 + targetStatement: + chapter: 19 + file: canonnbhd.tex + label: neckgeo + mtref: '19.5' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 8075 + sourceTransform: tex2mdx.strip_comments + start: 7413 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 58804e22eff2 +- b74cd360f72f +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/94418a3381ec.md b/projects/poincare-conjecture/.astrolabe/edges/94418a3381ec.md new file mode 100644 index 00000000..5db708d4 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/94418a3381ec.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 9 + file: temp2kappa.tex + label: 4times + mtref: '9.37' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 65333 + sourceTransform: tex2mdx.strip_comments + start: 64727 + targetStatement: + chapter: 9 + file: temp2kappa.tex + label: pp + mtref: '9.38' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 67234 + sourceTransform: tex2mdx.strip_comments + start: 66738 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 351728d82639 +- b71a179769b1 +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/9483a1f18fd4.md b/projects/poincare-conjecture/.astrolabe/edges/9483a1f18fd4.md new file mode 100644 index 00000000..60ee4405 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/9483a1f18fd4.md @@ -0,0 +1,80 @@ +--- +confidence: 0.8 +edgeClass: semantic +evidence: + proseTriggers: + - label: surgery + refSpan: + coordinateSpace: comment-stripped-tex + end: 12858 + file: intro.tex + sourceTransform: tex2mdx.strip_comments + start: 12845 + sentence: 'We immediately deduce Theorem~\ref{Theorem1} from Theorems~\ref{surgery} + and~\ref{finiteext} as follows: Let $M$ be a $3$-manifold satisfying the hypothesis + of Theorem~\ref{Theorem1}. Then there is a finite sequence $M=M_0,M_1,\ldots,M_k=\emptyset$ + such that for each $i,\ 1\le i\le k$, $M_i$ is obtained from $M_{i-1}$ by a + connected sum decomposition or $M_i$ is obtained from $M_{i-1}$ by removing + a component diffeomorphic to one of $S^2\times S^1$, $\Ar P^3\#\Ar P^3$, a non-orientable + $2$-sphere bundle over $S^1$, or a $3$-dimensional spherical space-form. Clearly, + it follows by downward induction on $i$ that each connected component of $M_i$ + is diffeomorphic to a connected sum of $3$-dimensional spherical space-forms, + copies of $S^2\times S^1$, and copies of the non-orientable $2$-sphere bundle + over $S^1$' + sentenceSpan: + coordinateSpace: comment-stripped-tex + end: 13608 + file: intro.tex + sourceTransform: tex2mdx.strip_comments + start: 12785 + sourceLabel: Theorem1 + sourceRefSpan: + coordinateSpace: comment-stripped-tex + end: 12830 + file: intro.tex + sourceTransform: tex2mdx.strip_comments + start: 12816 + sourceStrategy: refs-in-inferential-sentence + via: + - prose-deduce-from + sourceStatement: + chapter: 0 + file: intro.tex + label: Theorem1 + mtref: '0.1' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 2955 + sourceTransform: tex2mdx.strip_comments + start: 2574 + targetStatement: + chapter: 0 + file: intro.tex + label: surgery + mtref: '0.3' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 11419 + sourceTransform: tex2mdx.strip_comments + start: 10567 + type: inferential-prose-reference + via: + - prose-deduce-from +generator: tools/poincare_tex_extract.py +inference: inferred +kind: dependency +ref: +- 9999efa2be2c +- 847af3ecd4a8 +rel: depends +reviewStatus: accepted +scope: prose +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- prose-deduce-from +--- +Morgan--Tian high-confidence prose dependency. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/949ee4f91d86.md b/projects/poincare-conjecture/.astrolabe/edges/949ee4f91d86.md new file mode 100644 index 00000000..c752f816 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/949ee4f91d86.md @@ -0,0 +1,63 @@ +--- +confidence: 0.35 +edgeClass: prose +evidence: + mentionTriggers: + - label: noncompsol + refSpan: + coordinateSpace: comment-stripped-tex + end: 94147 + file: temp2kappa.tex + sourceTransform: tex2mdx.strip_comments + start: 94131 + sentence: Now that we have shown that every two-dimensional $\kappa$-solution + is a shrinking family of round surfaces, we can complete the proof of Proposition~\ref{noncompsol} + sentenceSpan: + coordinateSpace: comment-stripped-tex + end: 94147 + file: temp2kappa.tex + sourceTransform: tex2mdx.strip_comments + start: 93980 + sourceStrategy: nearest-previous-statement + via: prose + sourceStatement: + chapter: 9 + file: temp2kappa.tex + label: 2DGSS + mtref: '9.50' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 91949 + sourceTransform: tex2mdx.strip_comments + start: 91266 + targetStatement: + chapter: 9 + file: temp2kappa.tex + label: noncompsol + mtref: '9.46' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 82429 + sourceTransform: tex2mdx.strip_comments + start: 82233 + type: prose-mention-nearest-previous-statement + via: + - prose +generator: tools/poincare_tex_extract.py +inference: weak +kind: mention +ref: +- da767c71996b +- 03957890f08f +rel: mentions +reviewStatus: candidate +scope: prose +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- prose +--- +Morgan--Tian prose mention. The source is the nearest preceding statement in the same chapter segment. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/94d3c28dfd12.md b/projects/poincare-conjecture/.astrolabe/edges/94d3c28dfd12.md new file mode 100644 index 00000000..8ad74f06 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/94d3c28dfd12.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 1 + file: prelim.tex + label: Bianchi + mtref: '1.5' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 8462 + sourceTransform: tex2mdx.strip_comments + start: 7974 + targetStatement: + chapter: 1 + file: prelim.tex + label: '' + mtref: '1.6' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 8971 + sourceTransform: tex2mdx.strip_comments + start: 8513 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- c4848a1b6f80 +- 71e017db1c5b +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/94e0739f111f.md b/projects/poincare-conjecture/.astrolabe/edges/94e0739f111f.md new file mode 100644 index 00000000..f0223e25 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/94e0739f111f.md @@ -0,0 +1,45 @@ +--- +confidence: 0.2 +edgeClass: structural +evidence: + sourceStatement: + chapter: 7 + file: newcomp2.tex + label: '' + mtref: '7.4' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 8412 + sourceTransform: tex2mdx.strip_comments + start: 7565 + targetStatement: + chapter: 7 + file: newcomp2.tex + label: lipcomplete + mtref: '7.5' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 9437 + sourceTransform: tex2mdx.strip_comments + start: 8692 + type: chapter-reading-order-adjacency + via: + - chapter-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 0bfc692d99b1 +- 16fbe263a7c8 +rel: in-chapter +reviewStatus: accepted +scope: chapter +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- chapter-sequence +--- +Morgan--Tian reading-order adjacency across sections in a chapter. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/94e282d72d25.md b/projects/poincare-conjecture/.astrolabe/edges/94e282d72d25.md new file mode 100644 index 00000000..2ee89aef --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/94e282d72d25.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: '16.2' + proofSpan: + bodyEnd: 229182 + bodyStart: 185422 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 229193 + file: surgery.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 185409 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 205411 + file: surgery.tex + sourceTransform: tex2mdx.strip_comments + start: 205401 + via: proof + sourceStatement: + chapter: 17 + file: surgery.tex + label: extend + mtref: '17.1' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 185406 + sourceTransform: tex2mdx.strip_comments + start: 184285 + targetStatement: + chapter: 17 + file: surgery.tex + label: '16.2' + mtref: '17.2' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 186718 + sourceTransform: tex2mdx.strip_comments + start: 186200 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 254e8dae97e6 +- 778d690c4e7c +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/954200e9bb66.md b/projects/poincare-conjecture/.astrolabe/edges/954200e9bb66.md new file mode 100644 index 00000000..e2a1e4c2 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/954200e9bb66.md @@ -0,0 +1,56 @@ +--- +confidence: 0.85 +edgeClass: semantic +evidence: + containmentReason: target theorem-like statement occurs inside source proof + proofSpan: + bodyEnd: 109929 + bodyStart: 106397 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 109940 + file: energy1.tex + ownerLabel: goodapprox + sourceTransform: tex2mdx.strip_comments + start: 106384 + sourceStatement: + chapter: 18 + file: energy1.tex + label: goodapprox + mtref: '18.45' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 100508 + sourceTransform: tex2mdx.strip_comments + start: 99535 + targetStatement: + chapter: 18 + file: energy1.tex + label: '' + mtref: '18.50' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 107575 + sourceTransform: tex2mdx.strip_comments + start: 107348 + type: proof-contained-theorem-like-statement + via: + - proof-contained-statement +generator: tools/poincare_tex_extract.py +inference: inferred +kind: dependency +ref: +- 6bb9851ce528 +- cb69804a2166 +rel: depends +reviewStatus: accepted +scope: proof-containment +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof-contained-statement +--- +Morgan--Tian proof-contained statement dependency. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/956c07bd36e6.md b/projects/poincare-conjecture/.astrolabe/edges/956c07bd36e6.md new file mode 100644 index 00000000..5a63f200 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/956c07bd36e6.md @@ -0,0 +1,65 @@ +--- +confidence: 0.45 +edgeClass: semantic +evidence: + endpointSemanticDegreeBeforeTermBackfill: + source: 0 + target: 5 + selectionReason: semantic-connectivity-backfill + sourceMatchSpan: + coordinateSpace: comment-stripped-tex + end: 74478 + file: newcompar.tex + sourceTransform: tex2mdx.strip_comments + start: 74470 + sourceStatement: + chapter: 6 + file: newcompar.tex + label: '' + mtref: '6.46' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 74785 + sourceTransform: tex2mdx.strip_comments + start: 74434 + targetStatement: + chapter: 6 + file: newcompar.tex + label: '' + mtref: '6.7' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 10481 + sourceTransform: tex2mdx.strip_comments + start: 10161 + targetTermSpan: + coordinateSpace: comment-stripped-tex + end: 10318 + file: newcompar.tex + sourceTransform: tex2mdx.strip_comments + start: 10310 + term: geodesic + termAmbiguity: unique-in-chapter + termSource: italic + termSpecificity: not-covered-by-longer-defined-term + type: same-chapter-definition-term-match + via: + - definition-term:geodesic +generator: tools/poincare_tex_extract.py +inference: weak +kind: definition-use +ref: +- c958eb1d5427 +- d23dc29f2aae +rel: uses +reviewStatus: candidate +scope: same-chapter-term-match +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- definition-term:geodesic +--- +Morgan--Tian same-chapter definition term match. This is a low-confidence candidate edge for review. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/958f7d622c14.md b/projects/poincare-conjecture/.astrolabe/edges/958f7d622c14.md new file mode 100644 index 00000000..78db18dc --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/958f7d622c14.md @@ -0,0 +1,62 @@ +--- +confidence: 0.9 +edgeClass: structural +evidence: + containmentDepth: 2 + proofSpan: + bodyEnd: 220240 + bodyStart: 211487 + coordinateSpace: comment-stripped-tex + depth: 2 + end: 220251 + file: surgery.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 211474 + sourceStatement: + chapter: 17 + file: surgery.tex + label: lemstdcan + mtref: '17.7' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 211471 + sourceTransform: tex2mdx.strip_comments + start: 210838 + targetContainedSpan: + coordinateSpace: comment-stripped-tex + end: 212622 + file: surgery.tex + sourceTransform: tex2mdx.strip_comments + start: 212469 + targetStatement: + chapter: 17 + file: surgery.tex + label: '' + mtref: '17.8' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 212622 + sourceTransform: tex2mdx.strip_comments + start: 212469 + type: proof-containment + via: + - proof-contains +generator: tools/poincare_tex_extract.py +inference: structural +kind: containment +ref: +- 111551d7831b +- a01b63baabde +rel: contains +reviewStatus: accepted +scope: proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof-contains +--- +Morgan--Tian proof containment. The source is the statement whose proof contains the target statement. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/95a7efaf9d4b.md b/projects/poincare-conjecture/.astrolabe/edges/95a7efaf9d4b.md new file mode 100644 index 00000000..76c717c7 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/95a7efaf9d4b.md @@ -0,0 +1,65 @@ +--- +confidence: 0.45 +edgeClass: semantic +evidence: + endpointSemanticDegreeBeforeTermBackfill: + source: 0 + target: 4 + selectionReason: semantic-connectivity-backfill + sourceMatchSpan: + coordinateSpace: comment-stripped-tex + end: 56523 + file: newcompar.tex + sourceTransform: tex2mdx.strip_comments + start: 56515 + sourceStatement: + chapter: 6 + file: newcompar.tex + label: '' + mtref: '6.39' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 57207 + sourceTransform: tex2mdx.strip_comments + start: 56367 + targetStatement: + chapter: 6 + file: newcompar.tex + label: '' + mtref: '6.7' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 10481 + sourceTransform: tex2mdx.strip_comments + start: 10161 + targetTermSpan: + coordinateSpace: comment-stripped-tex + end: 10318 + file: newcompar.tex + sourceTransform: tex2mdx.strip_comments + start: 10310 + term: geodesic + termAmbiguity: unique-in-chapter + termSource: italic + termSpecificity: not-covered-by-longer-defined-term + type: same-chapter-definition-term-match + via: + - definition-term:geodesic +generator: tools/poincare_tex_extract.py +inference: weak +kind: definition-use +ref: +- 2e4da429dc95 +- d23dc29f2aae +rel: uses +reviewStatus: candidate +scope: same-chapter-term-match +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- definition-term:geodesic +--- +Morgan--Tian same-chapter definition term match. This is a low-confidence candidate edge for review. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/95bed3ac7c41.md b/projects/poincare-conjecture/.astrolabe/edges/95bed3ac7c41.md new file mode 100644 index 00000000..369b3a19 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/95bed3ac7c41.md @@ -0,0 +1,114 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: statement + label: smlmtflow + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 60448 + file: singlimit2.tex + sourceTransform: tex2mdx.strip_comments + start: 60433 + via: statement + - context: proof + label: smlmtflow + proofSpan: + bodyEnd: 62795 + bodyStart: 60476 + coordinateSpace: comment-stripped-tex + depth: 2 + end: 62806 + file: singlimit2.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 60463 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 60546 + file: singlimit2.tex + sourceTransform: tex2mdx.strip_comments + start: 60531 + via: proof + - context: proof + label: smlmtflow + proofSpan: + bodyEnd: 62795 + bodyStart: 60476 + coordinateSpace: comment-stripped-tex + depth: 2 + end: 62806 + file: singlimit2.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 60463 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 61007 + file: singlimit2.tex + sourceTransform: tex2mdx.strip_comments + start: 60992 + via: proof + - context: proof + label: smlmtflow + proofSpan: + bodyEnd: 62795 + bodyStart: 60476 + coordinateSpace: comment-stripped-tex + depth: 2 + end: 62806 + file: singlimit2.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 60463 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 62756 + file: singlimit2.tex + sourceTransform: tex2mdx.strip_comments + start: 62741 + via: proof + sourceStatement: + chapter: 11 + file: singlimit2.tex + label: hyp + mtref: '11.32' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 60461 + sourceTransform: tex2mdx.strip_comments + start: 60310 + targetStatement: + chapter: 11 + file: singlimit2.tex + label: smlmtflow + mtref: '11.1' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 3459 + sourceTransform: tex2mdx.strip_comments + start: 1498 + type: tex-reference + via: + - proof + - statement +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- dff170186455 +- d51aab8a8065 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +- statement +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/95d1da98461d.md b/projects/poincare-conjecture/.astrolabe/edges/95d1da98461d.md new file mode 100644 index 00000000..5a774e3b --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/95d1da98461d.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 1 + file: prelim.tex + label: divRic + mtref: '1.9' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 12596 + sourceTransform: tex2mdx.strip_comments + start: 12512 + targetStatement: + chapter: 1 + file: prelim.tex + label: lapformula + mtref: '1.10' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 14031 + sourceTransform: tex2mdx.strip_comments + start: 13808 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 9c83686198e8 +- 8d03f3ca0aea +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/95f56402dffc.md b/projects/poincare-conjecture/.astrolabe/edges/95f56402dffc.md new file mode 100644 index 00000000..839a735f --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/95f56402dffc.md @@ -0,0 +1,45 @@ +--- +confidence: 0.2 +edgeClass: structural +evidence: + sourceStatement: + chapter: 9 + file: temp2kappa.tex + label: '' + mtref: '9.36' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 62699 + sourceTransform: tex2mdx.strip_comments + start: 62344 + targetStatement: + chapter: 9 + file: temp2kappa.tex + label: 4times + mtref: '9.37' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 65333 + sourceTransform: tex2mdx.strip_comments + start: 64727 + type: chapter-reading-order-adjacency + via: + - chapter-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 64100c8689ff +- 351728d82639 +rel: in-chapter +reviewStatus: accepted +scope: chapter +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- chapter-sequence +--- +Morgan--Tian reading-order adjacency across sections in a chapter. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/95fb3ae6c7e0.md b/projects/poincare-conjecture/.astrolabe/edges/95fb3ae6c7e0.md new file mode 100644 index 00000000..774e0537 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/95fb3ae6c7e0.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 9 + file: temp2kappa.tex + label: '' + mtref: '9.3' + sort: example + span: + coordinateSpace: comment-stripped-tex + end: 4539 + sourceTransform: tex2mdx.strip_comments + start: 4142 + targetStatement: + chapter: 9 + file: temp2kappa.tex + label: '' + mtref: '9.4' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 4791 + sourceTransform: tex2mdx.strip_comments + start: 4628 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 41b9e2369bf0 +- 1b78c6bffb78 +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/960a2ffb89ce.md b/projects/poincare-conjecture/.astrolabe/edges/960a2ffb89ce.md new file mode 100644 index 00000000..66a5aea0 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/960a2ffb89ce.md @@ -0,0 +1,45 @@ +--- +confidence: 0.2 +edgeClass: structural +evidence: + sourceStatement: + chapter: 4 + file: maxprin.tex + label: '' + mtref: '4.5' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 6940 + sourceTransform: tex2mdx.strip_comments + start: 6465 + targetStatement: + chapter: 4 + file: maxprin.tex + label: '' + mtref: '4.6' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 7829 + sourceTransform: tex2mdx.strip_comments + start: 7225 + type: chapter-reading-order-adjacency + via: + - chapter-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 5db278e369ae +- f3441849c78b +rel: in-chapter +reviewStatus: accepted +scope: chapter +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- chapter-sequence +--- +Morgan--Tian reading-order adjacency across sections in a chapter. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/9637e7696748.md b/projects/poincare-conjecture/.astrolabe/edges/9637e7696748.md new file mode 100644 index 00000000..9322560b --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/9637e7696748.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 6 + file: newcompar.tex + label: fullmeasure + mtref: '6.67' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 101253 + sourceTransform: tex2mdx.strip_comments + start: 100568 + targetStatement: + chapter: 6 + file: newcompar.tex + label: twosets + mtref: '6.68' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 102215 + sourceTransform: tex2mdx.strip_comments + start: 101653 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 7aad50e49af2 +- a673ebd63624 +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/964c7016605b.md b/projects/poincare-conjecture/.astrolabe/edges/964c7016605b.md new file mode 100644 index 00000000..af5086d2 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/964c7016605b.md @@ -0,0 +1,55 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: statement + label: firstshi + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 42571 + file: flowbasics.tex + sourceTransform: tex2mdx.strip_comments + start: 42557 + via: statement + sourceStatement: + chapter: 3 + file: flowbasics.tex + label: '' + mtref: '3.30' + sort: remark + span: + coordinateSpace: comment-stripped-tex + end: 42583 + sourceTransform: tex2mdx.strip_comments + start: 42467 + targetStatement: + chapter: 3 + file: flowbasics.tex + label: firstshi + mtref: '3.27' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 39652 + sourceTransform: tex2mdx.strip_comments + start: 39155 + type: tex-reference + via: + - statement +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- b51fee7a4b41 +- dbe4fabd0076 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- statement +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/967ce38aee22.md b/projects/poincare-conjecture/.astrolabe/edges/967ce38aee22.md new file mode 100644 index 00000000..b44ed54a --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/967ce38aee22.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: stdkappa + proofSpan: + bodyEnd: 85838 + bodyStart: 83533 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 85849 + file: stdsoln.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 83520 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 84698 + file: stdsoln.tex + sourceTransform: tex2mdx.strip_comments + start: 84684 + via: proof + sourceStatement: + chapter: 12 + file: stdsoln.tex + label: R_0cannbhd + mtref: '12.28' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 83517 + sourceTransform: tex2mdx.strip_comments + start: 82969 + targetStatement: + chapter: 12 + file: stdsoln.tex + label: stdkappa + mtref: '12.13' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 32215 + sourceTransform: tex2mdx.strip_comments + start: 32043 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 85e3c925177e +- ed069175cdf9 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/96b3f446f259.md b/projects/poincare-conjecture/.astrolabe/edges/96b3f446f259.md new file mode 100644 index 00000000..65994724 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/96b3f446f259.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 12 + file: stdsoln.tex + label: '' + mtref: '12.29' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 86070 + sourceTransform: tex2mdx.strip_comments + start: 86020 + targetStatement: + chapter: 12 + file: stdsoln.tex + label: '' + mtref: '12.30' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 88901 + sourceTransform: tex2mdx.strip_comments + start: 88738 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 729d9a0428c4 +- cf6eed7b1da4 +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/9710c10fda67.md b/projects/poincare-conjecture/.astrolabe/edges/9710c10fda67.md new file mode 100644 index 00000000..fb29fa05 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/9710c10fda67.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: epstopology + proofSpan: + bodyEnd: 72988 + bodyStart: 69979 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 72999 + file: surgery.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 69966 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 70735 + file: surgery.tex + sourceTransform: tex2mdx.strip_comments + start: 70718 + via: proof + sourceStatement: + chapter: 15 + file: surgery.tex + label: surgerytoptype + mtref: '15.3' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 69964 + sourceTransform: tex2mdx.strip_comments + start: 69392 + targetStatement: + chapter: 19 + file: canonnbhd.tex + label: epstopology + mtref: '19.25' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 54460 + sourceTransform: tex2mdx.strip_comments + start: 53603 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 7946423e82a4 +- fa8d9a43f34a +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/971749aa6b61.md b/projects/poincare-conjecture/.astrolabe/edges/971749aa6b61.md new file mode 100644 index 00000000..14bd31b8 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/971749aa6b61.md @@ -0,0 +1,62 @@ +--- +confidence: 0.35 +edgeClass: prose +evidence: + mentionTriggers: + - label: calabi + refSpan: + coordinateSpace: comment-stripped-tex + end: 42919 + file: prelim.tex + sourceTransform: tex2mdx.strip_comments + start: 42907 + sentence: Note that the inequality in Remark~\ref{calabi} follows from this theorem. + sentenceSpan: + coordinateSpace: comment-stripped-tex + end: 42946 + file: prelim.tex + sourceTransform: tex2mdx.strip_comments + start: 42871 + sourceStrategy: nearest-previous-statement + via: prose + sourceStatement: + chapter: 1 + file: prelim.tex + label: riccurvcomp + mtref: '1.33' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 42870 + sourceTransform: tex2mdx.strip_comments + start: 42464 + targetStatement: + chapter: 1 + file: prelim.tex + label: calabi + mtref: '1.29' + sort: remark + span: + coordinateSpace: comment-stripped-tex + end: 39490 + sourceTransform: tex2mdx.strip_comments + start: 38777 + type: prose-mention-nearest-previous-statement + via: + - prose +generator: tools/poincare_tex_extract.py +inference: weak +kind: mention +ref: +- 569be40ea88e +- 9f5db77333af +rel: mentions +reviewStatus: candidate +scope: prose +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- prose +--- +Morgan--Tian prose mention. The source is the nearest preceding statement in the same chapter segment. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/97321c2f5384.md b/projects/poincare-conjecture/.astrolabe/edges/97321c2f5384.md new file mode 100644 index 00000000..58809001 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/97321c2f5384.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 4 + file: maxprin.tex + label: '' + mtref: '4.38' + sort: remark + span: + coordinateSpace: comment-stripped-tex + end: 52902 + sourceTransform: tex2mdx.strip_comments + start: 52808 + targetStatement: + chapter: 4 + file: maxprin.tex + label: posderiv + mtref: '4.39' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 53144 + sourceTransform: tex2mdx.strip_comments + start: 52904 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 2629aff77c4a +- 36dd2f6fa9c2 +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/9732ab32c786.md b/projects/poincare-conjecture/.astrolabe/edges/9732ab32c786.md new file mode 100644 index 00000000..fbd2527b --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/9732ab32c786.md @@ -0,0 +1,55 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: statement + label: asympscal + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 117763 + file: temp2kappa.tex + sourceTransform: tex2mdx.strip_comments + start: 117748 + via: statement + sourceStatement: + chapter: 9 + file: temp2kappa.tex + label: asympvol + mtref: '9.59' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 118207 + sourceTransform: tex2mdx.strip_comments + start: 117600 + targetStatement: + chapter: 9 + file: temp2kappa.tex + label: asympscal + mtref: '9.56' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 106666 + sourceTransform: tex2mdx.strip_comments + start: 106402 + type: tex-reference + via: + - statement +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 7468282bd017 +- 5d9f1b75170d +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- statement +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/973522ecf167.md b/projects/poincare-conjecture/.astrolabe/edges/973522ecf167.md new file mode 100644 index 00000000..765ab0d1 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/973522ecf167.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 18 + file: energy1.tex + label: '' + mtref: '18.23' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 58028 + sourceTransform: tex2mdx.strip_comments + start: 57637 + targetStatement: + chapter: 18 + file: energy1.tex + label: XI + mtref: '18.24' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 58810 + sourceTransform: tex2mdx.strip_comments + start: 58030 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 19d01820dad0 +- fd85732525c1 +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/973aeefc48ca.md b/projects/poincare-conjecture/.astrolabe/edges/973aeefc48ca.md new file mode 100644 index 00000000..75142530 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/973aeefc48ca.md @@ -0,0 +1,66 @@ +--- +confidence: 0.35 +edgeClass: prose +evidence: + mentionTriggers: + - label: A_0 + refSpan: + coordinateSpace: comment-stripped-tex + end: 64095 + file: surgery.tex + sourceTransform: tex2mdx.strip_comments + start: 64086 + sentence: We fix $0<\epsilon\le {\rm min}(1/200,\left(\sqrt{D}(A_0+5)\right)^{-1},\bar\epsilon_1/2,\bar + \epsilon'/2,\epsilon_0)$ where $\bar\epsilon_1$ is the constant from Proposition~\ref{narrows}, + $\bar\epsilon'$ is the constant from Theorem~\ref{kappasummary}, $\epsilon_0$ + is the constant from Section~\ref{10.1}, and $A_0$ and $D$ are the constants + from Lemma~\ref{A_0}. We fix $\beta<1/2$, the constant from Proposition~\ref{neckglue} + sentenceSpan: + coordinateSpace: comment-stripped-tex + end: 64162 + file: surgery.tex + sourceTransform: tex2mdx.strip_comments + start: 63728 + sourceStrategy: nearest-previous-statement + via: prose + sourceStatement: + chapter: 15 + file: surgery.tex + label: neckglue + mtref: '15.2' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 61465 + sourceTransform: tex2mdx.strip_comments + start: 60812 + targetStatement: + chapter: 12 + file: stdsoln.tex + label: A_0 + mtref: '12.3' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 6040 + sourceTransform: tex2mdx.strip_comments + start: 5568 + type: prose-mention-nearest-previous-statement + via: + - prose +generator: tools/poincare_tex_extract.py +inference: weak +kind: mention +ref: +- a992d6a89479 +- 0b07beec84c0 +rel: mentions +reviewStatus: candidate +scope: prose +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- prose +--- +Morgan--Tian prose mention. The source is the nearest preceding statement in the same chapter segment. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/973db2154da6.md b/projects/poincare-conjecture/.astrolabe/edges/973db2154da6.md new file mode 100644 index 00000000..e74ab2a8 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/973db2154da6.md @@ -0,0 +1,67 @@ +--- +confidence: 0.72 +edgeClass: semantic +evidence: + anaphoraTriggers: + - context: proof + direction: previous + phrase: previous proposition + proofSpan: + bodyEnd: 164964 + bodyStart: 164856 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 164975 + file: surgery.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 164843 + sort: proposition + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 164962 + file: surgery.tex + sourceTransform: tex2mdx.strip_comments + start: 164942 + via: local-anaphora-proof + sourceStatement: + chapter: 16 + file: surgery.tex + label: bardelta1 + mtref: '16.24' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 164841 + sourceTransform: tex2mdx.strip_comments + start: 164269 + targetStatement: + chapter: 16 + file: surgery.tex + label: Lgeoexist + mtref: '16.21' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 154713 + sourceTransform: tex2mdx.strip_comments + start: 152982 + type: local-anaphora-reference + via: + - local-anaphora-proof +generator: tools/poincare_tex_extract.py +inference: inferred +kind: dependency +ref: +- ae8837c505cc +- d98ad25d9bc3 +rel: depends +reviewStatus: accepted +scope: local-context +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- local-anaphora-proof +--- +Morgan--Tian local anaphora dependency. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/975b132ec189.md b/projects/poincare-conjecture/.astrolabe/edges/975b132ec189.md new file mode 100644 index 00000000..3bb9244c --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/975b132ec189.md @@ -0,0 +1,67 @@ +--- +confidence: 0.72 +edgeClass: semantic +evidence: + anaphoraTriggers: + - context: proof + direction: previous + phrase: previous claim + proofSpan: + bodyEnd: 229182 + bodyStart: 185422 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 229193 + file: surgery.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 185409 + sort: claim + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 220388 + file: surgery.tex + sourceTransform: tex2mdx.strip_comments + start: 220374 + via: local-anaphora-proof + sourceStatement: + chapter: 17 + file: surgery.tex + label: extend + mtref: '17.1' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 185406 + sourceTransform: tex2mdx.strip_comments + start: 184285 + targetStatement: + chapter: 17 + file: surgery.tex + label: '' + mtref: '17.8' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 212622 + sourceTransform: tex2mdx.strip_comments + start: 212469 + type: local-anaphora-reference + via: + - local-anaphora-proof +generator: tools/poincare_tex_extract.py +inference: inferred +kind: dependency +ref: +- 254e8dae97e6 +- a01b63baabde +rel: depends +reviewStatus: accepted +scope: local-context +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- local-anaphora-proof +--- +Morgan--Tian local anaphora dependency. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/9797023dde2e.md b/projects/poincare-conjecture/.astrolabe/edges/9797023dde2e.md new file mode 100644 index 00000000..bb3fc8f4 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/9797023dde2e.md @@ -0,0 +1,45 @@ +--- +confidence: 0.2 +edgeClass: structural +evidence: + sourceStatement: + chapter: 2 + file: prelim.tex + label: htpytype + mtref: '2.11' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 63443 + sourceTransform: tex2mdx.strip_comments + start: 63143 + targetStatement: + chapter: 2 + file: prelim.tex + label: '' + mtref: '2.12' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 65820 + sourceTransform: tex2mdx.strip_comments + start: 64807 + type: chapter-reading-order-adjacency + via: + - chapter-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 68624c3d26bd +- 7e6aacfbf7d2 +rel: in-chapter +reviewStatus: accepted +scope: chapter +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- chapter-sequence +--- +Morgan--Tian reading-order adjacency across sections in a chapter. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/9797fbc10416.md b/projects/poincare-conjecture/.astrolabe/edges/9797fbc10416.md new file mode 100644 index 00000000..97e12c45 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/9797fbc10416.md @@ -0,0 +1,62 @@ +--- +confidence: 0.35 +edgeClass: prose +evidence: + mentionTriggers: + - label: CSSHI + refSpan: + coordinateSpace: comment-stripped-tex + end: 7721 + file: energy1.tex + sourceTransform: tex2mdx.strip_comments + start: 7710 + sentence: see Lemmas~\ref{CSSHI} and~\ref{csshi1} for the precise statement + sentenceSpan: + coordinateSpace: comment-stripped-tex + end: 7764 + file: energy1.tex + sourceTransform: tex2mdx.strip_comments + start: 7698 + sourceStrategy: nearest-previous-statement + via: prose + sourceStatement: + chapter: 18 + file: energy1.tex + label: extinct + mtref: '18.1' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 1107 + sourceTransform: tex2mdx.strip_comments + start: 511 + targetStatement: + chapter: 18 + file: energy1.tex + label: CSSHI + mtref: '18.52' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 117869 + sourceTransform: tex2mdx.strip_comments + start: 117059 + type: prose-mention-nearest-previous-statement + via: + - prose +generator: tools/poincare_tex_extract.py +inference: weak +kind: mention +ref: +- 4b8a44aa316f +- 38712e8cea3d +rel: mentions +reviewStatus: candidate +scope: prose +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- prose +--- +Morgan--Tian prose mention. The source is the nearest preceding statement in the same chapter segment. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/97e128bccf16.md b/projects/poincare-conjecture/.astrolabe/edges/97e128bccf16.md new file mode 100644 index 00000000..6973ffa5 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/97e128bccf16.md @@ -0,0 +1,65 @@ +--- +confidence: 0.45 +edgeClass: semantic +evidence: + endpointSemanticDegreeBeforeTermBackfill: + source: 0 + target: 5 + selectionReason: semantic-connectivity-backfill + sourceMatchSpan: + coordinateSpace: comment-stripped-tex + end: 23660 + file: converge2.tex + sourceTransform: tex2mdx.strip_comments + start: 23637 + sourceStatement: + chapter: 5 + file: converge2.tex + label: '' + mtref: '5.13' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 23788 + sourceTransform: tex2mdx.strip_comments + start: 23341 + targetStatement: + chapter: 5 + file: converge2.tex + label: smoothconv + mtref: '5.3' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 4757 + sourceTransform: tex2mdx.strip_comments + start: 2949 + targetTermSpan: + coordinateSpace: comment-stripped-tex + end: 4694 + file: converge2.tex + sourceTransform: tex2mdx.strip_comments + start: 4671 + term: partial geometric limit + termAmbiguity: unique-in-chapter + termSource: italic + termSpecificity: not-covered-by-longer-defined-term + type: same-chapter-definition-term-match + via: + - definition-term:partial geometric limit +generator: tools/poincare_tex_extract.py +inference: weak +kind: definition-use +ref: +- 59dc7d470f3e +- 467f4b920031 +rel: uses +reviewStatus: candidate +scope: same-chapter-term-match +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- definition-term:partial geometric limit +--- +Morgan--Tian same-chapter definition term match. This is a low-confidence candidate edge for review. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/98130e744025.md b/projects/poincare-conjecture/.astrolabe/edges/98130e744025.md new file mode 100644 index 00000000..c115b8e3 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/98130e744025.md @@ -0,0 +1,56 @@ +--- +confidence: 0.85 +edgeClass: semantic +evidence: + containmentReason: target theorem-like statement occurs inside source proof + proofSpan: + bodyEnd: 47855 + bodyStart: 43596 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 47866 + file: singlimit2.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 43583 + sourceStatement: + chapter: 11 + file: singlimit2.tex + label: omegacanon + mtref: '11.23' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 43580 + sourceTransform: tex2mdx.strip_comments + start: 43054 + targetStatement: + chapter: 11 + file: singlimit2.tex + label: epslimit + mtref: '11.24' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 45867 + sourceTransform: tex2mdx.strip_comments + start: 45550 + type: proof-contained-theorem-like-statement + via: + - proof-contained-statement +generator: tools/poincare_tex_extract.py +inference: inferred +kind: dependency +ref: +- 0760630d3990 +- d74749898b25 +rel: depends +reviewStatus: accepted +scope: proof-containment +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof-contained-statement +--- +Morgan--Tian proof-contained statement dependency. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/983c6d1660cb.md b/projects/poincare-conjecture/.astrolabe/edges/983c6d1660cb.md new file mode 100644 index 00000000..d257de70 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/983c6d1660cb.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: Xcontainedin + proofSpan: + bodyEnd: 34517 + bodyStart: 26920 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 34528 + file: bddcurvbdddist.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 26907 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 30325 + file: bddcurvbdddist.tex + sourceTransform: tex2mdx.strip_comments + start: 30307 + via: proof + sourceStatement: + chapter: 10 + file: bddcurvbdddist.tex + label: Uinfty + mtref: '10.7' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 26826 + sourceTransform: tex2mdx.strip_comments + start: 25406 + targetStatement: + chapter: 19 + file: canonnbhd.tex + label: Xcontainedin + mtref: '19.21' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 35375 + sourceTransform: tex2mdx.strip_comments + start: 34343 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- e2e64cdb0b75 +- 0e9a1a167caa +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/98438d29e64a.md b/projects/poincare-conjecture/.astrolabe/edges/98438d29e64a.md new file mode 100644 index 00000000..901be80a --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/98438d29e64a.md @@ -0,0 +1,65 @@ +--- +confidence: 0.35 +edgeClass: prose +evidence: + mentionTriggers: + - label: shi + refSpan: + coordinateSpace: comment-stripped-tex + end: 24027 + file: bddcurvbdddist.tex + sourceTransform: tex2mdx.strip_comments + start: 24018 + sentence: By Shi's theorem (Theorem~\ref{shi}), the fact that each point of $U_n$ + is the center of a strong $\epsilon$-neck means that there is a bound, independent + of $n$, on all covariant derivatives of the curvature at any point of $U_n$ + in terms of the bound on the curvature at the center point + sentenceSpan: + coordinateSpace: comment-stripped-tex + end: 24281 + file: bddcurvbdddist.tex + sourceTransform: tex2mdx.strip_comments + start: 23991 + sourceStrategy: nearest-previous-statement + via: prose + sourceStatement: + chapter: 10 + file: bddcurvbdddist.tex + label: '' + mtref: '10.6' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 21531 + sourceTransform: tex2mdx.strip_comments + start: 21390 + targetStatement: + chapter: 3 + file: flowbasics.tex + label: shi + mtref: '3.28' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 40541 + sourceTransform: tex2mdx.strip_comments + start: 39904 + type: prose-mention-nearest-previous-statement + via: + - prose +generator: tools/poincare_tex_extract.py +inference: weak +kind: mention +ref: +- 6a13c64a90e4 +- ed4b72caf9f9 +rel: mentions +reviewStatus: candidate +scope: prose +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- prose +--- +Morgan--Tian prose mention. The source is the nearest preceding statement in the same chapter segment. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/9891449239aa.md b/projects/poincare-conjecture/.astrolabe/edges/9891449239aa.md new file mode 100644 index 00000000..c785437c --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/9891449239aa.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 18 + file: energy1.tex + label: '' + mtref: '18.31' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 73544 + sourceTransform: tex2mdx.strip_comments + start: 73317 + targetStatement: + chapter: 18 + file: energy1.tex + label: nonembeasycase + mtref: '18.32' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 74095 + sourceTransform: tex2mdx.strip_comments + start: 73662 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 9e694ee343bf +- 27b452d6cb48 +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/989281fc4c7b.md b/projects/poincare-conjecture/.astrolabe/edges/989281fc4c7b.md new file mode 100644 index 00000000..435c823a --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/989281fc4c7b.md @@ -0,0 +1,84 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: min-max + proofSpan: + bodyEnd: 120659 + bodyStart: 118129 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 120670 + file: newcompar.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 118116 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 118818 + file: newcompar.tex + sourceTransform: tex2mdx.strip_comments + start: 118805 + via: proof + - context: proof + label: min-max + proofSpan: + bodyEnd: 120659 + bodyStart: 118129 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 120670 + file: newcompar.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 118116 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 119179 + file: newcompar.tex + sourceTransform: tex2mdx.strip_comments + start: 119166 + via: proof + sourceStatement: + chapter: 6 + file: newcompar.tex + label: Jlimit1 + mtref: '6.79' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 118112 + sourceTransform: tex2mdx.strip_comments + start: 117801 + targetStatement: + chapter: 6 + file: newcompar.tex + label: min-max + mtref: '6.60' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 90048 + sourceTransform: tex2mdx.strip_comments + start: 89598 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- f1123507a2ea +- 333be2e58780 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/98a5b6b540c8.md b/projects/poincare-conjecture/.astrolabe/edges/98a5b6b540c8.md new file mode 100644 index 00000000..6d0cd955 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/98a5b6b540c8.md @@ -0,0 +1,45 @@ +--- +confidence: 0.2 +edgeClass: structural +evidence: + sourceStatement: + chapter: 12 + file: stdsoln.tex + label: '' + mtref: '12.18' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 47727 + sourceTransform: tex2mdx.strip_comments + start: 46850 + targetStatement: + chapter: 12 + file: stdsoln.tex + label: rcor + mtref: '12.19' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 50482 + sourceTransform: tex2mdx.strip_comments + start: 50049 + type: chapter-reading-order-adjacency + via: + - chapter-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 8734da813eff +- e353993489e9 +rel: in-chapter +reviewStatus: accepted +scope: chapter +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- chapter-sequence +--- +Morgan--Tian reading-order adjacency across sections in a chapter. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/98d3cfd1c53b.md b/projects/poincare-conjecture/.astrolabe/edges/98d3cfd1c53b.md new file mode 100644 index 00000000..a91e83c2 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/98d3cfd1c53b.md @@ -0,0 +1,45 @@ +--- +confidence: 0.2 +edgeClass: structural +evidence: + sourceStatement: + chapter: 14 + file: surgery.tex + label: '' + mtref: '14.6' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 43870 + sourceTransform: tex2mdx.strip_comments + start: 43585 + targetStatement: + chapter: 14 + file: surgery.tex + label: '' + mtref: '14.7' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 45463 + sourceTransform: tex2mdx.strip_comments + start: 44092 + type: chapter-reading-order-adjacency + via: + - chapter-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 27af18c808d2 +- 124fe3535807 +rel: in-chapter +reviewStatus: accepted +scope: chapter +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- chapter-sequence +--- +Morgan--Tian reading-order adjacency across sections in a chapter. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/98da80e45503.md b/projects/poincare-conjecture/.astrolabe/edges/98da80e45503.md new file mode 100644 index 00000000..5e5b078b --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/98da80e45503.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 3 + file: flowbasics.tex + label: firstshi + mtref: '3.27' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 39652 + sourceTransform: tex2mdx.strip_comments + start: 39155 + targetStatement: + chapter: 3 + file: flowbasics.tex + label: shi + mtref: '3.28' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 40541 + sourceTransform: tex2mdx.strip_comments + start: 39904 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- dbe4fabd0076 +- ed4b72caf9f9 +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/98f218baedd0.md b/projects/poincare-conjecture/.astrolabe/edges/98f218baedd0.md new file mode 100644 index 00000000..71ea5b81 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/98f218baedd0.md @@ -0,0 +1,56 @@ +--- +confidence: 0.85 +edgeClass: semantic +evidence: + containmentReason: target theorem-like statement occurs inside source proof + proofSpan: + bodyEnd: 33071 + bodyStart: 25905 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 33082 + file: temp2kappa.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 25892 + sourceStatement: + chapter: 9 + file: temp2kappa.tex + label: nablaconv + mtref: '9.21' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 25890 + sourceTransform: tex2mdx.strip_comments + start: 25631 + targetStatement: + chapter: 9 + file: temp2kappa.tex + label: '' + mtref: '9.22' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 30396 + sourceTransform: tex2mdx.strip_comments + start: 30212 + type: proof-contained-theorem-like-statement + via: + - proof-contained-statement +generator: tools/poincare_tex_extract.py +inference: inferred +kind: dependency +ref: +- 366e3fdadf6f +- 598f6059280c +rel: depends +reviewStatus: accepted +scope: proof-containment +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof-contained-statement +--- +Morgan--Tian proof-contained statement dependency. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/9907672e4af2.md b/projects/poincare-conjecture/.astrolabe/edges/9907672e4af2.md new file mode 100644 index 00000000..8dccf8a2 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/9907672e4af2.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 18 + file: energy1.tex + label: areachange + mtref: '18.51' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 113141 + sourceTransform: tex2mdx.strip_comments + start: 112738 + targetStatement: + chapter: 18 + file: energy1.tex + label: CSSHI + mtref: '18.52' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 117869 + sourceTransform: tex2mdx.strip_comments + start: 117059 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 52123e3c5a27 +- 38712e8cea3d +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/9919f77829a6.md b/projects/poincare-conjecture/.astrolabe/edges/9919f77829a6.md new file mode 100644 index 00000000..190d0ce8 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/9919f77829a6.md @@ -0,0 +1,45 @@ +--- +confidence: 0.2 +edgeClass: structural +evidence: + sourceStatement: + chapter: 1 + file: prelim.tex + label: lapformula + mtref: '1.10' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 14031 + sourceTransform: tex2mdx.strip_comments + start: 13808 + targetStatement: + chapter: 1 + file: prelim.tex + label: '' + mtref: '1.11' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 14976 + sourceTransform: tex2mdx.strip_comments + start: 14241 + type: chapter-reading-order-adjacency + via: + - chapter-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 8d03f3ca0aea +- 5982e4029637 +rel: in-chapter +reviewStatus: accepted +scope: chapter +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- chapter-sequence +--- +Morgan--Tian reading-order adjacency across sections in a chapter. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/994aeb450eb0.md b/projects/poincare-conjecture/.astrolabe/edges/994aeb450eb0.md new file mode 100644 index 00000000..ab0073d1 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/994aeb450eb0.md @@ -0,0 +1,45 @@ +--- +confidence: 0.2 +edgeClass: structural +evidence: + sourceStatement: + chapter: 19 + file: canonnbhd.tex + label: curveshort + mtref: '19.1' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 1520 + sourceTransform: tex2mdx.strip_comments + start: 793 + targetStatement: + chapter: 19 + file: canonnbhd.tex + label: neckcurv + mtref: '19.2' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 3124 + sourceTransform: tex2mdx.strip_comments + start: 2315 + type: chapter-reading-order-adjacency + via: + - chapter-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- fc6c6ae605d2 +- dc5709757703 +rel: in-chapter +reviewStatus: accepted +scope: chapter +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- chapter-sequence +--- +Morgan--Tian reading-order adjacency across sections in a chapter. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/996e01b4ed11.md b/projects/poincare-conjecture/.astrolabe/edges/996e01b4ed11.md new file mode 100644 index 00000000..1aecc831 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/996e01b4ed11.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: Hessineq + proofSpan: + bodyEnd: 125200 + bodyStart: 122543 + coordinateSpace: comment-stripped-tex + depth: 2 + end: 125211 + file: newcompar.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 122530 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 123543 + file: newcompar.tex + sourceTransform: tex2mdx.strip_comments + start: 123529 + via: proof + sourceStatement: + chapter: 6 + file: newcompar.tex + label: fmonotone + mtref: '6.81' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 122525 + sourceTransform: tex2mdx.strip_comments + start: 122141 + targetStatement: + chapter: 6 + file: newcompar.tex + label: Hessineq + mtref: '6.37' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 56065 + sourceTransform: tex2mdx.strip_comments + start: 55015 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 4dc40f37cd6d +- 88d4ef67440f +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/998c953473f7.md b/projects/poincare-conjecture/.astrolabe/edges/998c953473f7.md new file mode 100644 index 00000000..a2c9e719 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/998c953473f7.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 7 + file: newcomp2.tex + label: phisupport + mtref: '7.15' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 25949 + sourceTransform: tex2mdx.strip_comments + start: 25492 + targetStatement: + chapter: 7 + file: newcomp2.tex + label: '' + mtref: '7.16' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 30129 + sourceTransform: tex2mdx.strip_comments + start: 29149 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 5e0538ed62cf +- d427ad10ceee +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/99a1249aa05f.md b/projects/poincare-conjecture/.astrolabe/edges/99a1249aa05f.md new file mode 100644 index 00000000..7e31400a --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/99a1249aa05f.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: localprod + proofSpan: + bodyEnd: 161702 + bodyStart: 159636 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 161713 + file: temp2kappa.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 159623 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 160524 + file: temp2kappa.tex + sourceTransform: tex2mdx.strip_comments + start: 160509 + via: proof + sourceStatement: + chapter: 9 + file: temp2kappa.tex + label: '' + mtref: '9.84' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 159621 + sourceTransform: tex2mdx.strip_comments + start: 158978 + targetStatement: + chapter: 4 + file: maxprin.tex + label: localprod + mtref: '4.19' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 25705 + sourceTransform: tex2mdx.strip_comments + start: 25065 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 6bda3981dbaa +- 8ec855a2aab5 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/99f360a60cdf.md b/projects/poincare-conjecture/.astrolabe/edges/99f360a60cdf.md new file mode 100644 index 00000000..c4c15073 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/99f360a60cdf.md @@ -0,0 +1,45 @@ +--- +confidence: 0.2 +edgeClass: structural +evidence: + sourceStatement: + chapter: 10 + file: bddcurvbdddist.tex + label: deltanet + mtref: '10.28' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 52334 + sourceTransform: tex2mdx.strip_comments + start: 51848 + targetStatement: + chapter: 10 + file: bddcurvbdddist.tex + label: secondstatment + mtref: '10.29' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 53510 + sourceTransform: tex2mdx.strip_comments + start: 52640 + type: chapter-reading-order-adjacency + via: + - chapter-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- f7e1d840a49f +- b34d555cd7b5 +rel: in-chapter +reviewStatus: accepted +scope: chapter +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- chapter-sequence +--- +Morgan--Tian reading-order adjacency across sections in a chapter. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/9a279eb9578e.md b/projects/poincare-conjecture/.astrolabe/edges/9a279eb9578e.md new file mode 100644 index 00000000..a5b5aa16 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/9a279eb9578e.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: etimesdis + proofSpan: + bodyEnd: 47957 + bodyStart: 46596 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 47968 + file: temp2kappa.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 46583 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 47913 + file: temp2kappa.tex + sourceTransform: tex2mdx.strip_comments + start: 47898 + via: proof + sourceStatement: + chapter: 9 + file: temp2kappa.tex + label: weakneg + mtref: '9.30' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 46581 + sourceTransform: tex2mdx.strip_comments + start: 46159 + targetStatement: + chapter: 9 + file: temp2kappa.tex + label: etimesdis + mtref: '9.24' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 35801 + sourceTransform: tex2mdx.strip_comments + start: 35580 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 42d9f91e59f3 +- a8e10db32f90 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/9a33c03c9d84.md b/projects/poincare-conjecture/.astrolabe/edges/9a33c03c9d84.md new file mode 100644 index 00000000..38b44794 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/9a33c03c9d84.md @@ -0,0 +1,66 @@ +--- +confidence: 0.35 +edgeClass: prose +evidence: + mentionTriggers: + - label: delta0ri+1 + refSpan: + coordinateSpace: comment-stripped-tex + end: 165303 + file: surgery.tex + sourceTransform: tex2mdx.strip_comments + start: 165287 + sentence: Having found a compact subset of the open subset of smooth points of + ${\mathcal M}$ that contains all paths parameterized by backward time whose + ${\mathcal L}$-length is not too large, we are in a position to prove Proposition~\ref{delta0ri+1}, + which states the existence of a minimizing ${\mathcal L}$-geodesics in ${\mathcal + M}$ from $x$ and gives estimates on their ${\mathcal L}$-lengths. + sentenceSpan: + coordinateSpace: comment-stripped-tex + end: 165452 + file: surgery.tex + sourceTransform: tex2mdx.strip_comments + start: 165059 + sourceStrategy: nearest-previous-statement + via: prose + sourceStatement: + chapter: 16 + file: surgery.tex + label: bardelta1 + mtref: '16.24' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 164841 + sourceTransform: tex2mdx.strip_comments + start: 164269 + targetStatement: + chapter: 16 + file: surgery.tex + label: delta0ri+1 + mtref: '16.4' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 109162 + sourceTransform: tex2mdx.strip_comments + start: 107682 + type: prose-mention-nearest-previous-statement + via: + - prose +generator: tools/poincare_tex_extract.py +inference: weak +kind: mention +ref: +- ae8837c505cc +- 89a60eb69c3f +rel: mentions +reviewStatus: candidate +scope: prose +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- prose +--- +Morgan--Tian prose mention. The source is the nearest preceding statement in the same chapter segment. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/9a50ef4e2d2b.md b/projects/poincare-conjecture/.astrolabe/edges/9a50ef4e2d2b.md new file mode 100644 index 00000000..7345f197 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/9a50ef4e2d2b.md @@ -0,0 +1,45 @@ +--- +confidence: 0.2 +edgeClass: structural +evidence: + sourceStatement: + chapter: 1 + file: prelim.tex + label: '' + mtref: '1.18' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 20381 + sourceTransform: tex2mdx.strip_comments + start: 20241 + targetStatement: + chapter: 1 + file: prelim.tex + label: '' + mtref: '1.19' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 26794 + sourceTransform: tex2mdx.strip_comments + start: 26539 + type: chapter-reading-order-adjacency + via: + - chapter-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- e07da18ab307 +- b209b247be34 +rel: in-chapter +reviewStatus: accepted +scope: chapter +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- chapter-sequence +--- +Morgan--Tian reading-order adjacency across sections in a chapter. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/9a745c1baaa8.md b/projects/poincare-conjecture/.astrolabe/edges/9a745c1baaa8.md new file mode 100644 index 00000000..7b735c35 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/9a745c1baaa8.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 6 + file: newcompar.tex + label: min-max + mtref: '6.60' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 90048 + sourceTransform: tex2mdx.strip_comments + start: 89598 + targetStatement: + chapter: 6 + file: newcompar.tex + label: minmax + mtref: '6.61' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 91839 + sourceTransform: tex2mdx.strip_comments + start: 91251 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 333be2e58780 +- a7c2ebee07a3 +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/9aa48bc911d6.md b/projects/poincare-conjecture/.astrolabe/edges/9aa48bc911d6.md new file mode 100644 index 00000000..1fd9ca16 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/9aa48bc911d6.md @@ -0,0 +1,65 @@ +--- +confidence: 0.45 +edgeClass: semantic +evidence: + endpointSemanticDegreeBeforeTermBackfill: + source: 0 + target: 1 + selectionReason: semantic-connectivity-backfill + sourceMatchSpan: + coordinateSpace: comment-stripped-tex + end: 34332 + file: prelim.tex + sourceTransform: tex2mdx.strip_comments + start: 34323 + sourceStatement: + chapter: 1 + file: prelim.tex + label: '' + mtref: '1.26' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 34361 + sourceTransform: tex2mdx.strip_comments + start: 33952 + targetStatement: + chapter: 1 + file: prelim.tex + label: '' + mtref: '1.24' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 33683 + sourceTransform: tex2mdx.strip_comments + start: 33241 + targetTermSpan: + coordinateSpace: comment-stripped-tex + end: 33606 + file: prelim.tex + sourceTransform: tex2mdx.strip_comments + start: 33597 + term: cut locus + termAmbiguity: unique-in-chapter + termSource: italic + termSpecificity: no-longer-defined-term + type: same-chapter-definition-term-match + via: + - definition-term:cut locus +generator: tools/poincare_tex_extract.py +inference: weak +kind: definition-use +ref: +- 8ebd93835797 +- 19afab2e3115 +rel: uses +reviewStatus: candidate +scope: same-chapter-term-match +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- definition-term:cut locus +--- +Morgan--Tian same-chapter definition term match. This is a low-confidence candidate edge for review. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/9ad61fca6f6a.md b/projects/poincare-conjecture/.astrolabe/edges/9ad61fca6f6a.md new file mode 100644 index 00000000..ae548dd5 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/9ad61fca6f6a.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 3 + file: flowbasics.tex + label: ortho + mtref: '3.16' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 20255 + sourceTransform: tex2mdx.strip_comments + start: 19959 + targetStatement: + chapter: 3 + file: flowbasics.tex + label: '' + mtref: '3.17' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 21425 + sourceTransform: tex2mdx.strip_comments + start: 21379 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 29b84db22679 +- c67b6547955b +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/9adc41e82066.md b/projects/poincare-conjecture/.astrolabe/edges/9adc41e82066.md new file mode 100644 index 00000000..56b85b1a --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/9adc41e82066.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 8 + file: noncoll.tex + label: inside + mtref: '8.7' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 10980 + sourceTransform: tex2mdx.strip_comments + start: 10665 + targetStatement: + chapter: 8 + file: noncoll.tex + label: volvomp + mtref: '8.8' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 13642 + sourceTransform: tex2mdx.strip_comments + start: 13363 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 0a1eac346e21 +- 35010eaa585e +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/9b066ea1e6aa.md b/projects/poincare-conjecture/.astrolabe/edges/9b066ea1e6aa.md new file mode 100644 index 00000000..04a4933a --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/9b066ea1e6aa.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: coreint + proofSpan: + bodyEnd: 19521 + bodyStart: 13193 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 19532 + file: bddcurvbdddist.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 13180 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 13704 + file: bddcurvbdddist.tex + sourceTransform: tex2mdx.strip_comments + start: 13691 + via: proof + sourceStatement: + chapter: 10 + file: bddcurvbdddist.tex + label: '' + mtref: '10.4' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 13178 + sourceTransform: tex2mdx.strip_comments + start: 12275 + targetStatement: + chapter: 19 + file: canonnbhd.tex + label: coreint + mtref: '19.8' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 11926 + sourceTransform: tex2mdx.strip_comments + start: 11192 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 96519f84d54e +- 5ef5906bb370 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/9b136c63850e.md b/projects/poincare-conjecture/.astrolabe/edges/9b136c63850e.md new file mode 100644 index 00000000..c1b41dfd --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/9b136c63850e.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: cnnbhd2 + proofSpan: + bodyEnd: 33794 + bodyStart: 17659 + coordinateSpace: comment-stripped-tex + depth: 2 + end: 33805 + file: singlimit2.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 17646 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 18581 + file: singlimit2.tex + sourceTransform: tex2mdx.strip_comments + start: 18568 + via: proof + sourceStatement: + chapter: 11 + file: singlimit2.tex + label: maxT + mtref: '11.10' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 17643 + sourceTransform: tex2mdx.strip_comments + start: 17183 + targetStatement: + chapter: 11 + file: singlimit2.tex + label: cnnbhd2 + mtref: '11.3' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 8201 + sourceTransform: tex2mdx.strip_comments + start: 7592 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- d6919616448d +- 3989b6b34e08 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/9b16e3b69470.md b/projects/poincare-conjecture/.astrolabe/edges/9b16e3b69470.md new file mode 100644 index 00000000..061bc09a --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/9b16e3b69470.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 7 + file: newcomp2.tex + label: '' + mtref: '7.23' + sort: remark + span: + coordinateSpace: comment-stripped-tex + end: 37322 + sourceTransform: tex2mdx.strip_comments + start: 37227 + targetStatement: + chapter: 7 + file: newcomp2.tex + label: ltauineq + mtref: '7.24' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 38959 + sourceTransform: tex2mdx.strip_comments + start: 38702 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- f93a1c4b1057 +- 490290f4c322 +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/9b17d79be2e9.md b/projects/poincare-conjecture/.astrolabe/edges/9b17d79be2e9.md new file mode 100644 index 00000000..8523f5d1 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/9b17d79be2e9.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 6 + file: newcompar.tex + label: Llength + mtref: '6.2' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 4712 + sourceTransform: tex2mdx.strip_comments + start: 3854 + targetStatement: + chapter: 6 + file: newcompar.tex + label: '' + mtref: '6.3' + sort: example + span: + coordinateSpace: comment-stripped-tex + end: 6323 + sourceTransform: tex2mdx.strip_comments + start: 5528 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 09af70d51aac +- '867793768688' +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/9b205f4c6ff1.md b/projects/poincare-conjecture/.astrolabe/edges/9b205f4c6ff1.md new file mode 100644 index 00000000..cbd207bc --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/9b205f4c6ff1.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: Arnquantity + proofSpan: + bodyEnd: 117788 + bodyStart: 114121 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 117799 + file: newcompar.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 114108 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 117099 + file: newcompar.tex + sourceTransform: tex2mdx.strip_comments + start: 117082 + via: proof + sourceStatement: + chapter: 6 + file: newcompar.tex + label: Jlimit + mtref: '6.78' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 114106 + sourceTransform: tex2mdx.strip_comments + start: 113458 + targetStatement: + chapter: 6 + file: newcompar.tex + label: Arnquantity + mtref: '6.77' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 113400 + sourceTransform: tex2mdx.strip_comments + start: 113216 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 4e6330b2aa46 +- 9d4d83df4874 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/9b2b75e5cda4.md b/projects/poincare-conjecture/.astrolabe/edges/9b2b75e5cda4.md new file mode 100644 index 00000000..5f47e542 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/9b2b75e5cda4.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 18 + file: energy1.tex + label: locemb + mtref: '18.75' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 175828 + sourceTransform: tex2mdx.strip_comments + start: 175637 + targetStatement: + chapter: 18 + file: energy1.tex + label: limittri + mtref: '18.76' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 177088 + sourceTransform: tex2mdx.strip_comments + start: 176809 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 14aeee986b49 +- 2ff35cef8589 +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/9b393376177c.md b/projects/poincare-conjecture/.astrolabe/edges/9b393376177c.md new file mode 100644 index 00000000..8c60dcb7 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/9b393376177c.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 15 + file: surgery.tex + label: '14.26' + mtref: '15.11' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 93849 + sourceTransform: tex2mdx.strip_comments + start: 93211 + targetStatement: + chapter: 15 + file: surgery.tex + label: surgerydist + mtref: '15.12' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 97780 + sourceTransform: tex2mdx.strip_comments + start: 96249 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- c8e7931dbf65 +- 9021a4eca03f +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/9c1f090442d8.md b/projects/poincare-conjecture/.astrolabe/edges/9c1f090442d8.md new file mode 100644 index 00000000..e8e21bb7 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/9c1f090442d8.md @@ -0,0 +1,84 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: muarea + proofSpan: + bodyEnd: 145345 + bodyStart: 144076 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 145356 + file: energy1.tex + ownerLabel: XI + sourceTransform: tex2mdx.strip_comments + start: 144063 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 144161 + file: energy1.tex + sourceTransform: tex2mdx.strip_comments + start: 144149 + via: proof + - context: proof + label: muarea + proofSpan: + bodyEnd: 145345 + bodyStart: 144076 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 145356 + file: energy1.tex + ownerLabel: XI + sourceTransform: tex2mdx.strip_comments + start: 144063 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 145224 + file: energy1.tex + sourceTransform: tex2mdx.strip_comments + start: 145212 + via: proof + sourceStatement: + chapter: 18 + file: energy1.tex + label: XI + mtref: '18.24' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 58810 + sourceTransform: tex2mdx.strip_comments + start: 58030 + targetStatement: + chapter: 18 + file: energy1.tex + label: muarea + mtref: '18.58' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 138479 + sourceTransform: tex2mdx.strip_comments + start: 137860 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- fd85732525c1 +- 10dace7fb165 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/9c7dd87b9585.md b/projects/poincare-conjecture/.astrolabe/edges/9c7dd87b9585.md new file mode 100644 index 00000000..dc597f16 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/9c7dd87b9585.md @@ -0,0 +1,62 @@ +--- +confidence: 0.9 +edgeClass: structural +evidence: + containmentDepth: 2 + proofSpan: + bodyEnd: 53671 + bodyStart: 52566 + coordinateSpace: comment-stripped-tex + depth: 2 + end: 53682 + file: energy1.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 52553 + sourceStatement: + chapter: 18 + file: energy1.tex + label: '' + mtref: '18.19' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 52551 + sourceTransform: tex2mdx.strip_comments + start: 52480 + targetContainedSpan: + coordinateSpace: comment-stripped-tex + end: 53052 + file: energy1.tex + sourceTransform: tex2mdx.strip_comments + start: 52883 + targetStatement: + chapter: 18 + file: energy1.tex + label: '' + mtref: '18.20' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 53052 + sourceTransform: tex2mdx.strip_comments + start: 52883 + type: proof-containment + via: + - proof-contains +generator: tools/poincare_tex_extract.py +inference: structural +kind: containment +ref: +- 6a1625220073 +- a04af7434ef0 +rel: contains +reviewStatus: accepted +scope: proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof-contains +--- +Morgan--Tian proof containment. The source is the statement whose proof contains the target statement. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/9c9fd35ca24e.md b/projects/poincare-conjecture/.astrolabe/edges/9c9fd35ca24e.md new file mode 100644 index 00000000..fc328e04 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/9c9fd35ca24e.md @@ -0,0 +1,56 @@ +--- +confidence: 0.85 +edgeClass: semantic +evidence: + containmentReason: target theorem-like statement occurs inside source proof + proofSpan: + bodyEnd: 176513 + bodyStart: 172329 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 176524 + file: temp2kappa.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 172316 + sourceStatement: + chapter: 9 + file: temp2kappa.tex + label: thmep3 + mtref: '9.89' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 172313 + sourceTransform: tex2mdx.strip_comments + start: 171695 + targetStatement: + chapter: 9 + file: temp2kappa.tex + label: '' + mtref: '9.90' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 173901 + sourceTransform: tex2mdx.strip_comments + start: 173501 + type: proof-contained-theorem-like-statement + via: + - proof-contained-statement +generator: tools/poincare_tex_extract.py +inference: inferred +kind: dependency +ref: +- a4ea411d0765 +- 494bc110f8ab +rel: depends +reviewStatus: accepted +scope: proof-containment +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof-contained-statement +--- +Morgan--Tian proof-contained statement dependency. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/9ca4d3fbf5fa.md b/projects/poincare-conjecture/.astrolabe/edges/9ca4d3fbf5fa.md new file mode 100644 index 00000000..59b2455b --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/9ca4d3fbf5fa.md @@ -0,0 +1,76 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: statement + label: lowerbdd + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 41852 + file: temp2kappa.tex + sourceTransform: tex2mdx.strip_comments + start: 41838 + via: statement + - context: proof + label: lowerbdd + proofSpan: + bodyEnd: 42448 + bodyStart: 41878 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 42459 + file: temp2kappa.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 41865 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 42001 + file: temp2kappa.tex + sourceTransform: tex2mdx.strip_comments + start: 41987 + via: proof + sourceStatement: + chapter: 9 + file: temp2kappa.tex + label: '' + mtref: '9.26' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 41863 + sourceTransform: tex2mdx.strip_comments + start: 41621 + targetStatement: + chapter: 9 + file: temp2kappa.tex + label: lowerbdd + mtref: '9.25' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 36672 + sourceTransform: tex2mdx.strip_comments + start: 36453 + type: tex-reference + via: + - proof + - statement +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- d16f43d74c54 +- a6ee3b2ab615 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +- statement +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/9cad02faa52b.md b/projects/poincare-conjecture/.astrolabe/edges/9cad02faa52b.md new file mode 100644 index 00000000..117a074d --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/9cad02faa52b.md @@ -0,0 +1,55 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: statement + label: corequiv + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 12255 + file: intro.tex + sourceTransform: tex2mdx.strip_comments + start: 12241 + via: statement + sourceStatement: + chapter: 0 + file: intro.tex + label: finiteext + mtref: '0.4' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 12644 + sourceTransform: tex2mdx.strip_comments + start: 11822 + targetStatement: + chapter: 0 + file: intro.tex + label: corequiv + mtref: '0.5' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 14939 + sourceTransform: tex2mdx.strip_comments + start: 13757 + type: tex-reference + via: + - statement +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 127db1dc9315 +- 7078952cb9e4 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- statement +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/9cddf88fe2c8.md b/projects/poincare-conjecture/.astrolabe/edges/9cddf88fe2c8.md new file mode 100644 index 00000000..ce7135db --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/9cddf88fe2c8.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: localprod + proofSpan: + bodyEnd: 74813 + bodyStart: 59556 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 74824 + file: singlimit2.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 59543 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 66066 + file: singlimit2.tex + sourceTransform: tex2mdx.strip_comments + start: 66051 + via: proof + sourceStatement: + chapter: 11 + file: singlimit2.tex + label: hexist + mtref: '11.31' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 59267 + sourceTransform: tex2mdx.strip_comments + start: 58375 + targetStatement: + chapter: 4 + file: maxprin.tex + label: localprod + mtref: '4.19' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 25705 + sourceTransform: tex2mdx.strip_comments + start: 25065 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 074e02b067b8 +- 8ec855a2aab5 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/9cf36a80eb37.md b/projects/poincare-conjecture/.astrolabe/edges/9cf36a80eb37.md new file mode 100644 index 00000000..1f10716b --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/9cf36a80eb37.md @@ -0,0 +1,62 @@ +--- +confidence: 0.9 +edgeClass: structural +evidence: + containmentDepth: 1 + proofSpan: + bodyEnd: 54967 + bodyStart: 37263 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 54978 + file: energy1.tex + ownerLabel: W_2 + sourceTransform: tex2mdx.strip_comments + start: 37250 + sourceStatement: + chapter: 18 + file: energy1.tex + label: W_2 + mtref: '18.11' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 36484 + sourceTransform: tex2mdx.strip_comments + start: 36186 + targetContainedSpan: + coordinateSpace: comment-stripped-tex + end: 47275 + file: energy1.tex + sourceTransform: tex2mdx.strip_comments + start: 47013 + targetStatement: + chapter: 18 + file: energy1.tex + label: '' + mtref: '18.16' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 47275 + sourceTransform: tex2mdx.strip_comments + start: 47013 + type: proof-containment + via: + - proof-contains +generator: tools/poincare_tex_extract.py +inference: structural +kind: containment +ref: +- 7bb0fa0c9412 +- 39199318fe64 +rel: contains +reviewStatus: accepted +scope: proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof-contains +--- +Morgan--Tian proof containment. The source is the statement whose proof contains the target statement. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/9d0376a2e5ff.md b/projects/poincare-conjecture/.astrolabe/edges/9d0376a2e5ff.md new file mode 100644 index 00000000..24d7ab2a --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/9d0376a2e5ff.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: morethanone + proofSpan: + bodyEnd: 48301 + bodyStart: 48145 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 48312 + file: bddcurvbdddist.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 48132 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 48299 + file: bddcurvbdddist.tex + sourceTransform: tex2mdx.strip_comments + start: 48282 + via: proof + sourceStatement: + chapter: 10 + file: bddcurvbdddist.tex + label: notray + mtref: '10.24' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 48129 + sourceTransform: tex2mdx.strip_comments + start: 48006 + targetStatement: + chapter: 10 + file: bddcurvbdddist.tex + label: morethanone + mtref: '10.18' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 42343 + sourceTransform: tex2mdx.strip_comments + start: 42251 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 72be1154f65a +- dda8a7c7acb6 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/9d1b82cebd22.md b/projects/poincare-conjecture/.astrolabe/edges/9d1b82cebd22.md new file mode 100644 index 00000000..a1408e16 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/9d1b82cebd22.md @@ -0,0 +1,56 @@ +--- +confidence: 0.85 +edgeClass: semantic +evidence: + containmentReason: target theorem-like statement occurs inside source proof + proofSpan: + bodyEnd: 73361 + bodyStart: 70289 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 73372 + file: newcompar.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 70276 + sourceStatement: + chapter: 6 + file: newcompar.tex + label: triL + mtref: '6.43' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 70273 + sourceTransform: tex2mdx.strip_comments + start: 68899 + targetStatement: + chapter: 6 + file: newcompar.tex + label: Hclaim + mtref: '6.44' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 71895 + sourceTransform: tex2mdx.strip_comments + start: 71719 + type: proof-contained-theorem-like-statement + via: + - proof-contained-statement +generator: tools/poincare_tex_extract.py +inference: inferred +kind: dependency +ref: +- 882677eecdc1 +- 11e3d00eb60d +rel: depends +reviewStatus: accepted +scope: proof-containment +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof-contained-statement +--- +Morgan--Tian proof-contained statement dependency. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/9d36d721e6f1.md b/projects/poincare-conjecture/.astrolabe/edges/9d36d721e6f1.md new file mode 100644 index 00000000..3c9558bc --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/9d36d721e6f1.md @@ -0,0 +1,62 @@ +--- +confidence: 0.9 +edgeClass: structural +evidence: + containmentDepth: 1 + proofSpan: + bodyEnd: 68625 + bodyStart: 60090 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 68636 + file: newcompar.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 60077 + sourceStatement: + chapter: 6 + file: newcompar.tex + label: adapjab + mtref: '6.41' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 59462 + sourceTransform: tex2mdx.strip_comments + start: 58890 + targetContainedSpan: + coordinateSpace: comment-stripped-tex + end: 66455 + file: newcompar.tex + sourceTransform: tex2mdx.strip_comments + start: 66260 + targetStatement: + chapter: 6 + file: newcompar.tex + label: '' + mtref: '6.42' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 66455 + sourceTransform: tex2mdx.strip_comments + start: 66260 + type: proof-containment + via: + - proof-contains +generator: tools/poincare_tex_extract.py +inference: structural +kind: containment +ref: +- 72949e6350d4 +- b20ded1b2964 +rel: contains +reviewStatus: accepted +scope: proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof-contains +--- +Morgan--Tian proof containment. The source is the statement whose proof contains the target statement. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/9d536fff8cb8.md b/projects/poincare-conjecture/.astrolabe/edges/9d536fff8cb8.md new file mode 100644 index 00000000..36a76d4e --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/9d536fff8cb8.md @@ -0,0 +1,79 @@ +--- +confidence: 0.35 +edgeClass: prose +evidence: + mentionTriggers: + - label: MAIN + refSpan: + coordinateSpace: comment-stripped-tex + end: 237049 + file: surgery.tex + sourceTransform: tex2mdx.strip_comments + start: 237039 + sentence: This completes the proof of Theorem~\ref{MAIN}. + sentenceSpan: + coordinateSpace: comment-stripped-tex + end: 237050 + file: surgery.tex + sourceTransform: tex2mdx.strip_comments + start: 237002 + sourceStrategy: nearest-previous-statement + via: prose + - label: MAIN + refSpan: + coordinateSpace: comment-stripped-tex + end: 237146 + file: surgery.tex + sourceTransform: tex2mdx.strip_comments + start: 237136 + sentence: To sum up, we have sequences $\Delta$, ${\bf K}$ and ${\bf r}$ as given + in Theorem~\ref{MAIN} + sentenceSpan: + coordinateSpace: comment-stripped-tex + end: 237146 + file: surgery.tex + sourceTransform: tex2mdx.strip_comments + start: 237052 + sourceStrategy: nearest-previous-statement + via: prose + sourceStatement: + chapter: 17 + file: surgery.tex + label: bddsurgery + mtref: '17.12' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 234130 + sourceTransform: tex2mdx.strip_comments + start: 233569 + targetStatement: + chapter: 15 + file: surgery.tex + label: MAIN + mtref: '15.9' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 91186 + sourceTransform: tex2mdx.strip_comments + start: 90004 + type: prose-mention-nearest-previous-statement + via: + - prose +generator: tools/poincare_tex_extract.py +inference: weak +kind: mention +ref: +- 6efe562bcc06 +- 703e4cfeece7 +rel: mentions +reviewStatus: candidate +scope: prose +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- prose +--- +Morgan--Tian prose mention. The source is the nearest preceding statement in the same chapter segment. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/9d7608b1d716.md b/projects/poincare-conjecture/.astrolabe/edges/9d7608b1d716.md new file mode 100644 index 00000000..d23cdd25 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/9d7608b1d716.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 9 + file: temp2kappa.tex + label: scaleinv2 + mtref: '9.13' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 16522 + sourceTransform: tex2mdx.strip_comments + start: 16053 + targetStatement: + chapter: 9 + file: temp2kappa.tex + label: redvolconst + mtref: '9.14' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 17598 + sourceTransform: tex2mdx.strip_comments + start: 17342 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- dbf6a3c0bae7 +- e08895f2b0d8 +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/9d782fd866af.md b/projects/poincare-conjecture/.astrolabe/edges/9d782fd866af.md new file mode 100644 index 00000000..f2c18aa1 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/9d782fd866af.md @@ -0,0 +1,62 @@ +--- +confidence: 0.9 +edgeClass: structural +evidence: + containmentDepth: 2 + proofSpan: + bodyEnd: 33794 + bodyStart: 17659 + coordinateSpace: comment-stripped-tex + depth: 2 + end: 33805 + file: singlimit2.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 17646 + sourceStatement: + chapter: 11 + file: singlimit2.tex + label: maxT + mtref: '11.10' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 17643 + sourceTransform: tex2mdx.strip_comments + start: 17183 + targetContainedSpan: + coordinateSpace: comment-stripped-tex + end: 23634 + file: singlimit2.tex + sourceTransform: tex2mdx.strip_comments + start: 23355 + targetStatement: + chapter: 11 + file: singlimit2.tex + label: '' + mtref: '11.13' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 23634 + sourceTransform: tex2mdx.strip_comments + start: 23355 + type: proof-containment + via: + - proof-contains +generator: tools/poincare_tex_extract.py +inference: structural +kind: containment +ref: +- d6919616448d +- 385a31c5d191 +rel: contains +reviewStatus: accepted +scope: proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof-contains +--- +Morgan--Tian proof containment. The source is the statement whose proof contains the target statement. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/9dcd016f3fe4.md b/projects/poincare-conjecture/.astrolabe/edges/9dcd016f3fe4.md new file mode 100644 index 00000000..6c05e40e --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/9dcd016f3fe4.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 5 + file: converge2.tex + label: '' + mtref: '5.22' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 35882 + sourceTransform: tex2mdx.strip_comments + start: 35265 + targetStatement: + chapter: 5 + file: converge2.tex + label: '' + mtref: '5.23' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 36234 + sourceTransform: tex2mdx.strip_comments + start: 35949 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 5be0c15e901a +- cd2cfdd79bea +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/9dd2d1cb23fa.md b/projects/poincare-conjecture/.astrolabe/edges/9dd2d1cb23fa.md new file mode 100644 index 00000000..f05aa13c --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/9dd2d1cb23fa.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: asympGSS + proofSpan: + bodyEnd: 93966 + bodyStart: 91964 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 93977 + file: temp2kappa.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 91951 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 92490 + file: temp2kappa.tex + sourceTransform: tex2mdx.strip_comments + start: 92476 + via: proof + sourceStatement: + chapter: 9 + file: temp2kappa.tex + label: 2DGSS + mtref: '9.50' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 91949 + sourceTransform: tex2mdx.strip_comments + start: 91266 + targetStatement: + chapter: 9 + file: temp2kappa.tex + label: asympGSS + mtref: '9.11' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 13767 + sourceTransform: tex2mdx.strip_comments + start: 12464 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- da767c71996b +- 671402eb454b +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/9de2af505724.md b/projects/poincare-conjecture/.astrolabe/edges/9de2af505724.md new file mode 100644 index 00000000..6736184a --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/9de2af505724.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: 2ndmfdconv + proofSpan: + bodyEnd: 14787 + bodyStart: 6954 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 14798 + file: singlimit2.tex + ownerLabel: smlmtflow + sourceTransform: tex2mdx.strip_comments + start: 6941 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 9141 + file: singlimit2.tex + sourceTransform: tex2mdx.strip_comments + start: 9125 + via: proof + sourceStatement: + chapter: 11 + file: singlimit2.tex + label: smlmtflow + mtref: '11.1' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 3459 + sourceTransform: tex2mdx.strip_comments + start: 1498 + targetStatement: + chapter: 5 + file: converge2.tex + label: 2ndmfdconv + mtref: '5.10' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 18602 + sourceTransform: tex2mdx.strip_comments + start: 18149 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- d51aab8a8065 +- b45765a3070e +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/9e350e2286fd.md b/projects/poincare-conjecture/.astrolabe/edges/9e350e2286fd.md new file mode 100644 index 00000000..0ba8c4a8 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/9e350e2286fd.md @@ -0,0 +1,62 @@ +--- +confidence: 0.9 +edgeClass: structural +evidence: + containmentDepth: 1 + proofSpan: + bodyEnd: 171471 + bodyStart: 161354 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 171482 + file: energy1.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 161341 + sourceStatement: + chapter: 18 + file: energy1.tex + label: laest + mtref: '18.66' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 161338 + sourceTransform: tex2mdx.strip_comments + start: 161080 + targetContainedSpan: + coordinateSpace: comment-stripped-tex + end: 162289 + file: energy1.tex + sourceTransform: tex2mdx.strip_comments + start: 162047 + targetStatement: + chapter: 18 + file: energy1.tex + label: theta/Kbd + mtref: '18.67' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 162289 + sourceTransform: tex2mdx.strip_comments + start: 162047 + type: proof-containment + via: + - proof-contains +generator: tools/poincare_tex_extract.py +inference: structural +kind: containment +ref: +- 45c783d384c1 +- 512c74b9ad1f +rel: contains +reviewStatus: accepted +scope: proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof-contains +--- +Morgan--Tian proof containment. The source is the statement whose proof contains the target statement. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/9e4e5ba30938.md b/projects/poincare-conjecture/.astrolabe/edges/9e4e5ba30938.md new file mode 100644 index 00000000..a9e8b922 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/9e4e5ba30938.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 5 + file: converge2.tex + label: '' + mtref: '5.29' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 43638 + sourceTransform: tex2mdx.strip_comments + start: 43134 + targetStatement: + chapter: 5 + file: converge2.tex + label: '' + mtref: '5.30' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 43701 + sourceTransform: tex2mdx.strip_comments + start: 43640 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 97c2fdddbf04 +- cb86e24eee63 +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/9e7287a79bb9.md b/projects/poincare-conjecture/.astrolabe/edges/9e7287a79bb9.md new file mode 100644 index 00000000..3f3bc9f7 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/9e7287a79bb9.md @@ -0,0 +1,62 @@ +--- +confidence: 0.9 +edgeClass: structural +evidence: + containmentDepth: 1 + proofSpan: + bodyEnd: 36068 + bodyStart: 28129 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 36079 + file: flowbasics.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 28116 + sourceStatement: + chapter: 3 + file: flowbasics.tex + label: '' + mtref: '3.22' + sort: remark + span: + coordinateSpace: comment-stripped-tex + end: 28114 + sourceTransform: tex2mdx.strip_comments + start: 27844 + targetContainedSpan: + coordinateSpace: comment-stripped-tex + end: 29260 + file: flowbasics.tex + sourceTransform: tex2mdx.strip_comments + start: 29003 + targetStatement: + chapter: 3 + file: flowbasics.tex + label: '3.23' + mtref: '3.24' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 29260 + sourceTransform: tex2mdx.strip_comments + start: 29003 + type: proof-containment + via: + - proof-contains +generator: tools/poincare_tex_extract.py +inference: structural +kind: containment +ref: +- e485edf8d992 +- a43d37f014e8 +rel: contains +reviewStatus: accepted +scope: proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof-contains +--- +Morgan--Tian proof containment. The source is the statement whose proof contains the target statement. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/9ec4540b898a.md b/projects/poincare-conjecture/.astrolabe/edges/9ec4540b898a.md new file mode 100644 index 00000000..c6fc62d7 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/9ec4540b898a.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 9 + file: temp2kappa.tex + label: '' + mtref: '9.47' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 83801 + sourceTransform: tex2mdx.strip_comments + start: 83646 + targetStatement: + chapter: 9 + file: temp2kappa.tex + label: '' + mtref: '9.48' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 86082 + sourceTransform: tex2mdx.strip_comments + start: 86005 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- e3b91b4c2028 +- 6060c7910950 +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/9edec63bb471.md b/projects/poincare-conjecture/.astrolabe/edges/9edec63bb471.md new file mode 100644 index 00000000..a60a2577 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/9edec63bb471.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 6 + file: newcompar.tex + label: '' + mtref: '6.35' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 54314 + sourceTransform: tex2mdx.strip_comments + start: 53838 + targetStatement: + chapter: 6 + file: newcompar.tex + label: adaptform + mtref: '6.36' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 54877 + sourceTransform: tex2mdx.strip_comments + start: 54358 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 1ac0cdbfd850 +- e1277342360a +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/9ef3640981ec.md b/projects/poincare-conjecture/.astrolabe/edges/9ef3640981ec.md new file mode 100644 index 00000000..a8997d1f --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/9ef3640981ec.md @@ -0,0 +1,62 @@ +--- +confidence: 0.9 +edgeClass: structural +evidence: + containmentDepth: 1 + proofSpan: + bodyEnd: 169692 + bodyStart: 163171 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 169703 + file: temp2kappa.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 163158 + sourceStatement: + chapter: 9 + file: temp2kappa.tex + label: '' + mtref: '9.86' + sort: remark + span: + coordinateSpace: comment-stripped-tex + end: 163155 + sourceTransform: tex2mdx.strip_comments + start: 163012 + targetContainedSpan: + coordinateSpace: comment-stripped-tex + end: 167340 + file: temp2kappa.tex + sourceTransform: tex2mdx.strip_comments + start: 167251 + targetStatement: + chapter: 9 + file: temp2kappa.tex + label: '' + mtref: '9.87' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 167340 + sourceTransform: tex2mdx.strip_comments + start: 167251 + type: proof-containment + via: + - proof-contains +generator: tools/poincare_tex_extract.py +inference: structural +kind: containment +ref: +- 51da06702a9f +- b9f671d590f3 +rel: contains +reviewStatus: accepted +scope: proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof-contains +--- +Morgan--Tian proof containment. The source is the statement whose proof contains the target statement. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/9f0102f63770.md b/projects/poincare-conjecture/.astrolabe/edges/9f0102f63770.md new file mode 100644 index 00000000..512d59fa --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/9f0102f63770.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: L_0 + proofSpan: + bodyEnd: 146058 + bodyStart: 138906 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 146069 + file: surgery.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 138893 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 146056 + file: surgery.tex + sourceTransform: tex2mdx.strip_comments + start: 146047 + via: proof + sourceStatement: + chapter: 16 + file: surgery.tex + label: '' + mtref: '16.16' + sort: remark + span: + coordinateSpace: comment-stripped-tex + end: 138750 + sourceTransform: tex2mdx.strip_comments + start: 138540 + targetStatement: + chapter: 16 + file: surgery.tex + label: L_0 + mtref: '16.15' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 138538 + sourceTransform: tex2mdx.strip_comments + start: 137276 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 4ee1bc9acc48 +- 86043191a3c2 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/9f014f144121.md b/projects/poincare-conjecture/.astrolabe/edges/9f014f144121.md new file mode 100644 index 00000000..49f5e183 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/9f014f144121.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 4 + file: maxprin.tex + label: quad + mtref: '4.13' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 16605 + sourceTransform: tex2mdx.strip_comments + start: 15794 + targetStatement: + chapter: 4 + file: maxprin.tex + label: rm>0 + mtref: '4.14' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 16848 + sourceTransform: tex2mdx.strip_comments + start: 16607 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 0c451aa3e5cc +- 0db415314964 +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/9f04614a2c7d.md b/projects/poincare-conjecture/.astrolabe/edges/9f04614a2c7d.md new file mode 100644 index 00000000..b663ec6a --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/9f04614a2c7d.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 1 + file: prelim.tex + label: '' + mtref: '1.4' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 6875 + sourceTransform: tex2mdx.strip_comments + start: 6491 + targetStatement: + chapter: 1 + file: prelim.tex + label: Bianchi + mtref: '1.5' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 8462 + sourceTransform: tex2mdx.strip_comments + start: 7974 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 8d7c3bebc8b1 +- c4848a1b6f80 +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/9f1a7610e114.md b/projects/poincare-conjecture/.astrolabe/edges/9f1a7610e114.md new file mode 100644 index 00000000..1be65047 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/9f1a7610e114.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: fmonotone + proofSpan: + bodyEnd: 16746 + bodyStart: 15073 + coordinateSpace: comment-stripped-tex + depth: 3 + end: 16757 + file: noncoll.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 15060 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 15129 + file: noncoll.tex + sourceTransform: tex2mdx.strip_comments + start: 15114 + via: proof + sourceStatement: + chapter: 8 + file: noncoll.tex + label: large + mtref: '8.9' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 15058 + sourceTransform: tex2mdx.strip_comments + start: 14671 + targetStatement: + chapter: 6 + file: newcompar.tex + label: fmonotone + mtref: '6.81' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 122525 + sourceTransform: tex2mdx.strip_comments + start: 122141 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 4b41e7ff8317 +- 4dc40f37cd6d +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/9f3bb17b30f6.md b/projects/poincare-conjecture/.astrolabe/edges/9f3bb17b30f6.md new file mode 100644 index 00000000..d0c0eb16 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/9f3bb17b30f6.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 6 + file: newcompar.tex + label: '' + mtref: '6.66' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 100067 + sourceTransform: tex2mdx.strip_comments + start: 99998 + targetStatement: + chapter: 6 + file: newcompar.tex + label: fullmeasure + mtref: '6.67' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 101253 + sourceTransform: tex2mdx.strip_comments + start: 100568 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- f655ed1e0c1f +- 7aad50e49af2 +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/9f43cd02af14.md b/projects/poincare-conjecture/.astrolabe/edges/9f43cd02af14.md new file mode 100644 index 00000000..2f364e60 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/9f43cd02af14.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: stdsoln + proofSpan: + bodyEnd: 119675 + bodyStart: 117487 + coordinateSpace: comment-stripped-tex + depth: 2 + end: 119686 + file: surgery.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 117474 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 118015 + file: surgery.tex + sourceTransform: tex2mdx.strip_comments + start: 118002 + via: proof + sourceStatement: + chapter: 16 + file: surgery.tex + label: '' + mtref: '16.8' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 117471 + sourceTransform: tex2mdx.strip_comments + start: 116211 + targetStatement: + chapter: 12 + file: stdsoln.tex + label: stdsoln + mtref: '12.5' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 8086 + sourceTransform: tex2mdx.strip_comments + start: 6763 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 4b6ee3b7a329 +- f44d182ef27e +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/9f4e5c536dfb.md b/projects/poincare-conjecture/.astrolabe/edges/9f4e5c536dfb.md new file mode 100644 index 00000000..cbf55306 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/9f4e5c536dfb.md @@ -0,0 +1,65 @@ +--- +confidence: 0.45 +edgeClass: semantic +evidence: + endpointSemanticDegreeBeforeTermBackfill: + source: 1 + target: 0 + selectionReason: semantic-connectivity-backfill + sourceMatchSpan: + coordinateSpace: comment-stripped-tex + end: 14410 + file: prelim.tex + sourceTransform: tex2mdx.strip_comments + start: 14391 + sourceStatement: + chapter: 1 + file: prelim.tex + label: '' + mtref: '1.11' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 14976 + sourceTransform: tex2mdx.strip_comments + start: 14241 + targetStatement: + chapter: 1 + file: prelim.tex + label: '' + mtref: '1.6' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 8971 + sourceTransform: tex2mdx.strip_comments + start: 8513 + targetTermSpan: + coordinateSpace: comment-stripped-tex + end: 8557 + file: prelim.tex + sourceTransform: tex2mdx.strip_comments + start: 8538 + term: sectional curvature + termAmbiguity: unique-in-chapter + termSource: italic + termSpecificity: not-covered-by-longer-defined-term + type: same-chapter-definition-term-match + via: + - definition-term:sectional curvature +generator: tools/poincare_tex_extract.py +inference: weak +kind: definition-use +ref: +- 5982e4029637 +- 71e017db1c5b +rel: uses +reviewStatus: candidate +scope: same-chapter-term-match +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- definition-term:sectional curvature +--- +Morgan--Tian same-chapter definition term match. This is a low-confidence candidate edge for review. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/9f76a65841fa.md b/projects/poincare-conjecture/.astrolabe/edges/9f76a65841fa.md new file mode 100644 index 00000000..fbac6b0d --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/9f76a65841fa.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 9 + file: temp2kappa.tex + label: '' + mtref: '9.55' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 106400 + sourceTransform: tex2mdx.strip_comments + start: 106063 + targetStatement: + chapter: 9 + file: temp2kappa.tex + label: asympscal + mtref: '9.56' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 106666 + sourceTransform: tex2mdx.strip_comments + start: 106402 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 1998da8dab3d +- 5d9f1b75170d +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/9fb7ac7d7532.md b/projects/poincare-conjecture/.astrolabe/edges/9fb7ac7d7532.md new file mode 100644 index 00000000..39dd6a51 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/9fb7ac7d7532.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 16 + file: surgery.tex + label: '' + mtref: '16.10' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 125643 + sourceTransform: tex2mdx.strip_comments + start: 125039 + targetStatement: + chapter: 16 + file: surgery.tex + label: '' + mtref: '16.11' + sort: remark + span: + coordinateSpace: comment-stripped-tex + end: 129667 + sourceTransform: tex2mdx.strip_comments + start: 129477 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 683bb06c1059 +- 4e18ead9ab36 +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/9fb84d4c2244.md b/projects/poincare-conjecture/.astrolabe/edges/9fb84d4c2244.md new file mode 100644 index 00000000..e58b94a7 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/9fb84d4c2244.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 6 + file: newcompar.tex + label: linequal + mtref: '6.49' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 78401 + sourceTransform: tex2mdx.strip_comments + start: 77391 + targetStatement: + chapter: 6 + file: newcompar.tex + label: Ueqn + mtref: '6.50' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 79994 + sourceTransform: tex2mdx.strip_comments + start: 79183 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 2b8ab39364e7 +- 4917e81d7684 +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/9fee78a6d7c7.md b/projects/poincare-conjecture/.astrolabe/edges/9fee78a6d7c7.md new file mode 100644 index 00000000..232c626b --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/9fee78a6d7c7.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: distlim + proofSpan: + bodyEnd: 47390 + bodyStart: 46531 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 47401 + file: bddcurvbdddist.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 46518 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 46558 + file: bddcurvbdddist.tex + sourceTransform: tex2mdx.strip_comments + start: 46545 + via: proof + sourceStatement: + chapter: 10 + file: bddcurvbdddist.tex + label: '' + mtref: '10.22' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 46515 + sourceTransform: tex2mdx.strip_comments + start: 46350 + targetStatement: + chapter: 10 + file: bddcurvbdddist.tex + label: distlim + mtref: '10.15' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 41003 + sourceTransform: tex2mdx.strip_comments + start: 40751 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- ddf1aed4b3bb +- 7ef7dfbbe151 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/9ff9c2722977.md b/projects/poincare-conjecture/.astrolabe/edges/9ff9c2722977.md new file mode 100644 index 00000000..c88cb679 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/9ff9c2722977.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: lipcomplete + proofSpan: + bodyEnd: 28599 + bodyStart: 25964 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 28610 + file: newcomp2.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 25951 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 25997 + file: newcomp2.tex + sourceTransform: tex2mdx.strip_comments + start: 25980 + via: proof + sourceStatement: + chapter: 7 + file: newcomp2.tex + label: phisupport + mtref: '7.15' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 25949 + sourceTransform: tex2mdx.strip_comments + start: 25492 + targetStatement: + chapter: 7 + file: newcomp2.tex + label: lipcomplete + mtref: '7.5' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 9437 + sourceTransform: tex2mdx.strip_comments + start: 8692 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 5e0538ed62cf +- 16fbe263a7c8 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/a004404ea0c1.md b/projects/poincare-conjecture/.astrolabe/edges/a004404ea0c1.md new file mode 100644 index 00000000..a6d1e532 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/a004404ea0c1.md @@ -0,0 +1,62 @@ +--- +confidence: 0.9 +edgeClass: structural +evidence: + containmentDepth: 1 + proofSpan: + bodyEnd: 14787 + bodyStart: 6954 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 14798 + file: singlimit2.tex + ownerLabel: smlmtflow + sourceTransform: tex2mdx.strip_comments + start: 6941 + sourceStatement: + chapter: 11 + file: singlimit2.tex + label: smlmtflow + mtref: '11.1' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 3459 + sourceTransform: tex2mdx.strip_comments + start: 1498 + targetContainedSpan: + coordinateSpace: comment-stripped-tex + end: 8841 + file: singlimit2.tex + sourceTransform: tex2mdx.strip_comments + start: 8603 + targetStatement: + chapter: 11 + file: singlimit2.tex + label: '' + mtref: '11.4' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 8841 + sourceTransform: tex2mdx.strip_comments + start: 8603 + type: proof-containment + via: + - proof-contains +generator: tools/poincare_tex_extract.py +inference: structural +kind: containment +ref: +- d51aab8a8065 +- 57c8cc8a5611 +rel: contains +reviewStatus: accepted +scope: proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof-contains +--- +Morgan--Tian proof containment. The source is the statement whose proof contains the target statement. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/a01b19bb1fe3.md b/projects/poincare-conjecture/.astrolabe/edges/a01b19bb1fe3.md new file mode 100644 index 00000000..6a62ef56 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/a01b19bb1fe3.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 6 + file: newcompar.tex + label: '' + mtref: '6.58' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 86220 + sourceTransform: tex2mdx.strip_comments + start: 85532 + targetStatement: + chapter: 6 + file: newcompar.tex + label: lipaty + mtref: '6.59' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 87473 + sourceTransform: tex2mdx.strip_comments + start: 86282 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 0fc5cfacef3f +- 96ced8c484df +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/a02e7195740c.md b/projects/poincare-conjecture/.astrolabe/edges/a02e7195740c.md new file mode 100644 index 00000000..cf7188e3 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/a02e7195740c.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: newbasis + proofSpan: + bodyEnd: 18640 + bodyStart: 18457 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 18651 + file: surgery.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 18444 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 18621 + file: surgery.tex + sourceTransform: tex2mdx.strip_comments + start: 18607 + via: proof + sourceStatement: + chapter: 13 + file: surgery.tex + label: '13.10' + mtref: '13.10' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 18442 + sourceTransform: tex2mdx.strip_comments + start: 17697 + targetStatement: + chapter: 13 + file: surgery.tex + label: newbasis + mtref: '13.7' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 14007 + sourceTransform: tex2mdx.strip_comments + start: 13167 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 347896f0385b +- 52d3cb562bdc +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/a098eeec7f33.md b/projects/poincare-conjecture/.astrolabe/edges/a098eeec7f33.md new file mode 100644 index 00000000..3b17362f --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/a098eeec7f33.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 3 + file: flowbasics.tex + label: '3.23' + mtref: '3.24' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 29260 + sourceTransform: tex2mdx.strip_comments + start: 29003 + targetStatement: + chapter: 3 + file: flowbasics.tex + label: 2ndI.8.3 + mtref: '3.25' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 36538 + sourceTransform: tex2mdx.strip_comments + start: 36082 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- a43d37f014e8 +- d22d3cf47fbf +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/a0a5c44673a9.md b/projects/poincare-conjecture/.astrolabe/edges/a0a5c44673a9.md new file mode 100644 index 00000000..cb020e93 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/a0a5c44673a9.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 1 + file: prelim.tex + label: '' + mtref: '1.17' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 19309 + sourceTransform: tex2mdx.strip_comments + start: 19124 + targetStatement: + chapter: 1 + file: prelim.tex + label: '' + mtref: '1.18' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 20381 + sourceTransform: tex2mdx.strip_comments + start: 20241 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- c4ccdd484637 +- e07da18ab307 +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/a0c8b62402d4.md b/projects/poincare-conjecture/.astrolabe/edges/a0c8b62402d4.md new file mode 100644 index 00000000..a26384cb --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/a0c8b62402d4.md @@ -0,0 +1,56 @@ +--- +confidence: 0.85 +edgeClass: semantic +evidence: + containmentReason: target theorem-like statement occurs inside source proof + proofSpan: + bodyEnd: 33794 + bodyStart: 17659 + coordinateSpace: comment-stripped-tex + depth: 2 + end: 33805 + file: singlimit2.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 17646 + sourceStatement: + chapter: 11 + file: singlimit2.tex + label: maxT + mtref: '11.10' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 17643 + sourceTransform: tex2mdx.strip_comments + start: 17183 + targetStatement: + chapter: 11 + file: singlimit2.tex + label: outerbd + mtref: '11.14' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 25285 + sourceTransform: tex2mdx.strip_comments + start: 24900 + type: proof-contained-theorem-like-statement + via: + - proof-contained-statement +generator: tools/poincare_tex_extract.py +inference: inferred +kind: dependency +ref: +- d6919616448d +- 9a7e3fe2e5c4 +rel: depends +reviewStatus: accepted +scope: proof-containment +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof-contained-statement +--- +Morgan--Tian proof-contained statement dependency. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/a120a052eafe.md b/projects/poincare-conjecture/.astrolabe/edges/a120a052eafe.md new file mode 100644 index 00000000..a2f84e3d --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/a120a052eafe.md @@ -0,0 +1,56 @@ +--- +confidence: 0.85 +edgeClass: semantic +evidence: + containmentReason: target theorem-like statement occurs inside source proof + proofSpan: + bodyEnd: 220240 + bodyStart: 211487 + coordinateSpace: comment-stripped-tex + depth: 2 + end: 220251 + file: surgery.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 211474 + sourceStatement: + chapter: 17 + file: surgery.tex + label: lemstdcan + mtref: '17.7' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 211471 + sourceTransform: tex2mdx.strip_comments + start: 210838 + targetStatement: + chapter: 17 + file: surgery.tex + label: '' + mtref: '17.8' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 212622 + sourceTransform: tex2mdx.strip_comments + start: 212469 + type: proof-contained-theorem-like-statement + via: + - proof-contained-statement +generator: tools/poincare_tex_extract.py +inference: inferred +kind: dependency +ref: +- 111551d7831b +- a01b63baabde +rel: depends +reviewStatus: accepted +scope: proof-containment +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof-contained-statement +--- +Morgan--Tian proof-contained statement dependency. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/a12de4f8669e.md b/projects/poincare-conjecture/.astrolabe/edges/a12de4f8669e.md new file mode 100644 index 00000000..a3f908b5 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/a12de4f8669e.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: 2ndvariation + proofSpan: + bodyEnd: 68625 + bodyStart: 60090 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 68636 + file: newcompar.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 60077 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 61354 + file: newcompar.tex + sourceTransform: tex2mdx.strip_comments + start: 61336 + via: proof + sourceStatement: + chapter: 6 + file: newcompar.tex + label: adapjab + mtref: '6.41' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 59462 + sourceTransform: tex2mdx.strip_comments + start: 58890 + targetStatement: + chapter: 6 + file: newcompar.tex + label: 2ndvariation + mtref: '6.33' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 47655 + sourceTransform: tex2mdx.strip_comments + start: 46570 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 72949e6350d4 +- 3add96057999 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/a145839c34ec.md b/projects/poincare-conjecture/.astrolabe/edges/a145839c34ec.md new file mode 100644 index 00000000..d2e0c07d --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/a145839c34ec.md @@ -0,0 +1,64 @@ +--- +confidence: 0.35 +edgeClass: prose +evidence: + mentionTriggers: + - label: omega + refSpan: + coordinateSpace: comment-stripped-tex + end: 48217 + file: singlimit2.tex + sourceTransform: tex2mdx.strip_comments + start: 48206 + sentence: Since, by assumption, points at time before $T$ with scalar curvature + at least $r_0^{-2}$ have strong $(C,\epsilon)$-canonical neighborhoods, this + completes the proof of the fifth conclusion of Theorem~\ref{omega} + sentenceSpan: + coordinateSpace: comment-stripped-tex + end: 48217 + file: singlimit2.tex + sourceTransform: tex2mdx.strip_comments + start: 48003 + sourceStrategy: nearest-previous-statement + via: prose + sourceStatement: + chapter: 11 + file: singlimit2.tex + label: epslimit + mtref: '11.24' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 45867 + sourceTransform: tex2mdx.strip_comments + start: 45550 + targetStatement: + chapter: 11 + file: singlimit2.tex + label: omega + mtref: '11.19' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 38153 + sourceTransform: tex2mdx.strip_comments + start: 36599 + type: prose-mention-nearest-previous-statement + via: + - prose +generator: tools/poincare_tex_extract.py +inference: weak +kind: mention +ref: +- d74749898b25 +- af00774dbaa9 +rel: mentions +reviewStatus: candidate +scope: prose +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- prose +--- +Morgan--Tian prose mention. The source is the nearest preceding statement in the same chapter segment. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/a1877523570b.md b/projects/poincare-conjecture/.astrolabe/edges/a1877523570b.md new file mode 100644 index 00000000..2da8df6a --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/a1877523570b.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: stdsoln + proofSpan: + bodyEnd: 220240 + bodyStart: 211487 + coordinateSpace: comment-stripped-tex + depth: 2 + end: 220251 + file: surgery.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 211474 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 214506 + file: surgery.tex + sourceTransform: tex2mdx.strip_comments + start: 214493 + via: proof + sourceStatement: + chapter: 17 + file: surgery.tex + label: lemstdcan + mtref: '17.7' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 211471 + sourceTransform: tex2mdx.strip_comments + start: 210838 + targetStatement: + chapter: 12 + file: stdsoln.tex + label: stdsoln + mtref: '12.5' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 8086 + sourceTransform: tex2mdx.strip_comments + start: 6763 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 111551d7831b +- f44d182ef27e +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/a1880d1db7b0.md b/projects/poincare-conjecture/.astrolabe/edges/a1880d1db7b0.md new file mode 100644 index 00000000..d8697308 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/a1880d1db7b0.md @@ -0,0 +1,45 @@ +--- +confidence: 0.2 +edgeClass: structural +evidence: + sourceStatement: + chapter: 12 + file: stdsoln.tex + label: R_0cannbhd + mtref: '12.28' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 83517 + sourceTransform: tex2mdx.strip_comments + start: 82969 + targetStatement: + chapter: 12 + file: stdsoln.tex + label: '' + mtref: '12.29' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 86070 + sourceTransform: tex2mdx.strip_comments + start: 86020 + type: chapter-reading-order-adjacency + via: + - chapter-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 85e3c925177e +- 729d9a0428c4 +rel: in-chapter +reviewStatus: accepted +scope: chapter +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- chapter-sequence +--- +Morgan--Tian reading-order adjacency across sections in a chapter. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/a189bb380dd9.md b/projects/poincare-conjecture/.astrolabe/edges/a189bb380dd9.md new file mode 100644 index 00000000..26569ca7 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/a189bb380dd9.md @@ -0,0 +1,63 @@ +--- +confidence: 0.35 +edgeClass: prose +evidence: + mentionTriggers: + - label: partialflowlimit + refSpan: + coordinateSpace: comment-stripped-tex + end: 17751 + file: stdsoln.tex + sourceTransform: tex2mdx.strip_comments + start: 17729 + sentence: Thus, by Theorem~\ref{partialflowlimit}, after passing to a subsequence, + the flows $\psi_k^*g(t)$ converge to a limiting flow on $S^2\times \Ar$ + sentenceSpan: + coordinateSpace: comment-stripped-tex + end: 17857 + file: stdsoln.tex + sourceTransform: tex2mdx.strip_comments + start: 17710 + sourceStrategy: nearest-previous-statement + via: prose + sourceStatement: + chapter: 12 + file: stdsoln.tex + label: third + mtref: '12.6' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 12060 + sourceTransform: tex2mdx.strip_comments + start: 11922 + targetStatement: + chapter: 5 + file: converge2.tex + label: partialflowlimit + mtref: '5.14' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 25227 + sourceTransform: tex2mdx.strip_comments + start: 23995 + type: prose-mention-nearest-previous-statement + via: + - prose +generator: tools/poincare_tex_extract.py +inference: weak +kind: mention +ref: +- 2e8c17743da6 +- 4673eb9f4424 +rel: mentions +reviewStatus: candidate +scope: prose +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- prose +--- +Morgan--Tian prose mention. The source is the nearest preceding statement in the same chapter segment. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/a1a5da211ebb.md b/projects/poincare-conjecture/.astrolabe/edges/a1a5da211ebb.md new file mode 100644 index 00000000..9a952669 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/a1a5da211ebb.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: zkbound + proofSpan: + bodyEnd: 137114 + bodyStart: 136498 + coordinateSpace: comment-stripped-tex + depth: 2 + end: 137125 + file: temp2kappa.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 136485 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 136820 + file: temp2kappa.tex + sourceTransform: tex2mdx.strip_comments + start: 136807 + via: proof + sourceStatement: + chapter: 9 + file: temp2kappa.tex + label: bdnearxk + mtref: '9.69' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 136482 + sourceTransform: tex2mdx.strip_comments + start: 136184 + targetStatement: + chapter: 9 + file: temp2kappa.tex + label: zkbound + mtref: '9.68' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 134931 + sourceTransform: tex2mdx.strip_comments + start: 134780 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 7c6182738b78 +- f61f72020102 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/a20556f7518a.md b/projects/poincare-conjecture/.astrolabe/edges/a20556f7518a.md new file mode 100644 index 00000000..33056df8 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/a20556f7518a.md @@ -0,0 +1,62 @@ +--- +confidence: 0.9 +edgeClass: structural +evidence: + containmentDepth: 3 + proofSpan: + bodyEnd: 14349 + bodyStart: 5983 + coordinateSpace: comment-stripped-tex + depth: 3 + end: 14360 + file: noncoll.tex + ownerLabel: small + sourceTransform: tex2mdx.strip_comments + start: 5970 + sourceStatement: + chapter: 8 + file: noncoll.tex + label: small + mtref: '8.3' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 5471 + sourceTransform: tex2mdx.strip_comments + start: 5161 + targetContainedSpan: + coordinateSpace: comment-stripped-tex + end: 8866 + file: noncoll.tex + sourceTransform: tex2mdx.strip_comments + start: 8367 + targetStatement: + chapter: 8 + file: noncoll.tex + label: '' + mtref: '8.6' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 8866 + sourceTransform: tex2mdx.strip_comments + start: 8367 + type: proof-containment + via: + - proof-contains +generator: tools/poincare_tex_extract.py +inference: structural +kind: containment +ref: +- 26e1a6292fef +- 59189adac6be +rel: contains +reviewStatus: accepted +scope: proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof-contains +--- +Morgan--Tian proof containment. The source is the statement whose proof contains the target statement. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/a21cbb49229c.md b/projects/poincare-conjecture/.astrolabe/edges/a21cbb49229c.md new file mode 100644 index 00000000..d1296737 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/a21cbb49229c.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: surgery + proofSpan: + bodyEnd: 16423 + bodyStart: 14954 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 16434 + file: intro.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 14941 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 15000 + file: intro.tex + sourceTransform: tex2mdx.strip_comments + start: 14987 + via: proof + sourceStatement: + chapter: 0 + file: intro.tex + label: corequiv + mtref: '0.5' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 14939 + sourceTransform: tex2mdx.strip_comments + start: 13757 + targetStatement: + chapter: 0 + file: intro.tex + label: surgery + mtref: '0.3' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 11419 + sourceTransform: tex2mdx.strip_comments + start: 10567 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 7078952cb9e4 +- 847af3ecd4a8 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/a22d7cee77cf.md b/projects/poincare-conjecture/.astrolabe/edges/a22d7cee77cf.md new file mode 100644 index 00000000..c4f0a72f --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/a22d7cee77cf.md @@ -0,0 +1,62 @@ +--- +confidence: 0.9 +edgeClass: structural +evidence: + containmentDepth: 1 + proofSpan: + bodyEnd: 123626 + bodyStart: 120090 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 123637 + file: temp2kappa.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 120077 + sourceStatement: + chapter: 9 + file: temp2kappa.tex + label: volcomp + mtref: '9.60' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 120074 + sourceTransform: tex2mdx.strip_comments + start: 119331 + targetContainedSpan: + coordinateSpace: comment-stripped-tex + end: 121789 + file: temp2kappa.tex + sourceTransform: tex2mdx.strip_comments + start: 121292 + targetStatement: + chapter: 9 + file: temp2kappa.tex + label: '' + mtref: '9.61' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 121789 + sourceTransform: tex2mdx.strip_comments + start: 121292 + type: proof-containment + via: + - proof-contains +generator: tools/poincare_tex_extract.py +inference: structural +kind: containment +ref: +- 5cfd7c3aee29 +- d2259da4d67d +rel: contains +reviewStatus: accepted +scope: proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof-contains +--- +Morgan--Tian proof containment. The source is the statement whose proof contains the target statement. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/a23047833762.md b/projects/poincare-conjecture/.astrolabe/edges/a23047833762.md new file mode 100644 index 00000000..a16f2239 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/a23047833762.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: outerbd + proofSpan: + bodyEnd: 31467 + bodyStart: 29842 + coordinateSpace: comment-stripped-tex + depth: 3 + end: 31478 + file: singlimit2.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 29829 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 30703 + file: singlimit2.tex + sourceTransform: tex2mdx.strip_comments + start: 30690 + via: proof + sourceStatement: + chapter: 11 + file: singlimit2.tex + label: ancientlimit + mtref: '11.16' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 29827 + sourceTransform: tex2mdx.strip_comments + start: 29411 + targetStatement: + chapter: 11 + file: singlimit2.tex + label: outerbd + mtref: '11.14' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 25285 + sourceTransform: tex2mdx.strip_comments + start: 24900 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- a66fa0ea8720 +- 9a7e3fe2e5c4 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/a2b2d2cc0fae.md b/projects/poincare-conjecture/.astrolabe/edges/a2b2d2cc0fae.md new file mode 100644 index 00000000..1d47b1e1 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/a2b2d2cc0fae.md @@ -0,0 +1,45 @@ +--- +confidence: 0.2 +edgeClass: structural +evidence: + sourceStatement: + chapter: 9 + file: temp2kappa.tex + label: asympvol + mtref: '9.59' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 118207 + sourceTransform: tex2mdx.strip_comments + start: 117600 + targetStatement: + chapter: 9 + file: temp2kappa.tex + label: volcomp + mtref: '9.60' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 120074 + sourceTransform: tex2mdx.strip_comments + start: 119331 + type: chapter-reading-order-adjacency + via: + - chapter-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 7468282bd017 +- 5cfd7c3aee29 +rel: in-chapter +reviewStatus: accepted +scope: chapter +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- chapter-sequence +--- +Morgan--Tian reading-order adjacency across sections in a chapter. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/a2f9e9c13621.md b/projects/poincare-conjecture/.astrolabe/edges/a2f9e9c13621.md new file mode 100644 index 00000000..23299725 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/a2f9e9c13621.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: surgerydist + proofSpan: + bodyEnd: 56882 + bodyStart: 56642 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 56893 + file: energy1.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 56629 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 56709 + file: energy1.tex + sourceTransform: tex2mdx.strip_comments + start: 56692 + via: proof + sourceStatement: + chapter: 18 + file: energy1.tex + label: '' + mtref: '18.22' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 56627 + sourceTransform: tex2mdx.strip_comments + start: 56502 + targetStatement: + chapter: 15 + file: surgery.tex + label: surgerydist + mtref: '15.12' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 97780 + sourceTransform: tex2mdx.strip_comments + start: 96249 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 981cc21e82e4 +- 9021a4eca03f +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/a30b6327f8d1.md b/projects/poincare-conjecture/.astrolabe/edges/a30b6327f8d1.md new file mode 100644 index 00000000..b6ed1545 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/a30b6327f8d1.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 6 + file: newcompar.tex + label: defnD + mtref: '6.17' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 24744 + sourceTransform: tex2mdx.strip_comments + start: 23484 + targetStatement: + chapter: 6 + file: newcompar.tex + label: lcdiffeo + mtref: '6.18' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 25637 + sourceTransform: tex2mdx.strip_comments + start: 25268 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- df842ef2ab9f +- 948d50598f2f +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/a30dccac6d51.md b/projects/poincare-conjecture/.astrolabe/edges/a30dccac6d51.md new file mode 100644 index 00000000..60e5536f --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/a30dccac6d51.md @@ -0,0 +1,45 @@ +--- +confidence: 0.2 +edgeClass: structural +evidence: + sourceStatement: + chapter: 9 + file: temp2kappa.tex + label: kappa0 + mtref: '9.58' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 113708 + sourceTransform: tex2mdx.strip_comments + start: 113563 + targetStatement: + chapter: 9 + file: temp2kappa.tex + label: asympvol + mtref: '9.59' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 118207 + sourceTransform: tex2mdx.strip_comments + start: 117600 + type: chapter-reading-order-adjacency + via: + - chapter-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 9c54e0d0c7d8 +- 7468282bd017 +rel: in-chapter +reviewStatus: accepted +scope: chapter +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- chapter-sequence +--- +Morgan--Tian reading-order adjacency across sections in a chapter. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/a3592b067b28.md b/projects/poincare-conjecture/.astrolabe/edges/a3592b067b28.md new file mode 100644 index 00000000..a83cd7c8 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/a3592b067b28.md @@ -0,0 +1,62 @@ +--- +confidence: 0.9 +edgeClass: structural +evidence: + containmentDepth: 1 + proofSpan: + bodyEnd: 129463 + bodyStart: 113176 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 129474 + file: surgery.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 113163 + sourceStatement: + chapter: 16 + file: surgery.tex + label: altern + mtref: '16.5' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 112999 + sourceTransform: tex2mdx.strip_comments + start: 110961 + targetContainedSpan: + coordinateSpace: comment-stripped-tex + end: 114511 + file: surgery.tex + sourceTransform: tex2mdx.strip_comments + start: 113887 + targetStatement: + chapter: 16 + file: surgery.tex + label: '' + mtref: '16.6' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 114511 + sourceTransform: tex2mdx.strip_comments + start: 113887 + type: proof-containment + via: + - proof-contains +generator: tools/poincare_tex_extract.py +inference: structural +kind: containment +ref: +- ad4f67ea75bc +- ef94451b5785 +rel: contains +reviewStatus: accepted +scope: proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof-contains +--- +Morgan--Tian proof containment. The source is the statement whose proof contains the target statement. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/a35a5ba1707b.md b/projects/poincare-conjecture/.astrolabe/edges/a35a5ba1707b.md new file mode 100644 index 00000000..c9a19134 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/a35a5ba1707b.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 13 + file: surgery.tex + label: LOCALSURGERY + mtref: '13.2' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 6442 + sourceTransform: tex2mdx.strip_comments + start: 4875 + targetStatement: + chapter: 13 + file: surgery.tex + label: surgerycap + mtref: '13.3' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 6755 + sourceTransform: tex2mdx.strip_comments + start: 6605 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- cf61b731c6f2 +- 443843fff58d +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/a374763400ab.md b/projects/poincare-conjecture/.astrolabe/edges/a374763400ab.md new file mode 100644 index 00000000..dd83e18f --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/a374763400ab.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: Rhatmatrix + proofSpan: + bodyEnd: 20268 + bodyStart: 19525 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 20279 + file: surgery.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 19512 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 19873 + file: surgery.tex + sourceTransform: tex2mdx.strip_comments + start: 19857 + via: proof + sourceStatement: + chapter: 13 + file: surgery.tex + label: improve + mtref: '13.11' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 19510 + sourceTransform: tex2mdx.strip_comments + start: 18654 + targetStatement: + chapter: 13 + file: surgery.tex + label: Rhatmatrix + mtref: '13.9' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 17454 + sourceTransform: tex2mdx.strip_comments + start: 16730 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 41a5f172b2dc +- 0a41b5c61dd2 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/a37e481a435e.md b/projects/poincare-conjecture/.astrolabe/edges/a37e481a435e.md new file mode 100644 index 00000000..f99e5daa --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/a37e481a435e.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: curvshrexist + proofSpan: + bodyEnd: 133472 + bodyStart: 130609 + coordinateSpace: comment-stripped-tex + depth: 2 + end: 133483 + file: energy1.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 130596 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 133015 + file: energy1.tex + sourceTransform: tex2mdx.strip_comments + start: 132997 + via: proof + sourceStatement: + chapter: 18 + file: energy1.tex + label: limitcs + mtref: '18.56' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 130594 + sourceTransform: tex2mdx.strip_comments + start: 129800 + targetStatement: + chapter: 18 + file: energy1.tex + label: curvshrexist + mtref: '18.29' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 68623 + sourceTransform: tex2mdx.strip_comments + start: 68167 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 07ac984fbe88 +- 656c8fbce129 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/a399ab6990e1.md b/projects/poincare-conjecture/.astrolabe/edges/a399ab6990e1.md new file mode 100644 index 00000000..0a7ac26a --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/a399ab6990e1.md @@ -0,0 +1,55 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: statement + label: L_0 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 164417 + file: surgery.tex + sourceTransform: tex2mdx.strip_comments + start: 164408 + via: statement + sourceStatement: + chapter: 16 + file: surgery.tex + label: bardelta1 + mtref: '16.24' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 164841 + sourceTransform: tex2mdx.strip_comments + start: 164269 + targetStatement: + chapter: 16 + file: surgery.tex + label: L_0 + mtref: '16.15' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 138538 + sourceTransform: tex2mdx.strip_comments + start: 137276 + type: tex-reference + via: + - statement +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- ae8837c505cc +- 86043191a3c2 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- statement +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/a3b2ba3e14bb.md b/projects/poincare-conjecture/.astrolabe/edges/a3b2ba3e14bb.md new file mode 100644 index 00000000..00b638c2 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/a3b2ba3e14bb.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: posdistdec + proofSpan: + bodyEnd: 89985 + bodyStart: 86085 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 89996 + file: stdsoln.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 86072 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 89545 + file: stdsoln.tex + sourceTransform: tex2mdx.strip_comments + start: 89529 + via: proof + sourceStatement: + chapter: 12 + file: stdsoln.tex + label: '' + mtref: '12.29' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 86070 + sourceTransform: tex2mdx.strip_comments + start: 86020 + targetStatement: + chapter: 3 + file: flowbasics.tex + label: posdistdec + mtref: '3.14' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 17555 + sourceTransform: tex2mdx.strip_comments + start: 17308 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 729d9a0428c4 +- 51a2afee4465 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/a3b9c0677fa7.md b/projects/poincare-conjecture/.astrolabe/edges/a3b9c0677fa7.md new file mode 100644 index 00000000..65672a9b --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/a3b9c0677fa7.md @@ -0,0 +1,45 @@ +--- +confidence: 0.2 +edgeClass: structural +evidence: + sourceStatement: + chapter: 3 + file: flowbasics.tex + label: patch + mtref: '3.12' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 15371 + sourceTransform: tex2mdx.strip_comments + start: 14869 + targetStatement: + chapter: 3 + file: flowbasics.tex + label: thmRmevol + mtref: '3.13' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 16962 + sourceTransform: tex2mdx.strip_comments + start: 16120 + type: chapter-reading-order-adjacency + via: + - chapter-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- b6bc786f7f44 +- 1515151790b2 +rel: in-chapter +reviewStatus: accepted +scope: chapter +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- chapter-sequence +--- +Morgan--Tian reading-order adjacency across sections in a chapter. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/a3c449287c59.md b/projects/poincare-conjecture/.astrolabe/edges/a3c449287c59.md new file mode 100644 index 00000000..05562ffd --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/a3c449287c59.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 11 + file: singlimit2.tex + label: controlnbhd + mtref: '11.2' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 4360 + sourceTransform: tex2mdx.strip_comments + start: 3597 + targetStatement: + chapter: 11 + file: singlimit2.tex + label: cnnbhd2 + mtref: '11.3' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 8201 + sourceTransform: tex2mdx.strip_comments + start: 7592 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 0ddfe578961e +- 3989b6b34e08 +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/a3c472abbd03.md b/projects/poincare-conjecture/.astrolabe/edges/a3c472abbd03.md new file mode 100644 index 00000000..b369431a --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/a3c472abbd03.md @@ -0,0 +1,45 @@ +--- +confidence: 0.2 +edgeClass: structural +evidence: + sourceStatement: + chapter: 8 + file: noncoll.tex + label: step + mtref: '8.2' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 3313 + sourceTransform: tex2mdx.strip_comments + start: 3033 + targetStatement: + chapter: 8 + file: noncoll.tex + label: small + mtref: '8.3' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 5471 + sourceTransform: tex2mdx.strip_comments + start: 5161 + type: chapter-reading-order-adjacency + via: + - chapter-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 12d93b430400 +- 26e1a6292fef +rel: in-chapter +reviewStatus: accepted +scope: chapter +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- chapter-sequence +--- +Morgan--Tian reading-order adjacency across sections in a chapter. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/a3d8a4e7e2e1.md b/projects/poincare-conjecture/.astrolabe/edges/a3d8a4e7e2e1.md new file mode 100644 index 00000000..1d10e97c --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/a3d8a4e7e2e1.md @@ -0,0 +1,62 @@ +--- +confidence: 0.9 +edgeClass: structural +evidence: + containmentDepth: 1 + proofSpan: + bodyEnd: 184898 + bodyStart: 173854 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 184909 + file: energy1.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 173841 + sourceStatement: + chapter: 18 + file: energy1.tex + label: Dxacross + mtref: '18.74' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 173839 + sourceTransform: tex2mdx.strip_comments + start: 173679 + targetContainedSpan: + coordinateSpace: comment-stripped-tex + end: 180730 + file: energy1.tex + sourceTransform: tex2mdx.strip_comments + start: 180558 + targetStatement: + chapter: 18 + file: energy1.tex + label: '' + mtref: '18.78' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 180730 + sourceTransform: tex2mdx.strip_comments + start: 180558 + type: proof-containment + via: + - proof-contains +generator: tools/poincare_tex_extract.py +inference: structural +kind: containment +ref: +- 0d6f74e37efb +- 2b50ea294b54 +rel: contains +reviewStatus: accepted +scope: proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof-contains +--- +Morgan--Tian proof containment. The source is the statement whose proof contains the target statement. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/a3e35995bbf3.md b/projects/poincare-conjecture/.astrolabe/edges/a3e35995bbf3.md new file mode 100644 index 00000000..28d43cc6 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/a3e35995bbf3.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: minmax + proofSpan: + bodyEnd: 104354 + bodyStart: 102230 + coordinateSpace: comment-stripped-tex + depth: 2 + end: 104365 + file: newcompar.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 102217 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 103462 + file: newcompar.tex + sourceTransform: tex2mdx.strip_comments + start: 103450 + via: proof + sourceStatement: + chapter: 6 + file: newcompar.tex + label: twosets + mtref: '6.68' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 102215 + sourceTransform: tex2mdx.strip_comments + start: 101653 + targetStatement: + chapter: 6 + file: newcompar.tex + label: minmax + mtref: '6.61' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 91839 + sourceTransform: tex2mdx.strip_comments + start: 91251 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- a673ebd63624 +- a7c2ebee07a3 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/a3ecc93dee2f.md b/projects/poincare-conjecture/.astrolabe/edges/a3ecc93dee2f.md new file mode 100644 index 00000000..1d2c1a87 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/a3ecc93dee2f.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: L_0 + proofSpan: + bodyEnd: 157029 + bodyStart: 156024 + coordinateSpace: comment-stripped-tex + depth: 2 + end: 157040 + file: surgery.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 156011 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 157027 + file: surgery.tex + sourceTransform: tex2mdx.strip_comments + start: 157018 + via: proof + sourceStatement: + chapter: 16 + file: surgery.tex + label: '' + mtref: '16.22' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 156009 + sourceTransform: tex2mdx.strip_comments + start: 155858 + targetStatement: + chapter: 16 + file: surgery.tex + label: L_0 + mtref: '16.15' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 138538 + sourceTransform: tex2mdx.strip_comments + start: 137276 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- cfb464e3cf35 +- 86043191a3c2 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/a3eeb9e2665b.md b/projects/poincare-conjecture/.astrolabe/edges/a3eeb9e2665b.md new file mode 100644 index 00000000..d49419f8 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/a3eeb9e2665b.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: MP + proofSpan: + bodyEnd: 18062 + bodyStart: 16863 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 18073 + file: maxprin.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 16850 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 18042 + file: maxprin.tex + sourceTransform: tex2mdx.strip_comments + start: 18034 + via: proof + sourceStatement: + chapter: 4 + file: maxprin.tex + label: rm>0 + mtref: '4.14' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 16848 + sourceTransform: tex2mdx.strip_comments + start: 16607 + targetStatement: + chapter: 4 + file: maxprin.tex + label: MP + mtref: '4.7' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 8808 + sourceTransform: tex2mdx.strip_comments + start: 7939 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 0db415314964 +- c954e95b5d89 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/a411aaff789d.md b/projects/poincare-conjecture/.astrolabe/edges/a411aaff789d.md new file mode 100644 index 00000000..1f121fc7 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/a411aaff789d.md @@ -0,0 +1,65 @@ +--- +confidence: 0.35 +edgeClass: prose +evidence: + mentionTriggers: + - label: smlmtflow + refSpan: + coordinateSpace: comment-stripped-tex + end: 64536 + file: surgery.tex + sourceTransform: tex2mdx.strip_comments + start: 64521 + sentence: For all such $\epsilon$, Theorem~\ref{bcbd} holds for $\epsilon$ and + Proposition~\ref{narrows}, Proposition~\ref{canonvary} and Corollaries~\ref{kappacannbhd} + and~\ref{limitcannbhd} and Theorems~\ref{smlmtflow} and~\ref{kaplimit} hold + for $2\epsilon$ + sentenceSpan: + coordinateSpace: comment-stripped-tex + end: 64577 + file: surgery.tex + sourceTransform: tex2mdx.strip_comments + start: 64325 + sourceStrategy: nearest-previous-statement + via: prose + sourceStatement: + chapter: 15 + file: surgery.tex + label: neckglue + mtref: '15.2' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 61465 + sourceTransform: tex2mdx.strip_comments + start: 60812 + targetStatement: + chapter: 11 + file: singlimit2.tex + label: smlmtflow + mtref: '11.1' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 3459 + sourceTransform: tex2mdx.strip_comments + start: 1498 + type: prose-mention-nearest-previous-statement + via: + - prose +generator: tools/poincare_tex_extract.py +inference: weak +kind: mention +ref: +- a992d6a89479 +- d51aab8a8065 +rel: mentions +reviewStatus: candidate +scope: prose +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- prose +--- +Morgan--Tian prose mention. The source is the nearest preceding statement in the same chapter segment. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/a42cf8f8d3e4.md b/projects/poincare-conjecture/.astrolabe/edges/a42cf8f8d3e4.md new file mode 100644 index 00000000..c41ff324 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/a42cf8f8d3e4.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 1 + file: prelim.tex + label: conecurv + mtref: '1.15' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 18132 + sourceTransform: tex2mdx.strip_comments + start: 17714 + targetStatement: + chapter: 1 + file: prelim.tex + label: '' + mtref: '1.16' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 18438 + sourceTransform: tex2mdx.strip_comments + start: 18135 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 1b517dfd4bd7 +- ad532c16fad1 +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/a43702221a6c.md b/projects/poincare-conjecture/.astrolabe/edges/a43702221a6c.md new file mode 100644 index 00000000..6b1d7a1a --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/a43702221a6c.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 18 + file: energy1.tex + label: '1835' + mtref: '18.37' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 85554 + sourceTransform: tex2mdx.strip_comments + start: 85239 + targetStatement: + chapter: 18 + file: energy1.tex + label: lncurvgrowth + mtref: '18.38' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 87232 + sourceTransform: tex2mdx.strip_comments + start: 86861 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 88b8b3366b19 +- 156a3da68fbd +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/a457d29ddd55.md b/projects/poincare-conjecture/.astrolabe/edges/a457d29ddd55.md new file mode 100644 index 00000000..a62b2586 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/a457d29ddd55.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: item5 + proofSpan: + bodyEnd: 129463 + bodyStart: 113176 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 129474 + file: surgery.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 113163 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 122367 + file: surgery.tex + sourceTransform: tex2mdx.strip_comments + start: 122356 + via: proof + sourceStatement: + chapter: 16 + file: surgery.tex + label: altern + mtref: '16.5' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 112999 + sourceTransform: tex2mdx.strip_comments + start: 110961 + targetStatement: + chapter: 12 + file: stdsoln.tex + label: item5 + mtref: '12.7' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 18988 + sourceTransform: tex2mdx.strip_comments + start: 18657 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- ad4f67ea75bc +- 77ca4669ffb3 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/a4865d4226dc.md b/projects/poincare-conjecture/.astrolabe/edges/a4865d4226dc.md new file mode 100644 index 00000000..2d3587f7 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/a4865d4226dc.md @@ -0,0 +1,56 @@ +--- +confidence: 0.85 +edgeClass: semantic +evidence: + containmentReason: target theorem-like statement occurs inside source proof + proofSpan: + bodyEnd: 74813 + bodyStart: 59556 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 74824 + file: singlimit2.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 59543 + sourceStatement: + chapter: 11 + file: singlimit2.tex + label: hexist + mtref: '11.31' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 59267 + sourceTransform: tex2mdx.strip_comments + start: 58375 + targetStatement: + chapter: 11 + file: singlimit2.tex + label: hyp + mtref: '11.32' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 60461 + sourceTransform: tex2mdx.strip_comments + start: 60310 + type: proof-contained-theorem-like-statement + via: + - proof-contained-statement +generator: tools/poincare_tex_extract.py +inference: inferred +kind: dependency +ref: +- 074e02b067b8 +- dff170186455 +rel: depends +reviewStatus: accepted +scope: proof-containment +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof-contained-statement +--- +Morgan--Tian proof-contained statement dependency. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/a49198db53c8.md b/projects/poincare-conjecture/.astrolabe/edges/a49198db53c8.md new file mode 100644 index 00000000..50ae4cf5 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/a49198db53c8.md @@ -0,0 +1,45 @@ +--- +confidence: 0.2 +edgeClass: structural +evidence: + sourceStatement: + chapter: 5 + file: converge2.tex + label: conelimit + mtref: '5.31' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 45881 + sourceTransform: tex2mdx.strip_comments + start: 45285 + targetStatement: + chapter: 5 + file: converge2.tex + label: '' + mtref: '5.32' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 50532 + sourceTransform: tex2mdx.strip_comments + start: 49697 + type: chapter-reading-order-adjacency + via: + - chapter-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 88a75d6dd09e +- 666538bf577e +rel: in-chapter +reviewStatus: accepted +scope: chapter +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- chapter-sequence +--- +Morgan--Tian reading-order adjacency across sections in a chapter. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/a4bc74072052.md b/projects/poincare-conjecture/.astrolabe/edges/a4bc74072052.md new file mode 100644 index 00000000..27bf6f00 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/a4bc74072052.md @@ -0,0 +1,56 @@ +--- +confidence: 0.85 +edgeClass: semantic +evidence: + containmentReason: target theorem-like statement occurs inside source proof + proofSpan: + bodyEnd: 151923 + bodyStart: 149815 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 151934 + file: energy1.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 149802 + sourceStatement: + chapter: 18 + file: energy1.tex + label: '' + mtref: '18.61' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 149799 + sourceTransform: tex2mdx.strip_comments + start: 149158 + targetStatement: + chapter: 18 + file: energy1.tex + label: '' + mtref: '18.62' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 150673 + sourceTransform: tex2mdx.strip_comments + start: 150561 + type: proof-contained-theorem-like-statement + via: + - proof-contained-statement +generator: tools/poincare_tex_extract.py +inference: inferred +kind: dependency +ref: +- d91a6d9473a7 +- 541ad02dd5f8 +rel: depends +reviewStatus: accepted +scope: proof-containment +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof-contained-statement +--- +Morgan--Tian proof-contained statement dependency. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/a4d5ca8dddf8.md b/projects/poincare-conjecture/.astrolabe/edges/a4d5ca8dddf8.md new file mode 100644 index 00000000..e079b374 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/a4d5ca8dddf8.md @@ -0,0 +1,55 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: statement + label: smoothconv + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 24536 + file: converge2.tex + sourceTransform: tex2mdx.strip_comments + start: 24520 + via: statement + sourceStatement: + chapter: 5 + file: converge2.tex + label: partialflowlimit + mtref: '5.14' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 25227 + sourceTransform: tex2mdx.strip_comments + start: 23995 + targetStatement: + chapter: 5 + file: converge2.tex + label: smoothconv + mtref: '5.3' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 4757 + sourceTransform: tex2mdx.strip_comments + start: 2949 + type: tex-reference + via: + - statement +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 4673eb9f4424 +- 467f4b920031 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- statement +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/a51bfb63dad1.md b/projects/poincare-conjecture/.astrolabe/edges/a51bfb63dad1.md new file mode 100644 index 00000000..9574b34d --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/a51bfb63dad1.md @@ -0,0 +1,45 @@ +--- +confidence: 0.2 +edgeClass: structural +evidence: + sourceStatement: + chapter: 5 + file: converge2.tex + label: '' + mtref: '5.26' + sort: example + span: + coordinateSpace: comment-stripped-tex + end: 40373 + sourceTransform: tex2mdx.strip_comments + start: 40144 + targetStatement: + chapter: 5 + file: converge2.tex + label: '' + mtref: '5.27' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 40739 + sourceTransform: tex2mdx.strip_comments + start: 40490 + type: chapter-reading-order-adjacency + via: + - chapter-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 4f373966523b +- 9b970f13e747 +rel: in-chapter +reviewStatus: accepted +scope: chapter +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- chapter-sequence +--- +Morgan--Tian reading-order adjacency across sections in a chapter. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/a51d1cb5a653.md b/projects/poincare-conjecture/.astrolabe/edges/a51d1cb5a653.md new file mode 100644 index 00000000..a923e115 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/a51d1cb5a653.md @@ -0,0 +1,103 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: curveshort + proofSpan: + bodyEnd: 12140 + bodyStart: 9462 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 12151 + file: bddcurvbdddist.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 9449 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 9870 + file: bddcurvbdddist.tex + sourceTransform: tex2mdx.strip_comments + start: 9854 + via: proof + - context: proof + label: curveshort + proofSpan: + bodyEnd: 12140 + bodyStart: 9462 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 12151 + file: bddcurvbdddist.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 9449 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 10351 + file: bddcurvbdddist.tex + sourceTransform: tex2mdx.strip_comments + start: 10335 + via: proof + - context: proof + label: curveshort + proofSpan: + bodyEnd: 12140 + bodyStart: 9462 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 12151 + file: bddcurvbdddist.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 9449 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 11248 + file: bddcurvbdddist.tex + sourceTransform: tex2mdx.strip_comments + start: 11232 + via: proof + sourceStatement: + chapter: 10 + file: bddcurvbdddist.tex + label: '' + mtref: '10.3' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 9447 + sourceTransform: tex2mdx.strip_comments + start: 9272 + targetStatement: + chapter: 19 + file: canonnbhd.tex + label: curveshort + mtref: '19.1' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 1520 + sourceTransform: tex2mdx.strip_comments + start: 793 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 31216ebde119 +- fc6c6ae605d2 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/a59aefab2c04.md b/projects/poincare-conjecture/.astrolabe/edges/a59aefab2c04.md new file mode 100644 index 00000000..8b688461 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/a59aefab2c04.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 3 + file: flowbasics.tex + label: '' + mtref: '3.36' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 68154 + sourceTransform: tex2mdx.strip_comments + start: 67865 + targetStatement: + chapter: 3 + file: flowbasics.tex + label: '' + mtref: '3.37' + sort: remark + span: + coordinateSpace: comment-stripped-tex + end: 68588 + sourceTransform: tex2mdx.strip_comments + start: 68157 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 7048ed283c84 +- f4523c2b6f1e +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/a5ba3af87ba1.md b/projects/poincare-conjecture/.astrolabe/edges/a5ba3af87ba1.md new file mode 100644 index 00000000..fae185ea --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/a5ba3af87ba1.md @@ -0,0 +1,72 @@ +--- +confidence: 0.35 +edgeClass: prose +evidence: + mentionTriggers: + - label: surgery + refSpan: + coordinateSpace: comment-stripped-tex + end: 12858 + file: intro.tex + sourceTransform: tex2mdx.strip_comments + start: 12845 + sentence: 'We immediately deduce Theorem~\ref{Theorem1} from Theorems~\ref{surgery} + and~\ref{finiteext} as follows: Let $M$ be a $3$-manifold satisfying the hypothesis + of Theorem~\ref{Theorem1}. Then there is a finite sequence $M=M_0,M_1,\ldots,M_k=\emptyset$ + such that for each $i,\ 1\le i\le k$, $M_i$ is obtained from $M_{i-1}$ by a + connected sum decomposition or $M_i$ is obtained from $M_{i-1}$ by removing + a component diffeomorphic to one of $S^2\times S^1$, $\Ar P^3\#\Ar P^3$, a non-orientable + $2$-sphere bundle over $S^1$, or a $3$-dimensional spherical space-form. Clearly, + it follows by downward induction on $i$ that each connected component of $M_i$ + is diffeomorphic to a connected sum of $3$-dimensional spherical space-forms, + copies of $S^2\times S^1$, and copies of the non-orientable $2$-sphere bundle + over $S^1$' + sentenceSpan: + coordinateSpace: comment-stripped-tex + end: 13608 + file: intro.tex + sourceTransform: tex2mdx.strip_comments + start: 12785 + sourceStrategy: nearest-previous-statement + via: prose + sourceStatement: + chapter: 0 + file: intro.tex + label: finiteext + mtref: '0.4' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 12644 + sourceTransform: tex2mdx.strip_comments + start: 11822 + targetStatement: + chapter: 0 + file: intro.tex + label: surgery + mtref: '0.3' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 11419 + sourceTransform: tex2mdx.strip_comments + start: 10567 + type: prose-mention-nearest-previous-statement + via: + - prose +generator: tools/poincare_tex_extract.py +inference: weak +kind: mention +ref: +- 127db1dc9315 +- 847af3ecd4a8 +rel: mentions +reviewStatus: candidate +scope: prose +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- prose +--- +Morgan--Tian prose mention. The source is the nearest preceding statement in the same chapter segment. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/a5c1458b6071.md b/projects/poincare-conjecture/.astrolabe/edges/a5c1458b6071.md new file mode 100644 index 00000000..3a1bdbe5 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/a5c1458b6071.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 6 + file: newcompar.tex + label: Arnquantity + mtref: '6.77' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 113400 + sourceTransform: tex2mdx.strip_comments + start: 113216 + targetStatement: + chapter: 6 + file: newcompar.tex + label: Jlimit + mtref: '6.78' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 114106 + sourceTransform: tex2mdx.strip_comments + start: 113458 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 9d4d83df4874 +- 4e6330b2aa46 +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/a5c9387f4210.md b/projects/poincare-conjecture/.astrolabe/edges/a5c9387f4210.md new file mode 100644 index 00000000..1157c2a8 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/a5c9387f4210.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: tausqtau + proofSpan: + bodyEnd: 42448 + bodyStart: 41878 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 42459 + file: temp2kappa.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 41865 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 41902 + file: temp2kappa.tex + sourceTransform: tex2mdx.strip_comments + start: 41888 + via: proof + sourceStatement: + chapter: 9 + file: temp2kappa.tex + label: '' + mtref: '9.26' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 41863 + sourceTransform: tex2mdx.strip_comments + start: 41621 + targetStatement: + chapter: 9 + file: temp2kappa.tex + label: tausqtau + mtref: '9.15' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 17779 + sourceTransform: tex2mdx.strip_comments + start: 17653 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- d16f43d74c54 +- 88571a22a2bc +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/a5f5e7db4532.md b/projects/poincare-conjecture/.astrolabe/edges/a5f5e7db4532.md new file mode 100644 index 00000000..a594e655 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/a5f5e7db4532.md @@ -0,0 +1,62 @@ +--- +confidence: 0.9 +edgeClass: structural +evidence: + containmentDepth: 1 + proofSpan: + bodyEnd: 173338 + bodyStart: 165469 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 173349 + file: surgery.tex + ownerLabel: delta0ri+1 + sourceTransform: tex2mdx.strip_comments + start: 165456 + sourceStatement: + chapter: 16 + file: surgery.tex + label: delta0ri+1 + mtref: '16.4' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 109162 + sourceTransform: tex2mdx.strip_comments + start: 107682 + targetContainedSpan: + coordinateSpace: comment-stripped-tex + end: 167879 + file: surgery.tex + sourceTransform: tex2mdx.strip_comments + start: 167677 + targetStatement: + chapter: 16 + file: surgery.tex + label: '' + mtref: '16.25' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 167879 + sourceTransform: tex2mdx.strip_comments + start: 167677 + type: proof-containment + via: + - proof-contains +generator: tools/poincare_tex_extract.py +inference: structural +kind: containment +ref: +- 89a60eb69c3f +- fc5c13ed2e67 +rel: contains +reviewStatus: accepted +scope: proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof-contains +--- +Morgan--Tian proof containment. The source is the statement whose proof contains the target statement. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/a6107958029d.md b/projects/poincare-conjecture/.astrolabe/edges/a6107958029d.md new file mode 100644 index 00000000..4372e644 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/a6107958029d.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: compactkappa + proofSpan: + bodyEnd: 175215 + bodyStart: 173916 + coordinateSpace: comment-stripped-tex + depth: 2 + end: 175226 + file: temp2kappa.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 173903 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 174480 + file: temp2kappa.tex + sourceTransform: tex2mdx.strip_comments + start: 174462 + via: proof + sourceStatement: + chapter: 9 + file: temp2kappa.tex + label: '' + mtref: '9.90' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 173901 + sourceTransform: tex2mdx.strip_comments + start: 173501 + targetStatement: + chapter: 9 + file: temp2kappa.tex + label: compactkappa + mtref: '9.64' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 127434 + sourceTransform: tex2mdx.strip_comments + start: 127205 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 494bc110f8ab +- 49e865228b19 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/a6252571d404.md b/projects/poincare-conjecture/.astrolabe/edges/a6252571d404.md new file mode 100644 index 00000000..1bb4ca67 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/a6252571d404.md @@ -0,0 +1,79 @@ +--- +confidence: 0.35 +edgeClass: prose +evidence: + mentionTriggers: + - label: Theorem1 + refSpan: + coordinateSpace: comment-stripped-tex + end: 1193 + file: energy1.tex + sourceTransform: tex2mdx.strip_comments + start: 1179 + sentence: Let us quickly show how this theorem implies our main result Theorem~\ref{Theorem1}. + sentenceSpan: + coordinateSpace: comment-stripped-tex + end: 1194 + file: energy1.tex + sourceTransform: tex2mdx.strip_comments + start: 1109 + sourceStrategy: nearest-previous-statement + via: prose + - label: Theorem1 + refSpan: + coordinateSpace: comment-stripped-tex + end: 2428 + file: energy1.tex + sourceTransform: tex2mdx.strip_comments + start: 2414 + sentence: The rest of this chapter is devoted to the proof of Theorem~\ref{extinct} + which will then complete the proof of Theorem~\ref{Theorem1}. + sentenceSpan: + coordinateSpace: comment-stripped-tex + end: 2429 + file: energy1.tex + sourceTransform: tex2mdx.strip_comments + start: 2293 + sourceStrategy: nearest-previous-statement + via: prose + sourceStatement: + chapter: 18 + file: energy1.tex + label: extinct + mtref: '18.1' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 1107 + sourceTransform: tex2mdx.strip_comments + start: 511 + targetStatement: + chapter: 0 + file: intro.tex + label: Theorem1 + mtref: '0.1' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 2955 + sourceTransform: tex2mdx.strip_comments + start: 2574 + type: prose-mention-nearest-previous-statement + via: + - prose +generator: tools/poincare_tex_extract.py +inference: weak +kind: mention +ref: +- 4b8a44aa316f +- 9999efa2be2c +rel: mentions +reviewStatus: candidate +scope: prose +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- prose +--- +Morgan--Tian prose mention. The source is the nearest preceding statement in the same chapter segment. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/a65b438621f8.md b/projects/poincare-conjecture/.astrolabe/edges/a65b438621f8.md new file mode 100644 index 00000000..3b04266f --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/a65b438621f8.md @@ -0,0 +1,84 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: extinct + proofSpan: + bodyEnd: 2280 + bodyStart: 1209 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 2291 + file: energy1.tex + ownerLabel: Theorem1 + sourceTransform: tex2mdx.strip_comments + start: 1196 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 1266 + file: energy1.tex + sourceTransform: tex2mdx.strip_comments + start: 1253 + via: proof + - context: proof + label: extinct + proofSpan: + bodyEnd: 2280 + bodyStart: 1209 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 2291 + file: energy1.tex + ownerLabel: Theorem1 + sourceTransform: tex2mdx.strip_comments + start: 1196 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 1501 + file: energy1.tex + sourceTransform: tex2mdx.strip_comments + start: 1488 + via: proof + sourceStatement: + chapter: 0 + file: intro.tex + label: Theorem1 + mtref: '0.1' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 2955 + sourceTransform: tex2mdx.strip_comments + start: 2574 + targetStatement: + chapter: 18 + file: energy1.tex + label: extinct + mtref: '18.1' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 1107 + sourceTransform: tex2mdx.strip_comments + start: 511 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 9999efa2be2c +- 4b8a44aa316f +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/a6ab0c5b03f0.md b/projects/poincare-conjecture/.astrolabe/edges/a6ab0c5b03f0.md new file mode 100644 index 00000000..e7158f57 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/a6ab0c5b03f0.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: lineineq + proofSpan: + bodyEnd: 36616 + bodyStart: 35142 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 36627 + file: newcomp2.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 35129 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 35963 + file: newcomp2.tex + sourceTransform: tex2mdx.strip_comments + start: 35949 + via: proof + sourceStatement: + chapter: 7 + file: newcomp2.tex + label: '' + mtref: '7.20' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 35127 + sourceTransform: tex2mdx.strip_comments + start: 33923 + targetStatement: + chapter: 7 + file: newcomp2.tex + label: lineineq + mtref: '7.19' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 33702 + sourceTransform: tex2mdx.strip_comments + start: 33163 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 675c76833e83 +- 528f2ee171eb +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/a6eaf507bd23.md b/projects/poincare-conjecture/.astrolabe/edges/a6eaf507bd23.md new file mode 100644 index 00000000..bbd83f1b --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/a6eaf507bd23.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 11 + file: singlimit2.tex + label: omegacanon + mtref: '11.23' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 43580 + sourceTransform: tex2mdx.strip_comments + start: 43054 + targetStatement: + chapter: 11 + file: singlimit2.tex + label: epslimit + mtref: '11.24' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 45867 + sourceTransform: tex2mdx.strip_comments + start: 45550 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 0760630d3990 +- d74749898b25 +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/a6f2b25c8d2d.md b/projects/poincare-conjecture/.astrolabe/edges/a6f2b25c8d2d.md new file mode 100644 index 00000000..fbc7c6b5 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/a6f2b25c8d2d.md @@ -0,0 +1,62 @@ +--- +confidence: 0.9 +edgeClass: structural +evidence: + containmentDepth: 1 + proofSpan: + bodyEnd: 35466 + bodyStart: 32231 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 35477 + file: stdsoln.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 32218 + sourceStatement: + chapter: 12 + file: stdsoln.tex + label: stdkappa + mtref: '12.13' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 32215 + sourceTransform: tex2mdx.strip_comments + start: 32043 + targetContainedSpan: + coordinateSpace: comment-stripped-tex + end: 32654 + file: stdsoln.tex + sourceTransform: tex2mdx.strip_comments + start: 32551 + targetStatement: + chapter: 12 + file: stdsoln.tex + label: '' + mtref: '12.14' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 32654 + sourceTransform: tex2mdx.strip_comments + start: 32551 + type: proof-containment + via: + - proof-contains +generator: tools/poincare_tex_extract.py +inference: structural +kind: containment +ref: +- ed069175cdf9 +- f26ee81ded19 +rel: contains +reviewStatus: accepted +scope: proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof-contains +--- +Morgan--Tian proof containment. The source is the statement whose proof contains the target statement. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/a6fa1089685e.md b/projects/poincare-conjecture/.astrolabe/edges/a6fa1089685e.md new file mode 100644 index 00000000..ef05fd4f --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/a6fa1089685e.md @@ -0,0 +1,71 @@ +--- +confidence: 0.35 +edgeClass: prose +evidence: + mentionTriggers: + - label: lapformula + refSpan: + coordinateSpace: comment-stripped-tex + end: 82793 + file: temp2kappa.tex + sourceTransform: tex2mdx.strip_comments + start: 82777 + sentence: Taking the trace of the gradient shrinking soliton equation yields $$R+\triangle + f-\frac{n}{2}=0,$$ and consequently that $$ dR+d(\triangle f)=0.$$ Using Lemma~\ref{lapformula} + we rewrite this equation as \begin{equation}\label{Rfeqn} dR+\triangle(df)-{\rm + Ric}(\nabla f,\cdot)=0.\end{equation} On the other hand, taking the divergence + of the gradient shrinking soliton equation and using the fact that $\nabla^* + g=0$ gives $$\nabla^*{\rm Ric}+\nabla^*{\rm Hess}(f)=0.$$ Of course, $$\nabla^*{\rm + Hess}(f)=\nabla^*(\nabla\nabla f)=(\nabla^*\nabla)\nabla f=\triangle (df),$$ + so that $$\triangle(df)=-\nabla^*{\rm Ric}.$$ Plugging this into Equation~\ref{Rfeqn} + gives $$dR-\nabla^*{\rm Ric}-{\rm Ric}(\nabla f,\cdot)=0.$$ Now invoking Lemma~\ref{divRic} + we have \begin{equation}\label{dR} dR=2{\rm Ric}(\nabla f,\cdot).\end{equation} + sentenceSpan: + coordinateSpace: comment-stripped-tex + end: 83451 + file: temp2kappa.tex + sourceTransform: tex2mdx.strip_comments + start: 82616 + sourceStrategy: nearest-previous-statement + via: prose + sourceStatement: + chapter: 9 + file: temp2kappa.tex + label: noncompsol + mtref: '9.46' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 82429 + sourceTransform: tex2mdx.strip_comments + start: 82233 + targetStatement: + chapter: 1 + file: prelim.tex + label: lapformula + mtref: '1.10' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 14031 + sourceTransform: tex2mdx.strip_comments + start: 13808 + type: prose-mention-nearest-previous-statement + via: + - prose +generator: tools/poincare_tex_extract.py +inference: weak +kind: mention +ref: +- 03957890f08f +- 8d03f3ca0aea +rel: mentions +reviewStatus: candidate +scope: prose +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- prose +--- +Morgan--Tian prose mention. The source is the nearest preceding statement in the same chapter segment. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/a6fe9964c015.md b/projects/poincare-conjecture/.astrolabe/edges/a6fe9964c015.md new file mode 100644 index 00000000..14e6b0db --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/a6fe9964c015.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 5 + file: converge2.tex + label: GHconverge + mtref: '5.19' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 33874 + sourceTransform: tex2mdx.strip_comments + start: 33315 + targetStatement: + chapter: 5 + file: converge2.tex + label: '' + mtref: '5.20' + sort: example + span: + coordinateSpace: comment-stripped-tex + end: 34003 + sourceTransform: tex2mdx.strip_comments + start: 33876 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- d06299f3fe46 +- af79df2b9075 +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/a700c726d1b6.md b/projects/poincare-conjecture/.astrolabe/edges/a700c726d1b6.md new file mode 100644 index 00000000..2397a4dd --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/a700c726d1b6.md @@ -0,0 +1,55 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: statement + label: GSSclass + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 80698 + file: temp2kappa.tex + sourceTransform: tex2mdx.strip_comments + start: 80684 + via: statement + sourceStatement: + chapter: 9 + file: temp2kappa.tex + label: flatssplit + mtref: '9.45' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 81068 + sourceTransform: tex2mdx.strip_comments + start: 80573 + targetStatement: + chapter: 9 + file: temp2kappa.tex + label: GSSclass + mtref: '9.42' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 75321 + sourceTransform: tex2mdx.strip_comments + start: 74299 + type: tex-reference + via: + - statement +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- afc86a351f0c +- 2d68880771ff +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- statement +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/a717e14ce026.md b/projects/poincare-conjecture/.astrolabe/edges/a717e14ce026.md new file mode 100644 index 00000000..690c15de --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/a717e14ce026.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 12 + file: stdsoln.tex + label: rcor + mtref: '12.19' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 50482 + sourceTransform: tex2mdx.strip_comments + start: 50049 + targetStatement: + chapter: 12 + file: stdsoln.tex + label: '' + mtref: '12.20' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 51041 + sourceTransform: tex2mdx.strip_comments + start: 50934 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- e353993489e9 +- f917cf2b1576 +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/a726e5966b62.md b/projects/poincare-conjecture/.astrolabe/edges/a726e5966b62.md new file mode 100644 index 00000000..b9f53c25 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/a726e5966b62.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: prodatinf + proofSpan: + bodyEnd: 103549 + bodyStart: 101913 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 103560 + file: temp2kappa.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 101900 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 102285 + file: temp2kappa.tex + sourceTransform: tex2mdx.strip_comments + start: 102270 + via: proof + sourceStatement: + chapter: 9 + file: temp2kappa.tex + label: 3DGSSkappa + mtref: '9.53' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 101898 + sourceTransform: tex2mdx.strip_comments + start: 101535 + targetStatement: + chapter: 9 + file: temp2kappa.tex + label: prodatinf + mtref: '9.39' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 68632 + sourceTransform: tex2mdx.strip_comments + start: 67563 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 351e4ac14e4e +- 703d5ca9da05 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/a741b444ae1a.md b/projects/poincare-conjecture/.astrolabe/edges/a741b444ae1a.md new file mode 100644 index 00000000..2bb8a1db --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/a741b444ae1a.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: posderiv + proofSpan: + bodyEnd: 126183 + bodyStart: 124278 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 126194 + file: temp2kappa.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 124265 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 125417 + file: temp2kappa.tex + sourceTransform: tex2mdx.strip_comments + start: 125403 + via: proof + sourceStatement: + chapter: 9 + file: temp2kappa.tex + label: bdvolbdcurv + mtref: '9.62' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 124263 + sourceTransform: tex2mdx.strip_comments + start: 123826 + targetStatement: + chapter: 4 + file: maxprin.tex + label: posderiv + mtref: '4.39' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 53144 + sourceTransform: tex2mdx.strip_comments + start: 52904 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 3227c39e1892 +- 36dd2f6fa9c2 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/a741d72002c6.md b/projects/poincare-conjecture/.astrolabe/edges/a741d72002c6.md new file mode 100644 index 00000000..48f7bd37 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/a741d72002c6.md @@ -0,0 +1,45 @@ +--- +confidence: 0.2 +edgeClass: structural +evidence: + sourceStatement: + chapter: 6 + file: newcompar.tex + label: '' + mtref: '6.57' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 85285 + sourceTransform: tex2mdx.strip_comments + start: 85074 + targetStatement: + chapter: 6 + file: newcompar.tex + label: '' + mtref: '6.58' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 86220 + sourceTransform: tex2mdx.strip_comments + start: 85532 + type: chapter-reading-order-adjacency + via: + - chapter-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- a89a833cf7cb +- 0fc5cfacef3f +rel: in-chapter +reviewStatus: accepted +scope: chapter +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- chapter-sequence +--- +Morgan--Tian reading-order adjacency across sections in a chapter. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/a74e29a6b309.md b/projects/poincare-conjecture/.astrolabe/edges/a74e29a6b309.md new file mode 100644 index 00000000..2b60bca9 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/a74e29a6b309.md @@ -0,0 +1,84 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: s2isotopic + proofSpan: + bodyEnd: 21763 + bodyStart: 19629 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 21774 + file: canonnbhd.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 19616 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 19793 + file: canonnbhd.tex + sourceTransform: tex2mdx.strip_comments + start: 19777 + via: proof + - 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context: proof + label: line + proofSpan: + bodyEnd: 58686 + bodyStart: 55699 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 58697 + file: converge2.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 55686 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 58545 + file: converge2.tex + sourceTransform: tex2mdx.strip_comments + start: 58535 + via: proof + sourceStatement: + chapter: 5 + file: converge2.tex + label: topsplit + mtref: '5.35' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 55387 + sourceTransform: tex2mdx.strip_comments + start: 54529 + targetStatement: + chapter: 2 + file: prelim.tex + label: line + mtref: '2.14' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 67980 + sourceTransform: tex2mdx.strip_comments + start: 67775 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- e9a9172d16e2 +- 006e42ef5bb5 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/a7889258e8b5.md b/projects/poincare-conjecture/.astrolabe/edges/a7889258e8b5.md new file mode 100644 index 00000000..fa32aad2 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/a7889258e8b5.md @@ -0,0 +1,67 @@ +--- +confidence: 0.35 +edgeClass: prose +evidence: + mentionTriggers: + - label: flowlimit + refSpan: + coordinateSpace: comment-stripped-tex + end: 11065 + file: stdsoln.tex + sourceTransform: tex2mdx.strip_comments + start: 11050 + sentence: It now follows from Proposition~\ref{flowextend} and the fact that the + $T_n$ are maximal that $T_n>t_0$ for all $n$. Since the Riemann curvatures of + the $(S^3_{R_n},g_{R_n}(t)),\ 0\le t1$, we see that $${\mathcal L}(\tilde \mu_z)\le \sqrt{T}.$$ + Putting this together with the estimate, Equation~(\ref{Lgamma1}), for ${\mathcal + L}(\gamma_1)$ tells us that for each $z\in B(y,t'',r')$ we have $${\mathcal + L}(\gamma_1*\tilde\mu_z)\le 4(T_{i+1})^{1/2}+4(T_{i+1})^{3/2}\le L/2.$$ Hence, + by Proposition~\ref{delta0ri+1} and the choice of $L$, there is a minimizing + ${\mathcal L}$-geodesic from $x$ to each point of $B(y,t'',r')$ of length $\le + L/2$, and these geodesics are contained in the smooth part of ${\mathcal M}$ + sentenceSpan: + coordinateSpace: comment-stripped-tex + end: 182488 + file: surgery.tex + sourceTransform: tex2mdx.strip_comments + start: 181949 + sourceStrategy: nearest-previous-statement + via: prose + sourceStatement: + chapter: 16 + file: surgery.tex + label: '' + mtref: '16.28' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 176062 + sourceTransform: tex2mdx.strip_comments + start: 175778 + targetStatement: + chapter: 16 + file: surgery.tex + label: delta0ri+1 + mtref: '16.4' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 109162 + sourceTransform: tex2mdx.strip_comments + start: 107682 + type: prose-mention-nearest-previous-statement + via: + - prose +generator: tools/poincare_tex_extract.py +inference: weak +kind: mention +ref: +- 3e9580da0ff5 +- 89a60eb69c3f +rel: mentions +reviewStatus: candidate +scope: prose +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- prose +--- +Morgan--Tian prose mention. The source is the nearest preceding statement in the same chapter segment. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/ad129a61f03c.md b/projects/poincare-conjecture/.astrolabe/edges/ad129a61f03c.md new file mode 100644 index 00000000..bb0c80f0 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/ad129a61f03c.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: canonvary + proofSpan: + bodyEnd: 93873 + bodyStart: 90350 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 93884 + file: stdsoln.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 90337 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 90879 + file: stdsoln.tex + sourceTransform: tex2mdx.strip_comments + start: 90864 + via: proof + sourceStatement: + chapter: 12 + file: stdsoln.tex + label: Restim + mtref: '12.31' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 90335 + sourceTransform: tex2mdx.strip_comments + start: 90184 + targetStatement: + chapter: 9 + file: temp2kappa.tex + label: canonvary + mtref: '9.79' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 149236 + sourceTransform: tex2mdx.strip_comments + start: 147728 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- ffae4504318a +- 7c2758f0e2b3 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/ad1d3f413092.md b/projects/poincare-conjecture/.astrolabe/edges/ad1d3f413092.md new file mode 100644 index 00000000..194fcb82 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/ad1d3f413092.md @@ -0,0 +1,45 @@ +--- +confidence: 0.2 +edgeClass: structural +evidence: + sourceStatement: + chapter: 1 + file: prelim.tex + label: BishopGromov + mtref: '1.34' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 43978 + sourceTransform: tex2mdx.strip_comments + start: 43265 + targetStatement: + chapter: 1 + file: prelim.tex + label: injvol + mtref: '1.35' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 44668 + sourceTransform: tex2mdx.strip_comments + start: 44242 + type: chapter-reading-order-adjacency + via: + - chapter-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 1a1b2c4fdc9b +- 7fec5a940413 +rel: in-chapter +reviewStatus: accepted +scope: chapter +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- chapter-sequence +--- +Morgan--Tian reading-order adjacency across sections in a chapter. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/ad30051e2371.md b/projects/poincare-conjecture/.astrolabe/edges/ad30051e2371.md new file mode 100644 index 00000000..6d4649bd --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/ad30051e2371.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: altern + proofSpan: + bodyEnd: 220240 + bodyStart: 211487 + coordinateSpace: comment-stripped-tex + depth: 2 + end: 220251 + file: surgery.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 211474 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 214844 + file: surgery.tex + sourceTransform: tex2mdx.strip_comments + start: 214832 + via: proof + sourceStatement: + chapter: 17 + file: surgery.tex + label: lemstdcan + mtref: '17.7' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 211471 + sourceTransform: tex2mdx.strip_comments + start: 210838 + targetStatement: + chapter: 16 + file: surgery.tex + label: altern + mtref: '16.5' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 112999 + sourceTransform: tex2mdx.strip_comments + start: 110961 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 111551d7831b +- ad4f67ea75bc +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/ad364a4e59cb.md b/projects/poincare-conjecture/.astrolabe/edges/ad364a4e59cb.md new file mode 100644 index 00000000..f67efebd --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/ad364a4e59cb.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: strepsopen + proofSpan: + bodyEnd: 169692 + bodyStart: 163171 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 169703 + file: temp2kappa.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 163158 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 165856 + file: temp2kappa.tex + sourceTransform: tex2mdx.strip_comments + start: 165840 + via: proof + sourceStatement: + chapter: 9 + file: temp2kappa.tex + label: '' + mtref: '9.86' + sort: remark + span: + coordinateSpace: comment-stripped-tex + end: 163155 + sourceTransform: tex2mdx.strip_comments + start: 163012 + targetStatement: + chapter: 9 + file: temp2kappa.tex + label: strepsopen + mtref: '9.80' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 153741 + sourceTransform: tex2mdx.strip_comments + start: 153588 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 51da06702a9f +- 75b4c67c9141 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/ae06fcb25d4b.md b/projects/poincare-conjecture/.astrolabe/edges/ae06fcb25d4b.md new file mode 100644 index 00000000..7aed401f --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/ae06fcb25d4b.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 9 + file: temp2kappa.tex + label: compact2Dsol + mtref: '9.49' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 90897 + sourceTransform: tex2mdx.strip_comments + start: 90165 + targetStatement: + chapter: 9 + file: temp2kappa.tex + label: 2DGSS + mtref: '9.50' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 91949 + sourceTransform: tex2mdx.strip_comments + start: 91266 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- fd11f1f8def3 +- da767c71996b +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/ae1c3505b758.md b/projects/poincare-conjecture/.astrolabe/edges/ae1c3505b758.md new file mode 100644 index 00000000..f10949f9 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/ae1c3505b758.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: 2ndvari + proofSpan: + bodyEnd: 67066 + bodyStart: 66471 + coordinateSpace: comment-stripped-tex + depth: 2 + end: 67077 + file: newcompar.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 66458 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 66557 + file: newcompar.tex + sourceTransform: tex2mdx.strip_comments + start: 66544 + via: proof + sourceStatement: + chapter: 6 + file: newcompar.tex + label: '' + mtref: '6.42' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 66455 + sourceTransform: tex2mdx.strip_comments + start: 66260 + targetStatement: + chapter: 6 + file: newcompar.tex + label: 2ndvari + mtref: '6.14' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 17350 + sourceTransform: tex2mdx.strip_comments + start: 16270 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- b20ded1b2964 +- f48558c60ca2 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/ae58147ce153.md b/projects/poincare-conjecture/.astrolabe/edges/ae58147ce153.md new file mode 100644 index 00000000..abeea568 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/ae58147ce153.md @@ -0,0 +1,65 @@ +--- +confidence: 0.45 +edgeClass: semantic +evidence: + endpointSemanticDegreeBeforeTermBackfill: + source: 0 + target: 0 + selectionReason: semantic-connectivity-backfill + sourceMatchSpan: + coordinateSpace: comment-stripped-tex + end: 49389 + file: singlimit2.tex + sourceTransform: tex2mdx.strip_comments + start: 49366 + sourceStatement: + chapter: 11 + file: singlimit2.tex + label: '' + mtref: '11.26' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 49703 + sourceTransform: tex2mdx.strip_comments + start: 48874 + targetStatement: + chapter: 11 + file: singlimit2.tex + label: '' + mtref: '11.25' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 48871 + sourceTransform: tex2mdx.strip_comments + start: 48329 + targetTermSpan: + coordinateSpace: comment-stripped-tex + end: 48373 + file: singlimit2.tex + sourceTransform: tex2mdx.strip_comments + start: 48350 + term: strong horn + termAmbiguity: unique-in-chapter + termSource: italic + termSpecificity: no-longer-defined-term + type: same-chapter-definition-term-match + via: + - definition-term:strong horn +generator: tools/poincare_tex_extract.py +inference: weak +kind: definition-use +ref: +- 08c3ff9d0a1e +- da99c0859788 +rel: uses +reviewStatus: candidate +scope: same-chapter-term-match +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- definition-term:strong horn +--- +Morgan--Tian same-chapter definition term match. This is a low-confidence candidate edge for review. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/aeb5da271307.md b/projects/poincare-conjecture/.astrolabe/edges/aeb5da271307.md new file mode 100644 index 00000000..b7b7f61b --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/aeb5da271307.md @@ -0,0 +1,45 @@ +--- +confidence: 0.2 +edgeClass: structural +evidence: + sourceStatement: + chapter: 6 + file: newcompar.tex + label: tildelK + mtref: '6.54' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 82952 + sourceTransform: tex2mdx.strip_comments + start: 82766 + targetStatement: + chapter: 6 + file: newcompar.tex + label: '' + mtref: '6.55' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 84083 + sourceTransform: tex2mdx.strip_comments + start: 83631 + type: chapter-reading-order-adjacency + via: + - chapter-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 451b4cc590c1 +- 27c072f30ccf +rel: in-chapter +reviewStatus: accepted +scope: chapter +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- chapter-sequence +--- +Morgan--Tian reading-order adjacency across sections in a chapter. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/aece1879c136.md b/projects/poincare-conjecture/.astrolabe/edges/aece1879c136.md new file mode 100644 index 00000000..b52904de --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/aece1879c136.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 10 + file: bddcurvbdddist.tex + label: '' + mtref: '10.27' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 50344 + sourceTransform: tex2mdx.strip_comments + start: 50058 + targetStatement: + chapter: 10 + file: bddcurvbdddist.tex + label: deltanet + mtref: '10.28' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 52334 + sourceTransform: tex2mdx.strip_comments + start: 51848 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- aedf30fd9f7e +- f7e1d840a49f +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/aedd4f5c4d54.md b/projects/poincare-conjecture/.astrolabe/edges/aedd4f5c4d54.md new file mode 100644 index 00000000..7a95510d --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/aedd4f5c4d54.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 18 + file: energy1.tex + label: '' + mtref: '18.70' + sort: remark + span: + coordinateSpace: comment-stripped-tex + end: 168380 + sourceTransform: tex2mdx.strip_comments + start: 168035 + targetStatement: + chapter: 18 + file: energy1.tex + label: '' + mtref: '18.71' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 170254 + sourceTransform: tex2mdx.strip_comments + start: 170143 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- cbf568c0a596 +- e94b6de05447 +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/aeeff6fdb772.md b/projects/poincare-conjecture/.astrolabe/edges/aeeff6fdb772.md new file mode 100644 index 00000000..1fafe4e1 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/aeeff6fdb772.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 13 + file: surgery.tex + label: improve + mtref: '13.11' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 19510 + sourceTransform: tex2mdx.strip_comments + start: 18654 + targetStatement: + chapter: 13 + file: surgery.tex + label: smeigen + mtref: '13.12' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 20688 + sourceTransform: tex2mdx.strip_comments + start: 20353 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 41a5f172b2dc +- e70bacf8aab0 +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/af15662699bf.md b/projects/poincare-conjecture/.astrolabe/edges/af15662699bf.md new file mode 100644 index 00000000..f717fbe1 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/af15662699bf.md @@ -0,0 +1,56 @@ +--- +confidence: 0.85 +edgeClass: semantic +evidence: + containmentReason: target theorem-like statement occurs inside source proof + proofSpan: + bodyEnd: 169692 + bodyStart: 163171 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 169703 + file: temp2kappa.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 163158 + sourceStatement: + chapter: 9 + file: temp2kappa.tex + label: '' + mtref: '9.86' + sort: remark + span: + coordinateSpace: comment-stripped-tex + end: 163155 + sourceTransform: tex2mdx.strip_comments + start: 163012 + targetStatement: + chapter: 9 + file: temp2kappa.tex + label: '' + mtref: '9.87' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 167340 + sourceTransform: tex2mdx.strip_comments + start: 167251 + type: proof-contained-theorem-like-statement + via: + - proof-contained-statement +generator: tools/poincare_tex_extract.py +inference: inferred +kind: dependency +ref: +- 51da06702a9f +- b9f671d590f3 +rel: depends +reviewStatus: accepted +scope: proof-containment +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof-contained-statement +--- +Morgan--Tian proof-contained statement dependency. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/af16fdf8aa4a.md b/projects/poincare-conjecture/.astrolabe/edges/af16fdf8aa4a.md new file mode 100644 index 00000000..838459ee --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/af16fdf8aa4a.md @@ -0,0 +1,64 @@ +--- +confidence: 0.35 +edgeClass: prose +evidence: + mentionTriggers: + - label: Theorem1 + refSpan: + coordinateSpace: comment-stripped-tex + end: 22234 + file: intro.tex + sourceTransform: tex2mdx.strip_comments + start: 22220 + sentence: Rather, we content ourselves with presenting a proof of Theorem~\ref{Theorem1} + above which, as we have indicated, concerns initial Riemannian manifolds for + which the Ricci flow with surgery becomes extinct after finite time + sentenceSpan: + coordinateSpace: comment-stripped-tex + end: 22379 + file: intro.tex + sourceTransform: tex2mdx.strip_comments + start: 22155 + sourceStrategy: nearest-previous-statement + via: prose + sourceStatement: + chapter: 0 + file: intro.tex + label: '' + mtref: '0.6' + sort: remark + span: + coordinateSpace: comment-stripped-tex + end: 19243 + sourceTransform: tex2mdx.strip_comments + start: 16436 + targetStatement: + chapter: 0 + file: intro.tex + label: Theorem1 + mtref: '0.1' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 2955 + sourceTransform: tex2mdx.strip_comments + start: 2574 + type: prose-mention-nearest-previous-statement + via: + - prose +generator: tools/poincare_tex_extract.py +inference: weak +kind: mention +ref: +- 48f3d2b5dea3 +- 9999efa2be2c +rel: mentions +reviewStatus: candidate +scope: prose +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- prose +--- +Morgan--Tian prose mention. The source is the nearest preceding statement in the same chapter segment. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/af19f996b999.md b/projects/poincare-conjecture/.astrolabe/edges/af19f996b999.md new file mode 100644 index 00000000..dda9e055 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/af19f996b999.md @@ -0,0 +1,63 @@ +--- +confidence: 0.35 +edgeClass: prose +evidence: + mentionTriggers: + - label: nullspace + refSpan: + coordinateSpace: comment-stripped-tex + end: 18169 + file: stdsoln.tex + sourceTransform: tex2mdx.strip_comments + start: 18154 + sentence: According to Corollary~\ref{nullspace} this implies that the flow is + the product $(S^2,h(t))\times (\Ar,ds^2)$ + sentenceSpan: + coordinateSpace: comment-stripped-tex + end: 18241 + file: stdsoln.tex + sourceTransform: tex2mdx.strip_comments + start: 18130 + sourceStrategy: nearest-previous-statement + via: prose + sourceStatement: + chapter: 12 + file: stdsoln.tex + label: third + mtref: '12.6' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 12060 + sourceTransform: tex2mdx.strip_comments + start: 11922 + targetStatement: + chapter: 4 + file: maxprin.tex + label: nullspace + mtref: '4.20' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 31663 + sourceTransform: tex2mdx.strip_comments + start: 30958 + type: prose-mention-nearest-previous-statement + via: + - prose +generator: tools/poincare_tex_extract.py +inference: weak +kind: mention +ref: +- 2e8c17743da6 +- d21a675e4ea8 +rel: mentions +reviewStatus: candidate +scope: prose +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- prose +--- +Morgan--Tian prose mention. The source is the nearest preceding statement in the same chapter segment. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/af44e4321f4f.md b/projects/poincare-conjecture/.astrolabe/edges/af44e4321f4f.md new file mode 100644 index 00000000..3ab749ed --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/af44e4321f4f.md @@ -0,0 +1,63 @@ +--- +confidence: 0.35 +edgeClass: prose +evidence: + mentionTriggers: + - label: XI + refSpan: + coordinateSpace: comment-stripped-tex + end: 122722 + file: energy1.tex + sourceTransform: tex2mdx.strip_comments + start: 122714 + sentence: The next claim is the statement that a slightly stronger version of + Proposition~\ref{XI} holds for $p_1\widetilde\Gamma_c^\lambda(t)$. + sentenceSpan: + coordinateSpace: comment-stripped-tex + end: 122768 + file: energy1.tex + sourceTransform: tex2mdx.strip_comments + start: 122633 + sourceStrategy: nearest-previous-statement + via: prose + sourceStatement: + chapter: 18 + file: energy1.tex + label: CSSHI + mtref: '18.52' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 117869 + sourceTransform: tex2mdx.strip_comments + start: 117059 + targetStatement: + chapter: 18 + file: energy1.tex + label: XI + mtref: '18.24' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 58810 + sourceTransform: tex2mdx.strip_comments + start: 58030 + type: prose-mention-nearest-previous-statement + via: + - prose +generator: tools/poincare_tex_extract.py +inference: weak +kind: mention +ref: +- 38712e8cea3d +- fd85732525c1 +rel: mentions +reviewStatus: candidate +scope: prose +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- prose +--- +Morgan--Tian prose mention. The source is the nearest preceding statement in the same chapter segment. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/af57bc88232c.md b/projects/poincare-conjecture/.astrolabe/edges/af57bc88232c.md new file mode 100644 index 00000000..bc40a025 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/af57bc88232c.md @@ -0,0 +1,45 @@ +--- +confidence: 0.2 +edgeClass: structural +evidence: + sourceStatement: + chapter: 10 + file: bddcurvbdddist.tex + label: '' + mtref: '10.4' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 13178 + sourceTransform: tex2mdx.strip_comments + start: 12275 + targetStatement: + chapter: 10 + file: bddcurvbdddist.tex + label: '' + mtref: '10.5' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 20530 + sourceTransform: tex2mdx.strip_comments + start: 20337 + type: chapter-reading-order-adjacency + via: + - chapter-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 96519f84d54e +- 67090b75df54 +rel: in-chapter +reviewStatus: accepted +scope: chapter +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- chapter-sequence +--- +Morgan--Tian reading-order adjacency across sections in a chapter. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/af782306c0dc.md b/projects/poincare-conjecture/.astrolabe/edges/af782306c0dc.md new file mode 100644 index 00000000..f2d1672f --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/af782306c0dc.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: muepsbd + proofSpan: + bodyEnd: 181090 + bodyStart: 180824 + coordinateSpace: comment-stripped-tex + depth: 2 + end: 181101 + file: energy1.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 180811 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 180919 + file: energy1.tex + sourceTransform: tex2mdx.strip_comments + start: 180906 + via: proof + sourceStatement: + chapter: 18 + file: energy1.tex + label: '' + mtref: '18.79' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 180809 + sourceTransform: tex2mdx.strip_comments + start: 180734 + targetStatement: + chapter: 18 + file: energy1.tex + label: muepsbd + mtref: '18.72' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 171988 + sourceTransform: tex2mdx.strip_comments + start: 171484 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- e967f00df0a8 +- 9c8d4163f700 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/af7832c5013a.md b/projects/poincare-conjecture/.astrolabe/edges/af7832c5013a.md new file mode 100644 index 00000000..c75eab10 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/af7832c5013a.md @@ -0,0 +1,103 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: neckgeo + proofSpan: + bodyEnd: 40983 + bodyStart: 36070 + coordinateSpace: comment-stripped-tex + depth: 2 + end: 40994 + file: canonnbhd.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 36057 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 37773 + file: canonnbhd.tex + sourceTransform: tex2mdx.strip_comments + start: 37760 + via: proof + - context: proof + label: neckgeo + proofSpan: + bodyEnd: 40983 + bodyStart: 36070 + coordinateSpace: comment-stripped-tex + depth: 2 + end: 40994 + file: canonnbhd.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 36057 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 39996 + file: canonnbhd.tex + sourceTransform: tex2mdx.strip_comments + start: 39983 + via: proof + - context: proof + label: neckgeo + proofSpan: + bodyEnd: 40983 + bodyStart: 36070 + coordinateSpace: comment-stripped-tex + depth: 2 + end: 40994 + file: canonnbhd.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 36057 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 40382 + file: canonnbhd.tex + sourceTransform: tex2mdx.strip_comments + start: 40369 + via: proof + sourceStatement: + chapter: 19 + file: canonnbhd.tex + label: '' + mtref: '19.22' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 36055 + sourceTransform: tex2mdx.strip_comments + start: 35803 + targetStatement: + chapter: 19 + file: canonnbhd.tex + label: neckgeo + mtref: '19.5' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 8075 + sourceTransform: tex2mdx.strip_comments + start: 7413 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 9737d037f97b +- b74cd360f72f +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/afaf424e94f7.md b/projects/poincare-conjecture/.astrolabe/edges/afaf424e94f7.md new file mode 100644 index 00000000..b7bfd554 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/afaf424e94f7.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: delta0ri+1 + proofSpan: + bodyEnd: 171658 + bodyStart: 171512 + coordinateSpace: comment-stripped-tex + depth: 2 + end: 171669 + file: surgery.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 171499 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 171604 + file: surgery.tex + sourceTransform: tex2mdx.strip_comments + start: 171588 + via: proof + sourceStatement: + chapter: 16 + file: surgery.tex + label: lvalue + mtref: '16.26' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 171497 + sourceTransform: tex2mdx.strip_comments + start: 171318 + targetStatement: + chapter: 16 + file: surgery.tex + label: delta0ri+1 + mtref: '16.4' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 109162 + sourceTransform: tex2mdx.strip_comments + start: 107682 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 8cae73a6a02c +- 89a60eb69c3f +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/afb9700112e6.md b/projects/poincare-conjecture/.astrolabe/edges/afb9700112e6.md new file mode 100644 index 00000000..7abd4b3d --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/afb9700112e6.md @@ -0,0 +1,45 @@ +--- +confidence: 0.2 +edgeClass: structural +evidence: + sourceStatement: + chapter: 4 + file: maxprin.tex + label: ivey + mtref: '4.36' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 51946 + sourceTransform: tex2mdx.strip_comments + start: 51803 + targetStatement: + chapter: 4 + file: maxprin.tex + label: Harnack + mtref: '4.37' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 52804 + sourceTransform: tex2mdx.strip_comments + start: 52280 + type: chapter-reading-order-adjacency + via: + - chapter-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 7b914755dcf6 +- aa9d836895d8 +rel: in-chapter +reviewStatus: accepted +scope: chapter +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- chapter-sequence +--- +Morgan--Tian reading-order adjacency across sections in a chapter. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/aff39b27ea35.md b/projects/poincare-conjecture/.astrolabe/edges/aff39b27ea35.md new file mode 100644 index 00000000..cef0aa67 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/aff39b27ea35.md @@ -0,0 +1,63 @@ +--- +confidence: 0.35 +edgeClass: prose +evidence: + mentionTriggers: + - label: delta0ri+1 + refSpan: + coordinateSpace: comment-stripped-tex + end: 136883 + file: surgery.tex + sourceTransform: tex2mdx.strip_comments + start: 136867 + sentence: We suppose that these data satisfy the hypothesis of Proposition~\ref{delta0ri+1} + with respect to the given sequences and $r\ge r_{i+1}>0$ + sentenceSpan: + coordinateSpace: comment-stripped-tex + end: 136940 + file: surgery.tex + sourceTransform: tex2mdx.strip_comments + start: 136801 + sourceStrategy: nearest-previous-statement + via: prose + sourceStatement: + chapter: 16 + file: surgery.tex + label: '' + mtref: '16.14' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 134202 + sourceTransform: tex2mdx.strip_comments + start: 133745 + targetStatement: + chapter: 16 + file: surgery.tex + label: delta0ri+1 + mtref: '16.4' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 109162 + sourceTransform: tex2mdx.strip_comments + start: 107682 + type: prose-mention-nearest-previous-statement + via: + - prose +generator: tools/poincare_tex_extract.py +inference: weak +kind: mention +ref: +- e3c80c436275 +- 89a60eb69c3f +rel: mentions +reviewStatus: candidate +scope: prose +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- prose +--- +Morgan--Tian prose mention. The source is the nearest preceding statement in the same chapter segment. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/b005342258bc.md b/projects/poincare-conjecture/.astrolabe/edges/b005342258bc.md new file mode 100644 index 00000000..87d033b4 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/b005342258bc.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 2 + file: prelim.tex + label: anglemono + mtref: '2.5' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 54639 + sourceTransform: tex2mdx.strip_comments + start: 54438 + targetStatement: + chapter: 2 + file: prelim.tex + label: anglecompar + mtref: '2.6' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 55820 + sourceTransform: tex2mdx.strip_comments + start: 55235 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- c81962d31706 +- dc263832416c +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/b026b4c1d0b4.md b/projects/poincare-conjecture/.astrolabe/edges/b026b4c1d0b4.md new file mode 100644 index 00000000..fb721f61 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/b026b4c1d0b4.md @@ -0,0 +1,62 @@ +--- +confidence: 0.9 +edgeClass: structural +evidence: + containmentDepth: 1 + proofSpan: + bodyEnd: 61925 + bodyStart: 54414 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 61936 + file: temp2kappa.tex + ownerLabel: '9.1' + sourceTransform: tex2mdx.strip_comments + start: 54401 + sourceStatement: + chapter: 9 + file: temp2kappa.tex + label: '9.1' + mtref: '9.33' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 53478 + sourceTransform: tex2mdx.strip_comments + start: 52780 + targetContainedSpan: + coordinateSpace: comment-stripped-tex + end: 55633 + file: temp2kappa.tex + sourceTransform: tex2mdx.strip_comments + start: 55469 + targetStatement: + chapter: 9 + file: temp2kappa.tex + label: '11.20' + mtref: '9.34' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 55633 + sourceTransform: tex2mdx.strip_comments + start: 55469 + type: proof-containment + via: + - proof-contains +generator: tools/poincare_tex_extract.py +inference: structural +kind: containment +ref: +- ba1adbac0a75 +- 8283cc66c217 +rel: contains +reviewStatus: accepted +scope: proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof-contains +--- +Morgan--Tian proof containment. The source is the statement whose proof contains the target statement. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/b02fb9bd3a19.md b/projects/poincare-conjecture/.astrolabe/edges/b02fb9bd3a19.md new file mode 100644 index 00000000..1279e8e7 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/b02fb9bd3a19.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: S2times01 + proofSpan: + bodyEnd: 53552 + bodyStart: 35551 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 53563 + file: canonnbhd.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 35538 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 43320 + file: canonnbhd.tex + sourceTransform: tex2mdx.strip_comments + start: 43305 + via: proof + sourceStatement: + chapter: 19 + file: canonnbhd.tex + label: Xcontainedin + mtref: '19.21' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 35375 + sourceTransform: tex2mdx.strip_comments + start: 34343 + targetStatement: + chapter: 19 + file: canonnbhd.tex + label: S2times01 + mtref: '19.20' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 33333 + sourceTransform: tex2mdx.strip_comments + start: 32976 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 0e9a1a167caa +- aa8a682b4513 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/b04c73af9ac9.md b/projects/poincare-conjecture/.astrolabe/edges/b04c73af9ac9.md new file mode 100644 index 00000000..f687b593 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/b04c73af9ac9.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: infprod2 + proofSpan: + bodyEnd: 169692 + bodyStart: 163171 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 169703 + file: temp2kappa.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 163158 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 164548 + file: temp2kappa.tex + sourceTransform: tex2mdx.strip_comments + start: 164534 + via: proof + sourceStatement: + chapter: 9 + file: temp2kappa.tex + label: '' + mtref: '9.86' + sort: remark + span: + coordinateSpace: comment-stripped-tex + end: 163155 + sourceTransform: tex2mdx.strip_comments + start: 163012 + targetStatement: + chapter: 9 + file: temp2kappa.tex + label: infprod2 + mtref: '9.83' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 157798 + sourceTransform: tex2mdx.strip_comments + start: 157331 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 51da06702a9f +- 25e0946e7222 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/b06c4fda9008.md b/projects/poincare-conjecture/.astrolabe/edges/b06c4fda9008.md new file mode 100644 index 00000000..caee94c3 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/b06c4fda9008.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 11 + file: singlimit2.tex + label: hyp + mtref: '11.32' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 60461 + sourceTransform: tex2mdx.strip_comments + start: 60310 + targetStatement: + chapter: 11 + file: singlimit2.tex + label: '' + mtref: '11.33' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 61224 + sourceTransform: tex2mdx.strip_comments + start: 61026 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- dff170186455 +- ce620437e5d6 +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/b08a5ff6c713.md b/projects/poincare-conjecture/.astrolabe/edges/b08a5ff6c713.md new file mode 100644 index 00000000..2ceaa6d6 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/b08a5ff6c713.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 6 + file: newcompar.tex + label: '' + mtref: '6.8' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 11018 + sourceTransform: tex2mdx.strip_comments + start: 10755 + targetStatement: + chapter: 6 + file: newcompar.tex + label: '' + mtref: '6.9' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 11539 + sourceTransform: tex2mdx.strip_comments + start: 11300 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- b2c992468118 +- f09cbffcbdce +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/b0b22ee544f8.md b/projects/poincare-conjecture/.astrolabe/edges/b0b22ee544f8.md new file mode 100644 index 00000000..c0eb0191 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/b0b22ee544f8.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: I.8.3 + proofSpan: + bodyEnd: 37375 + bodyStart: 36553 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 37386 + file: flowbasics.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 36540 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 36745 + file: flowbasics.tex + sourceTransform: tex2mdx.strip_comments + start: 36734 + via: proof + sourceStatement: + chapter: 3 + file: flowbasics.tex + label: 2ndI.8.3 + mtref: '3.25' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 36538 + sourceTransform: tex2mdx.strip_comments + start: 36082 + targetStatement: + chapter: 3 + file: flowbasics.tex + label: I.8.3 + mtref: '3.21' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 27841 + sourceTransform: tex2mdx.strip_comments + start: 27149 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- d22d3cf47fbf +- 5d895f9c3918 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/b0c289b8f7f9.md b/projects/poincare-conjecture/.astrolabe/edges/b0c289b8f7f9.md new file mode 100644 index 00000000..9d9cbea7 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/b0c289b8f7f9.md @@ -0,0 +1,84 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: epschains + proofSpan: + bodyEnd: 19521 + bodyStart: 13193 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 19532 + file: bddcurvbdddist.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 13180 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 14859 + file: bddcurvbdddist.tex + sourceTransform: tex2mdx.strip_comments + start: 14844 + via: proof + - context: proof + label: epschains + proofSpan: + bodyEnd: 19521 + bodyStart: 13193 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 19532 + file: bddcurvbdddist.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 13180 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 15191 + file: bddcurvbdddist.tex + sourceTransform: tex2mdx.strip_comments + start: 15176 + via: proof + sourceStatement: + chapter: 10 + file: bddcurvbdddist.tex + label: '' + mtref: '10.4' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 13178 + sourceTransform: tex2mdx.strip_comments + start: 12275 + targetStatement: + chapter: 19 + file: canonnbhd.tex + label: epschains + mtref: '19.19' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 31004 + sourceTransform: tex2mdx.strip_comments + start: 30360 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 96519f84d54e +- ba681e15e1a6 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/b0d2581a638f.md b/projects/poincare-conjecture/.astrolabe/edges/b0d2581a638f.md new file mode 100644 index 00000000..5085dd36 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/b0d2581a638f.md @@ -0,0 +1,62 @@ +--- +confidence: 0.35 +edgeClass: prose +evidence: + mentionTriggers: + - label: weakeqn + refSpan: + coordinateSpace: comment-stripped-tex + end: 48046 + file: temp2kappa.tex + sourceTransform: tex2mdx.strip_comments + start: 48033 + sentence: \subsection{Completion of the proof of Proposition~\protect{\ref{weakeqn}}}\label{sect9.2.5} + sentenceSpan: + coordinateSpace: comment-stripped-tex + end: 48065 + file: temp2kappa.tex + sourceTransform: tex2mdx.strip_comments + start: 47971 + sourceStrategy: nearest-previous-statement + via: prose + sourceStatement: + chapter: 9 + file: temp2kappa.tex + label: weakneg + mtref: '9.30' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 46581 + sourceTransform: tex2mdx.strip_comments + start: 46159 + targetStatement: + chapter: 9 + file: temp2kappa.tex + label: weakeqn + mtref: '9.20' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 25064 + sourceTransform: tex2mdx.strip_comments + start: 24639 + type: prose-mention-nearest-previous-statement + via: + - prose +generator: tools/poincare_tex_extract.py +inference: weak +kind: mention +ref: +- 42d9f91e59f3 +- 2e9e6c89f40c +rel: mentions +reviewStatus: candidate +scope: prose +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- prose +--- +Morgan--Tian prose mention. The source is the nearest preceding statement in the same chapter segment. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/b0e5eb831f58.md b/projects/poincare-conjecture/.astrolabe/edges/b0e5eb831f58.md new file mode 100644 index 00000000..23ab654b --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/b0e5eb831f58.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: nabR + proofSpan: + bodyEnd: 13719 + bodyStart: 13658 + coordinateSpace: comment-stripped-tex + depth: 4 + end: 13730 + file: noncoll.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 13645 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 13717 + file: noncoll.tex + sourceTransform: tex2mdx.strip_comments + start: 13707 + via: proof + sourceStatement: + chapter: 8 + file: noncoll.tex + label: volvomp + mtref: '8.8' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 13642 + sourceTransform: tex2mdx.strip_comments + start: 13363 + targetStatement: + chapter: 8 + file: noncoll.tex + label: nabR + mtref: '8.5' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 6558 + sourceTransform: tex2mdx.strip_comments + start: 6089 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 35010eaa585e +- bf8f356cfd7a +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/b10326049bf3.md b/projects/poincare-conjecture/.astrolabe/edges/b10326049bf3.md new file mode 100644 index 00000000..34c7cd0e --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/b10326049bf3.md @@ -0,0 +1,56 @@ +--- +confidence: 0.85 +edgeClass: semantic +evidence: + containmentReason: target theorem-like statement occurs inside source proof + proofSpan: + bodyEnd: 229182 + bodyStart: 185422 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 229193 + file: surgery.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 185409 + sourceStatement: + chapter: 17 + file: surgery.tex + label: extend + mtref: '17.1' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 185406 + sourceTransform: tex2mdx.strip_comments + start: 184285 + targetStatement: + chapter: 17 + file: surgery.tex + label: lemstdcan + mtref: '17.7' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 211471 + sourceTransform: tex2mdx.strip_comments + start: 210838 + type: proof-contained-theorem-like-statement + via: + - proof-contained-statement +generator: tools/poincare_tex_extract.py +inference: inferred +kind: dependency +ref: +- 254e8dae97e6 +- 111551d7831b +rel: depends +reviewStatus: accepted +scope: proof-containment +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof-contained-statement +--- +Morgan--Tian proof-contained statement dependency. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/b13dbe98e621.md b/projects/poincare-conjecture/.astrolabe/edges/b13dbe98e621.md new file mode 100644 index 00000000..08dd2ff1 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/b13dbe98e621.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 17 + file: surgery.tex + label: '16.2' + mtref: '17.2' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 186718 + sourceTransform: tex2mdx.strip_comments + start: 186200 + targetStatement: + chapter: 17 + file: surgery.tex + label: '' + mtref: '17.3' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 199036 + sourceTransform: tex2mdx.strip_comments + start: 198777 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 778d690c4e7c +- d130b5adaed6 +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/b1a695222bcb.md b/projects/poincare-conjecture/.astrolabe/edges/b1a695222bcb.md new file mode 100644 index 00000000..ad71f32c --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/b1a695222bcb.md @@ -0,0 +1,56 @@ +--- +confidence: 0.85 +edgeClass: semantic +evidence: + containmentReason: target theorem-like statement occurs inside source proof + proofSpan: + bodyEnd: 61925 + bodyStart: 54414 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 61936 + file: temp2kappa.tex + ownerLabel: '9.1' + sourceTransform: tex2mdx.strip_comments + start: 54401 + sourceStatement: + chapter: 9 + file: temp2kappa.tex + label: '9.1' + mtref: '9.33' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 53478 + sourceTransform: tex2mdx.strip_comments + start: 52780 + targetStatement: + chapter: 9 + file: temp2kappa.tex + label: '' + mtref: '9.35' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 60315 + sourceTransform: tex2mdx.strip_comments + start: 60102 + type: proof-contained-theorem-like-statement + via: + - proof-contained-statement +generator: tools/poincare_tex_extract.py +inference: inferred +kind: dependency +ref: +- ba1adbac0a75 +- e8f087a9cbc6 +rel: depends +reviewStatus: accepted +scope: proof-containment +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof-contained-statement +--- +Morgan--Tian proof-contained statement dependency. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/b1d8b90cf593.md b/projects/poincare-conjecture/.astrolabe/edges/b1d8b90cf593.md new file mode 100644 index 00000000..564cacde --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/b1d8b90cf593.md @@ -0,0 +1,124 @@ +--- +confidence: 0.35 +edgeClass: prose +evidence: + mentionTriggers: + - label: XI + refSpan: + coordinateSpace: comment-stripped-tex + end: 111035 + file: energy1.tex + sourceTransform: tex2mdx.strip_comments + start: 111027 + sentence: We shall show that for $\lambda>0$ sufficiently small, the family $p_1\widetilde\Gamma^\lambda(t)$ + satisfies the conclusion of Proposition~\ref{XI} for the fixed $\Gamma$ and + $\zeta>0$ + sentenceSpan: + coordinateSpace: comment-stripped-tex + end: 111072 + file: energy1.tex + sourceTransform: tex2mdx.strip_comments + start: 110887 + sourceStrategy: nearest-previous-statement + via: prose + - label: XI + refSpan: + coordinateSpace: comment-stripped-tex + end: 111226 + file: energy1.tex + sourceTransform: tex2mdx.strip_comments + start: 111218 + sentence: First, we show that fixing one $c\in S^2$, for $\lambda$ sufficiently + small (depending on $c$) an analogue of Proposition~\ref{XI} holds for the one-parameter + family of loops $p_1\widetilde\Gamma_c^\lambda(t)$ + sentenceSpan: + coordinateSpace: comment-stripped-tex + end: 111305 + file: energy1.tex + sourceTransform: tex2mdx.strip_comments + start: 111094 + sourceStrategy: nearest-previous-statement + via: prose + - label: XI + refSpan: + coordinateSpace: comment-stripped-tex + end: 112033 + file: energy1.tex + sourceTransform: tex2mdx.strip_comments + start: 112025 + sentence: Then, using the result of a single $c$, we fix $\lambda>0$ sufficiently + small so that the analogue of Proposition~\ref{XI} for individual curves (or + rather the slightly stronger version of it) holds for every $\hat c\in {\mathcal + S}$. Then we complete the proof of Proposition~\ref{XI} using the fact that + for every $c$ the curve $\widetilde\Gamma(c)$ is sufficiently close to a curve + $\widetilde\Gamma(\hat c)$ associated to an element $\hat c\in {\mathcal S}$. + sentenceSpan: + coordinateSpace: comment-stripped-tex + end: 112373 + file: energy1.tex + sourceTransform: tex2mdx.strip_comments + start: 111910 + sourceStrategy: nearest-previous-statement + via: prose + - label: XI + refSpan: + coordinateSpace: comment-stripped-tex + end: 112196 + file: energy1.tex + sourceTransform: tex2mdx.strip_comments + start: 112188 + sentence: Then, using the result of a single $c$, we fix $\lambda>0$ sufficiently + small so that the analogue of Proposition~\ref{XI} for individual curves (or + rather the slightly stronger version of it) holds for every $\hat c\in {\mathcal + S}$. Then we complete the proof of Proposition~\ref{XI} using the fact that + for every $c$ the curve $\widetilde\Gamma(c)$ is sufficiently close to a curve + $\widetilde\Gamma(\hat c)$ associated to an element $\hat c\in {\mathcal S}$. + sentenceSpan: + coordinateSpace: comment-stripped-tex + end: 112373 + file: energy1.tex + sourceTransform: tex2mdx.strip_comments + start: 111910 + sourceStrategy: nearest-previous-statement + via: prose + sourceStatement: + chapter: 18 + file: energy1.tex + label: '' + mtref: '18.50' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 107575 + sourceTransform: tex2mdx.strip_comments + start: 107348 + targetStatement: + chapter: 18 + file: energy1.tex + label: XI + mtref: '18.24' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 58810 + sourceTransform: tex2mdx.strip_comments + start: 58030 + type: prose-mention-nearest-previous-statement + via: + - prose +generator: tools/poincare_tex_extract.py +inference: weak +kind: mention +ref: +- cb69804a2166 +- fd85732525c1 +rel: mentions +reviewStatus: candidate +scope: prose +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- prose +--- +Morgan--Tian prose mention. The source is the nearest preceding statement in the same chapter segment. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/b1eb2695af2c.md b/projects/poincare-conjecture/.astrolabe/edges/b1eb2695af2c.md new file mode 100644 index 00000000..e76689b6 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/b1eb2695af2c.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 11 + file: singlimit2.tex + label: '' + mtref: '11.25' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 48871 + sourceTransform: tex2mdx.strip_comments + start: 48329 + targetStatement: + chapter: 11 + file: singlimit2.tex + label: '' + mtref: '11.26' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 49703 + sourceTransform: tex2mdx.strip_comments + start: 48874 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- da99c0859788 +- 08c3ff9d0a1e +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/b20bf3044831.md b/projects/poincare-conjecture/.astrolabe/edges/b20bf3044831.md new file mode 100644 index 00000000..9323423f --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/b20bf3044831.md @@ -0,0 +1,64 @@ +--- +confidence: 0.35 +edgeClass: prose +evidence: + mentionTriggers: + - label: harmapflow + refSpan: + coordinateSpace: comment-stripped-tex + end: 81512 + file: stdsoln.tex + sourceTransform: tex2mdx.strip_comments + start: 81496 + sentence: Hence we apply Proposition~\ref{harmapflow} to construction solutions + $\psi_1$ and $\psi_2$ to the harmonic map flow for $g_1(t)$ and $g_2(t)$ with + $\psi_1$ and $\psi_2$ being the identity at time $t_0$ + sentenceSpan: + coordinateSpace: comment-stripped-tex + end: 81671 + file: stdsoln.tex + sourceTransform: tex2mdx.strip_comments + start: 81468 + sourceStrategy: nearest-previous-statement + via: prose + sourceStatement: + chapter: 12 + file: stdsoln.tex + label: duniq + mtref: '12.27' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 80569 + sourceTransform: tex2mdx.strip_comments + start: 80181 + targetStatement: + chapter: 12 + file: stdsoln.tex + label: harmapflow + mtref: '12.25' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 74100 + sourceTransform: tex2mdx.strip_comments + start: 73601 + type: prose-mention-nearest-previous-statement + via: + - prose +generator: tools/poincare_tex_extract.py +inference: weak +kind: mention +ref: +- a501760fb0e6 +- 5fbeadbba61c +rel: mentions +reviewStatus: candidate +scope: prose +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- prose +--- +Morgan--Tian prose mention. The source is the nearest preceding statement in the same chapter segment. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/b220f26e93ea.md b/projects/poincare-conjecture/.astrolabe/edges/b220f26e93ea.md new file mode 100644 index 00000000..e79023e8 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/b220f26e93ea.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: localMP + proofSpan: + bodyEnd: 23481 + bodyStart: 21942 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 23492 + file: maxprin.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 21929 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 23322 + file: maxprin.tex + sourceTransform: tex2mdx.strip_comments + start: 23309 + via: proof + sourceStatement: + chapter: 4 + file: maxprin.tex + label: SMPcurv + mtref: '4.17' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 21926 + sourceTransform: tex2mdx.strip_comments + start: 20592 + targetStatement: + chapter: 4 + file: maxprin.tex + label: localMP + mtref: '4.9' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 11555 + sourceTransform: tex2mdx.strip_comments + start: 10496 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- f696c0fa9db3 +- a89be7fdb431 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/b24729201992.md b/projects/poincare-conjecture/.astrolabe/edges/b24729201992.md new file mode 100644 index 00000000..b072dbaa --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/b24729201992.md @@ -0,0 +1,62 @@ +--- +confidence: 0.9 +edgeClass: structural +evidence: + containmentDepth: 1 + proofSpan: + bodyEnd: 54967 + bodyStart: 37263 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 54978 + file: energy1.tex + ownerLabel: W_2 + sourceTransform: tex2mdx.strip_comments + start: 37250 + sourceStatement: + chapter: 18 + file: energy1.tex + label: W_2 + mtref: '18.11' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 36484 + sourceTransform: tex2mdx.strip_comments + start: 36186 + targetContainedSpan: + coordinateSpace: comment-stripped-tex + end: 40079 + file: energy1.tex + sourceTransform: tex2mdx.strip_comments + start: 39789 + targetStatement: + chapter: 18 + file: energy1.tex + label: '' + mtref: '18.13' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 40079 + sourceTransform: tex2mdx.strip_comments + start: 39789 + type: proof-containment + via: + - proof-contains +generator: tools/poincare_tex_extract.py +inference: structural +kind: containment +ref: +- 7bb0fa0c9412 +- 1554c9624fa7 +rel: contains +reviewStatus: accepted +scope: proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof-contains +--- +Morgan--Tian proof containment. The source is the statement whose proof contains the target statement. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/b25ed1c28d09.md b/projects/poincare-conjecture/.astrolabe/edges/b25ed1c28d09.md new file mode 100644 index 00000000..af674111 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/b25ed1c28d09.md @@ -0,0 +1,62 @@ +--- +confidence: 0.9 +edgeClass: structural +evidence: + containmentDepth: 2 + proofSpan: + bodyEnd: 33794 + bodyStart: 17659 + coordinateSpace: comment-stripped-tex + depth: 2 + end: 33805 + file: singlimit2.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 17646 + sourceStatement: + chapter: 11 + file: singlimit2.tex + label: maxT + mtref: '11.10' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 17643 + sourceTransform: tex2mdx.strip_comments + start: 17183 + targetContainedSpan: + coordinateSpace: comment-stripped-tex + end: 19990 + file: singlimit2.tex + sourceTransform: tex2mdx.strip_comments + start: 19475 + targetStatement: + chapter: 11 + file: singlimit2.tex + label: compactbd + mtref: '11.11' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 19990 + sourceTransform: tex2mdx.strip_comments + start: 19475 + type: proof-containment + via: + - proof-contains +generator: tools/poincare_tex_extract.py +inference: structural +kind: containment +ref: +- d6919616448d +- cc3a26ca3809 +rel: contains +reviewStatus: accepted +scope: proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof-contains +--- +Morgan--Tian proof containment. The source is the statement whose proof contains the target statement. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/b25f00dd5de9.md b/projects/poincare-conjecture/.astrolabe/edges/b25f00dd5de9.md new file mode 100644 index 00000000..189cfed3 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/b25f00dd5de9.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 11 + file: singlimit2.tex + label: ancientlimit + mtref: '11.16' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 29827 + sourceTransform: tex2mdx.strip_comments + start: 29411 + targetStatement: + chapter: 11 + file: singlimit2.tex + label: '' + mtref: '11.17' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 31800 + sourceTransform: tex2mdx.strip_comments + start: 31539 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- a66fa0ea8720 +- b3b96a664796 +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/b271c33267ed.md b/projects/poincare-conjecture/.astrolabe/edges/b271c33267ed.md new file mode 100644 index 00000000..6f6b1f8a --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/b271c33267ed.md @@ -0,0 +1,68 @@ +--- +confidence: 0.35 +edgeClass: prose +evidence: + mentionTriggers: + - label: compact2D + refSpan: + coordinateSpace: comment-stripped-tex + end: 89406 + file: temp2kappa.tex + sourceTransform: tex2mdx.strip_comments + start: 89391 + sentence: Hence, by Corollary~\ref{compact2D} we see that $n$ cannot be equal + to two. Furthermore, by Proposition~\ref{prodatinf}, when $n=3$ there is another + subsequence $q_i$ tending to infinity in $M$ such that there is a geometric + limit $(M_\infty,g_\infty(t),(q_\infty,-1)), \ -\infty0 + proofSpan: + bodyEnd: 74813 + bodyStart: 59556 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 74824 + file: singlimit2.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 59543 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 66204 + file: singlimit2.tex + sourceTransform: tex2mdx.strip_comments + start: 66194 + via: proof + sourceStatement: + chapter: 11 + file: singlimit2.tex + label: hexist + mtref: '11.31' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 59267 + sourceTransform: tex2mdx.strip_comments + start: 58375 + targetStatement: + chapter: 4 + file: maxprin.tex + label: rm>0 + mtref: '4.14' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 16848 + sourceTransform: tex2mdx.strip_comments + start: 16607 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 074e02b067b8 +- 0db415314964 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/b31cd40205b3.md b/projects/poincare-conjecture/.astrolabe/edges/b31cd40205b3.md new file mode 100644 index 00000000..cee7e2fb --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/b31cd40205b3.md @@ -0,0 +1,45 @@ +--- +confidence: 0.2 +edgeClass: structural +evidence: + sourceStatement: + chapter: 10 + file: bddcurvbdddist.tex + label: pinchdef + mtref: '10.1' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 1390 + sourceTransform: tex2mdx.strip_comments + start: 647 + targetStatement: + chapter: 10 + file: bddcurvbdddist.tex + label: bcbd + mtref: '10.2' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 2942 + sourceTransform: tex2mdx.strip_comments + start: 2168 + type: chapter-reading-order-adjacency + via: + - chapter-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 7464980c4085 +- ecd62fc2ea2b +rel: in-chapter +reviewStatus: accepted +scope: chapter +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- chapter-sequence +--- +Morgan--Tian reading-order adjacency across sections in a chapter. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/b35d018d3d1d.md b/projects/poincare-conjecture/.astrolabe/edges/b35d018d3d1d.md new file mode 100644 index 00000000..98d6d56d --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/b35d018d3d1d.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 9 + file: temp2kappa.tex + label: gradshrink + mtref: '9.32' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 52719 + sourceTransform: tex2mdx.strip_comments + start: 52484 + targetStatement: + chapter: 9 + file: temp2kappa.tex + label: '9.1' + mtref: '9.33' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 53478 + sourceTransform: tex2mdx.strip_comments + start: 52780 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 07232fc42f3a +- ba1adbac0a75 +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/b36becd063dd.md b/projects/poincare-conjecture/.astrolabe/edges/b36becd063dd.md new file mode 100644 index 00000000..e653cec7 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/b36becd063dd.md @@ -0,0 +1,62 @@ +--- +confidence: 0.9 +edgeClass: structural +evidence: + containmentDepth: 1 + proofSpan: + bodyEnd: 47393 + bodyStart: 39592 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 47404 + file: maxprin.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 39579 + sourceStatement: + chapter: 4 + file: maxprin.tex + label: '' + mtref: '4.27' + sort: remark + span: + coordinateSpace: comment-stripped-tex + end: 39436 + sourceTransform: tex2mdx.strip_comments + start: 39187 + targetContainedSpan: + coordinateSpace: comment-stripped-tex + end: 41209 + file: maxprin.tex + sourceTransform: tex2mdx.strip_comments + start: 41046 + targetStatement: + chapter: 4 + file: maxprin.tex + label: '' + mtref: '4.28' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 41209 + sourceTransform: tex2mdx.strip_comments + start: 41046 + type: proof-containment + via: + - proof-contains +generator: tools/poincare_tex_extract.py +inference: structural +kind: containment +ref: +- 1160f53cc783 +- 0ac34e2feca6 +rel: contains +reviewStatus: accepted +scope: proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof-contains +--- +Morgan--Tian proof containment. The source is the statement whose proof contains the target statement. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/b3f3dc19f7b5.md b/projects/poincare-conjecture/.astrolabe/edges/b3f3dc19f7b5.md new file mode 100644 index 00000000..7848d7d0 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/b3f3dc19f7b5.md @@ -0,0 +1,45 @@ +--- +confidence: 0.2 +edgeClass: structural +evidence: + sourceStatement: + chapter: 2 + file: prelim.tex + label: anglecompar + mtref: '2.6' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 55820 + sourceTransform: tex2mdx.strip_comments + start: 55235 + targetStatement: + chapter: 2 + file: prelim.tex + label: soul + mtref: '2.7' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 57159 + sourceTransform: tex2mdx.strip_comments + start: 56544 + type: chapter-reading-order-adjacency + via: + - chapter-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- dc263832416c +- 2b5f18292e62 +rel: in-chapter +reviewStatus: accepted +scope: chapter +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- chapter-sequence +--- +Morgan--Tian reading-order adjacency across sections in a chapter. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/b420b4fbe569.md b/projects/poincare-conjecture/.astrolabe/edges/b420b4fbe569.md new file mode 100644 index 00000000..7b88a4dc --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/b420b4fbe569.md @@ -0,0 +1,63 @@ +--- +confidence: 0.35 +edgeClass: prose +evidence: + mentionTriggers: + - label: smeigen + refSpan: + coordinateSpace: comment-stripped-tex + end: 24916 + file: surgery.tex + sourceTransform: tex2mdx.strip_comments + start: 24903 + sentence: First of all in the preimage of the interval $[4,4+A_0,-r_0]$ this follows + from Corollary~\ref{smeigen} and the fact that $\eta g_0$ has non-negative curvature + sentenceSpan: + coordinateSpace: comment-stripped-tex + end: 24972 + file: surgery.tex + sourceTransform: tex2mdx.strip_comments + start: 24812 + sourceStrategy: nearest-previous-statement + via: prose + sourceStatement: + chapter: 13 + file: surgery.tex + label: '' + mtref: '13.14' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 23621 + sourceTransform: tex2mdx.strip_comments + start: 23086 + targetStatement: + chapter: 13 + file: surgery.tex + label: smeigen + mtref: '13.12' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 20688 + sourceTransform: tex2mdx.strip_comments + start: 20353 + type: prose-mention-nearest-previous-statement + via: + - prose +generator: tools/poincare_tex_extract.py +inference: weak +kind: mention +ref: +- c9d8cb9d34c3 +- e70bacf8aab0 +rel: mentions +reviewStatus: candidate +scope: prose +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- prose +--- +Morgan--Tian prose mention. The source is the nearest preceding statement in the same chapter segment. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/b4547f694686.md b/projects/poincare-conjecture/.astrolabe/edges/b4547f694686.md new file mode 100644 index 00000000..38f31480 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/b4547f694686.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 7 + file: newcomp2.tex + label: wineq + mtref: '7.22' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 37224 + sourceTransform: tex2mdx.strip_comments + start: 36952 + targetStatement: + chapter: 7 + file: newcomp2.tex + label: '' + mtref: '7.23' + sort: remark + span: + coordinateSpace: comment-stripped-tex + end: 37322 + sourceTransform: tex2mdx.strip_comments + start: 37227 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- de785906ce83 +- f93a1c4b1057 +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/b45db5f8a004.md b/projects/poincare-conjecture/.astrolabe/edges/b45db5f8a004.md new file mode 100644 index 00000000..37daa1a7 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/b45db5f8a004.md @@ -0,0 +1,56 @@ +--- +confidence: 0.85 +edgeClass: semantic +evidence: + containmentReason: target theorem-like statement occurs inside source proof + proofSpan: + bodyEnd: 171471 + bodyStart: 161354 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 171482 + file: energy1.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 161341 + sourceStatement: + chapter: 18 + file: energy1.tex + label: laest + mtref: '18.66' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 161338 + sourceTransform: tex2mdx.strip_comments + start: 161080 + targetStatement: + chapter: 18 + file: energy1.tex + label: '' + mtref: '18.71' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 170254 + sourceTransform: tex2mdx.strip_comments + start: 170143 + type: proof-contained-theorem-like-statement + via: + - proof-contained-statement +generator: tools/poincare_tex_extract.py +inference: inferred +kind: dependency +ref: +- 45c783d384c1 +- e94b6de05447 +rel: depends +reviewStatus: accepted +scope: proof-containment +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof-contained-statement +--- +Morgan--Tian proof-contained statement dependency. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/b468dd284022.md b/projects/poincare-conjecture/.astrolabe/edges/b468dd284022.md new file mode 100644 index 00000000..f804e048 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/b468dd284022.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 7 + file: newcomp2.tex + label: geoexist + mtref: '7.2' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 2760 + sourceTransform: tex2mdx.strip_comments + start: 2498 + targetStatement: + chapter: 7 + file: newcomp2.tex + label: '' + mtref: '7.3' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 5342 + sourceTransform: tex2mdx.strip_comments + start: 5096 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- cd062526b91a +- 44032fc43e5d +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/b484a13bfc6d.md b/projects/poincare-conjecture/.astrolabe/edges/b484a13bfc6d.md new file mode 100644 index 00000000..2499f3d9 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/b484a13bfc6d.md @@ -0,0 +1,65 @@ +--- +confidence: 0.45 +edgeClass: semantic +evidence: + endpointSemanticDegreeBeforeTermBackfill: + source: 0 + target: 2 + selectionReason: semantic-connectivity-backfill + sourceMatchSpan: + coordinateSpace: comment-stripped-tex + end: 44708 + file: surgery.tex + sourceTransform: tex2mdx.strip_comments + start: 44698 + sourceStatement: + chapter: 14 + file: surgery.tex + label: '' + mtref: '14.7' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 45463 + sourceTransform: tex2mdx.strip_comments + start: 44092 + targetStatement: + chapter: 14 + file: surgery.tex + label: '' + mtref: '14.1' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 28555 + sourceTransform: tex2mdx.strip_comments + start: 27727 + targetTermSpan: + coordinateSpace: comment-stripped-tex + end: 27760 + file: surgery.tex + sourceTransform: tex2mdx.strip_comments + start: 27750 + term: space time + termAmbiguity: unique-in-chapter + termSource: italic + termSpecificity: not-covered-by-longer-defined-term + type: same-chapter-definition-term-match + via: + - definition-term:space time +generator: tools/poincare_tex_extract.py +inference: weak +kind: definition-use +ref: +- 124fe3535807 +- 41da6871e864 +rel: uses +reviewStatus: candidate +scope: same-chapter-term-match +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- definition-term:space time +--- +Morgan--Tian same-chapter definition term match. This is a low-confidence candidate edge for review. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/b4b81a3478d0.md b/projects/poincare-conjecture/.astrolabe/edges/b4b81a3478d0.md new file mode 100644 index 00000000..23ebd94c --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/b4b81a3478d0.md @@ -0,0 +1,56 @@ +--- +confidence: 0.85 +edgeClass: semantic +evidence: + containmentReason: target theorem-like statement occurs inside source proof + proofSpan: + bodyEnd: 173338 + bodyStart: 165469 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 173349 + file: surgery.tex + ownerLabel: delta0ri+1 + sourceTransform: tex2mdx.strip_comments + start: 165456 + sourceStatement: + chapter: 16 + file: surgery.tex + label: delta0ri+1 + mtref: '16.4' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 109162 + sourceTransform: tex2mdx.strip_comments + start: 107682 + targetStatement: + chapter: 16 + file: surgery.tex + label: '' + mtref: '16.25' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 167879 + sourceTransform: tex2mdx.strip_comments + start: 167677 + type: proof-contained-theorem-like-statement + via: + - proof-contained-statement +generator: tools/poincare_tex_extract.py +inference: inferred +kind: dependency +ref: +- 89a60eb69c3f +- fc5c13ed2e67 +rel: depends +reviewStatus: accepted +scope: proof-containment +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof-contained-statement +--- +Morgan--Tian proof-contained statement dependency. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/b4df05be4dec.md b/projects/poincare-conjecture/.astrolabe/edges/b4df05be4dec.md new file mode 100644 index 00000000..ef1b1f63 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/b4df05be4dec.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: THM + proofSpan: + bodyEnd: 16850 + bodyStart: 3598 + coordinateSpace: comment-stripped-tex + depth: 2 + end: 16861 + file: noncoll.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 3585 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 4684 + file: noncoll.tex + sourceTransform: tex2mdx.strip_comments + start: 4675 + via: proof + sourceStatement: + chapter: 8 + file: noncoll.tex + label: step + mtref: '8.2' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 3313 + sourceTransform: tex2mdx.strip_comments + start: 3033 + targetStatement: + chapter: 8 + file: noncoll.tex + label: THM + mtref: '8.1' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 1970 + sourceTransform: tex2mdx.strip_comments + start: 682 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 12d93b430400 +- 3defa9b2209c +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/b4ed242ee2a8.md b/projects/poincare-conjecture/.astrolabe/edges/b4ed242ee2a8.md new file mode 100644 index 00000000..4ae391d2 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/b4ed242ee2a8.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 18 + file: energy1.tex + label: mugrowth + mtref: '18.43' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 93643 + sourceTransform: tex2mdx.strip_comments + start: 93070 + targetStatement: + chapter: 18 + file: energy1.tex + label: '' + mtref: '18.44' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 98203 + sourceTransform: tex2mdx.strip_comments + start: 97789 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 6590af0738c7 +- 08412f0ffb82 +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/b508de25be53.md b/projects/poincare-conjecture/.astrolabe/edges/b508de25be53.md new file mode 100644 index 00000000..9bef4804 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/b508de25be53.md @@ -0,0 +1,84 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: Hessin + proofSpan: + bodyEnd: 28599 + bodyStart: 25964 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 28610 + file: newcomp2.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 25951 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 27479 + file: newcomp2.tex + sourceTransform: tex2mdx.strip_comments + start: 27467 + via: proof-aux-label-proxy + - context: proof + label: Heqn + proofSpan: + bodyEnd: 28599 + bodyStart: 25964 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 28610 + file: newcomp2.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 25951 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 28414 + file: newcomp2.tex + sourceTransform: tex2mdx.strip_comments + start: 28404 + via: proof-aux-label-proxy + sourceStatement: + chapter: 7 + file: newcomp2.tex + label: phisupport + mtref: '7.15' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 25949 + sourceTransform: tex2mdx.strip_comments + start: 25492 + targetStatement: + chapter: 6 + file: newcompar.tex + label: Hessineq + mtref: '6.37' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 56065 + sourceTransform: tex2mdx.strip_comments + start: 55015 + type: tex-reference + via: + - proof-aux-label-proxy +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 5e0538ed62cf +- 88d4ef67440f +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof-aux-label-proxy +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/b517f7836c1b.md b/projects/poincare-conjecture/.astrolabe/edges/b517f7836c1b.md new file mode 100644 index 00000000..be58cf55 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/b517f7836c1b.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 12 + file: stdsoln.tex + label: stdsolncannbhd + mtref: '12.32' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 94479 + sourceTransform: tex2mdx.strip_comments + start: 93886 + targetStatement: + chapter: 12 + file: stdsoln.tex + label: '' + mtref: '12.33' + sort: remark + span: + coordinateSpace: comment-stripped-tex + end: 95195 + sourceTransform: tex2mdx.strip_comments + start: 94631 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 6ff28dba4e56 +- 246771506ee5 +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/b5928f60d3fe.md b/projects/poincare-conjecture/.astrolabe/edges/b5928f60d3fe.md new file mode 100644 index 00000000..6c80a029 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/b5928f60d3fe.md @@ -0,0 +1,65 @@ +--- +confidence: 0.45 +edgeClass: semantic +evidence: + endpointSemanticDegreeBeforeTermBackfill: + source: 0 + target: 1 + selectionReason: semantic-connectivity-backfill + sourceMatchSpan: + coordinateSpace: comment-stripped-tex + end: 43487 + file: converge2.tex + sourceTransform: tex2mdx.strip_comments + start: 43475 + sourceStatement: + chapter: 5 + file: converge2.tex + label: '' + mtref: '5.29' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 43638 + sourceTransform: tex2mdx.strip_comments + start: 43134 + targetStatement: + chapter: 5 + file: converge2.tex + label: '' + mtref: '5.27' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 40739 + sourceTransform: tex2mdx.strip_comments + start: 40490 + targetTermSpan: + coordinateSpace: comment-stripped-tex + end: 40522 + file: converge2.tex + sourceTransform: tex2mdx.strip_comments + start: 40510 + term: length space + termAmbiguity: unique-in-chapter + termSource: italic + termSpecificity: no-longer-defined-term + type: same-chapter-definition-term-match + via: + - definition-term:length space +generator: tools/poincare_tex_extract.py +inference: weak +kind: definition-use +ref: +- 97c2fdddbf04 +- 9b970f13e747 +rel: uses +reviewStatus: candidate +scope: same-chapter-term-match +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- definition-term:length space +--- +Morgan--Tian same-chapter definition term match. This is a low-confidence candidate edge for review. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/b5a1a8fc5bb0.md b/projects/poincare-conjecture/.astrolabe/edges/b5a1a8fc5bb0.md new file mode 100644 index 00000000..3bf7f3a8 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/b5a1a8fc5bb0.md @@ -0,0 +1,45 @@ +--- +confidence: 0.2 +edgeClass: structural +evidence: + sourceStatement: + chapter: 4 + file: maxprin.tex + label: flowtoround + mtref: '4.23' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 35932 + sourceTransform: tex2mdx.strip_comments + start: 34642 + targetStatement: + chapter: 4 + file: maxprin.tex + label: sphsf + mtref: '4.24' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 36469 + sourceTransform: tex2mdx.strip_comments + start: 36157 + type: chapter-reading-order-adjacency + via: + - chapter-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 86e5a5766bda +- 4f253de31e0c +rel: in-chapter +reviewStatus: accepted +scope: chapter +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- chapter-sequence +--- +Morgan--Tian reading-order adjacency across sections in a chapter. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/b5b0f66f384f.md b/projects/poincare-conjecture/.astrolabe/edges/b5b0f66f384f.md new file mode 100644 index 00000000..6a02302f --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/b5b0f66f384f.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 1 + file: prelim.tex + label: injvol + mtref: '1.35' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 44668 + sourceTransform: tex2mdx.strip_comments + start: 44242 + targetStatement: + chapter: 1 + file: prelim.tex + label: volinj + mtref: '1.36' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 46059 + sourceTransform: tex2mdx.strip_comments + start: 45642 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 7fec5a940413 +- f90cce212e88 +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/b5b9ae0a2bd8.md b/projects/poincare-conjecture/.astrolabe/edges/b5b9ae0a2bd8.md new file mode 100644 index 00000000..84eaa4f9 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/b5b9ae0a2bd8.md @@ -0,0 +1,55 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: statement + label: assumptions + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 38672 + file: singlimit2.tex + sourceTransform: tex2mdx.strip_comments + start: 38655 + via: statement + sourceStatement: + chapter: 11 + file: singlimit2.tex + label: Rproper + mtref: '11.21' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 39168 + sourceTransform: tex2mdx.strip_comments + start: 38496 + targetStatement: + chapter: 11 + file: singlimit2.tex + label: assumptions + mtref: '11.18' + sort: assumption + span: + coordinateSpace: comment-stripped-tex + end: 36515 + sourceTransform: tex2mdx.strip_comments + start: 35753 + type: tex-reference + via: + - statement +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- a99536ac8110 +- 3009be0a37fc +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- statement +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/b5c2c3ba8bed.md b/projects/poincare-conjecture/.astrolabe/edges/b5c2c3ba8bed.md new file mode 100644 index 00000000..4b512f3c --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/b5c2c3ba8bed.md @@ -0,0 +1,65 @@ +--- +confidence: 0.35 +edgeClass: prose +evidence: + mentionTriggers: + - label: fullmeasure + refSpan: + coordinateSpace: comment-stripped-tex + end: 182746 + file: surgery.tex + sourceTransform: tex2mdx.strip_comments + start: 182729 + sentence: Then, by Corollary~\ref{fullmeasure} (see also, Proposition~\ref{lips}), + the intersection, $B'$, of ${\mathcal U}_x$ with $B(y,t'',r')$ is an open subset + of full measure in $B(y,t'',r')$. Of course, ${\rm Vol}\,B'={\rm Vol}\,B(y,t'',r')\ge + \kappa_i(r')^3$ and the function $l_x$ is bounded by $L/2$ on $B'$ + sentenceSpan: + coordinateSpace: comment-stripped-tex + end: 183017 + file: surgery.tex + sourceTransform: tex2mdx.strip_comments + start: 182709 + sourceStrategy: nearest-previous-statement + via: prose + sourceStatement: + chapter: 16 + file: surgery.tex + label: '' + mtref: '16.28' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 176062 + sourceTransform: tex2mdx.strip_comments + start: 175778 + targetStatement: + chapter: 6 + file: newcompar.tex + label: fullmeasure + mtref: '6.67' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 101253 + sourceTransform: tex2mdx.strip_comments + start: 100568 + type: prose-mention-nearest-previous-statement + via: + - prose +generator: tools/poincare_tex_extract.py +inference: weak +kind: mention +ref: +- 3e9580da0ff5 +- 7aad50e49af2 +rel: mentions +reviewStatus: candidate +scope: prose +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- prose +--- +Morgan--Tian prose mention. The source is the nearest preceding statement in the same chapter segment. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/b5dad1c0fba2.md b/projects/poincare-conjecture/.astrolabe/edges/b5dad1c0fba2.md new file mode 100644 index 00000000..3466912d --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/b5dad1c0fba2.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: noloops + proofSpan: + bodyEnd: 184898 + bodyStart: 173854 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 184909 + file: energy1.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 173841 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 175577 + file: energy1.tex + sourceTransform: tex2mdx.strip_comments + start: 175564 + via: proof + sourceStatement: + chapter: 18 + file: energy1.tex + label: Dxacross + mtref: '18.74' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 173839 + sourceTransform: tex2mdx.strip_comments + start: 173679 + targetStatement: + chapter: 18 + file: energy1.tex + label: noloops + mtref: '18.73' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 172812 + sourceTransform: tex2mdx.strip_comments + start: 172656 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 0d6f74e37efb +- b6fb2b9c364b +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/b5e3a89c4fbc.md b/projects/poincare-conjecture/.astrolabe/edges/b5e3a89c4fbc.md new file mode 100644 index 00000000..0bc5ee23 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/b5e3a89c4fbc.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 17 + file: surgery.tex + label: '' + mtref: '17.10' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 223009 + sourceTransform: tex2mdx.strip_comments + start: 222555 + targetStatement: + chapter: 17 + file: surgery.tex + label: '' + mtref: '17.11' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 226210 + sourceTransform: tex2mdx.strip_comments + start: 226024 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 3b7da22bf357 +- 8dafe9c95b7f +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/b5e448a12e3c.md b/projects/poincare-conjecture/.astrolabe/edges/b5e448a12e3c.md new file mode 100644 index 00000000..c8b8fc94 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/b5e448a12e3c.md @@ -0,0 +1,76 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: statement + label: prodatinf + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 117795 + file: temp2kappa.tex + sourceTransform: tex2mdx.strip_comments + start: 117780 + via: statement + - context: proof + label: prodatinf + proofSpan: + bodyEnd: 119194 + bodyStart: 118223 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 119205 + file: temp2kappa.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 118210 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 118538 + file: temp2kappa.tex + sourceTransform: tex2mdx.strip_comments + start: 118523 + via: proof + sourceStatement: + chapter: 9 + file: temp2kappa.tex + label: asympvol + mtref: '9.59' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 118207 + sourceTransform: tex2mdx.strip_comments + start: 117600 + targetStatement: + chapter: 9 + file: temp2kappa.tex + label: prodatinf + mtref: '9.39' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 68632 + sourceTransform: tex2mdx.strip_comments + start: 67563 + type: tex-reference + via: + - proof + - statement +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 7468282bd017 +- 703d5ca9da05 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +- statement +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/b5f78676bab3.md b/projects/poincare-conjecture/.astrolabe/edges/b5f78676bab3.md new file mode 100644 index 00000000..51fd8558 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/b5f78676bab3.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: 4pi + proofSpan: + bodyEnd: 51188 + bodyStart: 48388 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 51199 + file: newcomp2.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 48375 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 48583 + file: newcomp2.tex + sourceTransform: tex2mdx.strip_comments + start: 48574 + via: proof + sourceStatement: + chapter: 7 + file: newcomp2.tex + label: 4picase + mtref: '7.27' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 48373 + sourceTransform: tex2mdx.strip_comments + start: 47991 + targetStatement: + chapter: 7 + file: newcomp2.tex + label: 4pi + mtref: '7.26' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 45413 + sourceTransform: tex2mdx.strip_comments + start: 44820 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 613fb3994956 +- ae6f1c6d00d9 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/b6278c15acb8.md b/projects/poincare-conjecture/.astrolabe/edges/b6278c15acb8.md new file mode 100644 index 00000000..6e9264ca --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/b6278c15acb8.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 9 + file: temp2kappa.tex + label: '' + mtref: '9.67' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 133378 + sourceTransform: tex2mdx.strip_comments + start: 133150 + targetStatement: + chapter: 9 + file: temp2kappa.tex + label: zkbound + mtref: '9.68' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 134931 + sourceTransform: tex2mdx.strip_comments + start: 134780 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- b8f989614649 +- f61f72020102 +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/b63c0a286c2d.md b/projects/poincare-conjecture/.astrolabe/edges/b63c0a286c2d.md new file mode 100644 index 00000000..9d420446 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/b63c0a286c2d.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: localprod + proofSpan: + bodyEnd: 71821 + bodyStart: 68779 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 71832 + file: temp2kappa.tex + ownerLabel: prodatinf + sourceTransform: tex2mdx.strip_comments + start: 68766 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 71698 + file: temp2kappa.tex + sourceTransform: tex2mdx.strip_comments + start: 71683 + via: proof + sourceStatement: + chapter: 9 + file: temp2kappa.tex + label: prodatinf + mtref: '9.39' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 68632 + sourceTransform: tex2mdx.strip_comments + start: 67563 + targetStatement: + chapter: 4 + file: maxprin.tex + label: localprod + mtref: '4.19' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 25705 + sourceTransform: tex2mdx.strip_comments + start: 25065 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 703d5ca9da05 +- 8ec855a2aab5 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/b658f4595997.md b/projects/poincare-conjecture/.astrolabe/edges/b658f4595997.md new file mode 100644 index 00000000..1a8b542b --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/b658f4595997.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: absconv + proofSpan: + bodyEnd: 49505 + bodyStart: 49313 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 49516 + file: temp2kappa.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 49300 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 49449 + file: temp2kappa.tex + sourceTransform: tex2mdx.strip_comments + start: 49436 + via: proof + sourceStatement: + chapter: 9 + file: temp2kappa.tex + label: '' + mtref: '9.31' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 49297 + sourceTransform: tex2mdx.strip_comments + start: 48976 + targetStatement: + chapter: 9 + file: temp2kappa.tex + label: absconv + mtref: '9.28' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 45006 + sourceTransform: tex2mdx.strip_comments + start: 44788 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- a8ab8a19cc61 +- 0e9703c15036 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/b65dc08ad90f.md b/projects/poincare-conjecture/.astrolabe/edges/b65dc08ad90f.md new file mode 100644 index 00000000..b901751f --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/b65dc08ad90f.md @@ -0,0 +1,55 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: statement + label: lips + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 85138 + file: newcompar.tex + sourceTransform: tex2mdx.strip_comments + start: 85128 + via: statement + sourceStatement: + chapter: 6 + file: newcompar.tex + label: '' + mtref: '6.57' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 85285 + sourceTransform: tex2mdx.strip_comments + start: 85074 + targetStatement: + chapter: 6 + file: newcompar.tex + label: lips + mtref: '6.56' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 85066 + sourceTransform: tex2mdx.strip_comments + start: 84130 + type: tex-reference + via: + - statement +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- a89a833cf7cb +- db4ced24b05f +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- statement +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/b6824e92f90a.md b/projects/poincare-conjecture/.astrolabe/edges/b6824e92f90a.md new file mode 100644 index 00000000..091a239b --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/b6824e92f90a.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: blowupposcurv + proofSpan: + bodyEnd: 14787 + bodyStart: 6954 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 14798 + file: singlimit2.tex + ownerLabel: smlmtflow + sourceTransform: tex2mdx.strip_comments + start: 6941 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 9494 + file: singlimit2.tex + sourceTransform: tex2mdx.strip_comments + start: 9475 + via: proof + sourceStatement: + chapter: 11 + file: singlimit2.tex + label: smlmtflow + mtref: '11.1' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 3459 + sourceTransform: tex2mdx.strip_comments + start: 1498 + targetStatement: + chapter: 5 + file: converge2.tex + label: blowupposcurv + mtref: '5.33' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 51450 + sourceTransform: tex2mdx.strip_comments + start: 50791 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- d51aab8a8065 +- 2ddfbc74b081 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/b68519a4a8ad.md b/projects/poincare-conjecture/.astrolabe/edges/b68519a4a8ad.md new file mode 100644 index 00000000..307b0b59 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/b68519a4a8ad.md @@ -0,0 +1,55 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: statement + label: wdiffeqn + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 138218 + file: energy1.tex + sourceTransform: tex2mdx.strip_comments + start: 138204 + via: statement-aux-label-proxy + sourceStatement: + chapter: 18 + file: energy1.tex + label: muarea + mtref: '18.58' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 138479 + sourceTransform: tex2mdx.strip_comments + start: 137860 + targetStatement: + chapter: 18 + file: energy1.tex + label: '' + mtref: '18.23' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 58028 + sourceTransform: tex2mdx.strip_comments + start: 57637 + type: tex-reference + via: + - statement-aux-label-proxy +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 10dace7fb165 +- 19d01820dad0 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- statement-aux-label-proxy +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/b68b72cba86d.md b/projects/poincare-conjecture/.astrolabe/edges/b68b72cba86d.md new file mode 100644 index 00000000..addad02b --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/b68b72cba86d.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: wdiffeqn + proofSpan: + bodyEnd: 145345 + bodyStart: 144076 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 145356 + file: energy1.tex + ownerLabel: XI + sourceTransform: tex2mdx.strip_comments + start: 144063 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 144701 + file: energy1.tex + sourceTransform: tex2mdx.strip_comments + start: 144687 + via: proof-aux-label-proxy + sourceStatement: + chapter: 18 + file: energy1.tex + label: XI + mtref: '18.24' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 58810 + sourceTransform: tex2mdx.strip_comments + start: 58030 + targetStatement: + chapter: 18 + file: energy1.tex + label: '' + mtref: '18.23' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 58028 + sourceTransform: tex2mdx.strip_comments + start: 57637 + type: tex-reference + via: + - proof-aux-label-proxy +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- fd85732525c1 +- 19d01820dad0 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof-aux-label-proxy +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/b6c5be519eee.md b/projects/poincare-conjecture/.astrolabe/edges/b6c5be519eee.md new file mode 100644 index 00000000..8f76f33e --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/b6c5be519eee.md @@ -0,0 +1,65 @@ +--- +confidence: 0.45 +edgeClass: semantic +evidence: + endpointSemanticDegreeBeforeTermBackfill: + source: 0 + target: 7 + selectionReason: semantic-connectivity-backfill + sourceMatchSpan: + coordinateSpace: comment-stripped-tex + end: 113065 + file: newcompar.tex + sourceTransform: tex2mdx.strip_comments + start: 113057 + sourceStatement: + chapter: 6 + file: newcompar.tex + label: '' + mtref: '6.76' + sort: example + span: + coordinateSpace: comment-stripped-tex + end: 113156 + sourceTransform: tex2mdx.strip_comments + start: 112685 + targetStatement: + chapter: 6 + file: newcompar.tex + label: '' + mtref: '6.7' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 10481 + sourceTransform: tex2mdx.strip_comments + start: 10161 + targetTermSpan: + coordinateSpace: comment-stripped-tex + end: 10318 + file: newcompar.tex + sourceTransform: tex2mdx.strip_comments + start: 10310 + term: geodesic + termAmbiguity: unique-in-chapter + termSource: italic + termSpecificity: not-covered-by-longer-defined-term + type: same-chapter-definition-term-match + via: + - definition-term:geodesic +generator: tools/poincare_tex_extract.py +inference: weak +kind: definition-use +ref: +- a6ea3cf8fb92 +- d23dc29f2aae +rel: uses +reviewStatus: candidate +scope: same-chapter-term-match +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- definition-term:geodesic +--- +Morgan--Tian same-chapter definition term match. This is a low-confidence candidate edge for review. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/b715fa1f46d4.md b/projects/poincare-conjecture/.astrolabe/edges/b715fa1f46d4.md new file mode 100644 index 00000000..5da46204 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/b715fa1f46d4.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 11 + file: singlimit2.tex + label: '' + mtref: '11.15' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 26714 + sourceTransform: tex2mdx.strip_comments + start: 26365 + targetStatement: + chapter: 11 + file: singlimit2.tex + label: ancientlimit + mtref: '11.16' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 29827 + sourceTransform: tex2mdx.strip_comments + start: 29411 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- b3fae89d3264 +- a66fa0ea8720 +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/b736674677f0.md b/projects/poincare-conjecture/.astrolabe/edges/b736674677f0.md new file mode 100644 index 00000000..ff4eb6a3 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/b736674677f0.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: Ltaueqn + proofSpan: + bodyEnd: 79076 + bodyStart: 78416 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 79087 + file: newcompar.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 78403 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 78469 + file: newcompar.tex + sourceTransform: tex2mdx.strip_comments + start: 78456 + via: proof-aux-label-proxy + sourceStatement: + chapter: 6 + file: newcompar.tex + label: linequal + mtref: '6.49' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 78401 + sourceTransform: tex2mdx.strip_comments + start: 77391 + targetStatement: + chapter: 6 + file: newcompar.tex + label: Ltau + mtref: '6.32' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 45443 + sourceTransform: tex2mdx.strip_comments + start: 45026 + type: tex-reference + via: + - proof-aux-label-proxy +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 2b8ab39364e7 +- b3bf2f526cfc +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof-aux-label-proxy +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/b74af2eafcef.md b/projects/poincare-conjecture/.astrolabe/edges/b74af2eafcef.md new file mode 100644 index 00000000..51377c2d --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/b74af2eafcef.md @@ -0,0 +1,62 @@ +--- +confidence: 0.9 +edgeClass: structural +evidence: + containmentDepth: 1 + proofSpan: + bodyEnd: 14787 + bodyStart: 6954 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 14798 + file: singlimit2.tex + ownerLabel: smlmtflow + sourceTransform: tex2mdx.strip_comments + start: 6941 + sourceStatement: + chapter: 11 + file: singlimit2.tex + label: smlmtflow + mtref: '11.1' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 3459 + sourceTransform: tex2mdx.strip_comments + start: 1498 + targetContainedSpan: + coordinateSpace: comment-stripped-tex + end: 10094 + file: singlimit2.tex + sourceTransform: tex2mdx.strip_comments + start: 9927 + targetStatement: + chapter: 11 + file: singlimit2.tex + label: '' + mtref: '11.6' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 10094 + sourceTransform: tex2mdx.strip_comments + start: 9927 + type: proof-containment + via: + - proof-contains +generator: tools/poincare_tex_extract.py +inference: structural +kind: containment +ref: +- d51aab8a8065 +- 9f8eade55fcd +rel: contains +reviewStatus: accepted +scope: proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof-contains +--- +Morgan--Tian proof containment. The source is the statement whose proof contains the target statement. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/b766bfab88b5.md b/projects/poincare-conjecture/.astrolabe/edges/b766bfab88b5.md new file mode 100644 index 00000000..bf504e05 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/b766bfab88b5.md @@ -0,0 +1,84 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: altern + proofSpan: + bodyEnd: 200569 + bodyStart: 199052 + coordinateSpace: comment-stripped-tex + depth: 3 + end: 200580 + file: surgery.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 199039 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 199400 + file: surgery.tex + sourceTransform: tex2mdx.strip_comments + start: 199388 + via: proof + - context: proof + label: altern + proofSpan: + bodyEnd: 200569 + bodyStart: 199052 + coordinateSpace: comment-stripped-tex + depth: 3 + end: 200580 + file: surgery.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 199039 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 200111 + file: surgery.tex + sourceTransform: tex2mdx.strip_comments + start: 200099 + via: proof + sourceStatement: + chapter: 17 + file: surgery.tex + label: '' + mtref: '17.3' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 199036 + sourceTransform: tex2mdx.strip_comments + start: 198777 + targetStatement: + chapter: 16 + file: surgery.tex + label: altern + mtref: '16.5' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 112999 + sourceTransform: tex2mdx.strip_comments + start: 110961 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- d130b5adaed6 +- ad4f67ea75bc +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/b769217da3ad.md b/projects/poincare-conjecture/.astrolabe/edges/b769217da3ad.md new file mode 100644 index 00000000..203c1365 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/b769217da3ad.md @@ -0,0 +1,62 @@ +--- +confidence: 0.9 +edgeClass: structural +evidence: + containmentDepth: 2 + proofSpan: + bodyEnd: 62795 + bodyStart: 60476 + coordinateSpace: comment-stripped-tex + depth: 2 + end: 62806 + file: singlimit2.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 60463 + sourceStatement: + chapter: 11 + file: singlimit2.tex + label: hyp + mtref: '11.32' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 60461 + sourceTransform: tex2mdx.strip_comments + start: 60310 + targetContainedSpan: + coordinateSpace: comment-stripped-tex + end: 61224 + file: singlimit2.tex + sourceTransform: tex2mdx.strip_comments + start: 61026 + targetStatement: + chapter: 11 + file: singlimit2.tex + label: '' + mtref: '11.33' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 61224 + sourceTransform: tex2mdx.strip_comments + start: 61026 + type: proof-containment + via: + - proof-contains +generator: tools/poincare_tex_extract.py +inference: structural +kind: containment +ref: +- dff170186455 +- ce620437e5d6 +rel: contains +reviewStatus: accepted +scope: proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof-contains +--- +Morgan--Tian proof containment. The source is the statement whose proof contains the target statement. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/b76c725ad487.md b/projects/poincare-conjecture/.astrolabe/edges/b76c725ad487.md new file mode 100644 index 00000000..13e53edc --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/b76c725ad487.md @@ -0,0 +1,45 @@ +--- +confidence: 0.2 +edgeClass: structural +evidence: + sourceStatement: + chapter: 12 + file: stdsoln.tex + label: '' + mtref: '12.24' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 62221 + sourceTransform: tex2mdx.strip_comments + start: 61942 + targetStatement: + chapter: 12 + file: stdsoln.tex + label: harmapflow + mtref: '12.25' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 74100 + sourceTransform: tex2mdx.strip_comments + start: 73601 + type: chapter-reading-order-adjacency + via: + - chapter-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 43ea5e1d04cf +- 5fbeadbba61c +rel: in-chapter +reviewStatus: accepted +scope: chapter +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- chapter-sequence +--- +Morgan--Tian reading-order adjacency across sections in a chapter. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/b7b658794f26.md b/projects/poincare-conjecture/.astrolabe/edges/b7b658794f26.md new file mode 100644 index 00000000..55125e7a --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/b7b658794f26.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 4 + file: maxprin.tex + label: '' + mtref: '4.28' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 41209 + sourceTransform: tex2mdx.strip_comments + start: 41046 + targetStatement: + chapter: 4 + file: maxprin.tex + label: '' + mtref: '4.29' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 42803 + sourceTransform: tex2mdx.strip_comments + start: 42730 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 0ac34e2feca6 +- cfd7d6954385 +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/b7bd3aba8451.md b/projects/poincare-conjecture/.astrolabe/edges/b7bd3aba8451.md new file mode 100644 index 00000000..dedaf1f9 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/b7bd3aba8451.md @@ -0,0 +1,56 @@ +--- +confidence: 0.85 +edgeClass: semantic +evidence: + containmentReason: target theorem-like statement occurs inside source proof + proofSpan: + bodyEnd: 54967 + bodyStart: 37263 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 54978 + file: energy1.tex + ownerLabel: W_2 + sourceTransform: tex2mdx.strip_comments + start: 37250 + sourceStatement: + chapter: 18 + file: energy1.tex + label: W_2 + mtref: '18.11' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 36484 + sourceTransform: tex2mdx.strip_comments + start: 36186 + targetStatement: + chapter: 18 + file: energy1.tex + label: spherethm + mtref: '18.14' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 43696 + sourceTransform: tex2mdx.strip_comments + start: 43231 + type: proof-contained-theorem-like-statement + via: + - proof-contained-statement +generator: tools/poincare_tex_extract.py +inference: inferred +kind: dependency +ref: +- 7bb0fa0c9412 +- a679e0086677 +rel: depends +reviewStatus: accepted +scope: proof-containment +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof-contained-statement +--- +Morgan--Tian proof-contained statement dependency. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/b85d5588ec07.md b/projects/poincare-conjecture/.astrolabe/edges/b85d5588ec07.md new file mode 100644 index 00000000..e01a26a7 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/b85d5588ec07.md @@ -0,0 +1,45 @@ +--- +confidence: 0.2 +edgeClass: structural +evidence: + sourceStatement: + chapter: 12 + file: stdsoln.tex + label: asympt + mtref: '12.23' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 61284 + sourceTransform: tex2mdx.strip_comments + start: 60562 + targetStatement: + chapter: 12 + file: stdsoln.tex + label: '' + mtref: '12.24' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 62221 + sourceTransform: tex2mdx.strip_comments + start: 61942 + type: chapter-reading-order-adjacency + via: + - chapter-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- ac4ac4fb06f8 +- 43ea5e1d04cf +rel: in-chapter +reviewStatus: accepted +scope: chapter +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- chapter-sequence +--- +Morgan--Tian reading-order adjacency across sections in a chapter. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/b871e8cc0eab.md b/projects/poincare-conjecture/.astrolabe/edges/b871e8cc0eab.md new file mode 100644 index 00000000..df3c2672 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/b871e8cc0eab.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: proposition + proofSpan: + bodyEnd: 168008 + bodyStart: 167355 + coordinateSpace: comment-stripped-tex + depth: 2 + end: 168019 + file: temp2kappa.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 167342 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 167974 + file: temp2kappa.tex + sourceTransform: tex2mdx.strip_comments + start: 167957 + via: proof + sourceStatement: + chapter: 9 + file: temp2kappa.tex + label: '' + mtref: '9.87' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 167340 + sourceTransform: tex2mdx.strip_comments + start: 167251 + targetStatement: + chapter: 9 + file: temp2kappa.tex + label: proposition + mtref: '9.65' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 127812 + sourceTransform: tex2mdx.strip_comments + start: 127549 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- b9f671d590f3 +- ee314aacd207 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/b874726cfeac.md b/projects/poincare-conjecture/.astrolabe/edges/b874726cfeac.md new file mode 100644 index 00000000..f08b38d9 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/b874726cfeac.md @@ -0,0 +1,98 @@ +--- +confidence: 0.35 +edgeClass: prose +evidence: + mentionTriggers: + - label: surgery + refSpan: + coordinateSpace: comment-stripped-tex + end: 20610 + file: intro.tex + sourceTransform: tex2mdx.strip_comments + start: 20597 + sentence: Perelman's proposed proof of Thurston's Geometrization Conjecture relies + in an essential way on Theorem~\ref{surgery}, namely the existence of Ricci + flow with surgery for all positive time + sentenceSpan: + coordinateSpace: comment-stripped-tex + end: 20681 + file: intro.tex + sourceTransform: tex2mdx.strip_comments + start: 20492 + sourceStrategy: nearest-previous-statement + via: prose + - label: surgery + refSpan: + coordinateSpace: comment-stripped-tex + end: 22068 + file: intro.tex + sourceTransform: tex2mdx.strip_comments + start: 22055 + sentence: In this book we do not attempt to explicate any of the results beyond + Theorem~\ref{surgery} described in the previous paragraph that are needed for + the Geometrization Conjecture + sentenceSpan: + coordinateSpace: comment-stripped-tex + end: 22154 + file: intro.tex + sourceTransform: tex2mdx.strip_comments + start: 21976 + sourceStrategy: nearest-previous-statement + via: prose + - label: surgery + refSpan: + coordinateSpace: comment-stripped-tex + end: 24022 + file: intro.tex + sourceTransform: tex2mdx.strip_comments + start: 24009 + sentence: At the end of Chapter 17 we have established Theorem~\ref{surgery} + sentenceSpan: + coordinateSpace: comment-stripped-tex + end: 24022 + file: intro.tex + sourceTransform: tex2mdx.strip_comments + start: 23955 + sourceStrategy: nearest-previous-statement + via: prose + sourceStatement: + chapter: 0 + file: intro.tex + label: '' + mtref: '0.6' + sort: remark + span: + coordinateSpace: comment-stripped-tex + end: 19243 + sourceTransform: tex2mdx.strip_comments + start: 16436 + targetStatement: + chapter: 0 + file: intro.tex + label: surgery + mtref: '0.3' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 11419 + sourceTransform: tex2mdx.strip_comments + start: 10567 + type: prose-mention-nearest-previous-statement + via: + - prose +generator: tools/poincare_tex_extract.py +inference: weak +kind: mention +ref: +- 48f3d2b5dea3 +- 847af3ecd4a8 +rel: mentions +reviewStatus: candidate +scope: prose +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- prose +--- +Morgan--Tian prose mention. The source is the nearest preceding statement in the same chapter segment. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/b881fb209824.md b/projects/poincare-conjecture/.astrolabe/edges/b881fb209824.md new file mode 100644 index 00000000..3e9e6f0d --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/b881fb209824.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 15 + file: surgery.tex + label: MAIN + mtref: '15.9' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 91186 + sourceTransform: tex2mdx.strip_comments + start: 90004 + targetStatement: + chapter: 15 + file: surgery.tex + label: RFSexists + mtref: '15.10' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 92173 + sourceTransform: tex2mdx.strip_comments + start: 91337 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 703e4cfeece7 +- c86aa999ce8a +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/b89002fe477f.md b/projects/poincare-conjecture/.astrolabe/edges/b89002fe477f.md new file mode 100644 index 00000000..578e2b1a --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/b89002fe477f.md @@ -0,0 +1,56 @@ +--- +confidence: 0.85 +edgeClass: semantic +evidence: + containmentReason: target theorem-like statement occurs inside source proof + proofSpan: + bodyEnd: 77920 + bodyStart: 74111 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 77931 + file: energy1.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 74098 + sourceStatement: + chapter: 18 + file: energy1.tex + label: nonembeasycase + mtref: '18.32' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 74095 + sourceTransform: tex2mdx.strip_comments + start: 73662 + targetStatement: + chapter: 18 + file: energy1.tex + label: '' + mtref: '18.33' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 74769 + sourceTransform: tex2mdx.strip_comments + start: 74223 + type: proof-contained-theorem-like-statement + via: + - proof-contained-statement +generator: tools/poincare_tex_extract.py +inference: inferred +kind: dependency +ref: +- 27b452d6cb48 +- f85de11f39d7 +rel: depends +reviewStatus: accepted +scope: proof-containment +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof-contained-statement +--- +Morgan--Tian proof-contained statement dependency. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/b8a4d8ca93ff.md b/projects/poincare-conjecture/.astrolabe/edges/b8a4d8ca93ff.md new file mode 100644 index 00000000..b93f154d --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/b8a4d8ca93ff.md @@ -0,0 +1,65 @@ +--- +confidence: 0.45 +edgeClass: semantic +evidence: + endpointSemanticDegreeBeforeTermBackfill: + source: 0 + target: 1 + selectionReason: semantic-connectivity-backfill + sourceMatchSpan: + coordinateSpace: comment-stripped-tex + end: 12300 + file: newcomp2.tex + sourceTransform: tex2mdx.strip_comments + start: 12270 + sourceStatement: + chapter: 7 + file: newcomp2.tex + label: weaksense + mtref: '7.8' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 12878 + sourceTransform: tex2mdx.strip_comments + start: 12168 + targetStatement: + chapter: 7 + file: newcomp2.tex + label: '' + mtref: '7.1' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 1366 + sourceTransform: tex2mdx.strip_comments + start: 781 + targetTermSpan: + coordinateSpace: comment-stripped-tex + end: 900 + file: newcomp2.tex + sourceTransform: tex2mdx.strip_comments + start: 871 + term: complete of bounded curvature + termAmbiguity: unique-in-chapter + termSource: italic + termSpecificity: no-longer-defined-term + type: same-chapter-definition-term-match + via: + - definition-term:complete of bounded curvature +generator: tools/poincare_tex_extract.py +inference: weak +kind: definition-use +ref: +- 1ef3a78b1911 +- 2336b554a054 +rel: uses +reviewStatus: candidate +scope: same-chapter-term-match +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- definition-term:complete of bounded curvature +--- +Morgan--Tian same-chapter definition term match. This is a low-confidence candidate edge for review. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/b8e73bf07126.md b/projects/poincare-conjecture/.astrolabe/edges/b8e73bf07126.md new file mode 100644 index 00000000..e0a721b4 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/b8e73bf07126.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 9 + file: temp2kappa.tex + label: noncompsol + mtref: '9.46' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 82429 + sourceTransform: tex2mdx.strip_comments + start: 82233 + targetStatement: + chapter: 9 + file: temp2kappa.tex + label: '' + mtref: '9.47' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 83801 + sourceTransform: tex2mdx.strip_comments + start: 83646 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 03957890f08f +- e3b91b4c2028 +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/b8f973487a8c.md b/projects/poincare-conjecture/.astrolabe/edges/b8f973487a8c.md new file mode 100644 index 00000000..46b0bc26 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/b8f973487a8c.md @@ -0,0 +1,45 @@ +--- +confidence: 0.2 +edgeClass: structural +evidence: + sourceStatement: + chapter: 10 + file: bddcurvbdddist.tex + label: '' + mtref: '10.6' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 21531 + sourceTransform: tex2mdx.strip_comments + start: 21390 + targetStatement: + chapter: 10 + file: bddcurvbdddist.tex + label: Uinfty + mtref: '10.7' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 26826 + sourceTransform: tex2mdx.strip_comments + start: 25406 + type: chapter-reading-order-adjacency + via: + - chapter-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 6a13c64a90e4 +- e2e64cdb0b75 +rel: in-chapter +reviewStatus: accepted +scope: chapter +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- chapter-sequence +--- +Morgan--Tian reading-order adjacency across sections in a chapter. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/b91b6012d5cb.md b/projects/poincare-conjecture/.astrolabe/edges/b91b6012d5cb.md new file mode 100644 index 00000000..cd922d1a --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/b91b6012d5cb.md @@ -0,0 +1,45 @@ +--- +confidence: 0.2 +edgeClass: structural +evidence: + sourceStatement: + chapter: 3 + file: flowbasics.tex + label: '' + mtref: '3.10' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 10220 + sourceTransform: tex2mdx.strip_comments + start: 10093 + targetStatement: + chapter: 3 + file: flowbasics.tex + label: compuniq + mtref: '3.11' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 10873 + sourceTransform: tex2mdx.strip_comments + start: 10413 + type: chapter-reading-order-adjacency + via: + - chapter-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- a560479acf35 +- e6b1bd9fa0d7 +rel: in-chapter +reviewStatus: accepted +scope: chapter +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- chapter-sequence +--- +Morgan--Tian reading-order adjacency across sections in a chapter. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/b922c0bff36c.md b/projects/poincare-conjecture/.astrolabe/edges/b922c0bff36c.md new file mode 100644 index 00000000..a3f2efc7 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/b922c0bff36c.md @@ -0,0 +1,56 @@ +--- +confidence: 0.85 +edgeClass: semantic +evidence: + containmentReason: target theorem-like statement occurs inside source proof + proofSpan: + bodyEnd: 34517 + bodyStart: 26920 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 34528 + file: bddcurvbdddist.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 26907 + sourceStatement: + chapter: 10 + file: bddcurvbdddist.tex + label: Uinfty + mtref: '10.7' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 26826 + sourceTransform: tex2mdx.strip_comments + start: 25406 + targetStatement: + chapter: 10 + file: bddcurvbdddist.tex + label: '' + mtref: '10.10' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 32796 + sourceTransform: tex2mdx.strip_comments + start: 32551 + type: proof-contained-theorem-like-statement + via: + - proof-contained-statement +generator: tools/poincare_tex_extract.py +inference: inferred +kind: dependency +ref: +- e2e64cdb0b75 +- 1418d3c286c7 +rel: depends +reviewStatus: accepted +scope: proof-containment +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof-contained-statement +--- +Morgan--Tian proof-contained statement dependency. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/b95adcb07c1a.md b/projects/poincare-conjecture/.astrolabe/edges/b95adcb07c1a.md new file mode 100644 index 00000000..81f729c1 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/b95adcb07c1a.md @@ -0,0 +1,65 @@ +--- +confidence: 0.45 +edgeClass: semantic +evidence: + endpointSemanticDegreeBeforeTermBackfill: + source: 1 + target: 0 + selectionReason: semantic-connectivity-backfill + sourceMatchSpan: + coordinateSpace: comment-stripped-tex + end: 125504 + file: newcompar.tex + sourceTransform: tex2mdx.strip_comments + start: 125490 + sourceStatement: + chapter: 6 + file: newcompar.tex + label: finiteredvol + mtref: '6.82' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 125562 + sourceTransform: tex2mdx.strip_comments + start: 125396 + targetStatement: + chapter: 6 + file: newcompar.tex + label: redvol + mtref: '6.70' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 107162 + sourceTransform: tex2mdx.strip_comments + start: 106757 + targetTermSpan: + coordinateSpace: comment-stripped-tex + end: 106898 + file: newcompar.tex + sourceTransform: tex2mdx.strip_comments + start: 106884 + term: reduced volume + termAmbiguity: unique-in-chapter + termSource: italic + termSpecificity: no-longer-defined-term + type: same-chapter-definition-term-match + via: + - definition-term:reduced volume +generator: tools/poincare_tex_extract.py +inference: weak +kind: definition-use +ref: +- e49b6b45c4d0 +- 6f08d8fab637 +rel: uses +reviewStatus: candidate +scope: same-chapter-term-match +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- definition-term:reduced volume +--- +Morgan--Tian same-chapter definition term match. This is a low-confidence candidate edge for review. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/b96834aa9ded.md b/projects/poincare-conjecture/.astrolabe/edges/b96834aa9ded.md new file mode 100644 index 00000000..1e7f485d --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/b96834aa9ded.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 18 + file: energy1.tex + label: homess + mtref: '18.7' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 22871 + sourceTransform: tex2mdx.strip_comments + start: 22605 + targetStatement: + chapter: 18 + file: energy1.tex + label: '' + mtref: '18.8' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 23392 + sourceTransform: tex2mdx.strip_comments + start: 23128 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 46f9d27654be +- 37efbeaa9eec +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/b9732622b09b.md b/projects/poincare-conjecture/.astrolabe/edges/b9732622b09b.md new file mode 100644 index 00000000..a82a2263 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/b9732622b09b.md @@ -0,0 +1,62 @@ +--- +confidence: 0.9 +edgeClass: structural +evidence: + containmentDepth: 3 + proofSpan: + bodyEnd: 14349 + bodyStart: 5983 + coordinateSpace: comment-stripped-tex + depth: 3 + end: 14360 + file: noncoll.tex + ownerLabel: small + sourceTransform: tex2mdx.strip_comments + start: 5970 + sourceStatement: + chapter: 8 + file: noncoll.tex + label: small + mtref: '8.3' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 5471 + sourceTransform: tex2mdx.strip_comments + start: 5161 + targetContainedSpan: + coordinateSpace: comment-stripped-tex + end: 6558 + file: noncoll.tex + sourceTransform: tex2mdx.strip_comments + start: 6089 + targetStatement: + chapter: 8 + file: noncoll.tex + label: nabR + mtref: '8.5' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 6558 + sourceTransform: tex2mdx.strip_comments + start: 6089 + type: proof-containment + via: + - proof-contains +generator: tools/poincare_tex_extract.py +inference: structural +kind: containment +ref: +- 26e1a6292fef +- bf8f356cfd7a +rel: contains +reviewStatus: accepted +scope: proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof-contains +--- +Morgan--Tian proof containment. The source is the statement whose proof contains the target statement. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/b9a1cb0ef591.md b/projects/poincare-conjecture/.astrolabe/edges/b9a1cb0ef591.md new file mode 100644 index 00000000..3fd09786 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/b9a1cb0ef591.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: theta/Kbd + proofSpan: + bodyEnd: 171471 + bodyStart: 161354 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 171482 + file: energy1.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 161341 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 171175 + file: energy1.tex + sourceTransform: tex2mdx.strip_comments + start: 171160 + via: proof + sourceStatement: + chapter: 18 + file: energy1.tex + label: laest + mtref: '18.66' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 161338 + sourceTransform: tex2mdx.strip_comments + start: 161080 + targetStatement: + chapter: 18 + file: energy1.tex + label: theta/Kbd + mtref: '18.67' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 162289 + sourceTransform: tex2mdx.strip_comments + start: 162047 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 45c783d384c1 +- 512c74b9ad1f +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/b9b3544e1952.md b/projects/poincare-conjecture/.astrolabe/edges/b9b3544e1952.md new file mode 100644 index 00000000..55a69e94 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/b9b3544e1952.md @@ -0,0 +1,56 @@ +--- +confidence: 0.85 +edgeClass: semantic +evidence: + containmentReason: target theorem-like statement occurs inside source proof + proofSpan: + bodyEnd: 53552 + bodyStart: 35551 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 53563 + file: canonnbhd.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 35538 + sourceStatement: + chapter: 19 + file: canonnbhd.tex + label: Xcontainedin + mtref: '19.21' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 35375 + sourceTransform: tex2mdx.strip_comments + start: 34343 + targetStatement: + chapter: 19 + file: canonnbhd.tex + label: cccomponent + mtref: '19.23' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 44166 + sourceTransform: tex2mdx.strip_comments + start: 43962 + type: proof-contained-theorem-like-statement + via: + - proof-contained-statement +generator: tools/poincare_tex_extract.py +inference: inferred +kind: dependency +ref: +- 0e9a1a167caa +- 327067f8535b +rel: depends +reviewStatus: accepted +scope: proof-containment +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof-contained-statement +--- +Morgan--Tian proof-contained statement dependency. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/b9c453aa3a68.md b/projects/poincare-conjecture/.astrolabe/edges/b9c453aa3a68.md new file mode 100644 index 00000000..93289918 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/b9c453aa3a68.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 16 + file: surgery.tex + label: altern + mtref: '16.5' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 112999 + sourceTransform: tex2mdx.strip_comments + start: 110961 + targetStatement: + chapter: 16 + file: surgery.tex + label: '' + mtref: '16.6' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 114511 + sourceTransform: tex2mdx.strip_comments + start: 113887 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- ad4f67ea75bc +- ef94451b5785 +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/b9f9f3959e8b.md b/projects/poincare-conjecture/.astrolabe/edges/b9f9f3959e8b.md new file mode 100644 index 00000000..65304538 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/b9f9f3959e8b.md @@ -0,0 +1,88 @@ +--- +confidence: 0.72 +edgeClass: semantic +evidence: + anaphoraTriggers: + - context: proof + direction: previous + phrase: previous proposition + proofSpan: + bodyEnd: 76413 + bodyStart: 74060 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 76424 + file: surgery.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 74047 + sort: proposition + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 74433 + file: surgery.tex + sourceTransform: tex2mdx.strip_comments + start: 74413 + via: local-anaphora-proof + - context: proof + direction: previous + phrase: previous proposition + proofSpan: + bodyEnd: 76413 + bodyStart: 74060 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 76424 + file: surgery.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 74047 + sort: proposition + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 76163 + file: surgery.tex + sourceTransform: tex2mdx.strip_comments + start: 76143 + via: local-anaphora-proof + sourceStatement: + chapter: 15 + file: surgery.tex + label: Tgood0good + mtref: '15.4' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 74045 + sourceTransform: tex2mdx.strip_comments + start: 73002 + targetStatement: + chapter: 15 + file: surgery.tex + label: surgerytoptype + mtref: '15.3' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 69964 + sourceTransform: tex2mdx.strip_comments + start: 69392 + type: local-anaphora-reference + via: + - local-anaphora-proof +generator: tools/poincare_tex_extract.py +inference: inferred +kind: dependency +ref: +- 7822d06083e0 +- 7946423e82a4 +rel: depends +reviewStatus: accepted +scope: local-context +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- local-anaphora-proof +--- +Morgan--Tian local anaphora dependency. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/ba2a9e166407.md b/projects/poincare-conjecture/.astrolabe/edges/ba2a9e166407.md new file mode 100644 index 00000000..1796c97e --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/ba2a9e166407.md @@ -0,0 +1,65 @@ +--- +confidence: 0.45 +edgeClass: semantic +evidence: + endpointSemanticDegreeBeforeTermBackfill: + source: 0 + target: 0 + selectionReason: semantic-connectivity-backfill + sourceMatchSpan: + coordinateSpace: comment-stripped-tex + end: 4024 + file: newcompar.tex + sourceTransform: tex2mdx.strip_comments + start: 3994 + sourceStatement: + chapter: 6 + file: newcompar.tex + label: Llength + mtref: '6.2' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 4712 + sourceTransform: tex2mdx.strip_comments + start: 3854 + targetStatement: + chapter: 6 + file: newcompar.tex + label: '' + mtref: '6.1' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 3679 + sourceTransform: tex2mdx.strip_comments + start: 3370 + targetTermSpan: + coordinateSpace: comment-stripped-tex + end: 3552 + file: newcompar.tex + sourceTransform: tex2mdx.strip_comments + start: 3522 + term: parameterized by backward time + termAmbiguity: unique-in-chapter + termSource: italic + termSpecificity: no-longer-defined-term + type: same-chapter-definition-term-match + via: + - definition-term:parameterized by backward time +generator: tools/poincare_tex_extract.py +inference: weak +kind: definition-use +ref: +- 09af70d51aac +- 6a961a764f44 +rel: uses +reviewStatus: candidate +scope: same-chapter-term-match +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- definition-term:parameterized by backward time +--- +Morgan--Tian same-chapter definition term match. This is a low-confidence candidate edge for review. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/ba34891faca7.md b/projects/poincare-conjecture/.astrolabe/edges/ba34891faca7.md new file mode 100644 index 00000000..a3850a70 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/ba34891faca7.md @@ -0,0 +1,55 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: statement + label: 2ndvari + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 18814 + file: newcompar.tex + sourceTransform: tex2mdx.strip_comments + start: 18801 + via: statement + sourceStatement: + chapter: 6 + file: newcompar.tex + label: variremark + mtref: '6.15' + sort: remark + span: + coordinateSpace: comment-stripped-tex + end: 19144 + sourceTransform: tex2mdx.strip_comments + start: 18610 + targetStatement: + chapter: 6 + file: newcompar.tex + label: 2ndvari + mtref: '6.14' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 17350 + sourceTransform: tex2mdx.strip_comments + start: 16270 + type: tex-reference + via: + - statement +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 2fef81872a01 +- f48558c60ca2 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- statement +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/ba418751c9b6.md b/projects/poincare-conjecture/.astrolabe/edges/ba418751c9b6.md new file mode 100644 index 00000000..16d99a4b --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/ba418751c9b6.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: neckgeo + proofSpan: + bodyEnd: 49745 + bodyStart: 44181 + coordinateSpace: comment-stripped-tex + depth: 2 + end: 49756 + file: canonnbhd.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 44168 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 48168 + file: canonnbhd.tex + sourceTransform: tex2mdx.strip_comments + start: 48155 + via: proof + sourceStatement: + chapter: 19 + file: canonnbhd.tex + label: cccomponent + mtref: '19.23' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 44166 + sourceTransform: tex2mdx.strip_comments + start: 43962 + targetStatement: + chapter: 19 + file: canonnbhd.tex + label: neckgeo + mtref: '19.5' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 8075 + sourceTransform: tex2mdx.strip_comments + start: 7413 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 327067f8535b +- b74cd360f72f +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/ba4d2de22e43.md b/projects/poincare-conjecture/.astrolabe/edges/ba4d2de22e43.md new file mode 100644 index 00000000..c2218f0e --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/ba4d2de22e43.md @@ -0,0 +1,62 @@ +--- +confidence: 0.9 +edgeClass: structural +evidence: + containmentDepth: 1 + proofSpan: + bodyEnd: 173338 + bodyStart: 165469 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 173349 + file: surgery.tex + ownerLabel: delta0ri+1 + sourceTransform: tex2mdx.strip_comments + start: 165456 + sourceStatement: + chapter: 16 + file: surgery.tex + label: delta0ri+1 + mtref: '16.4' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 109162 + sourceTransform: tex2mdx.strip_comments + start: 107682 + targetContainedSpan: + coordinateSpace: comment-stripped-tex + end: 171497 + file: surgery.tex + sourceTransform: tex2mdx.strip_comments + start: 171318 + targetStatement: + chapter: 16 + file: surgery.tex + label: lvalue + mtref: '16.26' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 171497 + sourceTransform: tex2mdx.strip_comments + start: 171318 + type: proof-containment + via: + - proof-contains +generator: tools/poincare_tex_extract.py +inference: structural +kind: containment +ref: +- 89a60eb69c3f +- 8cae73a6a02c +rel: contains +reviewStatus: accepted +scope: proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof-contains +--- +Morgan--Tian proof containment. The source is the statement whose proof contains the target statement. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/ba6653ba3204.md b/projects/poincare-conjecture/.astrolabe/edges/ba6653ba3204.md new file mode 100644 index 00000000..33d3ff6b --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/ba6653ba3204.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: lipcomplete + proofSpan: + bodyEnd: 43050 + bodyStart: 39413 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 43061 + file: newcomp2.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 39400 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 42035 + file: newcomp2.tex + sourceTransform: tex2mdx.strip_comments + start: 42018 + via: proof + sourceStatement: + chapter: 7 + file: newcomp2.tex + label: ltauineq + mtref: '7.24' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 38959 + sourceTransform: tex2mdx.strip_comments + start: 38702 + targetStatement: + chapter: 7 + file: newcomp2.tex + label: lipcomplete + mtref: '7.5' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 9437 + sourceTransform: tex2mdx.strip_comments + start: 8692 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 490290f4c322 +- 16fbe263a7c8 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/ba781f22fb98.md b/projects/poincare-conjecture/.astrolabe/edges/ba781f22fb98.md new file mode 100644 index 00000000..dc0b3f3d --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/ba781f22fb98.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 3 + file: flowbasics.tex + label: '' + mtref: '3.17' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 21425 + sourceTransform: tex2mdx.strip_comments + start: 21379 + targetStatement: + chapter: 3 + file: flowbasics.tex + label: M + mtref: '3.18' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 22220 + sourceTransform: tex2mdx.strip_comments + start: 21910 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- c67b6547955b +- d0267ffa1d98 +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/ba9ad895c227.md b/projects/poincare-conjecture/.astrolabe/edges/ba9ad895c227.md new file mode 100644 index 00000000..3a17c8e6 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/ba9ad895c227.md @@ -0,0 +1,45 @@ +--- +confidence: 0.2 +edgeClass: structural +evidence: + sourceStatement: + chapter: 16 + file: surgery.tex + label: '' + mtref: '16.2' + sort: remark + span: + coordinateSpace: comment-stripped-tex + end: 104034 + sourceTransform: tex2mdx.strip_comments + start: 103578 + targetStatement: + chapter: 16 + file: surgery.tex + label: '' + mtref: '16.3' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 105047 + sourceTransform: tex2mdx.strip_comments + start: 104658 + type: chapter-reading-order-adjacency + via: + - chapter-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- d93b1988d45d +- 7028ebd40f18 +rel: in-chapter +reviewStatus: accepted +scope: chapter +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- chapter-sequence +--- +Morgan--Tian reading-order adjacency across sections in a chapter. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/baa18afd8488.md b/projects/poincare-conjecture/.astrolabe/edges/baa18afd8488.md new file mode 100644 index 00000000..90ae918e --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/baa18afd8488.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 9 + file: temp2kappa.tex + label: asympGSS + mtref: '9.11' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 13767 + sourceTransform: tex2mdx.strip_comments + start: 12464 + targetStatement: + chapter: 9 + file: temp2kappa.tex + label: '' + mtref: '9.12' + sort: remark + span: + coordinateSpace: comment-stripped-tex + end: 14958 + sourceTransform: tex2mdx.strip_comments + start: 13925 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 671402eb454b +- bb839ebe8376 +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/bac5ba646cd1.md b/projects/poincare-conjecture/.astrolabe/edges/bac5ba646cd1.md new file mode 100644 index 00000000..f235afc8 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/bac5ba646cd1.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: maxT + proofSpan: + bodyEnd: 35374 + bodyStart: 16867 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 35385 + file: singlimit2.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 16854 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 33858 + file: singlimit2.tex + sourceTransform: tex2mdx.strip_comments + start: 33848 + via: proof + sourceStatement: + chapter: 11 + file: singlimit2.tex + label: '' + mtref: '11.9' + sort: remark + span: + coordinateSpace: comment-stripped-tex + end: 16852 + sourceTransform: tex2mdx.strip_comments + start: 16134 + targetStatement: + chapter: 11 + file: singlimit2.tex + label: maxT + mtref: '11.10' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 17643 + sourceTransform: tex2mdx.strip_comments + start: 17183 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 5e06d2593bfa +- d6919616448d +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/bac612381ac8.md b/projects/poincare-conjecture/.astrolabe/edges/bac612381ac8.md new file mode 100644 index 00000000..e1096b5a --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/bac612381ac8.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 18 + file: energy1.tex + label: BARMU + mtref: '18.59' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 140535 + sourceTransform: tex2mdx.strip_comments + start: 139920 + targetStatement: + chapter: 18 + file: energy1.tex + label: mulength + mtref: '18.60' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 141140 + sourceTransform: tex2mdx.strip_comments + start: 140651 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 8e814eab5ddf +- 52f36e456355 +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/bafe4bd4b959.md b/projects/poincare-conjecture/.astrolabe/edges/bafe4bd4b959.md new file mode 100644 index 00000000..d9d9df96 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/bafe4bd4b959.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: deltaprime + proofSpan: + bodyEnd: 136538 + bodyStart: 123471 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 136549 + file: energy1.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 123458 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 135880 + file: energy1.tex + sourceTransform: tex2mdx.strip_comments + start: 135864 + via: proof + sourceStatement: + chapter: 18 + file: energy1.tex + label: xistronger + mtref: '18.53' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 123454 + sourceTransform: tex2mdx.strip_comments + start: 122771 + targetStatement: + chapter: 18 + file: energy1.tex + label: deltaprime + mtref: '18.55' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 128310 + sourceTransform: tex2mdx.strip_comments + start: 127628 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 9df99ae6f4cb +- b87df0b67468 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/bb15e4732f80.md b/projects/poincare-conjecture/.astrolabe/edges/bb15e4732f80.md new file mode 100644 index 00000000..49d0ee2e --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/bb15e4732f80.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: kappa0r0t0 + proofSpan: + bodyEnd: 20911 + bodyStart: 18068 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 20922 + file: noncoll.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 18055 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 19287 + file: noncoll.tex + sourceTransform: tex2mdx.strip_comments + start: 19271 + via: proof + sourceStatement: + chapter: 8 + file: noncoll.tex + label: '' + mtref: '8.10' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 18052 + sourceTransform: tex2mdx.strip_comments + start: 17502 + targetStatement: + chapter: 4 + file: maxprin.tex + label: kappa0r0t0 + mtref: '4.11' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 13944 + sourceTransform: tex2mdx.strip_comments + start: 13445 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 198e7475ad2f +- c1330c5a8be3 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/bb35c7247c27.md b/projects/poincare-conjecture/.astrolabe/edges/bb35c7247c27.md new file mode 100644 index 00000000..01b29866 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/bb35c7247c27.md @@ -0,0 +1,76 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: statement + label: topiso + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 76969 + file: prelim.tex + sourceTransform: tex2mdx.strip_comments + start: 76957 + via: statement + - context: proof + label: topiso + proofSpan: + bodyEnd: 77881 + bodyStart: 77484 + coordinateSpace: comment-stripped-tex + depth: 2 + end: 77892 + file: prelim.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 77471 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 77555 + file: prelim.tex + sourceTransform: tex2mdx.strip_comments + start: 77543 + via: proof + sourceStatement: + chapter: 2 + file: prelim.tex + label: neckseparate + mtref: '2.20' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 77469 + sourceTransform: tex2mdx.strip_comments + start: 76866 + targetStatement: + chapter: 19 + file: canonnbhd.tex + label: topiso + mtref: '19.10' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 15949 + sourceTransform: tex2mdx.strip_comments + start: 15078 + type: tex-reference + via: + - proof + - statement +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 818772530e5a +- 04dd3c3ea1c8 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +- statement +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/bb375d005efb.md b/projects/poincare-conjecture/.astrolabe/edges/bb375d005efb.md new file mode 100644 index 00000000..cd7ae474 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/bb375d005efb.md @@ -0,0 +1,55 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: statement + label: areachange + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 124188 + file: energy1.tex + sourceTransform: tex2mdx.strip_comments + start: 124172 + via: statement + sourceStatement: + chapter: 18 + file: energy1.tex + label: C1C4delta + mtref: '18.54' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 124884 + sourceTransform: tex2mdx.strip_comments + start: 124119 + targetStatement: + chapter: 18 + file: energy1.tex + label: areachange + mtref: '18.51' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 113141 + sourceTransform: tex2mdx.strip_comments + start: 112738 + type: tex-reference + via: + - statement +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 7e0b4b1362c0 +- 52123e3c5a27 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- statement +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/bb41442ebfbe.md b/projects/poincare-conjecture/.astrolabe/edges/bb41442ebfbe.md new file mode 100644 index 00000000..a39d1846 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/bb41442ebfbe.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: canonvary + proofSpan: + bodyEnd: 47855 + bodyStart: 43596 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 47866 + file: singlimit2.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 43583 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 45168 + file: singlimit2.tex + sourceTransform: tex2mdx.strip_comments + start: 45153 + via: proof + sourceStatement: + chapter: 11 + file: singlimit2.tex + label: omegacanon + mtref: '11.23' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 43580 + sourceTransform: tex2mdx.strip_comments + start: 43054 + targetStatement: + chapter: 9 + file: temp2kappa.tex + label: canonvary + mtref: '9.79' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 149236 + sourceTransform: tex2mdx.strip_comments + start: 147728 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 0760630d3990 +- 7c2758f0e2b3 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/bc00e2ed9c69.md b/projects/poincare-conjecture/.astrolabe/edges/bc00e2ed9c69.md new file mode 100644 index 00000000..2dea10c6 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/bc00e2ed9c69.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: 1stcurvshr + proofSpan: + bodyEnd: 83378 + bodyStart: 81190 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 83389 + file: energy1.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 81177 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 81540 + file: energy1.tex + sourceTransform: tex2mdx.strip_comments + start: 81524 + via: proof + sourceStatement: + chapter: 18 + file: energy1.tex + label: 2ndcurvshr + mtref: '18.35' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 81175 + sourceTransform: tex2mdx.strip_comments + start: 80801 + targetStatement: + chapter: 18 + file: energy1.tex + label: 1stcurvshr + mtref: '18.34' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 78816 + sourceTransform: tex2mdx.strip_comments + start: 78146 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 75b98ac99c73 +- 4fc24fa1d31e +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/bc429cf84c21.md b/projects/poincare-conjecture/.astrolabe/edges/bc429cf84c21.md new file mode 100644 index 00000000..80ed1c96 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/bc429cf84c21.md @@ -0,0 +1,114 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: statement + label: altern + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 130483 + file: surgery.tex + sourceTransform: tex2mdx.strip_comments + start: 130471 + via: statement + - context: proof + label: altern + proofSpan: + bodyEnd: 136429 + bodyStart: 132110 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 136440 + file: surgery.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 132097 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 132394 + file: surgery.tex + sourceTransform: tex2mdx.strip_comments + start: 132382 + via: proof + - context: proof + label: altern + proofSpan: + bodyEnd: 136429 + bodyStart: 132110 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 136440 + file: surgery.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 132097 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 133110 + file: surgery.tex + sourceTransform: tex2mdx.strip_comments + start: 133098 + via: proof + - context: proof + label: altern + proofSpan: + bodyEnd: 136429 + bodyStart: 132110 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 136440 + file: surgery.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 132097 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 134536 + file: surgery.tex + sourceTransform: tex2mdx.strip_comments + start: 134524 + via: proof + sourceStatement: + chapter: 16 + file: surgery.tex + label: bigell + mtref: '16.13' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 131955 + sourceTransform: tex2mdx.strip_comments + start: 130235 + targetStatement: + chapter: 16 + file: surgery.tex + label: altern + mtref: '16.5' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 112999 + sourceTransform: tex2mdx.strip_comments + start: 110961 + type: tex-reference + via: + - proof + - statement +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 0a4d6f17810d +- ad4f67ea75bc +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +- statement +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/bc7c78dab928.md b/projects/poincare-conjecture/.astrolabe/edges/bc7c78dab928.md new file mode 100644 index 00000000..2b88ec94 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/bc7c78dab928.md @@ -0,0 +1,62 @@ +--- +confidence: 0.35 +edgeClass: prose +evidence: + mentionTriggers: + - label: Fm + refSpan: + coordinateSpace: comment-stripped-tex + end: 53252 + file: flowbasics.tex + sourceTransform: tex2mdx.strip_comments + start: 53244 + sentence: Now we turn to the proof of Claim~\ref{Fm}. + sentenceSpan: + coordinateSpace: comment-stripped-tex + end: 53253 + file: flowbasics.tex + sourceTransform: tex2mdx.strip_comments + start: 53209 + sourceStrategy: nearest-previous-statement + via: prose + sourceStatement: + chapter: 3 + file: flowbasics.tex + label: etaprop + mtref: '3.33' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 47994 + sourceTransform: tex2mdx.strip_comments + start: 47039 + targetStatement: + chapter: 3 + file: flowbasics.tex + label: Fm + mtref: '3.32' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 46079 + sourceTransform: tex2mdx.strip_comments + start: 45486 + type: prose-mention-nearest-previous-statement + via: + - prose +generator: tools/poincare_tex_extract.py +inference: weak +kind: mention +ref: +- 81f8de5102de +- e7fe06259f43 +rel: mentions +reviewStatus: candidate +scope: prose +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- prose +--- +Morgan--Tian prose mention. The source is the nearest preceding statement in the same chapter segment. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/bcb61c71e514.md b/projects/poincare-conjecture/.astrolabe/edges/bcb61c71e514.md new file mode 100644 index 00000000..db3b3bb8 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/bcb61c71e514.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 7 + file: newcomp2.tex + label: '' + mtref: '7.21' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 36936 + sourceTransform: tex2mdx.strip_comments + start: 36629 + targetStatement: + chapter: 7 + file: newcomp2.tex + label: wineq + mtref: '7.22' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 37224 + sourceTransform: tex2mdx.strip_comments + start: 36952 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 06ac7beab852 +- de785906ce83 +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/bccb1d7ae9e4.md b/projects/poincare-conjecture/.astrolabe/edges/bccb1d7ae9e4.md new file mode 100644 index 00000000..e97249e3 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/bccb1d7ae9e4.md @@ -0,0 +1,65 @@ +--- +confidence: 0.45 +edgeClass: semantic +evidence: + endpointSemanticDegreeBeforeTermBackfill: + source: 0 + target: 1 + selectionReason: semantic-connectivity-backfill + sourceMatchSpan: + coordinateSpace: comment-stripped-tex + end: 11464 + file: newcompar.tex + sourceTransform: tex2mdx.strip_comments + start: 11434 + sourceStatement: + chapter: 6 + file: newcompar.tex + label: '' + mtref: '6.9' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 11539 + sourceTransform: tex2mdx.strip_comments + start: 11300 + targetStatement: + chapter: 6 + file: newcompar.tex + label: '' + mtref: '6.1' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 3679 + sourceTransform: tex2mdx.strip_comments + start: 3370 + targetTermSpan: + coordinateSpace: comment-stripped-tex + end: 3552 + file: newcompar.tex + sourceTransform: tex2mdx.strip_comments + start: 3522 + term: parameterized by backward time + termAmbiguity: unique-in-chapter + termSource: italic + termSpecificity: no-longer-defined-term + type: same-chapter-definition-term-match + via: + - definition-term:parameterized by backward time +generator: tools/poincare_tex_extract.py +inference: weak +kind: definition-use +ref: +- f09cbffcbdce +- 6a961a764f44 +rel: uses +reviewStatus: candidate +scope: same-chapter-term-match +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- definition-term:parameterized by backward time +--- +Morgan--Tian same-chapter definition term match. This is a low-confidence candidate edge for review. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/bd776be8d1bc.md b/projects/poincare-conjecture/.astrolabe/edges/bd776be8d1bc.md new file mode 100644 index 00000000..f0be96d5 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/bd776be8d1bc.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 5 + file: converge2.tex + label: '' + mtref: '5.2' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 2539 + sourceTransform: tex2mdx.strip_comments + start: 2465 + targetStatement: + chapter: 5 + file: converge2.tex + label: smoothconv + mtref: '5.3' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 4757 + sourceTransform: tex2mdx.strip_comments + start: 2949 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 786d89b0be9f +- 467f4b920031 +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/bd977e0fa0de.md b/projects/poincare-conjecture/.astrolabe/edges/bd977e0fa0de.md new file mode 100644 index 00000000..324e7d74 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/bd977e0fa0de.md @@ -0,0 +1,65 @@ +--- +confidence: 0.45 +edgeClass: semantic +evidence: + endpointSemanticDegreeBeforeTermBackfill: + source: 0 + target: 5 + selectionReason: semantic-connectivity-backfill + sourceMatchSpan: + coordinateSpace: comment-stripped-tex + end: 52501 + file: surgery.tex + sourceTransform: tex2mdx.strip_comments + start: 52489 + sourceStatement: + chapter: 14 + file: surgery.tex + label: '' + mtref: '14.12' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 52840 + sourceTransform: tex2mdx.strip_comments + start: 52271 + targetStatement: + chapter: 14 + file: surgery.tex + label: '' + mtref: '14.1' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 28555 + sourceTransform: tex2mdx.strip_comments + start: 27727 + targetTermSpan: + coordinateSpace: comment-stripped-tex + end: 28217 + file: surgery.tex + sourceTransform: tex2mdx.strip_comments + start: 28205 + term: initial time + termAmbiguity: unique-in-chapter + termSource: italic + termSpecificity: no-longer-defined-term + type: same-chapter-definition-term-match + via: + - definition-term:initial time +generator: tools/poincare_tex_extract.py +inference: weak +kind: definition-use +ref: +- cff982ed3611 +- 41da6871e864 +rel: uses +reviewStatus: candidate +scope: same-chapter-term-match +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- definition-term:initial time +--- +Morgan--Tian same-chapter definition term match. This is a low-confidence candidate edge for review. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/bdd73db130be.md b/projects/poincare-conjecture/.astrolabe/edges/bdd73db130be.md new file mode 100644 index 00000000..608383f4 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/bdd73db130be.md @@ -0,0 +1,56 @@ +--- +confidence: 0.85 +edgeClass: semantic +evidence: + containmentReason: target theorem-like statement occurs inside source proof + proofSpan: + bodyEnd: 14349 + bodyStart: 5983 + coordinateSpace: comment-stripped-tex + depth: 3 + end: 14360 + file: noncoll.tex + ownerLabel: small + sourceTransform: tex2mdx.strip_comments + start: 5970 + sourceStatement: + chapter: 8 + file: noncoll.tex + label: small + mtref: '8.3' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 5471 + sourceTransform: tex2mdx.strip_comments + start: 5161 + targetStatement: + chapter: 8 + file: noncoll.tex + label: nabR + mtref: '8.5' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 6558 + sourceTransform: tex2mdx.strip_comments + start: 6089 + type: proof-contained-theorem-like-statement + via: + - proof-contained-statement +generator: tools/poincare_tex_extract.py +inference: inferred +kind: dependency +ref: +- 26e1a6292fef +- bf8f356cfd7a +rel: depends +reviewStatus: accepted +scope: proof-containment +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof-contained-statement +--- +Morgan--Tian proof-contained statement dependency. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/be05bf25c8bd.md b/projects/poincare-conjecture/.astrolabe/edges/be05bf25c8bd.md new file mode 100644 index 00000000..d231181f --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/be05bf25c8bd.md @@ -0,0 +1,45 @@ +--- +confidence: 0.2 +edgeClass: structural +evidence: + sourceStatement: + chapter: 10 + file: bddcurvbdddist.tex + label: '' + mtref: '10.30' + sort: remark + span: + coordinateSpace: comment-stripped-tex + end: 58650 + sourceTransform: tex2mdx.strip_comments + start: 58353 + targetStatement: + chapter: 10 + file: bddcurvbdddist.tex + label: '' + mtref: '10.31' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 61603 + sourceTransform: tex2mdx.strip_comments + start: 61488 + type: chapter-reading-order-adjacency + via: + - chapter-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- c397ced906ac +- 6d6b9e050054 +rel: in-chapter +reviewStatus: accepted +scope: chapter +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- chapter-sequence +--- +Morgan--Tian reading-order adjacency across sections in a chapter. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/be0abeb9b892.md b/projects/poincare-conjecture/.astrolabe/edges/be0abeb9b892.md new file mode 100644 index 00000000..11d08ab4 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/be0abeb9b892.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 4 + file: maxprin.tex + label: Harnack + mtref: '4.37' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 52804 + sourceTransform: tex2mdx.strip_comments + start: 52280 + targetStatement: + chapter: 4 + file: maxprin.tex + label: '' + mtref: '4.38' + sort: remark + span: + coordinateSpace: comment-stripped-tex + end: 52902 + sourceTransform: tex2mdx.strip_comments + start: 52808 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- aa9d836895d8 +- 2629aff77c4a +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/be4350244108.md b/projects/poincare-conjecture/.astrolabe/edges/be4350244108.md new file mode 100644 index 00000000..20cf5a46 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/be4350244108.md @@ -0,0 +1,62 @@ +--- +confidence: 0.35 +edgeClass: prose +evidence: + mentionTriggers: + - label: lformula + refSpan: + coordinateSpace: comment-stripped-tex + end: 21947 + file: newcomp2.tex + sourceTransform: tex2mdx.strip_comments + start: 21933 + sentence: \subsection{Extension of the other inequalities in Corollary~\protect{\ref{lformula}}} + sentenceSpan: + coordinateSpace: comment-stripped-tex + end: 21949 + file: newcomp2.tex + sourceTransform: tex2mdx.strip_comments + start: 21861 + sourceStrategy: nearest-previous-statement + via: prose + sourceStatement: + chapter: 7 + file: newcomp2.tex + label: compactcoro + mtref: '7.12' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 21856 + sourceTransform: tex2mdx.strip_comments + start: 20802 + targetStatement: + chapter: 6 + file: newcompar.tex + label: lformula + mtref: '6.51' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 81082 + sourceTransform: tex2mdx.strip_comments + start: 80194 + type: prose-mention-nearest-previous-statement + via: + - prose +generator: tools/poincare_tex_extract.py +inference: weak +kind: mention +ref: +- 71669e7fd5ec +- 967c0f8bb774 +rel: mentions +reviewStatus: candidate +scope: prose +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- prose +--- +Morgan--Tian prose mention. The source is the nearest preceding statement in the same chapter segment. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/be5f8702984e.md b/projects/poincare-conjecture/.astrolabe/edges/be5f8702984e.md new file mode 100644 index 00000000..4f3265ad --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/be5f8702984e.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: locstr + proofSpan: + bodyEnd: 43529 + bodyStart: 40472 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 43540 + file: newcompar.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 40459 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 41938 + file: newcompar.tex + sourceTransform: tex2mdx.strip_comments + start: 41926 + via: proof + sourceStatement: + chapter: 6 + file: newcompar.tex + label: inclusion + mtref: '6.30' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 40456 + sourceTransform: tex2mdx.strip_comments + start: 40281 + targetStatement: + chapter: 6 + file: newcompar.tex + label: locstr + mtref: '6.23' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 32019 + sourceTransform: tex2mdx.strip_comments + start: 31685 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 8b9b3403dbc1 +- 4180719ab2c8 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/be784350ae97.md b/projects/poincare-conjecture/.astrolabe/edges/be784350ae97.md new file mode 100644 index 00000000..5be2d742 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/be784350ae97.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: partialflowlimit + proofSpan: + bodyEnd: 34517 + bodyStart: 26920 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 34528 + file: bddcurvbdddist.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 26907 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 33342 + file: bddcurvbdddist.tex + sourceTransform: tex2mdx.strip_comments + start: 33320 + via: proof + sourceStatement: + chapter: 10 + file: bddcurvbdddist.tex + label: Uinfty + mtref: '10.7' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 26826 + sourceTransform: tex2mdx.strip_comments + start: 25406 + targetStatement: + chapter: 5 + file: converge2.tex + label: partialflowlimit + mtref: '5.14' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 25227 + sourceTransform: tex2mdx.strip_comments + start: 23995 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- e2e64cdb0b75 +- 4673eb9f4424 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/be871b547206.md b/projects/poincare-conjecture/.astrolabe/edges/be871b547206.md new file mode 100644 index 00000000..b5323463 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/be871b547206.md @@ -0,0 +1,64 @@ +--- +confidence: 0.35 +edgeClass: prose +evidence: + mentionTriggers: + - label: controlnbhd + refSpan: + coordinateSpace: comment-stripped-tex + end: 177874 + file: surgery.tex + sourceTransform: tex2mdx.strip_comments + start: 177857 + sentence: According to Lemma~\ref{controlnbhd} every point $z$ on a backward flow + line starting in $B$ and defined for time at most $ \Delta$ has the property + that $R(z)\le 2r_i^{-2}$ + sentenceSpan: + coordinateSpace: comment-stripped-tex + end: 178012 + file: surgery.tex + sourceTransform: tex2mdx.strip_comments + start: 177837 + sourceStrategy: nearest-previous-statement + via: prose + sourceStatement: + chapter: 16 + file: surgery.tex + label: '' + mtref: '16.28' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 176062 + sourceTransform: tex2mdx.strip_comments + start: 175778 + targetStatement: + chapter: 11 + file: singlimit2.tex + label: controlnbhd + mtref: '11.2' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 4360 + sourceTransform: tex2mdx.strip_comments + start: 3597 + type: prose-mention-nearest-previous-statement + via: + - prose +generator: tools/poincare_tex_extract.py +inference: weak +kind: mention +ref: +- 3e9580da0ff5 +- 0ddfe578961e +rel: mentions +reviewStatus: candidate +scope: prose +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- prose +--- +Morgan--Tian prose mention. The source is the nearest preceding statement in the same chapter segment. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/bf3482ff02a5.md b/projects/poincare-conjecture/.astrolabe/edges/bf3482ff02a5.md new file mode 100644 index 00000000..d9dc5cba --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/bf3482ff02a5.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 18 + file: energy1.tex + label: Wdefn + mtref: '18.17' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 49255 + sourceTransform: tex2mdx.strip_comments + start: 48287 + targetStatement: + chapter: 18 + file: energy1.tex + label: W_3fordiff + mtref: '18.18' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 51926 + sourceTransform: tex2mdx.strip_comments + start: 51222 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- d88ab31ec11f +- b187c06d1432 +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/bf39bcde7197.md b/projects/poincare-conjecture/.astrolabe/edges/bf39bcde7197.md new file mode 100644 index 00000000..2f429ed0 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/bf39bcde7197.md @@ -0,0 +1,133 @@ +--- +confidence: 0.35 +edgeClass: prose +evidence: + mentionTriggers: + - label: W_3fordiff + refSpan: + coordinateSpace: comment-stripped-tex + end: 56932 + file: energy1.tex + sourceTransform: tex2mdx.strip_comments + start: 56916 + sentence: To prove Proposition~\ref{W_3fordiff} and hence Theorem~\ref{extinct}, + it remains to prove the forward difference quotient statement for $W_\xi(t)$ + given in Proposition~\ref{W_3fordiff}. + sentenceSpan: + coordinateSpace: comment-stripped-tex + end: 57081 + file: energy1.tex + sourceTransform: tex2mdx.strip_comments + start: 56894 + sourceStrategy: nearest-previous-statement + via: prose + - label: W_3fordiff + refSpan: + coordinateSpace: comment-stripped-tex + end: 57080 + file: energy1.tex + sourceTransform: tex2mdx.strip_comments + start: 57064 + sentence: To prove Proposition~\ref{W_3fordiff} and hence Theorem~\ref{extinct}, + it remains to prove the forward difference quotient statement for $W_\xi(t)$ + given in Proposition~\ref{W_3fordiff}. + sentenceSpan: + coordinateSpace: comment-stripped-tex + end: 57081 + file: energy1.tex + sourceTransform: tex2mdx.strip_comments + start: 56894 + sourceStrategy: nearest-previous-statement + via: prose + - label: W_3fordiff + refSpan: + coordinateSpace: comment-stripped-tex + end: 57160 + file: energy1.tex + sourceTransform: tex2mdx.strip_comments + start: 57144 + sentence: \subsection{A further reduction of Proposition~\protect{\ref{W_3fordiff}}} + sentenceSpan: + coordinateSpace: comment-stripped-tex + end: 57162 + file: energy1.tex + sourceTransform: tex2mdx.strip_comments + start: 57087 + sourceStrategy: nearest-previous-statement + via: prose + - label: W_3fordiff + refSpan: + coordinateSpace: comment-stripped-tex + end: 57356 + file: energy1.tex + sourceTransform: tex2mdx.strip_comments + start: 57340 + sentence: We must construct an appropriate deformation of the family of loops + $\Gamma$ in order to establish Proposition~\ref{W_3fordiff} + sentenceSpan: + coordinateSpace: comment-stripped-tex + end: 57356 + file: energy1.tex + sourceTransform: tex2mdx.strip_comments + start: 57228 + sourceStrategy: nearest-previous-statement + via: prose + - label: W_3fordiff + refSpan: + coordinateSpace: comment-stripped-tex + end: 57570 + file: energy1.tex + sourceTransform: tex2mdx.strip_comments + start: 57554 + sentence: Now we are ready to state the more technical estimate for the evolution + of $W(\Gamma)$ under Ricci flow that will imply the forward difference quotient + result for $W_\xi(t)$ stated in Proposition~\ref{W_3fordiff} + sentenceSpan: + coordinateSpace: comment-stripped-tex + end: 57570 + file: energy1.tex + sourceTransform: tex2mdx.strip_comments + start: 57357 + sourceStrategy: nearest-previous-statement + via: prose + sourceStatement: + chapter: 18 + file: energy1.tex + label: '' + mtref: '18.22' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 56627 + sourceTransform: tex2mdx.strip_comments + start: 56502 + targetStatement: + chapter: 18 + file: energy1.tex + label: W_3fordiff + mtref: '18.18' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 51926 + sourceTransform: tex2mdx.strip_comments + start: 51222 + type: prose-mention-nearest-previous-statement + via: + - prose +generator: tools/poincare_tex_extract.py +inference: weak +kind: mention +ref: +- 981cc21e82e4 +- b187c06d1432 +rel: mentions +reviewStatus: candidate +scope: prose +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- prose +--- +Morgan--Tian prose mention. The source is the nearest preceding statement in the same chapter segment. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/bf579bf0008f.md b/projects/poincare-conjecture/.astrolabe/edges/bf579bf0008f.md new file mode 100644 index 00000000..0a3db5c3 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/bf579bf0008f.md @@ -0,0 +1,56 @@ +--- +confidence: 0.85 +edgeClass: semantic +evidence: + containmentReason: target theorem-like statement occurs inside source proof + proofSpan: + bodyEnd: 164255 + bodyStart: 154729 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 164266 + file: surgery.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 154716 + sourceStatement: + chapter: 16 + file: surgery.tex + label: Lgeoexist + mtref: '16.21' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 154713 + sourceTransform: tex2mdx.strip_comments + start: 152982 + targetStatement: + chapter: 16 + file: surgery.tex + label: '' + mtref: '16.23' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 158095 + sourceTransform: tex2mdx.strip_comments + start: 157903 + type: proof-contained-theorem-like-statement + via: + - proof-contained-statement +generator: tools/poincare_tex_extract.py +inference: inferred +kind: dependency +ref: +- d98ad25d9bc3 +- 5ba0ec2c68db +rel: depends +reviewStatus: accepted +scope: proof-containment +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof-contained-statement +--- +Morgan--Tian proof-contained statement dependency. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/bf8dc1b20da4.md b/projects/poincare-conjecture/.astrolabe/edges/bf8dc1b20da4.md new file mode 100644 index 00000000..3a5042b7 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/bf8dc1b20da4.md @@ -0,0 +1,45 @@ +--- +confidence: 0.2 +edgeClass: structural +evidence: + sourceStatement: + chapter: 4 + file: maxprin.tex + label: Ric>0 + mtref: '4.15' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 18295 + sourceTransform: tex2mdx.strip_comments + start: 18076 + targetStatement: + chapter: 4 + file: maxprin.tex + label: SMPheat + mtref: '4.16' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 20373 + sourceTransform: tex2mdx.strip_comments + start: 19770 + type: chapter-reading-order-adjacency + via: + - chapter-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- f8afc1c61977 +- 5b296315fdf9 +rel: in-chapter +reviewStatus: accepted +scope: chapter +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- chapter-sequence +--- +Morgan--Tian reading-order adjacency across sections in a chapter. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/bfb953cdbb69.md b/projects/poincare-conjecture/.astrolabe/edges/bfb953cdbb69.md new file mode 100644 index 00000000..e195d43a --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/bfb953cdbb69.md @@ -0,0 +1,65 @@ +--- +confidence: 0.45 +edgeClass: semantic +evidence: + endpointSemanticDegreeBeforeTermBackfill: + source: 0 + target: 2 + selectionReason: semantic-connectivity-backfill + sourceMatchSpan: + coordinateSpace: comment-stripped-tex + end: 14502 + file: newcompar.tex + sourceTransform: tex2mdx.strip_comments + start: 14494 + sourceStatement: + chapter: 6 + file: newcompar.tex + label: '' + mtref: '6.12' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 14734 + sourceTransform: tex2mdx.strip_comments + start: 14448 + targetStatement: + chapter: 6 + file: newcompar.tex + label: '' + mtref: '6.7' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 10481 + sourceTransform: tex2mdx.strip_comments + start: 10161 + targetTermSpan: + coordinateSpace: comment-stripped-tex + end: 10318 + file: newcompar.tex + sourceTransform: tex2mdx.strip_comments + start: 10310 + term: geodesic + termAmbiguity: unique-in-chapter + termSource: italic + termSpecificity: not-covered-by-longer-defined-term + type: same-chapter-definition-term-match + via: + - definition-term:geodesic +generator: tools/poincare_tex_extract.py +inference: weak +kind: definition-use +ref: +- 6b205e5b51a2 +- d23dc29f2aae +rel: uses +reviewStatus: candidate +scope: same-chapter-term-match +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- definition-term:geodesic +--- +Morgan--Tian same-chapter definition term match. This is a low-confidence candidate edge for review. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/c009550e2822.md b/projects/poincare-conjecture/.astrolabe/edges/c009550e2822.md new file mode 100644 index 00000000..e4fae465 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/c009550e2822.md @@ -0,0 +1,45 @@ +--- +confidence: 0.2 +edgeClass: structural +evidence: + sourceStatement: + chapter: 11 + file: singlimit2.tex + label: epslimit + mtref: '11.24' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 45867 + sourceTransform: tex2mdx.strip_comments + start: 45550 + targetStatement: + chapter: 11 + file: singlimit2.tex + label: '' + mtref: '11.25' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 48871 + sourceTransform: tex2mdx.strip_comments + start: 48329 + type: chapter-reading-order-adjacency + via: + - chapter-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- d74749898b25 +- da99c0859788 +rel: in-chapter +reviewStatus: accepted +scope: chapter +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- chapter-sequence +--- +Morgan--Tian reading-order adjacency across sections in a chapter. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/c040cb0048fd.md b/projects/poincare-conjecture/.astrolabe/edges/c040cb0048fd.md new file mode 100644 index 00000000..f10906d2 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/c040cb0048fd.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: firststrongmax + proofSpan: + bodyEnd: 30823 + bodyStart: 25720 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 30834 + file: maxprin.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 25707 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 25783 + file: maxprin.tex + sourceTransform: tex2mdx.strip_comments + start: 25763 + via: proof + sourceStatement: + chapter: 4 + file: maxprin.tex + label: localprod + mtref: '4.19' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 25705 + sourceTransform: tex2mdx.strip_comments + start: 25065 + targetStatement: + chapter: 4 + file: maxprin.tex + label: firststrongmax + mtref: '4.18' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 23916 + sourceTransform: tex2mdx.strip_comments + start: 23628 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 8ec855a2aab5 +- 016cf36686b2 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/c0776a47bbf7.md b/projects/poincare-conjecture/.astrolabe/edges/c0776a47bbf7.md new file mode 100644 index 00000000..e0ff5df9 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/c0776a47bbf7.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: DtildeL + proofSpan: + bodyEnd: 106548 + bodyStart: 105481 + coordinateSpace: comment-stripped-tex + depth: 2 + end: 106559 + file: newcompar.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 105468 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 105960 + file: newcompar.tex + sourceTransform: tex2mdx.strip_comments + start: 105947 + via: proof + sourceStatement: + chapter: 6 + file: newcompar.tex + label: '' + mtref: '6.69' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 105466 + sourceTransform: tex2mdx.strip_comments + start: 104872 + targetStatement: + chapter: 6 + file: newcompar.tex + label: DtildeL + mtref: '6.22' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 30026 + sourceTransform: tex2mdx.strip_comments + start: 29708 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- c25742cc5546 +- abf9cbb10c68 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/c08632c08606.md b/projects/poincare-conjecture/.astrolabe/edges/c08632c08606.md new file mode 100644 index 00000000..94becd94 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/c08632c08606.md @@ -0,0 +1,62 @@ +--- +confidence: 0.9 +edgeClass: structural +evidence: + containmentDepth: 1 + proofSpan: + bodyEnd: 229182 + bodyStart: 185422 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 229193 + file: surgery.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 185409 + sourceStatement: + chapter: 17 + file: surgery.tex + label: extend + mtref: '17.1' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 185406 + sourceTransform: tex2mdx.strip_comments + start: 184285 + targetContainedSpan: + coordinateSpace: comment-stripped-tex + end: 205281 + file: surgery.tex + sourceTransform: tex2mdx.strip_comments + start: 204957 + targetStatement: + chapter: 17 + file: surgery.tex + label: '' + mtref: '17.5' + sort: remark + span: + coordinateSpace: comment-stripped-tex + end: 205281 + sourceTransform: tex2mdx.strip_comments + start: 204957 + type: proof-containment + via: + - proof-contains +generator: tools/poincare_tex_extract.py +inference: structural +kind: containment +ref: +- 254e8dae97e6 +- 1596df6ade35 +rel: contains +reviewStatus: accepted +scope: proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof-contains +--- +Morgan--Tian proof containment. The source is the statement whose proof contains the target statement. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/c08c906fa2d6.md b/projects/poincare-conjecture/.astrolabe/edges/c08c906fa2d6.md new file mode 100644 index 00000000..adb20df9 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/c08c906fa2d6.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: stdsolncannbhd + proofSpan: + bodyEnd: 204939 + bodyStart: 186733 + coordinateSpace: comment-stripped-tex + depth: 2 + end: 204950 + file: surgery.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 186720 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 203520 + file: surgery.tex + sourceTransform: tex2mdx.strip_comments + start: 203500 + via: proof + sourceStatement: + chapter: 17 + file: surgery.tex + label: '16.2' + mtref: '17.2' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 186718 + sourceTransform: tex2mdx.strip_comments + start: 186200 + targetStatement: + chapter: 12 + file: stdsoln.tex + label: stdsolncannbhd + mtref: '12.32' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 94479 + sourceTransform: tex2mdx.strip_comments + start: 93886 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 778d690c4e7c +- 6ff28dba4e56 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/c09d1d8d14c1.md b/projects/poincare-conjecture/.astrolabe/edges/c09d1d8d14c1.md new file mode 100644 index 00000000..543d368e --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/c09d1d8d14c1.md @@ -0,0 +1,64 @@ +--- +confidence: 0.35 +edgeClass: prose +evidence: + mentionTriggers: + - label: extinct + refSpan: + coordinateSpace: comment-stripped-tex + end: 56964 + file: energy1.tex + sourceTransform: tex2mdx.strip_comments + start: 56951 + sentence: To prove Proposition~\ref{W_3fordiff} and hence Theorem~\ref{extinct}, + it remains to prove the forward difference quotient statement for $W_\xi(t)$ + given in Proposition~\ref{W_3fordiff}. + sentenceSpan: + coordinateSpace: comment-stripped-tex + end: 57081 + file: energy1.tex + sourceTransform: tex2mdx.strip_comments + start: 56894 + sourceStrategy: nearest-previous-statement + via: prose + sourceStatement: + chapter: 18 + file: energy1.tex + label: '' + mtref: '18.22' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 56627 + sourceTransform: tex2mdx.strip_comments + start: 56502 + targetStatement: + chapter: 18 + file: energy1.tex + label: extinct + mtref: '18.1' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 1107 + sourceTransform: tex2mdx.strip_comments + start: 511 + type: prose-mention-nearest-previous-statement + via: + - prose +generator: tools/poincare_tex_extract.py +inference: weak +kind: mention +ref: +- 981cc21e82e4 +- 4b8a44aa316f +rel: mentions +reviewStatus: candidate +scope: prose +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- prose +--- +Morgan--Tian prose mention. The source is the nearest preceding statement in the same chapter segment. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/c0a46cc0c5f8.md b/projects/poincare-conjecture/.astrolabe/edges/c0a46cc0c5f8.md new file mode 100644 index 00000000..c5b9d2bf --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/c0a46cc0c5f8.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 13 + file: surgery.tex + label: surgerycap + mtref: '13.3' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 6755 + sourceTransform: tex2mdx.strip_comments + start: 6605 + targetStatement: + chapter: 13 + file: surgery.tex + label: capball + mtref: '13.4' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 7464 + sourceTransform: tex2mdx.strip_comments + start: 6854 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 443843fff58d +- 752133843e1e +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/c0c4e6ad098c.md b/projects/poincare-conjecture/.astrolabe/edges/c0c4e6ad098c.md new file mode 100644 index 00000000..5f00ef3b --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/c0c4e6ad098c.md @@ -0,0 +1,62 @@ +--- +confidence: 0.9 +edgeClass: structural +evidence: + containmentDepth: 1 + proofSpan: + bodyEnd: 33071 + bodyStart: 25905 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 33082 + file: temp2kappa.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 25892 + sourceStatement: + chapter: 9 + file: temp2kappa.tex + label: nablaconv + mtref: '9.21' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 25890 + sourceTransform: tex2mdx.strip_comments + start: 25631 + targetContainedSpan: + coordinateSpace: comment-stripped-tex + end: 30396 + file: temp2kappa.tex + sourceTransform: tex2mdx.strip_comments + start: 30212 + targetStatement: + chapter: 9 + file: temp2kappa.tex + label: '' + mtref: '9.22' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 30396 + sourceTransform: tex2mdx.strip_comments + start: 30212 + type: proof-containment + via: + - proof-contains +generator: tools/poincare_tex_extract.py +inference: structural +kind: containment +ref: +- 366e3fdadf6f +- 598f6059280c +rel: contains +reviewStatus: accepted +scope: proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof-contains +--- +Morgan--Tian proof containment. The source is the statement whose proof contains the target statement. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/c0ce59e30f8d.md b/projects/poincare-conjecture/.astrolabe/edges/c0ce59e30f8d.md new file mode 100644 index 00000000..401adc11 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/c0ce59e30f8d.md @@ -0,0 +1,83 @@ +--- +confidence: 0.35 +edgeClass: prose +evidence: + mentionTriggers: + - label: hexist + refSpan: + coordinateSpace: comment-stripped-tex + end: 85726 + file: surgery.tex + sourceTransform: tex2mdx.strip_comments + start: 85714 + sentence: Set $\rho=\overline\delta(T)r_{i+1}$, and set $h(T)=h(\rho(T),\overline\delta(T))$ + as in Theorem~\ref{hexist}. Since $\overline\delta(T)\le \delta_0<1$, we see + that $\rho0 + mtref: '13.13' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 21433 + sourceTransform: tex2mdx.strip_comments + start: 21250 + targetStatement: + chapter: 13 + file: surgery.tex + label: '' + mtref: '13.14' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 23621 + sourceTransform: tex2mdx.strip_comments + start: 23086 + type: chapter-reading-order-adjacency + via: + - chapter-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- cc9110b4d2fe +- c9d8cb9d34c3 +rel: in-chapter +reviewStatus: accepted +scope: chapter +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- chapter-sequence +--- +Morgan--Tian reading-order adjacency across sections in a chapter. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/c66a58859677.md b/projects/poincare-conjecture/.astrolabe/edges/c66a58859677.md new file mode 100644 index 00000000..2c595970 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/c66a58859677.md @@ -0,0 +1,84 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: kaplimit + proofSpan: + bodyEnd: 89985 + bodyStart: 86085 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 89996 + file: stdsoln.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 86072 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 88409 + file: stdsoln.tex + sourceTransform: tex2mdx.strip_comments + start: 88395 + via: proof + - context: proof + label: kaplimit + proofSpan: + bodyEnd: 89985 + bodyStart: 86085 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 89996 + file: stdsoln.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 86072 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 88485 + file: stdsoln.tex + sourceTransform: tex2mdx.strip_comments + start: 88471 + via: proof + sourceStatement: + chapter: 12 + file: stdsoln.tex + label: '' + mtref: '12.29' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 86070 + sourceTransform: tex2mdx.strip_comments + start: 86020 + targetStatement: + chapter: 11 + file: singlimit2.tex + label: kaplimit + mtref: '11.8' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 16131 + sourceTransform: tex2mdx.strip_comments + start: 15040 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 729d9a0428c4 +- a9e023d082dc +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/c66b970d92b1.md b/projects/poincare-conjecture/.astrolabe/edges/c66b970d92b1.md new file mode 100644 index 00000000..10487c67 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/c66b970d92b1.md @@ -0,0 +1,64 @@ +--- +confidence: 0.35 +edgeClass: prose +evidence: + mentionTriggers: + - label: csforramps + refSpan: + coordinateSpace: comment-stripped-tex + end: 110440 + file: energy1.tex + sourceTransform: tex2mdx.strip_comments + start: 110424 + sentence: Fix $c\in S^2$, and let $\widetilde\Gamma_c^\lambda(t),\ t_0\le t\le + t_1$, be the curve-shrinking flow given in Lemma~\ref{csforramps} with initial + data the ramp $\widetilde\Gamma^\lambda(c)$ + sentenceSpan: + coordinateSpace: comment-stripped-tex + end: 110498 + file: energy1.tex + sourceTransform: tex2mdx.strip_comments + start: 110304 + sourceStrategy: nearest-previous-statement + via: prose + sourceStatement: + chapter: 18 + file: energy1.tex + label: '' + mtref: '18.50' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 107575 + sourceTransform: tex2mdx.strip_comments + start: 107348 + targetStatement: + chapter: 18 + file: energy1.tex + label: csforramps + mtref: '18.42' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 91212 + sourceTransform: tex2mdx.strip_comments + start: 90872 + type: prose-mention-nearest-previous-statement + via: + - prose +generator: tools/poincare_tex_extract.py +inference: weak +kind: mention +ref: +- cb69804a2166 +- 11ec367eae93 +rel: mentions +reviewStatus: candidate +scope: prose +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- prose +--- +Morgan--Tian prose mention. The source is the nearest preceding statement in the same chapter segment. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/c6749b2e5d84.md b/projects/poincare-conjecture/.astrolabe/edges/c6749b2e5d84.md new file mode 100644 index 00000000..042d7202 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/c6749b2e5d84.md @@ -0,0 +1,64 @@ +--- +confidence: 0.35 +edgeClass: prose +evidence: + mentionTriggers: + - label: unianc + refSpan: + coordinateSpace: comment-stripped-tex + end: 19476 + file: temp2kappa.tex + sourceTransform: tex2mdx.strip_comments + start: 19464 + sentence: Once we know that the $l_{x_k}$ are uniformly bounded on $B(q_k,-\tau_0,A)$, + it follows from Corollary~\ref{unianc} that $R_{g_k}$ are also uniformly bounded + on the $B(q_k,-\tau_0,A)$ + sentenceSpan: + coordinateSpace: comment-stripped-tex + end: 19544 + file: temp2kappa.tex + sourceTransform: tex2mdx.strip_comments + start: 19360 + sourceStrategy: nearest-previous-statement + via: prose + sourceStatement: + chapter: 9 + file: temp2kappa.tex + label: lxkest + mtref: '9.16' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 18673 + sourceTransform: tex2mdx.strip_comments + start: 18334 + targetStatement: + chapter: 9 + file: temp2kappa.tex + label: unianc + mtref: '9.10' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 10335 + sourceTransform: tex2mdx.strip_comments + start: 9624 + type: prose-mention-nearest-previous-statement + via: + - prose +generator: tools/poincare_tex_extract.py +inference: weak +kind: mention +ref: +- 611a73900deb +- 8b8e3a0f3e36 +rel: mentions +reviewStatus: candidate +scope: prose +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- prose +--- +Morgan--Tian prose mention. The source is the nearest preceding statement in the same chapter segment. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/c6b745d20d41.md b/projects/poincare-conjecture/.astrolabe/edges/c6b745d20d41.md new file mode 100644 index 00000000..ab66bb98 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/c6b745d20d41.md @@ -0,0 +1,56 @@ +--- +confidence: 0.85 +edgeClass: semantic +evidence: + containmentReason: target theorem-like statement occurs inside source proof + proofSpan: + bodyEnd: 46655 + bodyStart: 43350 + coordinateSpace: comment-stripped-tex + depth: 2 + end: 46666 + file: maxprin.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 43337 + sourceStatement: + chapter: 4 + file: maxprin.tex + label: '' + mtref: '4.30' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 43335 + sourceTransform: tex2mdx.strip_comments + start: 43182 + targetStatement: + chapter: 4 + file: maxprin.tex + label: '' + mtref: '4.31' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 44605 + sourceTransform: tex2mdx.strip_comments + start: 44568 + type: proof-contained-theorem-like-statement + via: + - proof-contained-statement +generator: tools/poincare_tex_extract.py +inference: inferred +kind: dependency +ref: +- d3f49c0de6a0 +- 61df8ef86274 +rel: depends +reviewStatus: accepted +scope: proof-containment +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof-contained-statement +--- +Morgan--Tian proof-contained statement dependency. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/c6c7ba4a16e2.md b/projects/poincare-conjecture/.astrolabe/edges/c6c7ba4a16e2.md new file mode 100644 index 00000000..9fc82210 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/c6c7ba4a16e2.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: posderiv + proofSpan: + bodyEnd: 136142 + bodyStart: 134948 + coordinateSpace: comment-stripped-tex + depth: 2 + end: 136153 + file: temp2kappa.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 134935 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 135630 + file: temp2kappa.tex + sourceTransform: tex2mdx.strip_comments + start: 135616 + via: proof + sourceStatement: + chapter: 9 + file: temp2kappa.tex + label: zkbound + mtref: '9.68' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 134931 + sourceTransform: tex2mdx.strip_comments + start: 134780 + targetStatement: + chapter: 4 + file: maxprin.tex + label: posderiv + mtref: '4.39' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 53144 + sourceTransform: tex2mdx.strip_comments + start: 52904 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- f61f72020102 +- 36dd2f6fa9c2 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/c6e656b1fa53.md b/projects/poincare-conjecture/.astrolabe/edges/c6e656b1fa53.md new file mode 100644 index 00000000..366c82bf --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/c6e656b1fa53.md @@ -0,0 +1,63 @@ +--- +confidence: 0.35 +edgeClass: prose +evidence: + mentionTriggers: + - label: basicconv + refSpan: + coordinateSpace: comment-stripped-tex + end: 16484 + file: converge2.tex + sourceTransform: tex2mdx.strip_comments + start: 16469 + sentence: Now as an immediate corollary of Theorem~\ref{basicconv} we have the + following. + sentenceSpan: + coordinateSpace: comment-stripped-tex + end: 16507 + file: converge2.tex + sourceTransform: tex2mdx.strip_comments + start: 16427 + sourceStrategy: nearest-previous-statement + via: prose + sourceStatement: + chapter: 5 + file: converge2.tex + label: '' + mtref: '5.8' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 16131 + sourceTransform: tex2mdx.strip_comments + start: 15866 + targetStatement: + chapter: 5 + file: converge2.tex + label: basicconv + mtref: '5.6' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 9928 + sourceTransform: tex2mdx.strip_comments + start: 8668 + type: prose-mention-nearest-previous-statement + via: + - prose +generator: tools/poincare_tex_extract.py +inference: weak +kind: mention +ref: +- 6ed4f379cb9f +- 74c7682a68ed +rel: mentions +reviewStatus: candidate +scope: prose +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- prose +--- +Morgan--Tian prose mention. The source is the nearest preceding statement in the same chapter segment. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/c6fb5b950979.md b/projects/poincare-conjecture/.astrolabe/edges/c6fb5b950979.md new file mode 100644 index 00000000..bc0fe5ef --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/c6fb5b950979.md @@ -0,0 +1,84 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: defntheta + proofSpan: + bodyEnd: 49439 + bodyStart: 48913 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 49450 + file: bddcurvbdddist.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 48900 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 49080 + file: bddcurvbdddist.tex + sourceTransform: tex2mdx.strip_comments + start: 49065 + via: proof + - context: proof + label: defntheta + proofSpan: + bodyEnd: 49439 + bodyStart: 48913 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 49450 + file: bddcurvbdddist.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 48900 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 49281 + file: bddcurvbdddist.tex + sourceTransform: tex2mdx.strip_comments + start: 49266 + via: proof + sourceStatement: + chapter: 10 + file: bddcurvbdddist.tex + label: distcomp + mtref: '10.25' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 48898 + sourceTransform: tex2mdx.strip_comments + start: 48362 + targetStatement: + chapter: 10 + file: bddcurvbdddist.tex + label: defntheta + mtref: '10.20' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 43420 + sourceTransform: tex2mdx.strip_comments + start: 42950 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 317c62593611 +- f4ee075394ef +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/c752d7f5bc87.md b/projects/poincare-conjecture/.astrolabe/edges/c752d7f5bc87.md new file mode 100644 index 00000000..f027c994 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/c752d7f5bc87.md @@ -0,0 +1,62 @@ +--- +confidence: 0.9 +edgeClass: structural +evidence: + containmentDepth: 2 + proofSpan: + bodyEnd: 46655 + bodyStart: 43350 + coordinateSpace: comment-stripped-tex + depth: 2 + end: 46666 + file: maxprin.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 43337 + sourceStatement: + chapter: 4 + file: maxprin.tex + label: '' + mtref: '4.30' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 43335 + sourceTransform: tex2mdx.strip_comments + start: 43182 + targetContainedSpan: + coordinateSpace: comment-stripped-tex + end: 44605 + file: maxprin.tex + sourceTransform: tex2mdx.strip_comments + start: 44568 + targetStatement: + chapter: 4 + file: maxprin.tex + label: '' + mtref: '4.31' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 44605 + sourceTransform: tex2mdx.strip_comments + start: 44568 + type: proof-containment + via: + - proof-contains +generator: tools/poincare_tex_extract.py +inference: structural +kind: containment +ref: +- d3f49c0de6a0 +- 61df8ef86274 +rel: contains +reviewStatus: accepted +scope: proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof-contains +--- +Morgan--Tian proof containment. The source is the statement whose proof contains the target statement. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/c75b9f8427df.md b/projects/poincare-conjecture/.astrolabe/edges/c75b9f8427df.md new file mode 100644 index 00000000..d9be13d1 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/c75b9f8427df.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: controlnbhd + proofSpan: + bodyEnd: 229182 + bodyStart: 185422 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 229193 + file: surgery.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 185409 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 228620 + file: surgery.tex + sourceTransform: tex2mdx.strip_comments + start: 228603 + via: proof + sourceStatement: + chapter: 17 + file: surgery.tex + label: extend + mtref: '17.1' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 185406 + sourceTransform: tex2mdx.strip_comments + start: 184285 + targetStatement: + chapter: 11 + file: singlimit2.tex + label: controlnbhd + mtref: '11.2' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 4360 + sourceTransform: tex2mdx.strip_comments + start: 3597 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 254e8dae97e6 +- 0ddfe578961e +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/c7899332c0a2.md b/projects/poincare-conjecture/.astrolabe/edges/c7899332c0a2.md new file mode 100644 index 00000000..d46a3240 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/c7899332c0a2.md @@ -0,0 +1,64 @@ +--- +confidence: 0.35 +edgeClass: prose +evidence: + mentionTriggers: + - label: distdecrease + refSpan: + coordinateSpace: comment-stripped-tex + end: 25874 + file: surgery.tex + sourceTransform: tex2mdx.strip_comments + start: 25856 + sentence: On the other hand, it follows from Lemma~\ref{distdecrease} that the + map from $([0,\delta^{-1}]\times S^2,R(x_0)\rho^* g)$ to $(B(p_0,4+A_0),\eta + g)$ is distance decreasing + sentenceSpan: + coordinateSpace: comment-stripped-tex + end: 25987 + file: surgery.tex + sourceTransform: tex2mdx.strip_comments + start: 25814 + sourceStrategy: nearest-previous-statement + via: prose + sourceStatement: + chapter: 13 + file: surgery.tex + label: '' + mtref: '13.14' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 23621 + sourceTransform: tex2mdx.strip_comments + start: 23086 + targetStatement: + chapter: 13 + file: surgery.tex + label: distdecrease + mtref: '13.1' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 3514 + sourceTransform: tex2mdx.strip_comments + start: 3099 + type: prose-mention-nearest-previous-statement + via: + - prose +generator: tools/poincare_tex_extract.py +inference: weak +kind: mention +ref: +- c9d8cb9d34c3 +- 6168ffcb3bed +rel: mentions +reviewStatus: candidate +scope: prose +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- prose +--- +Morgan--Tian prose mention. The source is the nearest preceding statement in the same chapter segment. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/c789a4857bde.md b/projects/poincare-conjecture/.astrolabe/edges/c789a4857bde.md new file mode 100644 index 00000000..e5d82831 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/c789a4857bde.md @@ -0,0 +1,65 @@ +--- +confidence: 0.45 +edgeClass: semantic +evidence: + endpointSemanticDegreeBeforeTermBackfill: + source: 3 + target: 0 + selectionReason: semantic-connectivity-backfill + sourceMatchSpan: + coordinateSpace: comment-stripped-tex + end: 6820 + file: stdsoln.tex + sourceTransform: tex2mdx.strip_comments + start: 6801 + sourceStatement: + chapter: 12 + file: stdsoln.tex + label: stdsoln + mtref: '12.5' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 8086 + sourceTransform: tex2mdx.strip_comments + start: 6763 + targetStatement: + chapter: 12 + file: stdsoln.tex + label: '' + mtref: '12.4' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 6719 + sourceTransform: tex2mdx.strip_comments + start: 6171 + targetTermSpan: + coordinateSpace: comment-stripped-tex + end: 6218 + file: stdsoln.tex + sourceTransform: tex2mdx.strip_comments + start: 6199 + term: standard ricci flow + termAmbiguity: unique-in-chapter + termSource: italic + termSpecificity: not-covered-by-longer-defined-term + type: same-chapter-definition-term-match + via: + - definition-term:standard ricci flow +generator: tools/poincare_tex_extract.py +inference: weak +kind: definition-use +ref: +- f44d182ef27e +- acada7b5d698 +rel: uses +reviewStatus: candidate +scope: same-chapter-term-match +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- definition-term:standard ricci flow +--- +Morgan--Tian same-chapter definition term match. This is a low-confidence candidate edge for review. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/c81d2c09ac57.md b/projects/poincare-conjecture/.astrolabe/edges/c81d2c09ac57.md new file mode 100644 index 00000000..34e055e4 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/c81d2c09ac57.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: posderiv + proofSpan: + bodyEnd: 132635 + bodyStart: 128771 + coordinateSpace: comment-stripped-tex + depth: 2 + end: 132646 + file: temp2kappa.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 128758 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 130500 + file: temp2kappa.tex + sourceTransform: tex2mdx.strip_comments + start: 130486 + via: proof + sourceStatement: + chapter: 9 + file: temp2kappa.tex + label: 2Rbound + mtref: '9.66' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 128756 + sourceTransform: tex2mdx.strip_comments + start: 128570 + targetStatement: + chapter: 4 + file: maxprin.tex + label: posderiv + mtref: '4.39' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 53144 + sourceTransform: tex2mdx.strip_comments + start: 52904 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- ea9b285e40cb +- 36dd2f6fa9c2 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/c82dd6bda08f.md b/projects/poincare-conjecture/.astrolabe/edges/c82dd6bda08f.md new file mode 100644 index 00000000..0ac5da84 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/c82dd6bda08f.md @@ -0,0 +1,64 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: statement + label: jaceqn + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 14178 + file: newcompar.tex + sourceTransform: tex2mdx.strip_comments + start: 14166 + via: statement-aux-label-proxy + - context: statement + label: jaceqn + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 14376 + file: newcompar.tex + sourceTransform: tex2mdx.strip_comments + start: 14364 + via: statement-aux-label-proxy + sourceStatement: + chapter: 6 + file: newcompar.tex + label: '' + mtref: '6.11' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 14389 + sourceTransform: tex2mdx.strip_comments + start: 13955 + targetStatement: + chapter: 6 + file: newcompar.tex + label: '' + mtref: '6.10' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 12573 + sourceTransform: tex2mdx.strip_comments + start: 11870 + type: tex-reference + via: + - statement-aux-label-proxy +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 200f280a8e3b +- deb9b83d83f2 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- statement-aux-label-proxy +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/c859a0cbc339.md b/projects/poincare-conjecture/.astrolabe/edges/c859a0cbc339.md new file mode 100644 index 00000000..30cf1a7e --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/c859a0cbc339.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 3 + file: flowbasics.tex + label: '' + mtref: '3.8' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 9455 + sourceTransform: tex2mdx.strip_comments + start: 9202 + targetStatement: + chapter: 3 + file: flowbasics.tex + label: GSSgeneration + mtref: '3.9' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 9814 + sourceTransform: tex2mdx.strip_comments + start: 9457 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 6ace86bb0b70 +- 8a667675febd +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/c8631fa72e55.md b/projects/poincare-conjecture/.astrolabe/edges/c8631fa72e55.md new file mode 100644 index 00000000..75181d87 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/c8631fa72e55.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: volinj + proofSpan: + bodyEnd: 15714 + bodyStart: 11152 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 15725 + file: converge2.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 11139 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 11375 + file: converge2.tex + sourceTransform: tex2mdx.strip_comments + start: 11363 + via: proof + sourceStatement: + chapter: 5 + file: converge2.tex + label: '' + mtref: '5.7' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 10866 + sourceTransform: tex2mdx.strip_comments + start: 10009 + targetStatement: + chapter: 1 + file: prelim.tex + label: volinj + mtref: '1.36' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 46059 + sourceTransform: tex2mdx.strip_comments + start: 45642 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- e5006fafe096 +- f90cce212e88 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/c880ef53087d.md b/projects/poincare-conjecture/.astrolabe/edges/c880ef53087d.md new file mode 100644 index 00000000..8bf173aa --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/c880ef53087d.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: asympvol + proofSpan: + bodyEnd: 89985 + bodyStart: 86085 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 89996 + file: stdsoln.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 86072 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 88669 + file: stdsoln.tex + sourceTransform: tex2mdx.strip_comments + start: 88655 + via: proof + sourceStatement: + chapter: 12 + file: stdsoln.tex + label: '' + mtref: '12.29' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 86070 + sourceTransform: tex2mdx.strip_comments + start: 86020 + targetStatement: + chapter: 9 + file: temp2kappa.tex + label: asympvol + mtref: '9.59' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 118207 + sourceTransform: tex2mdx.strip_comments + start: 117600 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 729d9a0428c4 +- 7468282bd017 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/c8819f02c1d9.md b/projects/poincare-conjecture/.astrolabe/edges/c8819f02c1d9.md new file mode 100644 index 00000000..123105c2 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/c8819f02c1d9.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: Harnack2 + proofSpan: + bodyEnd: 137790 + bodyStart: 127828 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 137801 + file: temp2kappa.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 127815 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 134439 + file: temp2kappa.tex + sourceTransform: tex2mdx.strip_comments + start: 134425 + via: proof + sourceStatement: + chapter: 9 + file: temp2kappa.tex + label: proposition + mtref: '9.65' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 127812 + sourceTransform: tex2mdx.strip_comments + start: 127549 + targetStatement: + chapter: 4 + file: maxprin.tex + label: Harnack2 + mtref: '4.40' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 53826 + sourceTransform: tex2mdx.strip_comments + start: 53456 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- ee314aacd207 +- 6b21448ff214 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/c894dd37cbd3.md b/projects/poincare-conjecture/.astrolabe/edges/c894dd37cbd3.md new file mode 100644 index 00000000..2e98312d --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/c894dd37cbd3.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 5 + file: converge2.tex + label: '' + mtref: '5.30' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 43701 + sourceTransform: tex2mdx.strip_comments + start: 43640 + targetStatement: + chapter: 5 + file: converge2.tex + label: conelimit + mtref: '5.31' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 45881 + sourceTransform: tex2mdx.strip_comments + start: 45285 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- cb86e24eee63 +- 88a75d6dd09e +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/c8d8e1d03c7a.md b/projects/poincare-conjecture/.astrolabe/edges/c8d8e1d03c7a.md new file mode 100644 index 00000000..1b29a112 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/c8d8e1d03c7a.md @@ -0,0 +1,45 @@ +--- +confidence: 0.2 +edgeClass: structural +evidence: + sourceStatement: + chapter: 18 + file: energy1.tex + label: '' + mtref: '18.15' + sort: remark + span: + coordinateSpace: comment-stripped-tex + end: 46028 + sourceTransform: tex2mdx.strip_comments + start: 45155 + targetStatement: + chapter: 18 + file: energy1.tex + label: '' + mtref: '18.16' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 47275 + sourceTransform: tex2mdx.strip_comments + start: 47013 + type: chapter-reading-order-adjacency + via: + - chapter-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- e800ffebd2ce +- 39199318fe64 +rel: in-chapter +reviewStatus: accepted +scope: chapter +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- chapter-sequence +--- +Morgan--Tian reading-order adjacency across sections in a chapter. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/c8e81cd5e069.md b/projects/poincare-conjecture/.astrolabe/edges/c8e81cd5e069.md new file mode 100644 index 00000000..8f6abe36 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/c8e81cd5e069.md @@ -0,0 +1,62 @@ +--- +confidence: 0.9 +edgeClass: structural +evidence: + containmentDepth: 1 + proofSpan: + bodyEnd: 184898 + bodyStart: 173854 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 184909 + file: energy1.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 173841 + sourceStatement: + chapter: 18 + file: energy1.tex + label: Dxacross + mtref: '18.74' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 173839 + sourceTransform: tex2mdx.strip_comments + start: 173679 + targetContainedSpan: + coordinateSpace: comment-stripped-tex + end: 177088 + file: energy1.tex + sourceTransform: tex2mdx.strip_comments + start: 176809 + targetStatement: + chapter: 18 + file: energy1.tex + label: limittri + mtref: '18.76' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 177088 + sourceTransform: tex2mdx.strip_comments + start: 176809 + type: proof-containment + via: + - proof-contains +generator: tools/poincare_tex_extract.py +inference: structural +kind: containment +ref: +- 0d6f74e37efb +- 2ff35cef8589 +rel: contains +reviewStatus: accepted +scope: proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof-contains +--- +Morgan--Tian proof containment. The source is the statement whose proof contains the target statement. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/c9291e1d4bc7.md b/projects/poincare-conjecture/.astrolabe/edges/c9291e1d4bc7.md new file mode 100644 index 00000000..04b9789a --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/c9291e1d4bc7.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: nuprime>0 + proofSpan: + bodyEnd: 24647 + bodyStart: 23636 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 24658 + file: surgery.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 23623 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 24601 + file: surgery.tex + sourceTransform: tex2mdx.strip_comments + start: 24586 + via: proof + sourceStatement: + chapter: 13 + file: surgery.tex + label: '' + mtref: '13.14' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 23621 + sourceTransform: tex2mdx.strip_comments + start: 23086 + targetStatement: + chapter: 13 + file: surgery.tex + label: nuprime>0 + mtref: '13.13' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 21433 + sourceTransform: tex2mdx.strip_comments + start: 21250 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- c9d8cb9d34c3 +- cc9110b4d2fe +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/c92d67bfe749.md b/projects/poincare-conjecture/.astrolabe/edges/c92d67bfe749.md new file mode 100644 index 00000000..e6d19887 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/c92d67bfe749.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 11 + file: singlimit2.tex + label: distch + mtref: '11.12' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 20248 + sourceTransform: tex2mdx.strip_comments + start: 20027 + targetStatement: + chapter: 11 + file: singlimit2.tex + label: '' + mtref: '11.13' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 23634 + sourceTransform: tex2mdx.strip_comments + start: 23355 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 002f3496c54c +- 385a31c5d191 +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/c966e7aad4d6.md b/projects/poincare-conjecture/.astrolabe/edges/c966e7aad4d6.md new file mode 100644 index 00000000..32d6f8a9 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/c966e7aad4d6.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: geoexist + proofSpan: + bodyEnd: 28599 + bodyStart: 25964 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 28610 + file: newcomp2.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 25951 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 26384 + file: newcomp2.tex + sourceTransform: tex2mdx.strip_comments + start: 26370 + via: proof + sourceStatement: + chapter: 7 + file: newcomp2.tex + label: phisupport + mtref: '7.15' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 25949 + sourceTransform: tex2mdx.strip_comments + start: 25492 + targetStatement: + chapter: 7 + file: newcomp2.tex + label: geoexist + mtref: '7.2' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 2760 + sourceTransform: tex2mdx.strip_comments + start: 2498 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 5e0538ed62cf +- cd062526b91a +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/c98a81eaee50.md b/projects/poincare-conjecture/.astrolabe/edges/c98a81eaee50.md new file mode 100644 index 00000000..d793a4b3 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/c98a81eaee50.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: kappa0 + proofSpan: + bodyEnd: 176513 + bodyStart: 172329 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 176524 + file: temp2kappa.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 172316 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 173343 + file: temp2kappa.tex + sourceTransform: tex2mdx.strip_comments + start: 173331 + via: proof + sourceStatement: + chapter: 9 + file: temp2kappa.tex + label: thmep3 + mtref: '9.89' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 172313 + sourceTransform: tex2mdx.strip_comments + start: 171695 + targetStatement: + chapter: 9 + file: temp2kappa.tex + label: kappa0 + mtref: '9.58' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 113708 + sourceTransform: tex2mdx.strip_comments + start: 113563 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- a4ea411d0765 +- 9c54e0d0c7d8 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/c994d0e047cb.md b/projects/poincare-conjecture/.astrolabe/edges/c994d0e047cb.md new file mode 100644 index 00000000..510f6e07 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/c994d0e047cb.md @@ -0,0 +1,62 @@ +--- +confidence: 0.35 +edgeClass: prose +evidence: + mentionTriggers: + - label: KAPPALIMIT + refSpan: + coordinateSpace: comment-stripped-tex + end: 183844 + file: surgery.tex + sourceTransform: tex2mdx.strip_comments + start: 183828 + sentence: This completes the proof of Proposition~\ref{KAPPALIMIT}. + sentenceSpan: + coordinateSpace: comment-stripped-tex + end: 183845 + file: surgery.tex + sourceTransform: tex2mdx.strip_comments + start: 183787 + sourceStrategy: nearest-previous-statement + via: prose + sourceStatement: + chapter: 16 + file: surgery.tex + label: '' + mtref: '16.28' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 176062 + sourceTransform: tex2mdx.strip_comments + start: 175778 + targetStatement: + chapter: 16 + file: surgery.tex + label: KAPPALIMIT + mtref: '16.1' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 103576 + sourceTransform: tex2mdx.strip_comments + start: 102336 + type: prose-mention-nearest-previous-statement + via: + - prose +generator: tools/poincare_tex_extract.py +inference: weak +kind: mention +ref: +- 3e9580da0ff5 +- 4ce8b5da9c58 +rel: mentions +reviewStatus: candidate +scope: prose +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- prose +--- +Morgan--Tian prose mention. The source is the nearest preceding statement in the same chapter segment. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/c9b319e56465.md b/projects/poincare-conjecture/.astrolabe/edges/c9b319e56465.md new file mode 100644 index 00000000..52b16845 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/c9b319e56465.md @@ -0,0 +1,56 @@ +--- +confidence: 0.85 +edgeClass: semantic +evidence: + containmentReason: target theorem-like statement occurs inside source proof + proofSpan: + bodyEnd: 29291 + bodyStart: 25300 + coordinateSpace: comment-stripped-tex + depth: 3 + end: 29302 + file: singlimit2.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 25287 + sourceStatement: + chapter: 11 + file: singlimit2.tex + label: outerbd + mtref: '11.14' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 25285 + sourceTransform: tex2mdx.strip_comments + start: 24900 + targetStatement: + chapter: 11 + file: singlimit2.tex + label: '' + mtref: '11.15' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 26714 + sourceTransform: tex2mdx.strip_comments + start: 26365 + type: proof-contained-theorem-like-statement + via: + - proof-contained-statement +generator: tools/poincare_tex_extract.py +inference: inferred +kind: dependency +ref: +- 9a7e3fe2e5c4 +- b3fae89d3264 +rel: depends +reviewStatus: accepted +scope: proof-containment +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof-contained-statement +--- +Morgan--Tian proof-contained statement dependency. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/c9cb855c12c6.md b/projects/poincare-conjecture/.astrolabe/edges/c9cb855c12c6.md new file mode 100644 index 00000000..a3f49f08 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/c9cb855c12c6.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 6 + file: newcompar.tex + label: L_x + mtref: '6.26' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 36613 + sourceTransform: tex2mdx.strip_comments + start: 36222 + targetStatement: + chapter: 6 + file: newcompar.tex + label: '' + mtref: '6.27' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 37340 + sourceTransform: tex2mdx.strip_comments + start: 36945 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 440e7d27aee8 +- 5578a00f3308 +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/c9f1f3f4bda5.md b/projects/poincare-conjecture/.astrolabe/edges/c9f1f3f4bda5.md new file mode 100644 index 00000000..1f312367 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/c9f1f3f4bda5.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 8 + file: noncoll.tex + label: '' + mtref: '8.6' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 8866 + sourceTransform: tex2mdx.strip_comments + start: 8367 + targetStatement: + chapter: 8 + file: noncoll.tex + label: inside + mtref: '8.7' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 10980 + sourceTransform: tex2mdx.strip_comments + start: 10665 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 59189adac6be +- 0a1eac346e21 +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/c9f326d011da.md b/projects/poincare-conjecture/.astrolabe/edges/c9f326d011da.md new file mode 100644 index 00000000..547522b0 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/c9f326d011da.md @@ -0,0 +1,122 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: Fm + proofSpan: + bodyEnd: 53194 + bodyStart: 43311 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 53205 + file: flowbasics.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 43298 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 46272 + file: flowbasics.tex + sourceTransform: tex2mdx.strip_comments + start: 46264 + via: proof + - context: proof + label: Fm + proofSpan: + bodyEnd: 53194 + bodyStart: 43311 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 53205 + file: flowbasics.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 43298 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 48538 + file: flowbasics.tex + sourceTransform: tex2mdx.strip_comments + start: 48530 + via: proof + - context: proof + label: Fm + proofSpan: + bodyEnd: 53194 + bodyStart: 43311 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 53205 + file: flowbasics.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 43298 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 49494 + file: flowbasics.tex + sourceTransform: tex2mdx.strip_comments + start: 49486 + via: proof + - context: proof + label: Fm + proofSpan: + bodyEnd: 53194 + bodyStart: 43311 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 53205 + file: flowbasics.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 43298 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 53192 + file: flowbasics.tex + sourceTransform: tex2mdx.strip_comments + start: 53184 + via: proof + sourceStatement: + chapter: 3 + file: flowbasics.tex + label: Cinftybd + mtref: '3.31' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 43134 + sourceTransform: tex2mdx.strip_comments + start: 42648 + targetStatement: + chapter: 3 + file: flowbasics.tex + label: Fm + mtref: '3.32' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 46079 + sourceTransform: tex2mdx.strip_comments + start: 45486 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- c7a0e64aa85c +- e7fe06259f43 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/c9f49a4d2787.md b/projects/poincare-conjecture/.astrolabe/edges/c9f49a4d2787.md new file mode 100644 index 00000000..680a4a07 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/c9f49a4d2787.md @@ -0,0 +1,45 @@ +--- +confidence: 0.2 +edgeClass: structural +evidence: + sourceStatement: + chapter: 6 + file: newcompar.tex + label: Ltau + mtref: '6.32' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 45443 + sourceTransform: tex2mdx.strip_comments + start: 45026 + targetStatement: + chapter: 6 + file: newcompar.tex + label: 2ndvariation + mtref: '6.33' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 47655 + sourceTransform: tex2mdx.strip_comments + start: 46570 + type: chapter-reading-order-adjacency + via: + - chapter-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- b3bf2f526cfc +- 3add96057999 +rel: in-chapter +reviewStatus: accepted +scope: chapter +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- chapter-sequence +--- +Morgan--Tian reading-order adjacency across sections in a chapter. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/ca36a28e47ec.md b/projects/poincare-conjecture/.astrolabe/edges/ca36a28e47ec.md new file mode 100644 index 00000000..5518b013 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/ca36a28e47ec.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 9 + file: temp2kappa.tex + label: compactkappa + mtref: '9.64' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 127434 + sourceTransform: tex2mdx.strip_comments + start: 127205 + targetStatement: + chapter: 9 + file: temp2kappa.tex + label: proposition + mtref: '9.65' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 127812 + sourceTransform: tex2mdx.strip_comments + start: 127549 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 49e865228b19 +- ee314aacd207 +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/ca39d46ebc18.md b/projects/poincare-conjecture/.astrolabe/edges/ca39d46ebc18.md new file mode 100644 index 00000000..943dcc65 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/ca39d46ebc18.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: 3Dinfty + proofSpan: + bodyEnd: 103549 + bodyStart: 101913 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 103560 + file: temp2kappa.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 101900 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 102746 + file: temp2kappa.tex + sourceTransform: tex2mdx.strip_comments + start: 102733 + via: proof + sourceStatement: + chapter: 9 + file: temp2kappa.tex + label: 3DGSSkappa + mtref: '9.53' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 101898 + sourceTransform: tex2mdx.strip_comments + start: 101535 + targetStatement: + chapter: 9 + file: temp2kappa.tex + label: 3Dinfty + mtref: '9.52' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 100781 + sourceTransform: tex2mdx.strip_comments + start: 100280 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 351e4ac14e4e +- f8ff249efb81 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/ca3a374b474c.md b/projects/poincare-conjecture/.astrolabe/edges/ca3a374b474c.md new file mode 100644 index 00000000..b750369b --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/ca3a374b474c.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: Harnack + proofSpan: + bodyEnd: 33794 + bodyStart: 17659 + coordinateSpace: comment-stripped-tex + depth: 2 + end: 33805 + file: singlimit2.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 17646 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 33637 + file: singlimit2.tex + sourceTransform: tex2mdx.strip_comments + start: 33624 + via: proof + sourceStatement: + chapter: 11 + file: singlimit2.tex + label: maxT + mtref: '11.10' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 17643 + sourceTransform: tex2mdx.strip_comments + start: 17183 + targetStatement: + chapter: 4 + file: maxprin.tex + label: Harnack + mtref: '4.37' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 52804 + sourceTransform: tex2mdx.strip_comments + start: 52280 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- d6919616448d +- aa9d836895d8 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/ca49fea9dd99.md b/projects/poincare-conjecture/.astrolabe/edges/ca49fea9dd99.md new file mode 100644 index 00000000..628908c7 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/ca49fea9dd99.md @@ -0,0 +1,45 @@ +--- +confidence: 0.2 +edgeClass: structural +evidence: + sourceStatement: + chapter: 14 + file: surgery.tex + label: '' + mtref: '14.12' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 52840 + sourceTransform: tex2mdx.strip_comments + start: 52271 + targetStatement: + chapter: 14 + file: surgery.tex + label: '' + mtref: '14.13' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 53922 + sourceTransform: tex2mdx.strip_comments + start: 53648 + type: chapter-reading-order-adjacency + via: + - chapter-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- cff982ed3611 +- c353298169dd +rel: in-chapter +reviewStatus: accepted +scope: chapter +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- chapter-sequence +--- +Morgan--Tian reading-order adjacency across sections in a chapter. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/cab8303850a0.md b/projects/poincare-conjecture/.astrolabe/edges/cab8303850a0.md new file mode 100644 index 00000000..97eaeca9 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/cab8303850a0.md @@ -0,0 +1,45 @@ +--- +confidence: 0.2 +edgeClass: structural +evidence: + sourceStatement: + chapter: 6 + file: newcompar.tex + label: posform + mtref: '6.21' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 29042 + sourceTransform: tex2mdx.strip_comments + start: 28516 + targetStatement: + chapter: 6 + file: newcompar.tex + label: DtildeL + mtref: '6.22' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 30026 + sourceTransform: tex2mdx.strip_comments + start: 29708 + type: chapter-reading-order-adjacency + via: + - chapter-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- faca28b0bd43 +- abf9cbb10c68 +rel: in-chapter +reviewStatus: accepted +scope: chapter +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- chapter-sequence +--- +Morgan--Tian reading-order adjacency across sections in a chapter. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/cabc9b69aba9.md b/projects/poincare-conjecture/.astrolabe/edges/cabc9b69aba9.md new file mode 100644 index 00000000..4899fa1e --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/cabc9b69aba9.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: canonvary + proofSpan: + bodyEnd: 34517 + bodyStart: 26920 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 34528 + file: bddcurvbdddist.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 26907 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 28096 + file: bddcurvbdddist.tex + sourceTransform: tex2mdx.strip_comments + start: 28081 + via: proof + sourceStatement: + chapter: 10 + file: bddcurvbdddist.tex + label: Uinfty + mtref: '10.7' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 26826 + sourceTransform: tex2mdx.strip_comments + start: 25406 + targetStatement: + chapter: 9 + file: temp2kappa.tex + label: canonvary + mtref: '9.79' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 149236 + sourceTransform: tex2mdx.strip_comments + start: 147728 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- e2e64cdb0b75 +- 7c2758f0e2b3 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/cac27d732827.md b/projects/poincare-conjecture/.astrolabe/edges/cac27d732827.md new file mode 100644 index 00000000..ecc52889 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/cac27d732827.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: gradshrink + proofSpan: + bodyEnd: 63561 + bodyStart: 62714 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 63572 + file: temp2kappa.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 62701 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 62901 + file: temp2kappa.tex + sourceTransform: tex2mdx.strip_comments + start: 62885 + via: proof + sourceStatement: + chapter: 9 + file: temp2kappa.tex + label: '' + mtref: '9.36' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 62699 + sourceTransform: tex2mdx.strip_comments + start: 62344 + targetStatement: + chapter: 9 + file: temp2kappa.tex + label: gradshrink + mtref: '9.32' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 52719 + sourceTransform: tex2mdx.strip_comments + start: 52484 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 64100c8689ff +- 07232fc42f3a +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/cae69859f696.md b/projects/poincare-conjecture/.astrolabe/edges/cae69859f696.md new file mode 100644 index 00000000..13d2a157 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/cae69859f696.md @@ -0,0 +1,45 @@ +--- +confidence: 0.2 +edgeClass: structural +evidence: + sourceStatement: + chapter: 10 + file: bddcurvbdddist.tex + label: bcbd + mtref: '10.2' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 2942 + sourceTransform: tex2mdx.strip_comments + start: 2168 + targetStatement: + chapter: 10 + file: bddcurvbdddist.tex + label: '' + mtref: '10.3' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 9447 + sourceTransform: tex2mdx.strip_comments + start: 9272 + type: chapter-reading-order-adjacency + via: + - chapter-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- ecd62fc2ea2b +- 31216ebde119 +rel: in-chapter +reviewStatus: accepted +scope: chapter +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- chapter-sequence +--- +Morgan--Tian reading-order adjacency across sections in a chapter. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/caefd13d7e5f.md b/projects/poincare-conjecture/.astrolabe/edges/caefd13d7e5f.md new file mode 100644 index 00000000..913ef5aa --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/caefd13d7e5f.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 11 + file: singlimit2.tex + label: '' + mtref: '11.13' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 23634 + sourceTransform: tex2mdx.strip_comments + start: 23355 + targetStatement: + chapter: 11 + file: singlimit2.tex + label: outerbd + mtref: '11.14' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 25285 + sourceTransform: tex2mdx.strip_comments + start: 24900 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 385a31c5d191 +- 9a7e3fe2e5c4 +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/cb079a88b462.md b/projects/poincare-conjecture/.astrolabe/edges/cb079a88b462.md new file mode 100644 index 00000000..cfcef455 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/cb079a88b462.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: Ccomponent + proofSpan: + bodyEnd: 204939 + bodyStart: 186733 + coordinateSpace: comment-stripped-tex + depth: 2 + end: 204950 + file: surgery.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 186720 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 189618 + file: surgery.tex + sourceTransform: tex2mdx.strip_comments + start: 189602 + via: proof + sourceStatement: + chapter: 17 + file: surgery.tex + label: '16.2' + mtref: '17.2' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 186718 + sourceTransform: tex2mdx.strip_comments + start: 186200 + targetStatement: + chapter: 9 + file: temp2kappa.tex + label: Ccomponent + mtref: '9.75' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 144233 + sourceTransform: tex2mdx.strip_comments + start: 143625 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 778d690c4e7c +- 50312568fd70 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/cb174f68fdcb.md b/projects/poincare-conjecture/.astrolabe/edges/cb174f68fdcb.md new file mode 100644 index 00000000..d90e9c10 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/cb174f68fdcb.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: topsplit + proofSpan: + bodyEnd: 71821 + bodyStart: 68779 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 71832 + file: temp2kappa.tex + ownerLabel: prodatinf + sourceTransform: tex2mdx.strip_comments + start: 68766 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 71211 + file: temp2kappa.tex + sourceTransform: tex2mdx.strip_comments + start: 71197 + via: proof + sourceStatement: + chapter: 9 + file: temp2kappa.tex + label: prodatinf + mtref: '9.39' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 68632 + sourceTransform: tex2mdx.strip_comments + start: 67563 + targetStatement: + chapter: 5 + file: converge2.tex + label: topsplit + mtref: '5.35' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 55387 + sourceTransform: tex2mdx.strip_comments + start: 54529 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 703d5ca9da05 +- e9a9172d16e2 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/cb53c11ca83e.md b/projects/poincare-conjecture/.astrolabe/edges/cb53c11ca83e.md new file mode 100644 index 00000000..d9fa6476 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/cb53c11ca83e.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: SMPcurv + proofSpan: + bodyEnd: 30823 + bodyStart: 25720 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 30834 + file: maxprin.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 25707 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 26713 + file: maxprin.tex + sourceTransform: tex2mdx.strip_comments + start: 26700 + via: proof + sourceStatement: + chapter: 4 + file: maxprin.tex + label: localprod + mtref: '4.19' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 25705 + sourceTransform: tex2mdx.strip_comments + start: 25065 + targetStatement: + chapter: 4 + file: maxprin.tex + label: SMPcurv + mtref: '4.17' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 21926 + sourceTransform: tex2mdx.strip_comments + start: 20592 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 8ec855a2aab5 +- f696c0fa9db3 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/cb687846ec81.md b/projects/poincare-conjecture/.astrolabe/edges/cb687846ec81.md new file mode 100644 index 00000000..29a2d15f --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/cb687846ec81.md @@ -0,0 +1,94 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: statement + label: smlmtflow + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 16226 + file: singlimit2.tex + sourceTransform: tex2mdx.strip_comments + start: 16211 + via: statement + - context: statement + label: smlmtflow + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 16312 + file: singlimit2.tex + sourceTransform: tex2mdx.strip_comments + start: 16297 + via: statement + - context: statement + label: smlmtflow + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 16392 + file: singlimit2.tex + sourceTransform: tex2mdx.strip_comments + start: 16377 + via: statement + - context: proof + label: smlmtflow + proofSpan: + bodyEnd: 35374 + bodyStart: 16867 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 35385 + file: singlimit2.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 16854 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 16894 + file: singlimit2.tex + sourceTransform: tex2mdx.strip_comments + start: 16879 + via: proof + sourceStatement: + chapter: 11 + file: singlimit2.tex + label: '' + mtref: '11.9' + sort: remark + span: + coordinateSpace: comment-stripped-tex + end: 16852 + sourceTransform: tex2mdx.strip_comments + start: 16134 + targetStatement: + chapter: 11 + file: singlimit2.tex + label: smlmtflow + mtref: '11.1' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 3459 + sourceTransform: tex2mdx.strip_comments + start: 1498 + type: tex-reference + via: + - proof + - statement +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 5e06d2593bfa +- d51aab8a8065 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +- statement +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/cb7d29d517a8.md b/projects/poincare-conjecture/.astrolabe/edges/cb7d29d517a8.md new file mode 100644 index 00000000..a8b594d7 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/cb7d29d517a8.md @@ -0,0 +1,55 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: statement + label: Theorem1 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 18508 + file: intro.tex + sourceTransform: tex2mdx.strip_comments + start: 18494 + via: statement + sourceStatement: + chapter: 0 + file: intro.tex + label: '' + mtref: '0.6' + sort: remark + span: + coordinateSpace: comment-stripped-tex + end: 19243 + sourceTransform: tex2mdx.strip_comments + start: 16436 + targetStatement: + chapter: 0 + file: intro.tex + label: Theorem1 + mtref: '0.1' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 2955 + sourceTransform: tex2mdx.strip_comments + start: 2574 + type: tex-reference + via: + - statement +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 48f3d2b5dea3 +- 9999efa2be2c +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- statement +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/cb93fa59a48b.md b/projects/poincare-conjecture/.astrolabe/edges/cb93fa59a48b.md new file mode 100644 index 00000000..3dc953ca --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/cb93fa59a48b.md @@ -0,0 +1,64 @@ +--- +confidence: 0.35 +edgeClass: prose +evidence: + mentionTriggers: + - label: fordiffmax + refSpan: + coordinateSpace: comment-stripped-tex + end: 30603 + file: stdsoln.tex + sourceTransform: tex2mdx.strip_comments + start: 30587 + sentence: By virtue of (\ref{eq.7}) and Proposition~\ref{fordiffmax}, the function + $h(t)$ satisfies $dh/dt\le Ch$ in the sense of forward difference quotients, + so that $d(e^{-Ct}h)/dt\le 0$, also in the sense of forward difference quotients + sentenceSpan: + coordinateSpace: comment-stripped-tex + end: 30775 + file: stdsoln.tex + sourceTransform: tex2mdx.strip_comments + start: 30544 + sourceStrategy: nearest-previous-statement + via: prose + sourceStatement: + chapter: 12 + file: stdsoln.tex + label: '' + mtref: '12.11' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 23953 + sourceTransform: tex2mdx.strip_comments + start: 23764 + targetStatement: + chapter: 2 + file: prelim.tex + label: fordiffmax + mtref: '2.23' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 83307 + sourceTransform: tex2mdx.strip_comments + start: 82254 + type: prose-mention-nearest-previous-statement + via: + - prose +generator: tools/poincare_tex_extract.py +inference: weak +kind: mention +ref: +- a5e421ef8adf +- 37d7d059a526 +rel: mentions +reviewStatus: candidate +scope: prose +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- prose +--- +Morgan--Tian prose mention. The source is the nearest preceding statement in the same chapter segment. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/cbac8e124dcb.md b/projects/poincare-conjecture/.astrolabe/edges/cbac8e124dcb.md new file mode 100644 index 00000000..2344c8b5 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/cbac8e124dcb.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 19 + file: canonnbhd.tex + label: sinside + mtref: '19.24' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 45026 + sourceTransform: tex2mdx.strip_comments + start: 44680 + targetStatement: + chapter: 19 + file: canonnbhd.tex + label: epstopology + mtref: '19.25' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 54460 + sourceTransform: tex2mdx.strip_comments + start: 53603 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 1b6c3ac94c37 +- fa8d9a43f34a +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/cc18dd5dfd6f.md b/projects/poincare-conjecture/.astrolabe/edges/cc18dd5dfd6f.md new file mode 100644 index 00000000..175390ef --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/cc18dd5dfd6f.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: neckgeo + proofSpan: + bodyEnd: 30261 + bodyStart: 28891 + coordinateSpace: comment-stripped-tex + depth: 2 + end: 30272 + file: bddcurvbdddist.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 28878 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 29299 + file: bddcurvbdddist.tex + sourceTransform: tex2mdx.strip_comments + start: 29286 + via: proof + sourceStatement: + chapter: 10 + file: bddcurvbdddist.tex + label: '' + mtref: '10.8' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 28876 + sourceTransform: tex2mdx.strip_comments + start: 28772 + targetStatement: + chapter: 19 + file: canonnbhd.tex + label: neckgeo + mtref: '19.5' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 8075 + sourceTransform: tex2mdx.strip_comments + start: 7413 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 2b1f8205acbe +- b74cd360f72f +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/cc1d34c65b29.md b/projects/poincare-conjecture/.astrolabe/edges/cc1d34c65b29.md new file mode 100644 index 00000000..a48c9ac9 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/cc1d34c65b29.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 9 + file: temp2kappa.tex + label: '' + mtref: '9.6' + sort: example + span: + coordinateSpace: comment-stripped-tex + end: 5640 + sourceTransform: tex2mdx.strip_comments + start: 5387 + targetStatement: + chapter: 9 + file: temp2kappa.tex + label: '' + mtref: '9.7' + sort: example + span: + coordinateSpace: comment-stripped-tex + end: 5908 + sourceTransform: tex2mdx.strip_comments + start: 5655 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 6345ee49174f +- 6257843023a2 +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/cc4799fcb1f1.md b/projects/poincare-conjecture/.astrolabe/edges/cc4799fcb1f1.md new file mode 100644 index 00000000..28450fdb --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/cc4799fcb1f1.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 1 + file: prelim.tex + label: '' + mtref: '1.28' + sort: exercise + span: + coordinateSpace: comment-stripped-tex + end: 38721 + sourceTransform: tex2mdx.strip_comments + start: 38523 + targetStatement: + chapter: 1 + file: prelim.tex + label: calabi + mtref: '1.29' + sort: remark + span: + coordinateSpace: comment-stripped-tex + end: 39490 + sourceTransform: tex2mdx.strip_comments + start: 38777 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 53758e75bb17 +- 9f5db77333af +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/cc610fecb695.md b/projects/poincare-conjecture/.astrolabe/edges/cc610fecb695.md new file mode 100644 index 00000000..e19ff78d --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/cc610fecb695.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 9 + file: temp2kappa.tex + label: linfest + mtref: '9.18' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 24152 + sourceTransform: tex2mdx.strip_comments + start: 23894 + targetStatement: + chapter: 9 + file: temp2kappa.tex + label: '' + mtref: '9.19' + sort: remark + span: + coordinateSpace: comment-stripped-tex + end: 24392 + sourceTransform: tex2mdx.strip_comments + start: 24247 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- ff5eee52ee0e +- adb6cc2705f2 +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/cc78c21c5dae.md b/projects/poincare-conjecture/.astrolabe/edges/cc78c21c5dae.md new file mode 100644 index 00000000..3e041058 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/cc78c21c5dae.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: pp + proofSpan: + bodyEnd: 71821 + bodyStart: 68779 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 71832 + file: temp2kappa.tex + ownerLabel: prodatinf + sourceTransform: tex2mdx.strip_comments + start: 68766 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 69223 + file: temp2kappa.tex + sourceTransform: tex2mdx.strip_comments + start: 69215 + via: proof + sourceStatement: + chapter: 9 + file: temp2kappa.tex + label: prodatinf + mtref: '9.39' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 68632 + sourceTransform: tex2mdx.strip_comments + start: 67563 + targetStatement: + chapter: 9 + file: temp2kappa.tex + label: pp + mtref: '9.38' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 67234 + sourceTransform: tex2mdx.strip_comments + start: 66738 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 703d5ca9da05 +- b71a179769b1 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/cc7e15ddf42f.md b/projects/poincare-conjecture/.astrolabe/edges/cc7e15ddf42f.md new file mode 100644 index 00000000..e33065d7 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/cc7e15ddf42f.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: firststrongmax + proofSpan: + bodyEnd: 34465 + bodyStart: 32850 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 34476 + file: maxprin.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 32837 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 32910 + file: maxprin.tex + sourceTransform: tex2mdx.strip_comments + start: 32890 + via: proof + sourceStatement: + chapter: 4 + file: maxprin.tex + label: nocones + mtref: '4.22' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 32834 + sourceTransform: tex2mdx.strip_comments + start: 32471 + targetStatement: + chapter: 4 + file: maxprin.tex + label: firststrongmax + mtref: '4.18' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 23916 + sourceTransform: tex2mdx.strip_comments + start: 23628 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 92b5c664399e +- 016cf36686b2 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/ccc7e8df4167.md b/projects/poincare-conjecture/.astrolabe/edges/ccc7e8df4167.md new file mode 100644 index 00000000..b2f59ba4 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/ccc7e8df4167.md @@ -0,0 +1,55 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: statement + label: 3DGSSkappa + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 14329 + file: temp2kappa.tex + sourceTransform: tex2mdx.strip_comments + start: 14313 + via: statement + sourceStatement: + chapter: 9 + file: temp2kappa.tex + label: '' + mtref: '9.12' + sort: remark + span: + coordinateSpace: comment-stripped-tex + end: 14958 + sourceTransform: tex2mdx.strip_comments + start: 13925 + targetStatement: + chapter: 9 + file: temp2kappa.tex + label: 3DGSSkappa + mtref: '9.53' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 101898 + sourceTransform: tex2mdx.strip_comments + start: 101535 + type: tex-reference + via: + - statement +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- bb839ebe8376 +- 351e4ac14e4e +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- statement +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/ccd49705f13d.md b/projects/poincare-conjecture/.astrolabe/edges/ccd49705f13d.md new file mode 100644 index 00000000..c72e8ce1 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/ccd49705f13d.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 6 + file: newcompar.tex + label: adapjab + mtref: '6.41' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 59462 + sourceTransform: tex2mdx.strip_comments + start: 58890 + targetStatement: + chapter: 6 + file: newcompar.tex + label: '' + mtref: '6.42' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 66455 + sourceTransform: tex2mdx.strip_comments + start: 66260 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 72949e6350d4 +- b20ded1b2964 +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/cceaee8b3cda.md b/projects/poincare-conjecture/.astrolabe/edges/cceaee8b3cda.md new file mode 100644 index 00000000..f4d27c21 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/cceaee8b3cda.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 11 + file: singlimit2.tex + label: hexist + mtref: '11.31' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 59267 + sourceTransform: tex2mdx.strip_comments + start: 58375 + targetStatement: + chapter: 11 + file: singlimit2.tex + label: hyp + mtref: '11.32' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 60461 + sourceTransform: tex2mdx.strip_comments + start: 60310 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 074e02b067b8 +- dff170186455 +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/cceb84623fc3.md b/projects/poincare-conjecture/.astrolabe/edges/cceb84623fc3.md new file mode 100644 index 00000000..f6193956 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/cceb84623fc3.md @@ -0,0 +1,45 @@ +--- +confidence: 0.2 +edgeClass: structural +evidence: + sourceStatement: + chapter: 9 + file: temp2kappa.tex + label: GSSclass + mtref: '9.42' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 75321 + sourceTransform: tex2mdx.strip_comments + start: 74299 + targetStatement: + chapter: 9 + file: temp2kappa.tex + label: '' + mtref: '9.43' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 77525 + sourceTransform: tex2mdx.strip_comments + start: 77226 + type: chapter-reading-order-adjacency + via: + - chapter-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 2d68880771ff +- 79547e2cdfd0 +rel: in-chapter +reviewStatus: accepted +scope: chapter +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- chapter-sequence +--- +Morgan--Tian reading-order adjacency across sections in a chapter. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/ccedc94dcac7.md b/projects/poincare-conjecture/.astrolabe/edges/ccedc94dcac7.md new file mode 100644 index 00000000..6be0cdfd --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/ccedc94dcac7.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: controlnbhd + proofSpan: + bodyEnd: 122171 + bodyStart: 120127 + coordinateSpace: comment-stripped-tex + depth: 2 + end: 122182 + file: surgery.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 120114 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 121057 + file: surgery.tex + sourceTransform: tex2mdx.strip_comments + start: 121040 + via: proof + sourceStatement: + chapter: 16 + file: surgery.tex + label: goodtotheta + mtref: '16.9' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 120111 + sourceTransform: tex2mdx.strip_comments + start: 119688 + targetStatement: + chapter: 11 + file: singlimit2.tex + label: controlnbhd + mtref: '11.2' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 4360 + sourceTransform: tex2mdx.strip_comments + start: 3597 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 712124d7ea02 +- 0ddfe578961e +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/ccf3d173dfcc.md b/projects/poincare-conjecture/.astrolabe/edges/ccf3d173dfcc.md new file mode 100644 index 00000000..7bb0470d --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/ccf3d173dfcc.md @@ -0,0 +1,95 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: statement + label: GSSclass + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 103853 + file: temp2kappa.tex + sourceTransform: tex2mdx.strip_comments + start: 103839 + via: statement + - context: proof + label: GSSclass + proofSpan: + bodyEnd: 105941 + bodyStart: 103880 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 105952 + file: temp2kappa.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 103867 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 105048 + file: temp2kappa.tex + sourceTransform: tex2mdx.strip_comments + start: 105034 + via: proof + - context: proof + label: GSSclass + proofSpan: + bodyEnd: 105941 + bodyStart: 103880 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 105952 + file: temp2kappa.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 103867 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 105467 + file: temp2kappa.tex + sourceTransform: tex2mdx.strip_comments + start: 105453 + via: proof + sourceStatement: + chapter: 9 + file: temp2kappa.tex + label: '' + mtref: '9.54' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 103864 + sourceTransform: tex2mdx.strip_comments + start: 103562 + targetStatement: + chapter: 9 + file: temp2kappa.tex + label: GSSclass + mtref: '9.42' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 75321 + sourceTransform: tex2mdx.strip_comments + start: 74299 + type: tex-reference + via: + - proof + - statement +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 92ee3ee82b58 +- 2d68880771ff +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +- statement +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/cd020619b041.md b/projects/poincare-conjecture/.astrolabe/edges/cd020619b041.md new file mode 100644 index 00000000..5ea620d3 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/cd020619b041.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 6 + file: newcompar.tex + label: '' + mtref: '6.52' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 81660 + sourceTransform: tex2mdx.strip_comments + start: 81446 + targetStatement: + chapter: 6 + file: newcompar.tex + label: tildeltau + mtref: '6.53' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 82438 + sourceTransform: tex2mdx.strip_comments + start: 82255 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 77ed5048ae37 +- 5e142c14df9d +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/cd115efe97c0.md b/projects/poincare-conjecture/.astrolabe/edges/cd115efe97c0.md new file mode 100644 index 00000000..52bebd49 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/cd115efe97c0.md @@ -0,0 +1,69 @@ +--- +confidence: 0.72 +edgeClass: semantic +evidence: + anaphoraTriggers: + - context: proof + direction: previous + phrase: 'previous + + proposition' + proofSpan: + bodyEnd: 11582 + bodyStart: 10351 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 11593 + file: temp2kappa.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 10338 + sort: proposition + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 10486 + file: temp2kappa.tex + sourceTransform: tex2mdx.strip_comments + start: 10466 + via: local-anaphora-proof + sourceStatement: + chapter: 9 + file: temp2kappa.tex + label: unianc + mtref: '9.10' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 10335 + sourceTransform: tex2mdx.strip_comments + start: 9624 + targetStatement: + chapter: 9 + file: temp2kappa.tex + label: uniformlip + mtref: '9.9' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 7575 + sourceTransform: tex2mdx.strip_comments + start: 6851 + type: local-anaphora-reference + via: + - local-anaphora-proof +generator: tools/poincare_tex_extract.py +inference: inferred +kind: dependency +ref: +- 8b8e3a0f3e36 +- 9387b1b04824 +rel: depends +reviewStatus: accepted +scope: local-context +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- local-anaphora-proof +--- +Morgan--Tian local anaphora dependency. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/cd360016b261.md b/projects/poincare-conjecture/.astrolabe/edges/cd360016b261.md new file mode 100644 index 00000000..8172db2d --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/cd360016b261.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 3 + file: flowbasics.tex + label: Cinftybd + mtref: '3.31' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 43134 + sourceTransform: tex2mdx.strip_comments + start: 42648 + targetStatement: + chapter: 3 + file: flowbasics.tex + label: Fm + mtref: '3.32' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 46079 + sourceTransform: tex2mdx.strip_comments + start: 45486 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- c7a0e64aa85c +- e7fe06259f43 +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/cd9299b7c5fd.md b/projects/poincare-conjecture/.astrolabe/edges/cd9299b7c5fd.md new file mode 100644 index 00000000..80c75275 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/cd9299b7c5fd.md @@ -0,0 +1,45 @@ +--- +confidence: 0.2 +edgeClass: structural +evidence: + sourceStatement: + chapter: 3 + file: flowbasics.tex + label: strongepsneck + mtref: '3.6' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 6733 + sourceTransform: tex2mdx.strip_comments + start: 5816 + targetStatement: + chapter: 3 + file: flowbasics.tex + label: solitonclaim + mtref: '3.7' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 8537 + sourceTransform: tex2mdx.strip_comments + start: 8364 + type: chapter-reading-order-adjacency + via: + - chapter-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 6d6227668950 +- 7fdbbb57c26b +rel: in-chapter +reviewStatus: accepted +scope: chapter +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- chapter-sequence +--- +Morgan--Tian reading-order adjacency across sections in a chapter. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/cdcc2ef61062.md b/projects/poincare-conjecture/.astrolabe/edges/cdcc2ef61062.md new file mode 100644 index 00000000..f13dc43a --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/cdcc2ef61062.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 18 + file: energy1.tex + label: uniformc + mtref: '18.48' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 102679 + sourceTransform: tex2mdx.strip_comments + start: 102426 + targetStatement: + chapter: 18 + file: energy1.tex + label: Gammaapprox + mtref: '18.49' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 104039 + sourceTransform: tex2mdx.strip_comments + start: 103290 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 028d3b5182b5 +- 7a8f55be175e +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/ce01eeae0fb6.md b/projects/poincare-conjecture/.astrolabe/edges/ce01eeae0fb6.md new file mode 100644 index 00000000..05948931 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/ce01eeae0fb6.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 9 + file: temp2kappa.tex + label: absconv1 + mtref: '9.27' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 44147 + sourceTransform: tex2mdx.strip_comments + start: 43269 + targetStatement: + chapter: 9 + file: temp2kappa.tex + label: absconv + mtref: '9.28' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 45006 + sourceTransform: tex2mdx.strip_comments + start: 44788 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 1766f37a13cf +- 0e9703c15036 +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/ce17e57b21f3.md b/projects/poincare-conjecture/.astrolabe/edges/ce17e57b21f3.md new file mode 100644 index 00000000..1c695a13 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/ce17e57b21f3.md @@ -0,0 +1,65 @@ +--- +confidence: 0.45 +edgeClass: semantic +evidence: + endpointSemanticDegreeBeforeTermBackfill: + source: 0 + target: 7 + selectionReason: semantic-connectivity-backfill + sourceMatchSpan: + coordinateSpace: comment-stripped-tex + end: 71303 + file: flowbasics.tex + sourceTransform: tex2mdx.strip_comments + start: 71284 + sourceStatement: + chapter: 3 + file: flowbasics.tex + label: '' + mtref: '3.40' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 71681 + sourceTransform: tex2mdx.strip_comments + start: 70994 + targetStatement: + chapter: 3 + file: flowbasics.tex + label: '' + mtref: '3.1' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 943 + sourceTransform: tex2mdx.strip_comments + start: 265 + targetTermSpan: + coordinateSpace: comment-stripped-tex + end: 306 + file: flowbasics.tex + sourceTransform: tex2mdx.strip_comments + start: 287 + term: ricci flow equation + termAmbiguity: unique-in-chapter + termSource: italic + termSpecificity: no-longer-defined-term + type: same-chapter-definition-term-match + via: + - definition-term:ricci flow equation +generator: tools/poincare_tex_extract.py +inference: weak +kind: definition-use +ref: +- 1aaf4983e041 +- 7e9caf7ba4da +rel: uses +reviewStatus: candidate +scope: same-chapter-term-match +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- definition-term:ricci flow equation +--- +Morgan--Tian same-chapter definition term match. This is a low-confidence candidate edge for review. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/ce2665235f52.md b/projects/poincare-conjecture/.astrolabe/edges/ce2665235f52.md new file mode 100644 index 00000000..d8f3deb0 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/ce2665235f52.md @@ -0,0 +1,67 @@ +--- +confidence: 0.35 +edgeClass: prose +evidence: + mentionTriggers: + - label: flowextend + refSpan: + coordinateSpace: comment-stripped-tex + end: 10768 + file: stdsoln.tex + sourceTransform: tex2mdx.strip_comments + start: 10752 + sentence: It now follows from Proposition~\ref{flowextend} and the fact that the + $T_n$ are maximal that $T_n>t_0$ for all $n$. Since the Riemann curvatures of + the $(S^3_{R_n},g_{R_n}(t)),\ 0\le t0 + mtref: '4.14' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 16848 + sourceTransform: tex2mdx.strip_comments + start: 16607 + targetStatement: + chapter: 4 + file: maxprin.tex + label: Ric>0 + mtref: '4.15' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 18295 + sourceTransform: tex2mdx.strip_comments + start: 18076 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 0db415314964 +- f8afc1c61977 +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/d49aaab4bdc9.md b/projects/poincare-conjecture/.astrolabe/edges/d49aaab4bdc9.md new file mode 100644 index 00000000..4e381ca9 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/d49aaab4bdc9.md @@ -0,0 +1,45 @@ +--- +confidence: 0.2 +edgeClass: structural +evidence: + sourceStatement: + chapter: 3 + file: flowbasics.tex + label: '' + mtref: '3.37' + sort: remark + span: + coordinateSpace: comment-stripped-tex + end: 68588 + sourceTransform: tex2mdx.strip_comments + start: 68157 + targetStatement: + chapter: 3 + file: flowbasics.tex + label: '' + mtref: '3.38' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 70309 + sourceTransform: tex2mdx.strip_comments + start: 68864 + type: chapter-reading-order-adjacency + via: + - chapter-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- f4523c2b6f1e +- faf16f1a175d +rel: in-chapter +reviewStatus: accepted +scope: chapter +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- chapter-sequence +--- +Morgan--Tian reading-order adjacency across sections in a chapter. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/d4e5310db1ca.md b/projects/poincare-conjecture/.astrolabe/edges/d4e5310db1ca.md new file mode 100644 index 00000000..7f5354b3 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/d4e5310db1ca.md @@ -0,0 +1,62 @@ +--- +confidence: 0.9 +edgeClass: structural +evidence: + containmentDepth: 1 + proofSpan: + bodyEnd: 146058 + bodyStart: 138906 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 146069 + file: surgery.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 138893 + sourceStatement: + chapter: 16 + file: surgery.tex + label: '' + mtref: '16.16' + sort: remark + span: + coordinateSpace: comment-stripped-tex + end: 138750 + sourceTransform: tex2mdx.strip_comments + start: 138540 + targetContainedSpan: + coordinateSpace: comment-stripped-tex + end: 140633 + file: surgery.tex + sourceTransform: tex2mdx.strip_comments + start: 140555 + targetStatement: + chapter: 16 + file: surgery.tex + label: '' + mtref: '16.17' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 140633 + sourceTransform: tex2mdx.strip_comments + start: 140555 + type: proof-containment + via: + - proof-contains +generator: tools/poincare_tex_extract.py +inference: structural +kind: containment +ref: +- 4ee1bc9acc48 +- 9b7f421ab0b9 +rel: contains +reviewStatus: accepted +scope: proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof-contains +--- +Morgan--Tian proof containment. The source is the statement whose proof contains the target statement. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/d50b7b254e19.md b/projects/poincare-conjecture/.astrolabe/edges/d50b7b254e19.md new file mode 100644 index 00000000..ed11739d --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/d50b7b254e19.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 12 + file: stdsoln.tex + label: '12.7' + mtref: '12.15' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 39561 + sourceTransform: tex2mdx.strip_comments + start: 39191 + targetStatement: + chapter: 12 + file: stdsoln.tex + label: dtuniqtouniq + mtref: '12.16' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 40918 + sourceTransform: tex2mdx.strip_comments + start: 40307 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- a8675ea40ba2 +- 5b79e248c5cb +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/d50fc1f75763.md b/projects/poincare-conjecture/.astrolabe/edges/d50fc1f75763.md new file mode 100644 index 00000000..554ced90 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/d50fc1f75763.md @@ -0,0 +1,45 @@ +--- +confidence: 0.2 +edgeClass: structural +evidence: + sourceStatement: + chapter: 11 + file: singlimit2.tex + label: 2epshorns + mtref: '11.30' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 53428 + sourceTransform: tex2mdx.strip_comments + start: 52907 + targetStatement: + chapter: 11 + file: singlimit2.tex + label: hexist + mtref: '11.31' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 59267 + sourceTransform: tex2mdx.strip_comments + start: 58375 + type: chapter-reading-order-adjacency + via: + - chapter-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 01f0d3baa28f +- 074e02b067b8 +rel: in-chapter +reviewStatus: accepted +scope: chapter +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- chapter-sequence +--- +Morgan--Tian reading-order adjacency across sections in a chapter. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/d5143d275500.md b/projects/poincare-conjecture/.astrolabe/edges/d5143d275500.md new file mode 100644 index 00000000..b79c9b9f --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/d5143d275500.md @@ -0,0 +1,84 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: lips + proofSpan: + bodyEnd: 89483 + bodyStart: 87584 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 89494 + file: newcompar.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 87571 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 87646 + file: newcompar.tex + sourceTransform: tex2mdx.strip_comments + start: 87636 + via: proof + - context: proof + label: lips + proofSpan: + bodyEnd: 89483 + bodyStart: 87584 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 89494 + file: newcompar.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 87571 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 87743 + file: newcompar.tex + sourceTransform: tex2mdx.strip_comments + start: 87733 + via: proof + sourceStatement: + chapter: 6 + file: newcompar.tex + label: lipaty + mtref: '6.59' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 87473 + sourceTransform: tex2mdx.strip_comments + start: 86282 + targetStatement: + chapter: 6 + file: newcompar.tex + label: lips + mtref: '6.56' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 85066 + sourceTransform: tex2mdx.strip_comments + start: 84130 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 96ced8c484df +- db4ced24b05f +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/d51c1f39062e.md b/projects/poincare-conjecture/.astrolabe/edges/d51c1f39062e.md new file mode 100644 index 00000000..241e3ef5 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/d51c1f39062e.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 18 + file: energy1.tex + label: '' + mtref: '18.25' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 59849 + sourceTransform: tex2mdx.strip_comments + start: 59775 + targetStatement: + chapter: 18 + file: energy1.tex + label: vclaim + mtref: '18.26' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 60140 + sourceTransform: tex2mdx.strip_comments + start: 59893 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- fba418d1df15 +- eaf31795ca89 +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/d522a00a7bdc.md b/projects/poincare-conjecture/.astrolabe/edges/d522a00a7bdc.md new file mode 100644 index 00000000..7a837f2f --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/d522a00a7bdc.md @@ -0,0 +1,64 @@ +--- +confidence: 0.35 +edgeClass: prose +evidence: + mentionTriggers: + - label: Lgeoexist + refSpan: + coordinateSpace: comment-stripped-tex + end: 182529 + file: surgery.tex + sourceTransform: tex2mdx.strip_comments + start: 182514 + sentence: In fact, by Proposition~\ref{Lgeoexist} there is a compact subset $Y$ + of the open subset of smooth points of ${\mathcal M}$ that contains all the + minimizing ${\mathcal L}$-geodesics from $x$ to points of $B(y,t'',r')$. + sentenceSpan: + coordinateSpace: comment-stripped-tex + end: 182708 + file: surgery.tex + sourceTransform: tex2mdx.strip_comments + start: 182489 + sourceStrategy: nearest-previous-statement + via: prose + sourceStatement: + chapter: 16 + file: surgery.tex + label: '' + mtref: '16.28' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 176062 + sourceTransform: tex2mdx.strip_comments + start: 175778 + targetStatement: + chapter: 16 + file: surgery.tex + label: Lgeoexist + mtref: '16.21' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 154713 + sourceTransform: tex2mdx.strip_comments + start: 152982 + type: prose-mention-nearest-previous-statement + via: + - prose +generator: tools/poincare_tex_extract.py +inference: weak +kind: mention +ref: +- 3e9580da0ff5 +- d98ad25d9bc3 +rel: mentions +reviewStatus: candidate +scope: prose +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- prose +--- +Morgan--Tian prose mention. The source is the nearest preceding statement in the same chapter segment. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/d54a13a8a5fa.md b/projects/poincare-conjecture/.astrolabe/edges/d54a13a8a5fa.md new file mode 100644 index 00000000..e8b9f819 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/d54a13a8a5fa.md @@ -0,0 +1,56 @@ +--- +confidence: 0.85 +edgeClass: semantic +evidence: + containmentReason: target theorem-like statement occurs inside source proof + proofSpan: + bodyEnd: 204939 + bodyStart: 186733 + coordinateSpace: comment-stripped-tex + depth: 2 + end: 204950 + file: surgery.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 186720 + sourceStatement: + chapter: 17 + file: surgery.tex + label: '16.2' + mtref: '17.2' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 186718 + sourceTransform: tex2mdx.strip_comments + start: 186200 + targetStatement: + chapter: 17 + file: surgery.tex + label: '' + mtref: '17.4' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 201153 + sourceTransform: tex2mdx.strip_comments + start: 201044 + type: proof-contained-theorem-like-statement + via: + - proof-contained-statement +generator: tools/poincare_tex_extract.py +inference: inferred +kind: dependency +ref: +- 778d690c4e7c +- 9b87a4e6674a +rel: depends +reviewStatus: accepted +scope: proof-containment +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof-contained-statement +--- +Morgan--Tian proof-contained statement dependency. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/d56af9f40587.md b/projects/poincare-conjecture/.astrolabe/edges/d56af9f40587.md new file mode 100644 index 00000000..e67bd113 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/d56af9f40587.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 16 + file: surgery.tex + label: defnoverlinedelta0 + mtref: '16.12' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 130201 + sourceTransform: tex2mdx.strip_comments + start: 129888 + targetStatement: + chapter: 16 + file: surgery.tex + label: bigell + mtref: '16.13' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 131955 + sourceTransform: tex2mdx.strip_comments + start: 130235 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- b92e1a990021 +- 0a4d6f17810d +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/d5874355ce09.md b/projects/poincare-conjecture/.astrolabe/edges/d5874355ce09.md new file mode 100644 index 00000000..3ad8aeac --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/d5874355ce09.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: 2ndvarj + proofSpan: + bodyEnd: 29412 + bodyStart: 29057 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 29423 + file: newcompar.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 29044 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 29096 + file: newcompar.tex + sourceTransform: tex2mdx.strip_comments + start: 29083 + via: proof + sourceStatement: + chapter: 6 + file: newcompar.tex + label: posform + mtref: '6.21' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 29042 + sourceTransform: tex2mdx.strip_comments + start: 28516 + targetStatement: + chapter: 6 + file: newcompar.tex + label: 2ndvarj + mtref: '6.13' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 16190 + sourceTransform: tex2mdx.strip_comments + start: 15621 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- faca28b0bd43 +- 26b7f69e803e +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/d5a2052a7365.md b/projects/poincare-conjecture/.astrolabe/edges/d5a2052a7365.md new file mode 100644 index 00000000..3310f15a --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/d5a2052a7365.md @@ -0,0 +1,45 @@ +--- +confidence: 0.2 +edgeClass: structural +evidence: + sourceStatement: + chapter: 3 + file: flowbasics.tex + label: '' + mtref: '3.20' + sort: remark + span: + coordinateSpace: comment-stripped-tex + end: 26907 + sourceTransform: tex2mdx.strip_comments + start: 26767 + targetStatement: + chapter: 3 + file: flowbasics.tex + label: I.8.3 + mtref: '3.21' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 27841 + sourceTransform: tex2mdx.strip_comments + start: 27149 + type: chapter-reading-order-adjacency + via: + - chapter-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 27bef66b1ee8 +- 5d895f9c3918 +rel: in-chapter +reviewStatus: accepted +scope: chapter +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- chapter-sequence +--- +Morgan--Tian reading-order adjacency across sections in a chapter. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/d5dac5ec1627.md b/projects/poincare-conjecture/.astrolabe/edges/d5dac5ec1627.md new file mode 100644 index 00000000..5363c890 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/d5dac5ec1627.md @@ -0,0 +1,55 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: statement + label: bigell + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 150957 + file: surgery.tex + sourceTransform: tex2mdx.strip_comments + start: 150945 + via: statement + sourceStatement: + chapter: 16 + file: surgery.tex + label: '' + mtref: '16.20' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 152979 + sourceTransform: tex2mdx.strip_comments + start: 150862 + targetStatement: + chapter: 16 + file: surgery.tex + label: bigell + mtref: '16.13' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 131955 + sourceTransform: tex2mdx.strip_comments + start: 130235 + type: tex-reference + via: + - statement +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 1ff58113d89b +- 0a4d6f17810d +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- statement +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/d5df1d56af9d.md b/projects/poincare-conjecture/.astrolabe/edges/d5df1d56af9d.md new file mode 100644 index 00000000..334cc5a0 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/d5df1d56af9d.md @@ -0,0 +1,65 @@ +--- +confidence: 0.45 +edgeClass: semantic +evidence: + endpointSemanticDegreeBeforeTermBackfill: + source: 3 + target: 0 + selectionReason: semantic-connectivity-backfill + sourceMatchSpan: + coordinateSpace: comment-stripped-tex + end: 13715 + file: maxprin.tex + sourceTransform: tex2mdx.strip_comments + start: 13686 + sourceStatement: + chapter: 4 + file: maxprin.tex + label: kappa0r0t0 + mtref: '4.11' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 13944 + sourceTransform: tex2mdx.strip_comments + start: 13445 + targetStatement: + chapter: 4 + file: maxprin.tex + label: norminitcond + mtref: '4.10' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 13043 + sourceTransform: tex2mdx.strip_comments + start: 12395 + targetTermSpan: + coordinateSpace: comment-stripped-tex + end: 12509 + file: maxprin.tex + sourceTransform: tex2mdx.strip_comments + start: 12480 + term: normalized initial conditions + termAmbiguity: unique-in-chapter + termSource: italic + termSpecificity: no-longer-defined-term + type: same-chapter-definition-term-match + via: + - definition-term:normalized initial conditions +generator: tools/poincare_tex_extract.py +inference: weak +kind: definition-use +ref: +- c1330c5a8be3 +- 12a9b4e99e24 +rel: uses +reviewStatus: candidate +scope: same-chapter-term-match +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- definition-term:normalized initial conditions +--- +Morgan--Tian same-chapter definition term match. This is a low-confidence candidate edge for review. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/d62bf0394d5a.md b/projects/poincare-conjecture/.astrolabe/edges/d62bf0394d5a.md new file mode 100644 index 00000000..68b095e0 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/d62bf0394d5a.md @@ -0,0 +1,63 @@ +--- +confidence: 0.35 +edgeClass: prose +evidence: + mentionTriggers: + - label: GSSclass + refSpan: + coordinateSpace: comment-stripped-tex + end: 94260 + file: temp2kappa.tex + sourceTransform: tex2mdx.strip_comments + start: 94246 + sentence: Let $(M,g)$ be a non-compact manifold of positive curvature satisfying + the hypotheses of Theorem~\ref{GSSclass} + sentenceSpan: + coordinateSpace: comment-stripped-tex + end: 94260 + file: temp2kappa.tex + sourceTransform: tex2mdx.strip_comments + start: 94148 + sourceStrategy: nearest-previous-statement + via: prose + sourceStatement: + chapter: 9 + file: temp2kappa.tex + label: 2DGSS + mtref: '9.50' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 91949 + sourceTransform: tex2mdx.strip_comments + start: 91266 + targetStatement: + chapter: 9 + file: temp2kappa.tex + label: GSSclass + mtref: '9.42' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 75321 + sourceTransform: tex2mdx.strip_comments + start: 74299 + type: prose-mention-nearest-previous-statement + via: + - prose +generator: tools/poincare_tex_extract.py +inference: weak +kind: mention +ref: +- da767c71996b +- 2d68880771ff +rel: mentions +reviewStatus: candidate +scope: prose +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- prose +--- +Morgan--Tian prose mention. The source is the nearest preceding statement in the same chapter segment. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/d695ec4c8857.md b/projects/poincare-conjecture/.astrolabe/edges/d695ec4c8857.md new file mode 100644 index 00000000..8327b819 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/d695ec4c8857.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: flowlimit + proofSpan: + bodyEnd: 132635 + bodyStart: 128771 + coordinateSpace: comment-stripped-tex + depth: 2 + end: 132646 + file: temp2kappa.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 128758 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 130682 + file: temp2kappa.tex + sourceTransform: tex2mdx.strip_comments + start: 130667 + via: proof + sourceStatement: + chapter: 9 + file: temp2kappa.tex + label: 2Rbound + mtref: '9.66' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 128756 + sourceTransform: tex2mdx.strip_comments + start: 128570 + targetStatement: + chapter: 5 + file: converge2.tex + label: flowlimit + mtref: '5.15' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 27817 + sourceTransform: tex2mdx.strip_comments + start: 26565 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- ea9b285e40cb +- a0212d0eefcf +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/d6b10654a0af.md b/projects/poincare-conjecture/.astrolabe/edges/d6b10654a0af.md new file mode 100644 index 00000000..3a56e937 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/d6b10654a0af.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: decrease + proofSpan: + bodyEnd: 3822 + bodyStart: 3529 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 3833 + file: surgery.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 3516 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 3675 + file: surgery.tex + sourceTransform: tex2mdx.strip_comments + start: 3661 + via: proof + sourceStatement: + chapter: 13 + file: surgery.tex + label: distdecrease + mtref: '13.1' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 3514 + sourceTransform: tex2mdx.strip_comments + start: 3099 + targetStatement: + chapter: 12 + file: stdsoln.tex + label: decrease + mtref: '12.37' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 98416 + sourceTransform: tex2mdx.strip_comments + start: 97969 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 6168ffcb3bed +- 545ca0a10566 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/d6c60d2a8394.md b/projects/poincare-conjecture/.astrolabe/edges/d6c60d2a8394.md new file mode 100644 index 00000000..2b0c4ae2 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/d6c60d2a8394.md @@ -0,0 +1,62 @@ +--- +confidence: 0.35 +edgeClass: prose +evidence: + mentionTriggers: + - label: DtildeL + refSpan: + coordinateSpace: comment-stripped-tex + end: 39255 + file: newcompar.tex + sourceTransform: tex2mdx.strip_comments + start: 39242 + sentence: 'According to Lemma~\ref{DtildeL} we have:' + sentenceSpan: + coordinateSpace: comment-stripped-tex + end: 39264 + file: newcompar.tex + sourceTransform: tex2mdx.strip_comments + start: 39222 + sourceStrategy: nearest-previous-statement + via: prose + sourceStatement: + chapter: 6 + file: newcompar.tex + label: diffeo + mtref: '6.28' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 38046 + sourceTransform: tex2mdx.strip_comments + start: 37549 + targetStatement: + chapter: 6 + file: newcompar.tex + label: DtildeL + mtref: '6.22' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 30026 + sourceTransform: tex2mdx.strip_comments + start: 29708 + type: prose-mention-nearest-previous-statement + via: + - prose +generator: tools/poincare_tex_extract.py +inference: weak +kind: mention +ref: +- 17254b50b27a +- abf9cbb10c68 +rel: mentions +reviewStatus: candidate +scope: prose +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- prose +--- +Morgan--Tian prose mention. The source is the nearest preceding statement in the same chapter segment. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/d6dbb93249b9.md b/projects/poincare-conjecture/.astrolabe/edges/d6dbb93249b9.md new file mode 100644 index 00000000..cd57e8ad --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/d6dbb93249b9.md @@ -0,0 +1,56 @@ +--- +confidence: 0.85 +edgeClass: semantic +evidence: + containmentReason: target theorem-like statement occurs inside source proof + proofSpan: + bodyEnd: 229182 + bodyStart: 185422 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 229193 + file: surgery.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 185409 + sourceStatement: + chapter: 17 + file: surgery.tex + label: extend + mtref: '17.1' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 185406 + sourceTransform: tex2mdx.strip_comments + start: 184285 + targetStatement: + chapter: 17 + file: surgery.tex + label: '' + mtref: '17.10' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 223009 + sourceTransform: tex2mdx.strip_comments + start: 222555 + type: proof-contained-theorem-like-statement + via: + - proof-contained-statement +generator: tools/poincare_tex_extract.py +inference: inferred +kind: dependency +ref: +- 254e8dae97e6 +- 3b7da22bf357 +rel: depends +reviewStatus: accepted +scope: proof-containment +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof-contained-statement +--- +Morgan--Tian proof-contained statement dependency. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/d6fda0bde8bd.md b/projects/poincare-conjecture/.astrolabe/edges/d6fda0bde8bd.md new file mode 100644 index 00000000..603fdd71 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/d6fda0bde8bd.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: splitting + proofSpan: + bodyEnd: 65413 + bodyStart: 63751 + coordinateSpace: comment-stripped-tex + depth: 2 + end: 65424 + file: singlimit2.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 63738 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 65162 + file: singlimit2.tex + sourceTransform: tex2mdx.strip_comments + start: 65147 + via: proof + sourceStatement: + chapter: 11 + file: singlimit2.tex + label: '' + mtref: '11.34' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 63736 + sourceTransform: tex2mdx.strip_comments + start: 63514 + targetStatement: + chapter: 2 + file: prelim.tex + label: splitting + mtref: '2.13' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 67619 + sourceTransform: tex2mdx.strip_comments + start: 67365 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- ee7e22fb33f6 +- e5b196773169 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/d71455541c5b.md b/projects/poincare-conjecture/.astrolabe/edges/d71455541c5b.md new file mode 100644 index 00000000..df376b43 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/d71455541c5b.md @@ -0,0 +1,62 @@ +--- +confidence: 0.9 +edgeClass: structural +evidence: + containmentDepth: 1 + proofSpan: + bodyEnd: 136538 + bodyStart: 123471 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 136549 + file: energy1.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 123458 + sourceStatement: + chapter: 18 + file: energy1.tex + label: xistronger + mtref: '18.53' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 123454 + sourceTransform: tex2mdx.strip_comments + start: 122771 + targetContainedSpan: + coordinateSpace: comment-stripped-tex + end: 128310 + file: energy1.tex + sourceTransform: tex2mdx.strip_comments + start: 127628 + targetStatement: + chapter: 18 + file: energy1.tex + label: deltaprime + mtref: '18.55' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 128310 + sourceTransform: tex2mdx.strip_comments + start: 127628 + type: proof-containment + via: + - proof-contains +generator: tools/poincare_tex_extract.py +inference: structural +kind: containment +ref: +- 9df99ae6f4cb +- b87df0b67468 +rel: contains +reviewStatus: accepted +scope: proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof-contains +--- +Morgan--Tian proof containment. The source is the statement whose proof contains the target statement. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/d74df2188c1f.md b/projects/poincare-conjecture/.astrolabe/edges/d74df2188c1f.md new file mode 100644 index 00000000..382f8a3b --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/d74df2188c1f.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: Restim + proofSpan: + bodyEnd: 96054 + bodyStart: 95212 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 96065 + file: stdsoln.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 95199 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 95542 + file: stdsoln.tex + sourceTransform: tex2mdx.strip_comments + start: 95530 + via: proof + sourceStatement: + chapter: 12 + file: stdsoln.tex + label: '' + mtref: '12.33' + sort: remark + span: + coordinateSpace: comment-stripped-tex + end: 95195 + sourceTransform: tex2mdx.strip_comments + start: 94631 + targetStatement: + chapter: 12 + file: stdsoln.tex + label: Restim + mtref: '12.31' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 90335 + sourceTransform: tex2mdx.strip_comments + start: 90184 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 246771506ee5 +- ffae4504318a +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/d7579ccd927b.md b/projects/poincare-conjecture/.astrolabe/edges/d7579ccd927b.md new file mode 100644 index 00000000..02952ff2 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/d7579ccd927b.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: narrows + proofSpan: + bodyEnd: 33794 + bodyStart: 17659 + coordinateSpace: comment-stripped-tex + depth: 2 + end: 33805 + file: singlimit2.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 17646 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 32762 + file: singlimit2.tex + sourceTransform: tex2mdx.strip_comments + start: 32749 + via: proof + sourceStatement: + chapter: 11 + file: singlimit2.tex + label: maxT + mtref: '11.10' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 17643 + sourceTransform: tex2mdx.strip_comments + start: 17183 + targetStatement: + chapter: 2 + file: prelim.tex + label: narrows + mtref: '2.19' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 76622 + sourceTransform: tex2mdx.strip_comments + start: 76377 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- d6919616448d +- cb9ca6d6db9f +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/d759d877ab34.md b/projects/poincare-conjecture/.astrolabe/edges/d759d877ab34.md new file mode 100644 index 00000000..74ec9fbe --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/d759d877ab34.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 11 + file: singlimit2.tex + label: '' + mtref: '11.4' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 8841 + sourceTransform: tex2mdx.strip_comments + start: 8603 + targetStatement: + chapter: 11 + file: singlimit2.tex + label: '' + mtref: '11.5' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 9079 + sourceTransform: tex2mdx.strip_comments + start: 8940 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 57c8cc8a5611 +- 61143f134fcf +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/d796c29db966.md b/projects/poincare-conjecture/.astrolabe/edges/d796c29db966.md new file mode 100644 index 00000000..f302700c --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/d796c29db966.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 18 + file: energy1.tex + label: '' + mtref: '18.46' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 101100 + sourceTransform: tex2mdx.strip_comments + start: 100594 + targetStatement: + chapter: 18 + file: energy1.tex + label: onec + mtref: '18.47' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 101619 + sourceTransform: tex2mdx.strip_comments + start: 101151 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 3b2ad0db7f6e +- 981d63c19e50 +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/d7a2403b3bf3.md b/projects/poincare-conjecture/.astrolabe/edges/d7a2403b3bf3.md new file mode 100644 index 00000000..eb5d6a65 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/d7a2403b3bf3.md @@ -0,0 +1,55 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: statement + label: EulerLag + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 10379 + file: newcompar.tex + sourceTransform: tex2mdx.strip_comments + start: 10365 + via: statement-aux-label-proxy + sourceStatement: + chapter: 6 + file: newcompar.tex + label: '' + mtref: '6.7' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 10481 + sourceTransform: tex2mdx.strip_comments + start: 10161 + targetStatement: + chapter: 6 + file: newcompar.tex + label: EL + mtref: '6.4' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 6659 + sourceTransform: tex2mdx.strip_comments + start: 6364 + type: tex-reference + via: + - statement-aux-label-proxy +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- d23dc29f2aae +- 75695d1b1b54 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- statement-aux-label-proxy +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/d7d27706e72f.md b/projects/poincare-conjecture/.astrolabe/edges/d7d27706e72f.md new file mode 100644 index 00000000..2fc4b749 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/d7d27706e72f.md @@ -0,0 +1,45 @@ +--- +confidence: 0.2 +edgeClass: structural +evidence: + sourceStatement: + chapter: 8 + file: noncoll.tex + label: large + mtref: '8.9' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 15058 + sourceTransform: tex2mdx.strip_comments + start: 14671 + targetStatement: + chapter: 8 + file: noncoll.tex + label: '' + mtref: '8.10' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 18052 + sourceTransform: tex2mdx.strip_comments + start: 17502 + type: chapter-reading-order-adjacency + via: + - chapter-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 4b41e7ff8317 +- 198e7475ad2f +rel: in-chapter +reviewStatus: accepted +scope: chapter +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- chapter-sequence +--- +Morgan--Tian reading-order adjacency across sections in a chapter. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/d7fc9013401c.md b/projects/poincare-conjecture/.astrolabe/edges/d7fc9013401c.md new file mode 100644 index 00000000..4ed8f667 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/d7fc9013401c.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: 2Rbound + proofSpan: + bodyEnd: 137114 + bodyStart: 136498 + coordinateSpace: comment-stripped-tex + depth: 2 + end: 137125 + file: temp2kappa.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 136485 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 136754 + file: temp2kappa.tex + sourceTransform: tex2mdx.strip_comments + start: 136741 + via: proof + sourceStatement: + chapter: 9 + file: temp2kappa.tex + label: bdnearxk + mtref: '9.69' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 136482 + sourceTransform: tex2mdx.strip_comments + start: 136184 + targetStatement: + chapter: 9 + file: temp2kappa.tex + label: 2Rbound + mtref: '9.66' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 128756 + sourceTransform: tex2mdx.strip_comments + start: 128570 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 7c6182738b78 +- ea9b285e40cb +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/d800eacf4e60.md b/projects/poincare-conjecture/.astrolabe/edges/d800eacf4e60.md new file mode 100644 index 00000000..91c3b0a8 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/d800eacf4e60.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: fordiffmax + proofSpan: + bodyEnd: 4162 + bodyStart: 4085 + coordinateSpace: comment-stripped-tex + depth: 2 + end: 4173 + file: maxprin.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 4072 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 4160 + file: maxprin.tex + sourceTransform: tex2mdx.strip_comments + start: 4144 + via: proof + sourceStatement: + chapter: 4 + file: maxprin.tex + label: Rminprime + mtref: '4.3' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 4069 + sourceTransform: tex2mdx.strip_comments + start: 3817 + targetStatement: + chapter: 2 + file: prelim.tex + label: fordiffmax + mtref: '2.23' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 83307 + sourceTransform: tex2mdx.strip_comments + start: 82254 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 84e573c0f2b2 +- 37d7d059a526 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/d80f128d162c.md b/projects/poincare-conjecture/.astrolabe/edges/d80f128d162c.md new file mode 100644 index 00000000..edb2a3d9 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/d80f128d162c.md @@ -0,0 +1,65 @@ +--- +confidence: 0.35 +edgeClass: prose +evidence: + mentionTriggers: + - label: flowtoround + refSpan: + coordinateSpace: comment-stripped-tex + end: 180518 + file: surgery.tex + sourceTransform: tex2mdx.strip_comments + start: 180501 + sentence: If $y$ is contained in a component of $M_{t''}$ of positive sectional + curvature, then by Hamilton's result, Theorem~\ref{flowtoround}, under Ricci + flow the component of $M_{t''}$ containing $y$ flows forward as a family of + components of positive sectional curvature until it disappears + sentenceSpan: + coordinateSpace: comment-stripped-tex + end: 180672 + file: surgery.tex + sourceTransform: tex2mdx.strip_comments + start: 180382 + sourceStrategy: nearest-previous-statement + via: prose + sourceStatement: + chapter: 16 + file: surgery.tex + label: '' + mtref: '16.28' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 176062 + sourceTransform: tex2mdx.strip_comments + start: 175778 + targetStatement: + chapter: 4 + file: maxprin.tex + label: flowtoround + mtref: '4.23' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 35932 + sourceTransform: tex2mdx.strip_comments + start: 34642 + type: prose-mention-nearest-previous-statement + via: + - prose +generator: tools/poincare_tex_extract.py +inference: weak +kind: mention +ref: +- 3e9580da0ff5 +- 86e5a5766bda +rel: mentions +reviewStatus: candidate +scope: prose +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- prose +--- +Morgan--Tian prose mention. The source is the nearest preceding statement in the same chapter segment. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/d82cbcd26c92.md b/projects/poincare-conjecture/.astrolabe/edges/d82cbcd26c92.md new file mode 100644 index 00000000..e13e9e11 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/d82cbcd26c92.md @@ -0,0 +1,45 @@ +--- +confidence: 0.2 +edgeClass: structural +evidence: + sourceStatement: + chapter: 16 + file: surgery.tex + label: '' + mtref: '16.11' + sort: remark + span: + coordinateSpace: comment-stripped-tex + end: 129667 + sourceTransform: tex2mdx.strip_comments + start: 129477 + targetStatement: + chapter: 16 + file: surgery.tex + label: defnoverlinedelta0 + mtref: '16.12' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 130201 + sourceTransform: tex2mdx.strip_comments + start: 129888 + type: chapter-reading-order-adjacency + via: + - chapter-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 4e18ead9ab36 +- b92e1a990021 +rel: in-chapter +reviewStatus: accepted +scope: chapter +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- chapter-sequence +--- +Morgan--Tian reading-order adjacency across sections in a chapter. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/d8754d5f9d91.md b/projects/poincare-conjecture/.astrolabe/edges/d8754d5f9d91.md new file mode 100644 index 00000000..0ca167f3 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/d8754d5f9d91.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 14 + file: surgery.tex + label: defnkappa + mtref: '14.15' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 55165 + sourceTransform: tex2mdx.strip_comments + start: 54665 + targetStatement: + chapter: 14 + file: surgery.tex + label: '' + mtref: '14.16' + sort: remark + span: + coordinateSpace: comment-stripped-tex + end: 55977 + sourceTransform: tex2mdx.strip_comments + start: 55168 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 75f3f7263208 +- 33d69fbf2cfb +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/d87699540bb6.md b/projects/poincare-conjecture/.astrolabe/edges/d87699540bb6.md new file mode 100644 index 00000000..0427f46a --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/d87699540bb6.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: pinch + proofSpan: + bodyEnd: 47393 + bodyStart: 39592 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 47404 + file: maxprin.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 39579 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 47391 + file: maxprin.tex + sourceTransform: tex2mdx.strip_comments + start: 47380 + via: proof + sourceStatement: + chapter: 4 + file: maxprin.tex + label: '' + mtref: '4.27' + sort: remark + span: + coordinateSpace: comment-stripped-tex + end: 39436 + sourceTransform: tex2mdx.strip_comments + start: 39187 + targetStatement: + chapter: 4 + file: maxprin.tex + label: pinch + mtref: '4.26' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 38776 + sourceTransform: tex2mdx.strip_comments + start: 38259 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 1160f53cc783 +- 92827a66b5dd +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/d88a1842ac92.md b/projects/poincare-conjecture/.astrolabe/edges/d88a1842ac92.md new file mode 100644 index 00000000..ed286a85 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/d88a1842ac92.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 18 + file: energy1.tex + label: Gammaapprox + mtref: '18.49' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 104039 + sourceTransform: tex2mdx.strip_comments + start: 103290 + targetStatement: + chapter: 18 + file: energy1.tex + label: '' + mtref: '18.50' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 107575 + sourceTransform: tex2mdx.strip_comments + start: 107348 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 7a8f55be175e +- cb69804a2166 +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/d893ce3518f4.md b/projects/poincare-conjecture/.astrolabe/edges/d893ce3518f4.md new file mode 100644 index 00000000..c6062ffe --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/d893ce3518f4.md @@ -0,0 +1,103 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: kaplimit + proofSpan: + bodyEnd: 85838 + bodyStart: 83533 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 85849 + file: stdsoln.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 83520 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 84430 + file: stdsoln.tex + sourceTransform: tex2mdx.strip_comments + start: 84416 + via: proof + - context: proof + label: kaplimit + proofSpan: + bodyEnd: 85838 + bodyStart: 83533 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 85849 + file: stdsoln.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 83520 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 85211 + file: stdsoln.tex + sourceTransform: tex2mdx.strip_comments + start: 85197 + via: proof + - context: proof + label: kaplimit + proofSpan: + bodyEnd: 85838 + bodyStart: 83533 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 85849 + file: stdsoln.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 83520 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 85439 + file: stdsoln.tex + sourceTransform: tex2mdx.strip_comments + start: 85425 + via: proof + sourceStatement: + chapter: 12 + file: stdsoln.tex + label: R_0cannbhd + mtref: '12.28' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 83517 + sourceTransform: tex2mdx.strip_comments + start: 82969 + targetStatement: + chapter: 11 + file: singlimit2.tex + label: kaplimit + mtref: '11.8' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 16131 + sourceTransform: tex2mdx.strip_comments + start: 15040 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 85e3c925177e +- a9e023d082dc +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/d899d224dcce.md b/projects/poincare-conjecture/.astrolabe/edges/d899d224dcce.md new file mode 100644 index 00000000..bbe151a1 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/d899d224dcce.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: rampcomp + proofSpan: + bodyEnd: 92953 + bodyStart: 91228 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 92964 + file: energy1.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 91215 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 91860 + file: energy1.tex + sourceTransform: tex2mdx.strip_comments + start: 91846 + via: proof + sourceStatement: + chapter: 18 + file: energy1.tex + label: csforramps + mtref: '18.42' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 91212 + sourceTransform: tex2mdx.strip_comments + start: 90872 + targetStatement: + chapter: 18 + file: energy1.tex + label: rampcomp + mtref: '18.39' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 88812 + sourceTransform: tex2mdx.strip_comments + start: 88619 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 11ec367eae93 +- 2f115406780a +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/d89be154d525.md b/projects/poincare-conjecture/.astrolabe/edges/d89be154d525.md new file mode 100644 index 00000000..385822bc --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/d89be154d525.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: pi2 + proofSpan: + bodyEnd: 45142 + bodyStart: 43711 + coordinateSpace: comment-stripped-tex + depth: 2 + end: 45153 + file: energy1.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 43698 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 43949 + file: energy1.tex + sourceTransform: tex2mdx.strip_comments + start: 43940 + via: proof + sourceStatement: + chapter: 18 + file: energy1.tex + label: spherethm + mtref: '18.14' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 43696 + sourceTransform: tex2mdx.strip_comments + start: 43231 + targetStatement: + chapter: 18 + file: energy1.tex + label: pi2 + mtref: '18.9' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 30087 + sourceTransform: tex2mdx.strip_comments + start: 29657 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- a679e0086677 +- 4415092c2079 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/d8d70e3298d4.md b/projects/poincare-conjecture/.astrolabe/edges/d8d70e3298d4.md new file mode 100644 index 00000000..af7cdf34 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/d8d70e3298d4.md @@ -0,0 +1,56 @@ +--- +confidence: 0.85 +edgeClass: semantic +evidence: + containmentReason: target theorem-like statement occurs inside source proof + proofSpan: + bodyEnd: 177272 + bodyStart: 175079 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 177283 + file: surgery.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 175066 + sourceStatement: + chapter: 16 + file: surgery.tex + label: tau0cl + mtref: '16.27' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 175063 + sourceTransform: tex2mdx.strip_comments + start: 174895 + targetStatement: + chapter: 16 + file: surgery.tex + label: '' + mtref: '16.28' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 176062 + sourceTransform: tex2mdx.strip_comments + start: 175778 + type: proof-contained-theorem-like-statement + via: + - proof-contained-statement +generator: tools/poincare_tex_extract.py +inference: inferred +kind: dependency +ref: +- 8aac9c08c058 +- 3e9580da0ff5 +rel: depends +reviewStatus: accepted +scope: proof-containment +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof-contained-statement +--- +Morgan--Tian proof-contained statement dependency. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/d8e844f6c70a.md b/projects/poincare-conjecture/.astrolabe/edges/d8e844f6c70a.md new file mode 100644 index 00000000..a4892fb4 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/d8e844f6c70a.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 4 + file: maxprin.tex + label: '' + mtref: '4.35' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 50442 + sourceTransform: tex2mdx.strip_comments + start: 50109 + targetStatement: + chapter: 4 + file: maxprin.tex + label: ivey + mtref: '4.36' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 51946 + sourceTransform: tex2mdx.strip_comments + start: 51803 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- c7dbaa8c2ca5 +- 7b914755dcf6 +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/d8ecfe6c7734.md b/projects/poincare-conjecture/.astrolabe/edges/d8ecfe6c7734.md new file mode 100644 index 00000000..af991df0 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/d8ecfe6c7734.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 6 + file: newcompar.tex + label: '' + mtref: '6.64' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 96436 + sourceTransform: tex2mdx.strip_comments + start: 96257 + targetStatement: + chapter: 6 + file: newcompar.tex + label: '' + mtref: '6.65' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 98830 + sourceTransform: tex2mdx.strip_comments + start: 98666 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 30414a5a14a9 +- e27e5f5d2bf7 +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/d926272f90b9.md b/projects/poincare-conjecture/.astrolabe/edges/d926272f90b9.md new file mode 100644 index 00000000..d0d9527f --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/d926272f90b9.md @@ -0,0 +1,68 @@ +--- +confidence: 0.35 +edgeClass: prose +evidence: + mentionTriggers: + - label: asympGSS + refSpan: + coordinateSpace: comment-stripped-tex + end: 72440 + file: temp2kappa.tex + sourceTransform: tex2mdx.strip_comments + start: 72426 + sentence: In this section we fix $\kappa>0$ and we classify all $\kappa$-solutions + $(M,g_\infty(t)),\ -\infty0$ depending + only on $\kappa_i$, $r'$, $\epsilon$ and $L$ such that $x$ is $\kappa$ non-collapsed + on scales $\le \epsilon$ + sentenceSpan: + coordinateSpace: comment-stripped-tex + end: 183201 + file: surgery.tex + sourceTransform: tex2mdx.strip_comments + start: 183018 + sourceStrategy: nearest-previous-statement + via: prose + sourceStatement: + chapter: 16 + file: surgery.tex + label: '' + mtref: '16.28' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 176062 + sourceTransform: tex2mdx.strip_comments + start: 175778 + targetStatement: + chapter: 8 + file: noncoll.tex + label: THM + mtref: '8.1' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 1970 + sourceTransform: tex2mdx.strip_comments + start: 682 + type: prose-mention-nearest-previous-statement + via: + - prose +generator: tools/poincare_tex_extract.py +inference: weak +kind: mention +ref: +- 3e9580da0ff5 +- 3defa9b2209c +rel: mentions +reviewStatus: candidate +scope: prose +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- prose +--- +Morgan--Tian prose mention. The source is the nearest preceding statement in the same chapter segment. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/de64379cd46d.md b/projects/poincare-conjecture/.astrolabe/edges/de64379cd46d.md new file mode 100644 index 00000000..08c15a52 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/de64379cd46d.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 18 + file: energy1.tex + label: '' + mtref: '18.71' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 170254 + sourceTransform: tex2mdx.strip_comments + start: 170143 + targetStatement: + chapter: 18 + file: energy1.tex + label: muepsbd + mtref: '18.72' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 171988 + sourceTransform: tex2mdx.strip_comments + start: 171484 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- e94b6de05447 +- 9c8d4163f700 +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/de6c02b7aff7.md b/projects/poincare-conjecture/.astrolabe/edges/de6c02b7aff7.md new file mode 100644 index 00000000..acafe178 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/de6c02b7aff7.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 5 + file: converge2.tex + label: flowlimit + mtref: '5.15' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 27817 + sourceTransform: tex2mdx.strip_comments + start: 26565 + targetStatement: + chapter: 5 + file: converge2.tex + label: '' + mtref: '5.16' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 28735 + sourceTransform: tex2mdx.strip_comments + start: 28434 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- a0212d0eefcf +- 00e5410e6aa1 +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/deb2a79cec96.md b/projects/poincare-conjecture/.astrolabe/edges/deb2a79cec96.md new file mode 100644 index 00000000..c1b7d775 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/deb2a79cec96.md @@ -0,0 +1,65 @@ +--- +confidence: 0.35 +edgeClass: prose +evidence: + mentionTriggers: + - label: bcbd + refSpan: + coordinateSpace: comment-stripped-tex + end: 64369 + file: surgery.tex + sourceTransform: tex2mdx.strip_comments + start: 64359 + sentence: For all such $\epsilon$, Theorem~\ref{bcbd} holds for $\epsilon$ and + Proposition~\ref{narrows}, Proposition~\ref{canonvary} and Corollaries~\ref{kappacannbhd} + and~\ref{limitcannbhd} and Theorems~\ref{smlmtflow} and~\ref{kaplimit} hold + for $2\epsilon$ + sentenceSpan: + coordinateSpace: comment-stripped-tex + end: 64577 + file: surgery.tex + sourceTransform: tex2mdx.strip_comments + start: 64325 + sourceStrategy: nearest-previous-statement + via: prose + sourceStatement: + chapter: 15 + file: surgery.tex + label: neckglue + mtref: '15.2' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 61465 + sourceTransform: tex2mdx.strip_comments + start: 60812 + targetStatement: + chapter: 10 + file: bddcurvbdddist.tex + label: bcbd + mtref: '10.2' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 2942 + sourceTransform: tex2mdx.strip_comments + start: 2168 + type: prose-mention-nearest-previous-statement + via: + - prose +generator: tools/poincare_tex_extract.py +inference: weak +kind: mention +ref: +- a992d6a89479 +- ecd62fc2ea2b +rel: mentions +reviewStatus: candidate +scope: prose +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- prose +--- +Morgan--Tian prose mention. The source is the nearest preceding statement in the same chapter segment. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/deb331dc8130.md b/projects/poincare-conjecture/.astrolabe/edges/deb331dc8130.md new file mode 100644 index 00000000..f7cfa9d9 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/deb331dc8130.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: fmonotone + proofSpan: + bodyEnd: 47272 + bodyStart: 45428 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 47283 + file: newcomp2.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 45415 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 45980 + file: newcomp2.tex + sourceTransform: tex2mdx.strip_comments + start: 45965 + via: proof + sourceStatement: + chapter: 7 + file: newcomp2.tex + label: 4pi + mtref: '7.26' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 45413 + sourceTransform: tex2mdx.strip_comments + start: 44820 + targetStatement: + chapter: 6 + file: newcompar.tex + label: fmonotone + mtref: '6.81' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 122525 + sourceTransform: tex2mdx.strip_comments + start: 122141 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- ae6f1c6d00d9 +- 4dc40f37cd6d +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/deb55833d538.md b/projects/poincare-conjecture/.astrolabe/edges/deb55833d538.md new file mode 100644 index 00000000..454f7474 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/deb55833d538.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: canonvary + proofSpan: + bodyEnd: 11439 + bodyStart: 10110 + coordinateSpace: comment-stripped-tex + depth: 2 + end: 11450 + file: singlimit2.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 10097 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 11293 + file: singlimit2.tex + sourceTransform: tex2mdx.strip_comments + start: 11278 + via: proof + sourceStatement: + chapter: 11 + file: singlimit2.tex + label: '' + mtref: '11.6' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 10094 + sourceTransform: tex2mdx.strip_comments + start: 9927 + targetStatement: + chapter: 9 + file: temp2kappa.tex + label: canonvary + mtref: '9.79' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 149236 + sourceTransform: tex2mdx.strip_comments + start: 147728 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 9f8eade55fcd +- 7c2758f0e2b3 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/debaafdbe077.md b/projects/poincare-conjecture/.astrolabe/edges/debaafdbe077.md new file mode 100644 index 00000000..d58d9acf --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/debaafdbe077.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 2 + file: prelim.tex + label: line + mtref: '2.14' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 67980 + sourceTransform: tex2mdx.strip_comments + start: 67775 + targetStatement: + chapter: 2 + file: prelim.tex + label: '' + mtref: '2.15' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 69160 + sourceTransform: tex2mdx.strip_comments + start: 68871 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 006e42ef5bb5 +- 0d534e8417eb +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/def4aa8b19ed.md b/projects/poincare-conjecture/.astrolabe/edges/def4aa8b19ed.md new file mode 100644 index 00000000..34301314 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/def4aa8b19ed.md @@ -0,0 +1,45 @@ +--- +confidence: 0.2 +edgeClass: structural +evidence: + sourceStatement: + chapter: 12 + file: stdsoln.tex + label: harmapflow + mtref: '12.25' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 74100 + sourceTransform: tex2mdx.strip_comments + start: 73601 + targetStatement: + chapter: 12 + file: stdsoln.tex + label: RDuniq + mtref: '12.26' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 75403 + sourceTransform: tex2mdx.strip_comments + start: 74267 + type: chapter-reading-order-adjacency + via: + - chapter-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 5fbeadbba61c +- c5e225b067f1 +rel: in-chapter +reviewStatus: accepted +scope: chapter +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- chapter-sequence +--- +Morgan--Tian reading-order adjacency across sections in a chapter. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/defa9fafbceb.md b/projects/poincare-conjecture/.astrolabe/edges/defa9fafbceb.md new file mode 100644 index 00000000..bfac71ab --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/defa9fafbceb.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: patch + proofSpan: + bodyEnd: 29700 + bodyStart: 28750 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 29711 + file: converge2.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 28737 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 29698 + file: converge2.tex + sourceTransform: tex2mdx.strip_comments + start: 29687 + via: proof + sourceStatement: + chapter: 5 + file: converge2.tex + label: '' + mtref: '5.16' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 28735 + sourceTransform: tex2mdx.strip_comments + start: 28434 + targetStatement: + chapter: 3 + file: flowbasics.tex + label: patch + mtref: '3.12' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 15371 + sourceTransform: tex2mdx.strip_comments + start: 14869 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 00e5410e6aa1 +- b6bc786f7f44 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/df5f5228dd94.md b/projects/poincare-conjecture/.astrolabe/edges/df5f5228dd94.md new file mode 100644 index 00000000..e6d5cfda --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/df5f5228dd94.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: xicomp + proofSpan: + bodyEnd: 136538 + bodyStart: 123471 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 136549 + file: energy1.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 123458 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 134195 + file: energy1.tex + sourceTransform: tex2mdx.strip_comments + start: 134183 + via: proof + sourceStatement: + chapter: 18 + file: energy1.tex + label: xistronger + mtref: '18.53' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 123454 + sourceTransform: tex2mdx.strip_comments + start: 122771 + targetStatement: + chapter: 18 + file: energy1.tex + label: xicomp + mtref: '18.57' + sort: remark + span: + coordinateSpace: comment-stripped-tex + end: 133717 + sourceTransform: tex2mdx.strip_comments + start: 133485 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 9df99ae6f4cb +- 012eefbbc44a +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/df6c4b115ed7.md b/projects/poincare-conjecture/.astrolabe/edges/df6c4b115ed7.md new file mode 100644 index 00000000..814a5e0d --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/df6c4b115ed7.md @@ -0,0 +1,84 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: Gammaapprox + proofSpan: + bodyEnd: 109929 + bodyStart: 106397 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 109940 + file: energy1.tex + ownerLabel: goodapprox + sourceTransform: tex2mdx.strip_comments + start: 106384 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 106560 + file: energy1.tex + sourceTransform: tex2mdx.strip_comments + start: 106543 + via: proof + - context: proof + label: Gammaapprox + proofSpan: + bodyEnd: 109929 + bodyStart: 106397 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 109940 + file: energy1.tex + ownerLabel: goodapprox + sourceTransform: tex2mdx.strip_comments + start: 106384 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 106907 + file: energy1.tex + sourceTransform: tex2mdx.strip_comments + start: 106890 + via: proof + sourceStatement: + chapter: 18 + file: energy1.tex + label: goodapprox + mtref: '18.45' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 100508 + sourceTransform: tex2mdx.strip_comments + start: 99535 + targetStatement: + chapter: 18 + file: energy1.tex + label: Gammaapprox + mtref: '18.49' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 104039 + sourceTransform: tex2mdx.strip_comments + start: 103290 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 6bb9851ce528 +- 7a8f55be175e +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/dfb4893161d9.md b/projects/poincare-conjecture/.astrolabe/edges/dfb4893161d9.md new file mode 100644 index 00000000..67557d09 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/dfb4893161d9.md @@ -0,0 +1,65 @@ +--- +confidence: 0.45 +edgeClass: semantic +evidence: + endpointSemanticDegreeBeforeTermBackfill: + source: 0 + target: 0 + selectionReason: semantic-connectivity-backfill + sourceMatchSpan: + coordinateSpace: comment-stripped-tex + end: 6896 + file: surgery.tex + sourceTransform: tex2mdx.strip_comments + start: 6885 + sourceStatement: + chapter: 13 + file: surgery.tex + label: capball + mtref: '13.4' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 7464 + sourceTransform: tex2mdx.strip_comments + start: 6854 + targetStatement: + chapter: 13 + file: surgery.tex + label: surgerycap + mtref: '13.3' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 6755 + sourceTransform: tex2mdx.strip_comments + start: 6605 + targetTermSpan: + coordinateSpace: comment-stripped-tex + end: 6720 + file: surgery.tex + sourceTransform: tex2mdx.strip_comments + start: 6709 + term: surgery cap + termAmbiguity: unique-in-chapter + termSource: italic + termSpecificity: no-longer-defined-term + type: same-chapter-definition-term-match + via: + - definition-term:surgery cap +generator: tools/poincare_tex_extract.py +inference: weak +kind: definition-use +ref: +- 752133843e1e +- 443843fff58d +rel: uses +reviewStatus: candidate +scope: same-chapter-term-match +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- definition-term:surgery cap +--- +Morgan--Tian same-chapter definition term match. This is a low-confidence candidate edge for review. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/dfed488e8672.md b/projects/poincare-conjecture/.astrolabe/edges/dfed488e8672.md new file mode 100644 index 00000000..fb65790e --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/dfed488e8672.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 16 + file: surgery.tex + label: KAPPALIMIT + mtref: '16.1' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 103576 + sourceTransform: tex2mdx.strip_comments + start: 102336 + targetStatement: + chapter: 16 + file: surgery.tex + label: '' + mtref: '16.2' + sort: remark + span: + coordinateSpace: comment-stripped-tex + end: 104034 + sourceTransform: tex2mdx.strip_comments + start: 103578 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 4ce8b5da9c58 +- d93b1988d45d +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/e04cf9d38fea.md b/projects/poincare-conjecture/.astrolabe/edges/e04cf9d38fea.md new file mode 100644 index 00000000..6dcb9400 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/e04cf9d38fea.md @@ -0,0 +1,85 @@ +--- +confidence: 0.35 +edgeClass: prose +evidence: + mentionTriggers: + - label: shortpi2triv + refSpan: + coordinateSpace: comment-stripped-tex + end: 65069 + file: energy1.tex + sourceTransform: tex2mdx.strip_comments + start: 65051 + sentence: 'Then using Lemma~\ref{shortpi2triv} and Claim~\ref{vclaim}, we fix + $\zeta$ with $0<\zeta<\eta/2$ such that: \begin{enumerate} \item[(i)] If $\Gamma\colon + S^2\to \Lambda {\mathcal X}(t_1)$ is a family of loops and each loop in the + family is of length less than $\zeta$, then the family is homotopically trivial. + \item[(ii)] For any $a\in [0,W_\xi(t_0)+2\zeta]$ we have $w_a(t_1)0 + mtref: '4.14' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 16848 + sourceTransform: tex2mdx.strip_comments + start: 16607 + targetStatement: + chapter: 4 + file: maxprin.tex + label: quad + mtref: '4.13' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 16605 + sourceTransform: tex2mdx.strip_comments + start: 15794 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 0db415314964 +- 0c451aa3e5cc +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/e3388a498e70.md b/projects/poincare-conjecture/.astrolabe/edges/e3388a498e70.md new file mode 100644 index 00000000..479d1c8a --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/e3388a498e70.md @@ -0,0 +1,56 @@ +--- +confidence: 0.85 +edgeClass: semantic +evidence: + containmentReason: target theorem-like statement occurs inside source proof + proofSpan: + bodyEnd: 14349 + bodyStart: 5983 + coordinateSpace: comment-stripped-tex + depth: 3 + end: 14360 + file: noncoll.tex + ownerLabel: small + sourceTransform: tex2mdx.strip_comments + start: 5970 + sourceStatement: + chapter: 8 + file: noncoll.tex + label: small + mtref: '8.3' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 5471 + sourceTransform: tex2mdx.strip_comments + start: 5161 + targetStatement: + chapter: 8 + file: noncoll.tex + label: '' + mtref: '8.6' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 8866 + sourceTransform: tex2mdx.strip_comments + start: 8367 + type: proof-contained-theorem-like-statement + via: + - proof-contained-statement +generator: tools/poincare_tex_extract.py +inference: inferred +kind: dependency +ref: +- 26e1a6292fef +- 59189adac6be +rel: depends +reviewStatus: accepted +scope: proof-containment +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof-contained-statement +--- +Morgan--Tian proof-contained statement dependency. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/e3479114557e.md b/projects/poincare-conjecture/.astrolabe/edges/e3479114557e.md new file mode 100644 index 00000000..25343e6c --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/e3479114557e.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 18 + file: energy1.tex + label: '' + mtref: '18.90' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 204623 + sourceTransform: tex2mdx.strip_comments + start: 204360 + targetStatement: + chapter: 18 + file: energy1.tex + label: D_4cor + mtref: '18.91' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 205302 + sourceTransform: tex2mdx.strip_comments + start: 204940 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- f7b6896c9e7d +- 55b7b71de6e4 +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/e357b13dcb9c.md b/projects/poincare-conjecture/.astrolabe/edges/e357b13dcb9c.md new file mode 100644 index 00000000..a784a946 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/e357b13dcb9c.md @@ -0,0 +1,56 @@ +--- +confidence: 0.85 +edgeClass: semantic +evidence: + containmentReason: target theorem-like statement occurs inside source proof + proofSpan: + bodyEnd: 34517 + bodyStart: 26920 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 34528 + file: bddcurvbdddist.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 26907 + sourceStatement: + chapter: 10 + file: bddcurvbdddist.tex + label: Uinfty + mtref: '10.7' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 26826 + sourceTransform: tex2mdx.strip_comments + start: 25406 + targetStatement: + chapter: 10 + file: bddcurvbdddist.tex + label: '' + mtref: '10.8' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 28876 + sourceTransform: tex2mdx.strip_comments + start: 28772 + type: proof-contained-theorem-like-statement + via: + - proof-contained-statement +generator: tools/poincare_tex_extract.py +inference: inferred +kind: dependency +ref: +- e2e64cdb0b75 +- 2b1f8205acbe +rel: depends +reviewStatus: accepted +scope: proof-containment +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof-contained-statement +--- +Morgan--Tian proof-contained statement dependency. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/e3930f5ba651.md b/projects/poincare-conjecture/.astrolabe/edges/e3930f5ba651.md new file mode 100644 index 00000000..ae7867f6 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/e3930f5ba651.md @@ -0,0 +1,122 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: hexist + proofSpan: + bodyEnd: 76057 + bodyStart: 75138 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 76068 + file: singlimit2.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 75125 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 75194 + file: singlimit2.tex + sourceTransform: tex2mdx.strip_comments + start: 75182 + via: proof + - context: proof + label: hexist + proofSpan: + bodyEnd: 76057 + bodyStart: 75138 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 76068 + file: singlimit2.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 75125 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 75328 + file: singlimit2.tex + sourceTransform: tex2mdx.strip_comments + start: 75316 + via: proof + - context: proof + label: hexist + proofSpan: + bodyEnd: 76057 + bodyStart: 75138 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 76068 + file: singlimit2.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 75125 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 75429 + file: singlimit2.tex + sourceTransform: tex2mdx.strip_comments + start: 75417 + via: proof + - context: proof + label: hexist + proofSpan: + bodyEnd: 76057 + bodyStart: 75138 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 76068 + file: singlimit2.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 75125 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 75702 + file: singlimit2.tex + sourceTransform: tex2mdx.strip_comments + start: 75690 + via: proof + sourceStatement: + chapter: 11 + file: singlimit2.tex + label: hmonotone + mtref: '11.36' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 75122 + sourceTransform: tex2mdx.strip_comments + start: 74827 + targetStatement: + chapter: 11 + file: singlimit2.tex + label: hexist + mtref: '11.31' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 59267 + sourceTransform: tex2mdx.strip_comments + start: 58375 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- ea6d9419271a +- 074e02b067b8 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/e3965344a2ff.md b/projects/poincare-conjecture/.astrolabe/edges/e3965344a2ff.md new file mode 100644 index 00000000..30c7dc52 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/e3965344a2ff.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: 2ndvari + proofSpan: + bodyEnd: 52568 + bodyStart: 50431 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 52579 + file: newcompar.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 50418 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 50464 + file: newcompar.tex + sourceTransform: tex2mdx.strip_comments + start: 50451 + via: proof + sourceStatement: + chapter: 6 + file: newcompar.tex + label: '' + mtref: '6.34' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 48328 + sourceTransform: tex2mdx.strip_comments + start: 47847 + targetStatement: + chapter: 6 + file: newcompar.tex + label: 2ndvari + mtref: '6.14' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 17350 + sourceTransform: tex2mdx.strip_comments + start: 16270 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- a75b1ef40db8 +- f48558c60ca2 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/e3a5922ad5ee.md b/projects/poincare-conjecture/.astrolabe/edges/e3a5922ad5ee.md new file mode 100644 index 00000000..abfe207a --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/e3a5922ad5ee.md @@ -0,0 +1,55 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: statement + label: weak + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 24724 + file: newcomp2.tex + sourceTransform: tex2mdx.strip_comments + start: 24714 + via: statement + sourceStatement: + chapter: 7 + file: newcomp2.tex + label: '' + mtref: '7.14' + sort: remark + span: + coordinateSpace: comment-stripped-tex + end: 25116 + sourceTransform: tex2mdx.strip_comments + start: 23417 + targetStatement: + chapter: 7 + file: newcomp2.tex + label: weak + mtref: '7.13' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 23409 + sourceTransform: tex2mdx.strip_comments + start: 22180 + type: tex-reference + via: + - statement +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 5df4eb648fc5 +- 7d5d94efe4e9 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- statement +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/e3aeef275c02.md b/projects/poincare-conjecture/.astrolabe/edges/e3aeef275c02.md new file mode 100644 index 00000000..4d25d149 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/e3aeef275c02.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: thetaconv + proofSpan: + bodyEnd: 58340 + bodyStart: 53526 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 58351 + file: bddcurvbdddist.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 53513 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 56613 + file: bddcurvbdddist.tex + sourceTransform: tex2mdx.strip_comments + start: 56598 + via: proof + sourceStatement: + chapter: 10 + file: bddcurvbdddist.tex + label: secondstatment + mtref: '10.29' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 53510 + sourceTransform: tex2mdx.strip_comments + start: 52640 + targetStatement: + chapter: 10 + file: bddcurvbdddist.tex + label: thetaconv + mtref: '10.21' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 44375 + sourceTransform: tex2mdx.strip_comments + start: 43422 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- b34d555cd7b5 +- 5b34d0c84cbb +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/e3cae3f1b964.md b/projects/poincare-conjecture/.astrolabe/edges/e3cae3f1b964.md new file mode 100644 index 00000000..1d16f973 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/e3cae3f1b964.md @@ -0,0 +1,56 @@ +--- +confidence: 0.85 +edgeClass: semantic +evidence: + containmentReason: target theorem-like statement occurs inside source proof + proofSpan: + bodyEnd: 100555 + bodyStart: 97739 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 100566 + file: newcompar.tex + ownerLabel: lipaty + sourceTransform: tex2mdx.strip_comments + start: 97726 + sourceStatement: + chapter: 6 + file: newcompar.tex + label: lipaty + mtref: '6.59' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 87473 + sourceTransform: tex2mdx.strip_comments + start: 86282 + targetStatement: + chapter: 6 + file: newcompar.tex + label: '' + mtref: '6.66' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 100067 + sourceTransform: tex2mdx.strip_comments + start: 99998 + type: proof-contained-theorem-like-statement + via: + - proof-contained-statement +generator: tools/poincare_tex_extract.py +inference: inferred +kind: dependency +ref: +- 96ced8c484df +- f655ed1e0c1f +rel: depends +reviewStatus: accepted +scope: proof-containment +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof-contained-statement +--- +Morgan--Tian proof-contained statement dependency. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/e3de5e1123ca.md b/projects/poincare-conjecture/.astrolabe/edges/e3de5e1123ca.md new file mode 100644 index 00000000..10e94006 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/e3de5e1123ca.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: 2DGSS + proofSpan: + bodyEnd: 119194 + bodyStart: 118223 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 119205 + file: temp2kappa.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 118210 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 118322 + file: temp2kappa.tex + sourceTransform: tex2mdx.strip_comments + start: 118311 + via: proof + sourceStatement: + chapter: 9 + file: temp2kappa.tex + label: asympvol + mtref: '9.59' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 118207 + sourceTransform: tex2mdx.strip_comments + start: 117600 + targetStatement: + chapter: 9 + file: temp2kappa.tex + label: 2DGSS + mtref: '9.50' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 91949 + sourceTransform: tex2mdx.strip_comments + start: 91266 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 7468282bd017 +- da767c71996b +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/e43f21b4a20e.md b/projects/poincare-conjecture/.astrolabe/edges/e43f21b4a20e.md new file mode 100644 index 00000000..b6e5a206 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/e43f21b4a20e.md @@ -0,0 +1,75 @@ +--- +confidence: 0.35 +edgeClass: prose +evidence: + mentionTriggers: + - label: curvflowcompare + refSpan: + coordinateSpace: comment-stripped-tex + end: 218499 + file: energy1.tex + sourceTransform: tex2mdx.strip_comments + start: 218478 + sentence: Now we wish to compute $\int_{I'\times\{t\}}|Z|^2dz.$ To do this we + first note that using the definition of $Y$, and arguing as in the proof of + the first equation in of Lemma~\ref{1stcurvshr} we have we have \begin{eqnarray*}\frac{\partial}{\partial + t}|Z|^2 & = & -2{\rm Ric}(Z,Z)+2\langle \nabla_YZ,Z\rangle \\ & = & -2{\rm Ric}(Z,Z)+2\langle + \nabla_ZY,Z\rangle \end{eqnarray*} Direct computation shows that $$2\langle + \nabla_Z\left(\frac{\nabla_ZZ}{|Z|^2}\right),Z\rangle =Z\left(\frac{Z(|Z|^2)}{|Z|^2}\right)-2\frac{|\nabla_ZZ|^2}{|Z|^4}|Z|^2.$$ + Thus from the Claim~\ref{curvflowcompare}, we have \begin{eqnarray}\label{Z_t} + \frac{\partial}{\partial t}|Z|^2 & = & 2\langle\nabla_ZY,Z\rangle -2{\rm Ric}(Z,Z)\nonumber + \\ & = & Z\left(\frac{Z(|Z|^2)}{|Z|^2}\right)-2\frac{|\nabla_ZZ|^2}{|Z|^4}|Z|^2-2\langle + \nabla_Z(\psi(Z)Z),Z\rangle-2{\rm Ric}(Z,Z) \nonumber \\ & = & Z\left(\frac{Z(|Z|^2)}{|Z|^2}\right)-2|Y|^2|Z|^2 + +V, \end{eqnarray} where $$V=-4|Z|^2\langle Y,\psi(Z)Z\rangle-2\psi^2(Z)|Z|^4-2\langle + \nabla_Z(\psi(Z)Z),Z\rangle-2{\rm Ric}(Z,Z).$$ By Item (1) in Claim~\ref{psiclaim} + and Item (4) in Claim~\ref{1891} we have \begin{equation}\label{Vest} |V|0 + mtref: '13.13' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 21433 + sourceTransform: tex2mdx.strip_comments + start: 21250 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- e70bacf8aab0 +- cc9110b4d2fe +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/e6d0ad2b91b0.md b/projects/poincare-conjecture/.astrolabe/edges/e6d0ad2b91b0.md new file mode 100644 index 00000000..6c8ace40 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/e6d0ad2b91b0.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 6 + file: newcompar.tex + label: lips + mtref: '6.56' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 85066 + sourceTransform: tex2mdx.strip_comments + start: 84130 + targetStatement: + chapter: 6 + file: newcompar.tex + label: '' + mtref: '6.57' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 85285 + sourceTransform: tex2mdx.strip_comments + start: 85074 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- db4ced24b05f +- a89a833cf7cb +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/e7513fd5888f.md b/projects/poincare-conjecture/.astrolabe/edges/e7513fd5888f.md new file mode 100644 index 00000000..44571110 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/e7513fd5888f.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 18 + file: energy1.tex + label: '' + mtref: '18.10' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 31561 + sourceTransform: tex2mdx.strip_comments + start: 31075 + targetStatement: + chapter: 18 + file: energy1.tex + label: W_2 + mtref: '18.11' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 36484 + sourceTransform: tex2mdx.strip_comments + start: 36186 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- e63bb828c601 +- 7bb0fa0c9412 +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/e7673246fa8c.md b/projects/poincare-conjecture/.astrolabe/edges/e7673246fa8c.md new file mode 100644 index 00000000..34cf6640 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/e7673246fa8c.md @@ -0,0 +1,57 @@ +--- +confidence: 0.72 +edgeClass: semantic +evidence: + anaphoraTriggers: + - context: statement + direction: previous + phrase: previous lemma + sort: lemma + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 43707 + file: surgery.tex + sourceTransform: tex2mdx.strip_comments + start: 43693 + via: local-anaphora-statement + sourceStatement: + chapter: 14 + file: surgery.tex + label: '' + mtref: '14.6' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 43870 + sourceTransform: tex2mdx.strip_comments + start: 43585 + targetStatement: + chapter: 14 + file: surgery.tex + label: '' + mtref: '14.5' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 43469 + sourceTransform: tex2mdx.strip_comments + start: 42757 + type: local-anaphora-reference + via: + - local-anaphora-statement +generator: tools/poincare_tex_extract.py +inference: inferred +kind: dependency +ref: +- 27af18c808d2 +- 6564ce56e2fd +rel: depends +reviewStatus: accepted +scope: local-context +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- local-anaphora-statement +--- +Morgan--Tian local anaphora dependency. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/e7686d1f819e.md b/projects/poincare-conjecture/.astrolabe/edges/e7686d1f819e.md new file mode 100644 index 00000000..249de62e --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/e7686d1f819e.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 5 + file: converge2.tex + label: '' + mtref: '5.20' + sort: example + span: + coordinateSpace: comment-stripped-tex + end: 34003 + sourceTransform: tex2mdx.strip_comments + start: 33876 + targetStatement: + chapter: 5 + file: converge2.tex + label: seqconv + mtref: '5.21' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 34707 + sourceTransform: tex2mdx.strip_comments + start: 34006 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- af79df2b9075 +- 7744e53809c5 +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/e78b57ecc673.md b/projects/poincare-conjecture/.astrolabe/edges/e78b57ecc673.md new file mode 100644 index 00000000..d78f644c --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/e78b57ecc673.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 9 + file: temp2kappa.tex + label: asympscal + mtref: '9.56' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 106666 + sourceTransform: tex2mdx.strip_comments + start: 106402 + targetStatement: + chapter: 9 + file: temp2kappa.tex + label: '' + mtref: '9.57' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 111901 + sourceTransform: tex2mdx.strip_comments + start: 111822 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 5d9f1b75170d +- 6448a516850a +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/e7943bd9fb18.md b/projects/poincare-conjecture/.astrolabe/edges/e7943bd9fb18.md new file mode 100644 index 00000000..2623e503 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/e7943bd9fb18.md @@ -0,0 +1,90 @@ +--- +confidence: 0.35 +edgeClass: prose +evidence: + mentionTriggers: + - label: 2epshorns + refSpan: + coordinateSpace: comment-stripped-tex + end: 85832 + file: surgery.tex + sourceTransform: tex2mdx.strip_comments + start: 85817 + sentence: According to Lemma~\ref{2epshorns} there are finitely many components + of $\Omega(T)$ that meet $\Omega_\rho(T)$. Let $\Omega^{\rm big}(T)$ be the + disjoint union of all the components of $\Omega(T)$ that meet $\Omega_\rho(T)$. + Lemma~\ref{2epshorns} also tells us that $\Omega^{\rm big}(T)$ contains a finite + collection of disjoint $2\epsilon$-horns with boundary contained in $\Omega_{\rho/2C}$, + and the complement of the union of the interiors of these horns is a compact + submanifold with boundary containing $\Omega_\rho$ + sentenceSpan: + coordinateSpace: comment-stripped-tex + end: 86320 + file: surgery.tex + sourceTransform: tex2mdx.strip_comments + start: 85797 + sourceStrategy: nearest-previous-statement + via: prose + - label: 2epshorns + refSpan: + coordinateSpace: comment-stripped-tex + end: 86045 + file: surgery.tex + sourceTransform: tex2mdx.strip_comments + start: 86030 + sentence: According to Lemma~\ref{2epshorns} there are finitely many components + of $\Omega(T)$ that meet $\Omega_\rho(T)$. Let $\Omega^{\rm big}(T)$ be the + disjoint union of all the components of $\Omega(T)$ that meet $\Omega_\rho(T)$. + Lemma~\ref{2epshorns} also tells us that $\Omega^{\rm big}(T)$ contains a finite + collection of disjoint $2\epsilon$-horns with boundary contained in $\Omega_{\rho/2C}$, + and the complement of the union of the interiors of these horns is a compact + submanifold with boundary containing $\Omega_\rho$ + sentenceSpan: + coordinateSpace: comment-stripped-tex + end: 86320 + file: surgery.tex + sourceTransform: tex2mdx.strip_comments + start: 85797 + sourceStrategy: nearest-previous-statement + via: prose + sourceStatement: + chapter: 15 + file: surgery.tex + label: surgeryparams + mtref: '15.7' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 80864 + sourceTransform: tex2mdx.strip_comments + start: 79853 + targetStatement: + chapter: 11 + file: singlimit2.tex + label: 2epshorns + mtref: '11.30' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 53428 + sourceTransform: tex2mdx.strip_comments + start: 52907 + type: prose-mention-nearest-previous-statement + via: + - prose +generator: tools/poincare_tex_extract.py +inference: weak +kind: mention +ref: +- 22c40a5664ac +- 01f0d3baa28f +rel: mentions +reviewStatus: candidate +scope: prose +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- prose +--- +Morgan--Tian prose mention. The source is the nearest preceding statement in the same chapter segment. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/e7b833c58345.md b/projects/poincare-conjecture/.astrolabe/edges/e7b833c58345.md new file mode 100644 index 00000000..ae3aa4d9 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/e7b833c58345.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: DLJacobi + proofSpan: + bodyEnd: 31451 + bodyStart: 30042 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 31462 + file: newcompar.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 30029 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 31153 + file: newcompar.tex + sourceTransform: tex2mdx.strip_comments + start: 31139 + via: proof + sourceStatement: + chapter: 6 + file: newcompar.tex + label: DtildeL + mtref: '6.22' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 30026 + sourceTransform: tex2mdx.strip_comments + start: 29708 + targetStatement: + chapter: 6 + file: newcompar.tex + label: DLJacobi + mtref: '6.19' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 26981 + sourceTransform: tex2mdx.strip_comments + start: 26317 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- abf9cbb10c68 +- d2e61ac52277 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/e7e23b2a1dfe.md b/projects/poincare-conjecture/.astrolabe/edges/e7e23b2a1dfe.md new file mode 100644 index 00000000..e512e10d --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/e7e23b2a1dfe.md @@ -0,0 +1,45 @@ +--- +confidence: 0.2 +edgeClass: structural +evidence: + sourceStatement: + chapter: 4 + file: maxprin.tex + label: localMP + mtref: '4.9' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 11555 + sourceTransform: tex2mdx.strip_comments + start: 10496 + targetStatement: + chapter: 4 + file: maxprin.tex + label: norminitcond + mtref: '4.10' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 13043 + sourceTransform: tex2mdx.strip_comments + start: 12395 + type: chapter-reading-order-adjacency + via: + - chapter-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- a89be7fdb431 +- 12a9b4e99e24 +rel: in-chapter +reviewStatus: accepted +scope: chapter +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- chapter-sequence +--- +Morgan--Tian reading-order adjacency across sections in a chapter. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/e80f9f61511b.md b/projects/poincare-conjecture/.astrolabe/edges/e80f9f61511b.md new file mode 100644 index 00000000..7b4d723b --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/e80f9f61511b.md @@ -0,0 +1,128 @@ +--- +confidence: 0.35 +edgeClass: prose +evidence: + mentionTriggers: + - label: KAPPALIMIT + refSpan: + coordinateSpace: comment-stripped-tex + end: 229610 + file: surgery.tex + sourceTransform: tex2mdx.strip_comments + start: 229594 + sentence: Now we turn to the proof of Theorem~\ref{MAIN}. Given surgery parameter + sequences $$\Delta_i=\{\delta_0,\ldots,\delta_i\}$$ $${\bf r_i}=\{r_0,\ldots,r_i\}$$ + $${\bf K_i}=\{\kappa_0,\ldots,\kappa_i\},$$ we let $r_{i+1}$ and $\delta_{i+1}$ + be as in Proposition~\ref{extend} and then set $\kappa_{i+1}=\kappa(r_{i+1})$ + as in Proposition~\ref{KAPPALIMIT}. Set $${\bf r_{i+1}}=\{{\bf r_i},r_{i+1}\}$$ + $${\bf K_{i+1}}=\{{\bf K_i},\kappa_{i+1}\}$$ $$\Delta_{i+1}=\{\delta_0,\ldots,\delta_{i-1},\delta_{i+1},\delta_{i+1}\}.$$ + Of course, these are also surgery parameter sequences. + sentenceSpan: + coordinateSpace: comment-stripped-tex + end: 229833 + file: surgery.tex + sourceTransform: tex2mdx.strip_comments + start: 229258 + sourceStrategy: nearest-previous-statement + via: prose + - label: KAPPALIMIT + refSpan: + coordinateSpace: comment-stripped-tex + end: 230783 + file: surgery.tex + sourceTransform: tex2mdx.strip_comments + start: 230767 + sentence: By Proposition~\ref{KAPPALIMIT} and Proposition~\ref{extend} this extension + will continue to satisfy the conclusions of Theorem~\ref{MAIN} on its entire + interval of definition + sentenceSpan: + coordinateSpace: comment-stripped-tex + end: 230927 + file: surgery.tex + sourceTransform: tex2mdx.strip_comments + start: 230751 + sourceStrategy: nearest-previous-statement + via: prose + - label: KAPPALIMIT + refSpan: + coordinateSpace: comment-stripped-tex + end: 232953 + file: surgery.tex + sourceTransform: tex2mdx.strip_comments + start: 232937 + sentence: Since the surgery control parameter $\overline\delta(t)$ is at most + $\delta(r_{i+1})$, the constant from Proposition~\ref{KAPPALIMIT}, for all $t\in + [T_{i-1},T]$, since $T\le T_{i+1}$, and since the restriction of $({\mathcal + M},G)$ to ${\bf t}^{-1}([0,T_i))$ satisfies Proposition~\ref{KAPPALIMIT}, we + see by Proposition~\ref{extend} that the extended Ricci flow with surgery satisfies + the conclusion of Theorem~\ref{MAIN} on its entire time interval of definition. + sentenceSpan: + coordinateSpace: comment-stripped-tex + end: 233287 + file: surgery.tex + sourceTransform: tex2mdx.strip_comments + start: 232818 + sourceStrategy: nearest-previous-statement + via: prose + - label: KAPPALIMIT + refSpan: + coordinateSpace: comment-stripped-tex + end: 233119 + file: surgery.tex + sourceTransform: tex2mdx.strip_comments + start: 233103 + sentence: Since the surgery control parameter $\overline\delta(t)$ is at most + $\delta(r_{i+1})$, the constant from Proposition~\ref{KAPPALIMIT}, for all $t\in + [T_{i-1},T]$, since $T\le T_{i+1}$, and since the restriction of $({\mathcal + M},G)$ to ${\bf t}^{-1}([0,T_i))$ satisfies Proposition~\ref{KAPPALIMIT}, we + see by Proposition~\ref{extend} that the extended Ricci flow with surgery satisfies + the conclusion of Theorem~\ref{MAIN} on its entire time interval of definition. + sentenceSpan: + coordinateSpace: comment-stripped-tex + end: 233287 + file: surgery.tex + sourceTransform: tex2mdx.strip_comments + start: 232818 + sourceStrategy: nearest-previous-statement + via: prose + sourceStatement: + chapter: 17 + file: surgery.tex + label: '' + mtref: '17.11' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 226210 + sourceTransform: tex2mdx.strip_comments + start: 226024 + targetStatement: + chapter: 16 + file: surgery.tex + label: KAPPALIMIT + mtref: '16.1' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 103576 + sourceTransform: tex2mdx.strip_comments + start: 102336 + type: prose-mention-nearest-previous-statement + via: + - prose +generator: tools/poincare_tex_extract.py +inference: weak +kind: mention +ref: +- 8dafe9c95b7f +- 4ce8b5da9c58 +rel: mentions +reviewStatus: candidate +scope: prose +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- prose +--- +Morgan--Tian prose mention. The source is the nearest preceding statement in the same chapter segment. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/e85830db6cb7.md b/projects/poincare-conjecture/.astrolabe/edges/e85830db6cb7.md new file mode 100644 index 00000000..7868756b --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/e85830db6cb7.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 3 + file: flowbasics.tex + label: posdistdec + mtref: '3.14' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 17555 + sourceTransform: tex2mdx.strip_comments + start: 17308 + targetStatement: + chapter: 3 + file: flowbasics.tex + label: '' + mtref: '3.15' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 18558 + sourceTransform: tex2mdx.strip_comments + start: 18005 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 51a2afee4465 +- 41667bcb0cab +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/e8643aa42c8b.md b/projects/poincare-conjecture/.astrolabe/edges/e8643aa42c8b.md new file mode 100644 index 00000000..7254342c --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/e8643aa42c8b.md @@ -0,0 +1,45 @@ +--- +confidence: 0.2 +edgeClass: structural +evidence: + sourceStatement: + chapter: 9 + file: temp2kappa.tex + label: pp + mtref: '9.38' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 67234 + sourceTransform: tex2mdx.strip_comments + start: 66738 + targetStatement: + chapter: 9 + file: temp2kappa.tex + label: prodatinf + mtref: '9.39' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 68632 + sourceTransform: tex2mdx.strip_comments + start: 67563 + type: chapter-reading-order-adjacency + via: + - chapter-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- b71a179769b1 +- 703d5ca9da05 +rel: in-chapter +reviewStatus: accepted +scope: chapter +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- chapter-sequence +--- +Morgan--Tian reading-order adjacency across sections in a chapter. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/e8782c6be4c0.md b/projects/poincare-conjecture/.astrolabe/edges/e8782c6be4c0.md new file mode 100644 index 00000000..b0b3b55c --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/e8782c6be4c0.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: varareaform + proofSpan: + bodyEnd: 54967 + bodyStart: 37263 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 54978 + file: energy1.tex + ownerLabel: W_2 + sourceTransform: tex2mdx.strip_comments + start: 37250 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 39208 + file: energy1.tex + sourceTransform: tex2mdx.strip_comments + start: 39191 + via: proof + sourceStatement: + chapter: 18 + file: energy1.tex + label: W_2 + mtref: '18.11' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 36484 + sourceTransform: tex2mdx.strip_comments + start: 36186 + targetStatement: + chapter: 18 + file: energy1.tex + label: varareaform + mtref: '18.12' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 37506 + sourceTransform: tex2mdx.strip_comments + start: 37347 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 7bb0fa0c9412 +- db9ef2c8d5c1 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/e87a8657196f.md b/projects/poincare-conjecture/.astrolabe/edges/e87a8657196f.md new file mode 100644 index 00000000..26d7361b --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/e87a8657196f.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: pincha + proofSpan: + bodyEnd: 49002 + bodyStart: 48683 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 49013 + file: maxprin.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 48670 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 48849 + file: maxprin.tex + sourceTransform: tex2mdx.strip_comments + start: 48837 + via: proof + sourceStatement: + chapter: 4 + file: maxprin.tex + label: '' + mtref: '4.33' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 48667 + sourceTransform: tex2mdx.strip_comments + start: 48350 + targetStatement: + chapter: 4 + file: maxprin.tex + label: pincha + mtref: '4.32' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 48151 + sourceTransform: tex2mdx.strip_comments + start: 47478 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 8b5afdc35b56 +- fc44d8653f35 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/e8a91724dc6f.md b/projects/poincare-conjecture/.astrolabe/edges/e8a91724dc6f.md new file mode 100644 index 00000000..a287dbca --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/e8a91724dc6f.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: controlnbhd + proofSpan: + bodyEnd: 47855 + bodyStart: 43596 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 47866 + file: singlimit2.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 43583 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 44301 + file: singlimit2.tex + sourceTransform: tex2mdx.strip_comments + start: 44284 + via: proof + sourceStatement: + chapter: 11 + file: singlimit2.tex + label: omegacanon + mtref: '11.23' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 43580 + sourceTransform: tex2mdx.strip_comments + start: 43054 + targetStatement: + chapter: 11 + file: singlimit2.tex + label: controlnbhd + mtref: '11.2' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 4360 + sourceTransform: tex2mdx.strip_comments + start: 3597 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 0760630d3990 +- 0ddfe578961e +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/e8e9567f9f17.md b/projects/poincare-conjecture/.astrolabe/edges/e8e9567f9f17.md new file mode 100644 index 00000000..a0757b88 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/e8e9567f9f17.md @@ -0,0 +1,63 @@ +--- +confidence: 0.35 +edgeClass: prose +evidence: + mentionTriggers: + - label: redvolconst + refSpan: + coordinateSpace: comment-stripped-tex + end: 63846 + file: temp2kappa.tex + sourceTransform: tex2mdx.strip_comments + start: 63829 + sentence: This contradicts Corollary~\ref{redvolconst}, and the contradiction + establishes that $(M_\infty,g_\infty(t))$ is not flat for any $t<0$ + sentenceSpan: + coordinateSpace: comment-stripped-tex + end: 63937 + file: temp2kappa.tex + sourceTransform: tex2mdx.strip_comments + start: 63801 + sourceStrategy: nearest-previous-statement + via: prose + sourceStatement: + chapter: 9 + file: temp2kappa.tex + label: '' + mtref: '9.36' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 62699 + sourceTransform: tex2mdx.strip_comments + start: 62344 + targetStatement: + chapter: 9 + file: temp2kappa.tex + label: redvolconst + mtref: '9.14' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 17598 + sourceTransform: tex2mdx.strip_comments + start: 17342 + type: prose-mention-nearest-previous-statement + via: + - prose +generator: tools/poincare_tex_extract.py +inference: weak +kind: mention +ref: +- 64100c8689ff +- e08895f2b0d8 +rel: mentions +reviewStatus: candidate +scope: prose +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- prose +--- +Morgan--Tian prose mention. The source is the nearest preceding statement in the same chapter segment. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/e9095ff120b2.md b/projects/poincare-conjecture/.astrolabe/edges/e9095ff120b2.md new file mode 100644 index 00000000..10f5eb5a --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/e9095ff120b2.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: compactcase + proofSpan: + bodyEnd: 117032 + bodyStart: 113723 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 117043 + file: temp2kappa.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 113710 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 113819 + file: temp2kappa.tex + sourceTransform: tex2mdx.strip_comments + start: 113802 + via: proof + sourceStatement: + chapter: 9 + file: temp2kappa.tex + label: kappa0 + mtref: '9.58' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 113708 + sourceTransform: tex2mdx.strip_comments + start: 113563 + targetStatement: + chapter: 9 + file: temp2kappa.tex + label: compactcase + mtref: '9.44' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 78969 + sourceTransform: tex2mdx.strip_comments + start: 78694 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 9c54e0d0c7d8 +- 01028c93347d +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/e931a9b16336.md b/projects/poincare-conjecture/.astrolabe/edges/e931a9b16336.md new file mode 100644 index 00000000..5694864c --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/e931a9b16336.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 12 + file: stdsoln.tex + label: '' + mtref: '12.33' + sort: remark + span: + coordinateSpace: comment-stripped-tex + end: 95195 + sourceTransform: tex2mdx.strip_comments + start: 94631 + targetStatement: + chapter: 12 + file: stdsoln.tex + label: stdsolncap + mtref: '12.34' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 96335 + sourceTransform: tex2mdx.strip_comments + start: 96067 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 246771506ee5 +- 20ae3cd37b93 +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/e964fd17a750.md b/projects/poincare-conjecture/.astrolabe/edges/e964fd17a750.md new file mode 100644 index 00000000..3aec0294 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/e964fd17a750.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: Fm + proofSpan: + bodyEnd: 64108 + bodyStart: 53268 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 64119 + file: flowbasics.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 53255 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 64106 + file: flowbasics.tex + sourceTransform: tex2mdx.strip_comments + start: 64098 + via: proof + sourceStatement: + chapter: 3 + file: flowbasics.tex + label: etaprop + mtref: '3.33' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 47994 + sourceTransform: tex2mdx.strip_comments + start: 47039 + targetStatement: + chapter: 3 + file: flowbasics.tex + label: Fm + mtref: '3.32' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 46079 + sourceTransform: tex2mdx.strip_comments + start: 45486 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 81f8de5102de +- e7fe06259f43 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/e9838bc855d0.md b/projects/poincare-conjecture/.astrolabe/edges/e9838bc855d0.md new file mode 100644 index 00000000..fac098c5 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/e9838bc855d0.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: MAIN + proofSpan: + bodyEnd: 2280 + bodyStart: 1209 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 2291 + file: energy1.tex + ownerLabel: Theorem1 + sourceTransform: tex2mdx.strip_comments + start: 1196 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 1430 + file: energy1.tex + sourceTransform: tex2mdx.strip_comments + start: 1420 + via: proof + sourceStatement: + chapter: 0 + file: intro.tex + label: Theorem1 + mtref: '0.1' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 2955 + sourceTransform: tex2mdx.strip_comments + start: 2574 + targetStatement: + chapter: 15 + file: surgery.tex + label: MAIN + mtref: '15.9' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 91186 + sourceTransform: tex2mdx.strip_comments + start: 90004 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 9999efa2be2c +- 703e4cfeece7 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/e9c90e5ba95a.md b/projects/poincare-conjecture/.astrolabe/edges/e9c90e5ba95a.md new file mode 100644 index 00000000..a56eb7d4 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/e9c90e5ba95a.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 1 + file: prelim.tex + label: '' + mtref: '1.24' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 33683 + sourceTransform: tex2mdx.strip_comments + start: 33241 + targetStatement: + chapter: 1 + file: prelim.tex + label: '' + mtref: '1.25' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 33904 + sourceTransform: tex2mdx.strip_comments + start: 33797 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 19afab2e3115 +- 26a127323e5a +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/e9ca92b45614.md b/projects/poincare-conjecture/.astrolabe/edges/e9ca92b45614.md new file mode 100644 index 00000000..eb927f30 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/e9ca92b45614.md @@ -0,0 +1,65 @@ +--- +confidence: 0.45 +edgeClass: semantic +evidence: + endpointSemanticDegreeBeforeTermBackfill: + source: 0 + target: 1 + selectionReason: semantic-connectivity-backfill + sourceMatchSpan: + coordinateSpace: comment-stripped-tex + end: 26578 + file: prelim.tex + sourceTransform: tex2mdx.strip_comments + start: 26570 + sourceStatement: + chapter: 1 + file: prelim.tex + label: '' + mtref: '1.19' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 26794 + sourceTransform: tex2mdx.strip_comments + start: 26539 + targetStatement: + chapter: 1 + file: prelim.tex + label: '' + mtref: '1.17' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 19309 + sourceTransform: tex2mdx.strip_comments + start: 19124 + targetTermSpan: + coordinateSpace: comment-stripped-tex + end: 19240 + file: prelim.tex + sourceTransform: tex2mdx.strip_comments + start: 19232 + term: geodesic + termAmbiguity: unique-in-chapter + termSource: italic + termSpecificity: no-longer-defined-term + type: same-chapter-definition-term-match + via: + - definition-term:geodesic +generator: tools/poincare_tex_extract.py +inference: weak +kind: definition-use +ref: +- b209b247be34 +- c4ccdd484637 +rel: uses +reviewStatus: candidate +scope: same-chapter-term-match +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- definition-term:geodesic +--- +Morgan--Tian same-chapter definition term match. This is a low-confidence candidate edge for review. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/e9fbb110ffd3.md b/projects/poincare-conjecture/.astrolabe/edges/e9fbb110ffd3.md new file mode 100644 index 00000000..8bf0f65a --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/e9fbb110ffd3.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 9 + file: temp2kappa.tex + label: '' + mtref: '9.61' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 121789 + sourceTransform: tex2mdx.strip_comments + start: 121292 + targetStatement: + chapter: 9 + file: temp2kappa.tex + label: bdvolbdcurv + mtref: '9.62' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 124263 + sourceTransform: tex2mdx.strip_comments + start: 123826 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- d2259da4d67d +- 3227c39e1892 +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/ea07a20788a9.md b/projects/poincare-conjecture/.astrolabe/edges/ea07a20788a9.md new file mode 100644 index 00000000..c75b3bdc --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/ea07a20788a9.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 2 + file: prelim.tex + label: '' + mtref: '2.17' + sort: remark + span: + coordinateSpace: comment-stripped-tex + end: 73683 + sourceTransform: tex2mdx.strip_comments + start: 72978 + targetStatement: + chapter: 2 + file: prelim.tex + label: epsneck + mtref: '2.18' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 76303 + sourceTransform: tex2mdx.strip_comments + start: 73778 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- bdfb2841e15e +- d91d54ddbd1c +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/ea2eab402ece.md b/projects/poincare-conjecture/.astrolabe/edges/ea2eab402ece.md new file mode 100644 index 00000000..7c697fcc --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/ea2eab402ece.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: unianc + proofSpan: + bodyEnd: 19105 + bodyStart: 18688 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 19116 + file: temp2kappa.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 18675 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 18715 + file: temp2kappa.tex + sourceTransform: tex2mdx.strip_comments + start: 18703 + via: proof + sourceStatement: + chapter: 9 + file: temp2kappa.tex + label: lxkest + mtref: '9.16' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 18673 + sourceTransform: tex2mdx.strip_comments + start: 18334 + targetStatement: + chapter: 9 + file: temp2kappa.tex + label: unianc + mtref: '9.10' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 10335 + sourceTransform: tex2mdx.strip_comments + start: 9624 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 611a73900deb +- 8b8e3a0f3e36 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/ea38a832b1dd.md b/projects/poincare-conjecture/.astrolabe/edges/ea38a832b1dd.md new file mode 100644 index 00000000..eb21cdbd --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/ea38a832b1dd.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: C0claim + proofSpan: + bodyEnd: 204347 + bodyStart: 202741 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 204358 + file: energy1.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 202728 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 203529 + file: energy1.tex + sourceTransform: tex2mdx.strip_comments + start: 203516 + via: proof + sourceStatement: + chapter: 18 + file: energy1.tex + label: claim1885 + mtref: '18.89' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 202725 + sourceTransform: tex2mdx.strip_comments + start: 202260 + targetStatement: + chapter: 18 + file: energy1.tex + label: C0claim + mtref: '18.87' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 198926 + sourceTransform: tex2mdx.strip_comments + start: 198558 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 1f7d2e438fef +- 1b8d34cae3a8 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/ea58b2122e63.md b/projects/poincare-conjecture/.astrolabe/edges/ea58b2122e63.md new file mode 100644 index 00000000..22314062 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/ea58b2122e63.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: quad + proofSpan: + bodyEnd: 19584 + bodyStart: 18311 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 19595 + file: maxprin.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 18298 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 19062 + file: maxprin.tex + sourceTransform: tex2mdx.strip_comments + start: 19052 + via: proof + sourceStatement: + chapter: 4 + file: maxprin.tex + label: Ric>0 + mtref: '4.15' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 18295 + sourceTransform: tex2mdx.strip_comments + start: 18076 + targetStatement: + chapter: 4 + file: maxprin.tex + label: quad + mtref: '4.13' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 16605 + sourceTransform: tex2mdx.strip_comments + start: 15794 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- f8afc1c61977 +- 0c451aa3e5cc +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/ea7cc57e90be.md b/projects/poincare-conjecture/.astrolabe/edges/ea7cc57e90be.md new file mode 100644 index 00000000..c0d06547 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/ea7cc57e90be.md @@ -0,0 +1,62 @@ +--- +confidence: 0.9 +edgeClass: structural +evidence: + containmentDepth: 1 + proofSpan: + bodyEnd: 137790 + bodyStart: 127828 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 137801 + file: temp2kappa.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 127815 + sourceStatement: + chapter: 9 + file: temp2kappa.tex + label: proposition + mtref: '9.65' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 127812 + sourceTransform: tex2mdx.strip_comments + start: 127549 + targetContainedSpan: + coordinateSpace: comment-stripped-tex + end: 133378 + file: temp2kappa.tex + sourceTransform: tex2mdx.strip_comments + start: 133150 + targetStatement: + chapter: 9 + file: temp2kappa.tex + label: '' + mtref: '9.67' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 133378 + sourceTransform: tex2mdx.strip_comments + start: 133150 + type: proof-containment + via: + - proof-contains +generator: tools/poincare_tex_extract.py +inference: structural +kind: containment +ref: +- ee314aacd207 +- b8f989614649 +rel: contains +reviewStatus: accepted +scope: proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof-contains +--- +Morgan--Tian proof containment. The source is the statement whose proof contains the target statement. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/eac8a0586e50.md b/projects/poincare-conjecture/.astrolabe/edges/eac8a0586e50.md new file mode 100644 index 00000000..26c8e013 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/eac8a0586e50.md @@ -0,0 +1,84 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: pinch + proofSpan: + bodyEnd: 51714 + bodyStart: 50457 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 51725 + file: maxprin.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 50444 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 50652 + file: maxprin.tex + sourceTransform: tex2mdx.strip_comments + start: 50641 + via: proof + - context: proof + label: pinch + proofSpan: + bodyEnd: 51714 + bodyStart: 50457 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 51725 + file: maxprin.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 50444 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 51076 + file: maxprin.tex + sourceTransform: tex2mdx.strip_comments + start: 51065 + via: proof + sourceStatement: + chapter: 4 + file: maxprin.tex + label: '' + mtref: '4.35' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 50442 + sourceTransform: tex2mdx.strip_comments + start: 50109 + targetStatement: + chapter: 4 + file: maxprin.tex + label: pinch + mtref: '4.26' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 38776 + sourceTransform: tex2mdx.strip_comments + start: 38259 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- c7dbaa8c2ca5 +- 92827a66b5dd +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/ead6edb9471b.md b/projects/poincare-conjecture/.astrolabe/edges/ead6edb9471b.md new file mode 100644 index 00000000..60a697ac --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/ead6edb9471b.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 4 + file: maxprin.tex + label: localprod + mtref: '4.19' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 25705 + sourceTransform: tex2mdx.strip_comments + start: 25065 + targetStatement: + chapter: 4 + file: maxprin.tex + label: nullspace + mtref: '4.20' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 31663 + sourceTransform: tex2mdx.strip_comments + start: 30958 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 8ec855a2aab5 +- d21a675e4ea8 +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/eaded5ca5033.md b/projects/poincare-conjecture/.astrolabe/edges/eaded5ca5033.md new file mode 100644 index 00000000..5ffdbfdb --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/eaded5ca5033.md @@ -0,0 +1,45 @@ +--- +confidence: 0.2 +edgeClass: structural +evidence: + sourceStatement: + chapter: 3 + file: flowbasics.tex + label: '' + mtref: '3.35' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 66625 + sourceTransform: tex2mdx.strip_comments + start: 65944 + targetStatement: + chapter: 3 + file: flowbasics.tex + label: '' + mtref: '3.36' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 68154 + sourceTransform: tex2mdx.strip_comments + start: 67865 + type: chapter-reading-order-adjacency + via: + - chapter-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- ec0b9637d440 +- 7048ed283c84 +rel: in-chapter +reviewStatus: accepted +scope: chapter +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- chapter-sequence +--- +Morgan--Tian reading-order adjacency across sections in a chapter. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/eaf4e6fcb818.md b/projects/poincare-conjecture/.astrolabe/edges/eaf4e6fcb818.md new file mode 100644 index 00000000..93ed3444 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/eaf4e6fcb818.md @@ -0,0 +1,65 @@ +--- +confidence: 0.45 +edgeClass: semantic +evidence: + endpointSemanticDegreeBeforeTermBackfill: + source: 0 + target: 1 + selectionReason: semantic-connectivity-backfill + sourceMatchSpan: + coordinateSpace: comment-stripped-tex + end: 31349 + file: newcomp2.tex + sourceTransform: tex2mdx.strip_comments + start: 31336 + sourceStatement: + chapter: 7 + file: newcomp2.tex + label: '' + mtref: '7.17' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 31480 + sourceTransform: tex2mdx.strip_comments + start: 31161 + targetStatement: + chapter: 7 + file: newcomp2.tex + label: '' + mtref: '7.7' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 12044 + sourceTransform: tex2mdx.strip_comments + start: 11683 + targetTermSpan: + coordinateSpace: comment-stripped-tex + end: 11799 + file: newcomp2.tex + sourceTransform: tex2mdx.strip_comments + start: 11786 + term: upper barrier + termAmbiguity: unique-in-chapter + termSource: italic + termSpecificity: no-longer-defined-term + type: same-chapter-definition-term-match + via: + - definition-term:upper barrier +generator: tools/poincare_tex_extract.py +inference: weak +kind: definition-use +ref: +- b58a798d5eea +- d50ccf232266 +rel: uses +reviewStatus: candidate +scope: same-chapter-term-match +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- definition-term:upper barrier +--- +Morgan--Tian same-chapter definition term match. This is a low-confidence candidate edge for review. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/eb05f51782aa.md b/projects/poincare-conjecture/.astrolabe/edges/eb05f51782aa.md new file mode 100644 index 00000000..f74097fa --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/eb05f51782aa.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 6 + file: newcompar.tex + label: '' + mtref: '6.65' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 98830 + sourceTransform: tex2mdx.strip_comments + start: 98666 + targetStatement: + chapter: 6 + file: newcompar.tex + label: '' + mtref: '6.66' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 100067 + sourceTransform: tex2mdx.strip_comments + start: 99998 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- e27e5f5d2bf7 +- f655ed1e0c1f +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/eb1dd1812938.md b/projects/poincare-conjecture/.astrolabe/edges/eb1dd1812938.md new file mode 100644 index 00000000..7daa1072 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/eb1dd1812938.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 11 + file: singlimit2.tex + label: compactbd + mtref: '11.11' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 19990 + sourceTransform: tex2mdx.strip_comments + start: 19475 + targetStatement: + chapter: 11 + file: singlimit2.tex + label: distch + mtref: '11.12' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 20248 + sourceTransform: tex2mdx.strip_comments + start: 20027 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- cc3a26ca3809 +- 002f3496c54c +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/eb4706c9ef0b.md b/projects/poincare-conjecture/.astrolabe/edges/eb4706c9ef0b.md new file mode 100644 index 00000000..b1b5ace8 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/eb4706c9ef0b.md @@ -0,0 +1,80 @@ +--- +confidence: 0.35 +edgeClass: prose +evidence: + mentionTriggers: + - label: XI + refSpan: + coordinateSpace: comment-stripped-tex + end: 136623 + file: energy1.tex + sourceTransform: tex2mdx.strip_comments + start: 136615 + sentence: \subsection{The completion of the proof of Proposition~\protect{\ref{XI}}} + sentenceSpan: + coordinateSpace: comment-stripped-tex + end: 136625 + file: energy1.tex + sourceTransform: tex2mdx.strip_comments + start: 136550 + sourceStrategy: nearest-previous-statement + via: prose + - label: XI + refSpan: + coordinateSpace: comment-stripped-tex + end: 136752 + file: energy1.tex + sourceTransform: tex2mdx.strip_comments + start: 136744 + sentence: Now we wish to pass from Lemma~\ref{xistronger} which deals with an + individual $c\in S^2$ to a proof of Proposition~\ref{XI} which deals with the + entire family $\widetilde\Gamma$ + sentenceSpan: + coordinateSpace: comment-stripped-tex + end: 136806 + file: energy1.tex + sourceTransform: tex2mdx.strip_comments + start: 136626 + sourceStrategy: nearest-previous-statement + via: prose + sourceStatement: + chapter: 18 + file: energy1.tex + label: xicomp + mtref: '18.57' + sort: remark + span: + coordinateSpace: comment-stripped-tex + end: 133717 + sourceTransform: tex2mdx.strip_comments + start: 133485 + targetStatement: + chapter: 18 + file: energy1.tex + label: XI + mtref: '18.24' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 58810 + sourceTransform: tex2mdx.strip_comments + start: 58030 + type: prose-mention-nearest-previous-statement + via: + - prose +generator: tools/poincare_tex_extract.py +inference: weak +kind: mention +ref: +- 012eefbbc44a +- fd85732525c1 +rel: mentions +reviewStatus: candidate +scope: prose +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- prose +--- +Morgan--Tian prose mention. The source is the nearest preceding statement in the same chapter segment. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/eb4dfe5c19f3.md b/projects/poincare-conjecture/.astrolabe/edges/eb4dfe5c19f3.md new file mode 100644 index 00000000..ae817987 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/eb4dfe5c19f3.md @@ -0,0 +1,45 @@ +--- +confidence: 0.2 +edgeClass: structural +evidence: + sourceStatement: + chapter: 11 + file: singlimit2.tex + label: '' + mtref: '11.17' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 31800 + sourceTransform: tex2mdx.strip_comments + start: 31539 + targetStatement: + chapter: 11 + file: singlimit2.tex + label: assumptions + mtref: '11.18' + sort: assumption + span: + coordinateSpace: comment-stripped-tex + end: 36515 + sourceTransform: tex2mdx.strip_comments + start: 35753 + type: chapter-reading-order-adjacency + via: + - chapter-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- b3b96a664796 +- 3009be0a37fc +rel: in-chapter +reviewStatus: accepted +scope: chapter +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- chapter-sequence +--- +Morgan--Tian reading-order adjacency across sections in a chapter. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/eb62f562b17c.md b/projects/poincare-conjecture/.astrolabe/edges/eb62f562b17c.md new file mode 100644 index 00000000..0437eba7 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/eb62f562b17c.md @@ -0,0 +1,62 @@ +--- +confidence: 0.9 +edgeClass: structural +evidence: + containmentDepth: 1 + proofSpan: + bodyEnd: 54967 + bodyStart: 37263 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 54978 + file: energy1.tex + ownerLabel: W_2 + sourceTransform: tex2mdx.strip_comments + start: 37250 + sourceStatement: + chapter: 18 + file: energy1.tex + label: W_2 + mtref: '18.11' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 36484 + sourceTransform: tex2mdx.strip_comments + start: 36186 + targetContainedSpan: + coordinateSpace: comment-stripped-tex + end: 43696 + file: energy1.tex + sourceTransform: tex2mdx.strip_comments + start: 43231 + targetStatement: + chapter: 18 + file: energy1.tex + label: spherethm + mtref: '18.14' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 43696 + sourceTransform: tex2mdx.strip_comments + start: 43231 + type: proof-containment + via: + - proof-contains +generator: tools/poincare_tex_extract.py +inference: structural +kind: containment +ref: +- 7bb0fa0c9412 +- a679e0086677 +rel: contains +reviewStatus: accepted +scope: proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof-contains +--- +Morgan--Tian proof containment. The source is the statement whose proof contains the target statement. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/eb9193ce05de.md b/projects/poincare-conjecture/.astrolabe/edges/eb9193ce05de.md new file mode 100644 index 00000000..9a05d34d --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/eb9193ce05de.md @@ -0,0 +1,62 @@ +--- +confidence: 0.9 +edgeClass: structural +evidence: + containmentDepth: 1 + proofSpan: + bodyEnd: 137790 + bodyStart: 127828 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 137801 + file: temp2kappa.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 127815 + sourceStatement: + chapter: 9 + file: temp2kappa.tex + label: proposition + mtref: '9.65' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 127812 + sourceTransform: tex2mdx.strip_comments + start: 127549 + targetContainedSpan: + coordinateSpace: comment-stripped-tex + end: 136482 + file: temp2kappa.tex + sourceTransform: tex2mdx.strip_comments + start: 136184 + targetStatement: + chapter: 9 + file: temp2kappa.tex + label: bdnearxk + mtref: '9.69' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 136482 + sourceTransform: tex2mdx.strip_comments + start: 136184 + type: proof-containment + via: + - proof-contains +generator: tools/poincare_tex_extract.py +inference: structural +kind: containment +ref: +- ee314aacd207 +- 7c6182738b78 +rel: contains +reviewStatus: accepted +scope: proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof-contains +--- +Morgan--Tian proof containment. The source is the statement whose proof contains the target statement. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/eb9656659b43.md b/projects/poincare-conjecture/.astrolabe/edges/eb9656659b43.md new file mode 100644 index 00000000..219bf212 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/eb9656659b43.md @@ -0,0 +1,45 @@ +--- +confidence: 0.2 +edgeClass: structural +evidence: + sourceStatement: + chapter: 6 + file: newcompar.tex + label: '' + mtref: '6.42' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 66455 + sourceTransform: tex2mdx.strip_comments + start: 66260 + targetStatement: + chapter: 6 + file: newcompar.tex + label: triL + mtref: '6.43' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 70273 + sourceTransform: tex2mdx.strip_comments + start: 68899 + type: chapter-reading-order-adjacency + via: + - chapter-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- b20ded1b2964 +- 882677eecdc1 +rel: in-chapter +reviewStatus: accepted +scope: chapter +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- chapter-sequence +--- +Morgan--Tian reading-order adjacency across sections in a chapter. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/ebacdee5f190.md b/projects/poincare-conjecture/.astrolabe/edges/ebacdee5f190.md new file mode 100644 index 00000000..ae18dad9 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/ebacdee5f190.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: shi + proofSpan: + bodyEnd: 136142 + bodyStart: 134948 + coordinateSpace: comment-stripped-tex + depth: 2 + end: 136153 + file: temp2kappa.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 134935 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 135169 + file: temp2kappa.tex + sourceTransform: tex2mdx.strip_comments + start: 135160 + via: proof + sourceStatement: + chapter: 9 + file: temp2kappa.tex + label: zkbound + mtref: '9.68' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 134931 + sourceTransform: tex2mdx.strip_comments + start: 134780 + targetStatement: + chapter: 3 + file: flowbasics.tex + label: shi + mtref: '3.28' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 40541 + sourceTransform: tex2mdx.strip_comments + start: 39904 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- f61f72020102 +- ed4b72caf9f9 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/ebb3b0a513ab.md b/projects/poincare-conjecture/.astrolabe/edges/ebb3b0a513ab.md new file mode 100644 index 00000000..5f819bf2 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/ebb3b0a513ab.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: limitcs + proofSpan: + bodyEnd: 136538 + bodyStart: 123471 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 136549 + file: energy1.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 123458 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 134171 + file: energy1.tex + sourceTransform: tex2mdx.strip_comments + start: 134158 + via: proof + sourceStatement: + chapter: 18 + file: energy1.tex + label: xistronger + mtref: '18.53' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 123454 + sourceTransform: tex2mdx.strip_comments + start: 122771 + targetStatement: + chapter: 18 + file: energy1.tex + label: limitcs + mtref: '18.56' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 130594 + sourceTransform: tex2mdx.strip_comments + start: 129800 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 9df99ae6f4cb +- 07ac984fbe88 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/ebd84a7254fc.md b/projects/poincare-conjecture/.astrolabe/edges/ebd84a7254fc.md new file mode 100644 index 00000000..9e15d11a --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/ebd84a7254fc.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 3 + file: flowbasics.tex + label: '' + mtref: '3.40' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 71681 + sourceTransform: tex2mdx.strip_comments + start: 70994 + targetStatement: + chapter: 3 + file: flowbasics.tex + label: '' + mtref: '3.41' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 72509 + sourceTransform: tex2mdx.strip_comments + start: 71686 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 1aaf4983e041 +- 0f8fc616f97c +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/ebdfc940c2b6.md b/projects/poincare-conjecture/.astrolabe/edges/ebdfc940c2b6.md new file mode 100644 index 00000000..90dcd7ec --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/ebdfc940c2b6.md @@ -0,0 +1,64 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: statement + label: Mevol + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 16008 + file: maxprin.tex + sourceTransform: tex2mdx.strip_comments + start: 15997 + via: statement + - context: statement + label: Mevol + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 16070 + file: maxprin.tex + sourceTransform: tex2mdx.strip_comments + start: 16059 + via: statement + sourceStatement: + chapter: 4 + file: maxprin.tex + label: quad + mtref: '4.13' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 16605 + sourceTransform: tex2mdx.strip_comments + start: 15794 + targetStatement: + chapter: 3 + file: flowbasics.tex + label: Mevol + mtref: '3.19' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 24163 + sourceTransform: tex2mdx.strip_comments + start: 23286 + type: tex-reference + via: + - statement +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 0c451aa3e5cc +- 883dba87d14c +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- statement +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/ec3feb4b6127.md b/projects/poincare-conjecture/.astrolabe/edges/ec3feb4b6127.md new file mode 100644 index 00000000..c4ffe0e2 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/ec3feb4b6127.md @@ -0,0 +1,103 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: onec + proofSpan: + bodyEnd: 103276 + bodyStart: 102694 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 103287 + file: energy1.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 102681 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 102936 + file: energy1.tex + sourceTransform: tex2mdx.strip_comments + start: 102926 + via: proof + - 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proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 028d3b5182b5 +- 981d63c19e50 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/ec44bb7d4f39.md b/projects/poincare-conjecture/.astrolabe/edges/ec44bb7d4f39.md new file mode 100644 index 00000000..1eab4d13 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/ec44bb7d4f39.md @@ -0,0 +1,62 @@ +--- +confidence: 0.35 +edgeClass: prose +evidence: + mentionTriggers: + - label: XI + refSpan: + coordinateSpace: comment-stripped-tex + end: 140636 + file: energy1.tex + sourceTransform: tex2mdx.strip_comments + start: 140628 + sentence: Here we finish the proof of Proposition~\ref{XI} assuming it. + sentenceSpan: + coordinateSpace: comment-stripped-tex + end: 140649 + file: energy1.tex + sourceTransform: tex2mdx.strip_comments + start: 140587 + sourceStrategy: nearest-previous-statement + via: prose + sourceStatement: + chapter: 18 + file: energy1.tex + label: BARMU + mtref: '18.59' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 140535 + sourceTransform: tex2mdx.strip_comments + start: 139920 + targetStatement: + chapter: 18 + file: energy1.tex + label: XI + mtref: '18.24' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 58810 + sourceTransform: tex2mdx.strip_comments + start: 58030 + type: prose-mention-nearest-previous-statement + via: + - prose +generator: tools/poincare_tex_extract.py +inference: weak +kind: mention +ref: +- 8e814eab5ddf +- fd85732525c1 +rel: mentions +reviewStatus: candidate +scope: prose +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- prose +--- +Morgan--Tian prose mention. 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proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 01f0d3baa28f +- 0e9a1a167caa +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/ec743b72d8fb.md b/projects/poincare-conjecture/.astrolabe/edges/ec743b72d8fb.md new file mode 100644 index 00000000..45f9f96c --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/ec743b72d8fb.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 5 + file: converge2.tex + label: epsnet + mtref: '5.17' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 31206 + sourceTransform: tex2mdx.strip_comments + start: 30954 + targetStatement: + chapter: 5 + file: converge2.tex + label: '' + mtref: '5.18' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 31734 + sourceTransform: tex2mdx.strip_comments + start: 31648 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 9baa1c0fea07 +- 243579230a8c +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/ec87ee083249.md b/projects/poincare-conjecture/.astrolabe/edges/ec87ee083249.md new file mode 100644 index 00000000..ddc10156 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/ec87ee083249.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: LOCALSURGERY + proofSpan: + bodyEnd: 209500 + bodyStart: 208091 + coordinateSpace: comment-stripped-tex + depth: 2 + end: 209511 + file: surgery.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 208078 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 208343 + file: surgery.tex + sourceTransform: tex2mdx.strip_comments + start: 208325 + via: proof + sourceStatement: + chapter: 17 + file: surgery.tex + label: '' + mtref: '17.6' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 208075 + sourceTransform: tex2mdx.strip_comments + start: 207857 + targetStatement: + chapter: 13 + file: surgery.tex + label: LOCALSURGERY + mtref: '13.2' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 6442 + sourceTransform: tex2mdx.strip_comments + start: 4875 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 8de850960905 +- cf61b731c6f2 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/ec8a97ab82fa.md b/projects/poincare-conjecture/.astrolabe/edges/ec8a97ab82fa.md new file mode 100644 index 00000000..255fbacf --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/ec8a97ab82fa.md @@ -0,0 +1,122 @@ +--- +confidence: 0.35 +edgeClass: prose +evidence: + mentionTriggers: + - label: MAIN + refSpan: + coordinateSpace: comment-stripped-tex + end: 4199 + file: energy1.tex + sourceTransform: tex2mdx.strip_comments + start: 4189 + sentence: This result holds for all Ricci flows with surgery satisfying the conclusion + of Theorem~\ref{MAIN}. + sentenceSpan: + coordinateSpace: comment-stripped-tex + end: 4200 + file: energy1.tex + sourceTransform: tex2mdx.strip_comments + start: 4100 + sourceStrategy: nearest-previous-statement + via: prose + - label: MAIN + refSpan: + coordinateSpace: comment-stripped-tex + end: 9579 + file: energy1.tex + sourceTransform: tex2mdx.strip_comments + start: 9569 + sentence: In the next section we deal with $\pi_2$ and, using $W_2$, we show that + given a Ricci flow with surgery as in Theorem~\ref{MAIN} there is $T_1<\infty$ + such that for all $T\ge T_1$ every connected component of $M_T$ has trivial + $\pi_2$ + sentenceSpan: + coordinateSpace: comment-stripped-tex + end: 9685 + file: energy1.tex + sourceTransform: tex2mdx.strip_comments + start: 9450 + sourceStrategy: nearest-previous-statement + via: prose + - label: MAIN + refSpan: + coordinateSpace: comment-stripped-tex + end: 13212 + file: energy1.tex + sourceTransform: tex2mdx.strip_comments + start: 13202 + sentence: Thus, by Theorem~\ref{MAIN}, for any compact $3$-manifold $M$ whose + fundamental group is a free product of finite groups and infinite cyclic groups + and for any normalized metric $g(0)$ on $M$ there is a Ricci flow with surgery + $({\mathcal M},G)$ defined for all time $t\in [0,\infty)$ satisfying the conclusion + of Theorem~\ref{MAIN} with $(M,g(0))$ as the initial conditions. + sentenceSpan: + coordinateSpace: comment-stripped-tex + end: 13561 + file: energy1.tex + sourceTransform: tex2mdx.strip_comments + start: 13184 + sourceStrategy: nearest-previous-statement + via: prose + - label: MAIN + refSpan: + coordinateSpace: comment-stripped-tex + end: 13518 + file: energy1.tex + sourceTransform: tex2mdx.strip_comments + start: 13508 + sentence: Thus, by Theorem~\ref{MAIN}, for any compact $3$-manifold $M$ whose + fundamental group is a free product of finite groups and infinite cyclic groups + and for any normalized metric $g(0)$ on $M$ there is a Ricci flow with surgery + $({\mathcal M},G)$ defined for all time $t\in [0,\infty)$ satisfying the conclusion + of Theorem~\ref{MAIN} with $(M,g(0))$ as the initial conditions. + sentenceSpan: + coordinateSpace: comment-stripped-tex + end: 13561 + file: energy1.tex + sourceTransform: tex2mdx.strip_comments + start: 13184 + sourceStrategy: nearest-previous-statement + via: prose + sourceStatement: + chapter: 18 + file: energy1.tex + label: extinct + mtref: '18.1' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 1107 + sourceTransform: tex2mdx.strip_comments + start: 511 + targetStatement: + chapter: 15 + file: surgery.tex + label: MAIN + mtref: '15.9' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 91186 + sourceTransform: tex2mdx.strip_comments + start: 90004 + type: prose-mention-nearest-previous-statement + via: + - prose +generator: tools/poincare_tex_extract.py +inference: weak +kind: mention +ref: +- 4b8a44aa316f +- 703e4cfeece7 +rel: mentions +reviewStatus: candidate +scope: prose +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- prose +--- +Morgan--Tian prose mention. The source is the nearest preceding statement in the same chapter segment. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/ec8c0f251faa.md b/projects/poincare-conjecture/.astrolabe/edges/ec8c0f251faa.md new file mode 100644 index 00000000..2f7bd4a6 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/ec8c0f251faa.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: bcbd + proofSpan: + bodyEnd: 23189 + bodyStart: 20006 + coordinateSpace: comment-stripped-tex + depth: 3 + end: 23200 + file: singlimit2.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 19993 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 23150 + file: singlimit2.tex + sourceTransform: tex2mdx.strip_comments + start: 23140 + via: proof + sourceStatement: + chapter: 11 + file: singlimit2.tex + label: compactbd + mtref: '11.11' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 19990 + sourceTransform: tex2mdx.strip_comments + start: 19475 + targetStatement: + chapter: 10 + file: bddcurvbdddist.tex + label: bcbd + mtref: '10.2' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 2942 + sourceTransform: tex2mdx.strip_comments + start: 2168 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- cc3a26ca3809 +- ecd62fc2ea2b +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/ed1dc29211c2.md b/projects/poincare-conjecture/.astrolabe/edges/ed1dc29211c2.md new file mode 100644 index 00000000..da001fed --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/ed1dc29211c2.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: adapjab + proofSpan: + bodyEnd: 73361 + bodyStart: 70289 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 73372 + file: newcompar.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 70276 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 73307 + file: newcompar.tex + sourceTransform: tex2mdx.strip_comments + start: 73294 + via: proof + sourceStatement: + chapter: 6 + file: newcompar.tex + label: triL + mtref: '6.43' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 70273 + sourceTransform: tex2mdx.strip_comments + start: 68899 + targetStatement: + chapter: 6 + file: newcompar.tex + label: adapjab + mtref: '6.41' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 59462 + sourceTransform: tex2mdx.strip_comments + start: 58890 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 882677eecdc1 +- 72949e6350d4 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/ed225d3b2c1d.md b/projects/poincare-conjecture/.astrolabe/edges/ed225d3b2c1d.md new file mode 100644 index 00000000..cf3185f9 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/ed225d3b2c1d.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: narrows + proofSpan: + bodyEnd: 35374 + bodyStart: 16867 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 35385 + file: singlimit2.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 16854 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 34601 + file: singlimit2.tex + sourceTransform: tex2mdx.strip_comments + start: 34588 + via: proof + sourceStatement: + chapter: 11 + file: singlimit2.tex + label: '' + mtref: '11.9' + sort: remark + span: + coordinateSpace: comment-stripped-tex + end: 16852 + sourceTransform: tex2mdx.strip_comments + start: 16134 + targetStatement: + chapter: 2 + file: prelim.tex + label: narrows + mtref: '2.19' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 76622 + sourceTransform: tex2mdx.strip_comments + start: 76377 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 5e06d2593bfa +- cb9ca6d6db9f +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/ed3abf14e9bb.md b/projects/poincare-conjecture/.astrolabe/edges/ed3abf14e9bb.md new file mode 100644 index 00000000..292ac408 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/ed3abf14e9bb.md @@ -0,0 +1,65 @@ +--- +confidence: 0.45 +edgeClass: semantic +evidence: + endpointSemanticDegreeBeforeTermBackfill: + source: 4 + target: 0 + selectionReason: semantic-connectivity-backfill + sourceMatchSpan: + coordinateSpace: comment-stripped-tex + end: 2701 + file: bddcurvbdddist.tex + sourceTransform: tex2mdx.strip_comments + start: 2678 + sourceStatement: + chapter: 10 + file: bddcurvbdddist.tex + label: bcbd + mtref: '10.2' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 2942 + sourceTransform: tex2mdx.strip_comments + start: 2168 + targetStatement: + chapter: 10 + file: bddcurvbdddist.tex + label: pinchdef + mtref: '10.1' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 1390 + sourceTransform: tex2mdx.strip_comments + start: 647 + targetTermSpan: + coordinateSpace: comment-stripped-tex + end: 1119 + file: bddcurvbdddist.tex + sourceTransform: tex2mdx.strip_comments + start: 1096 + term: pinched toward positive + termAmbiguity: unique-in-chapter + termSource: italic + termSpecificity: no-longer-defined-term + type: same-chapter-definition-term-match + via: + - definition-term:pinched toward positive +generator: tools/poincare_tex_extract.py +inference: weak +kind: definition-use +ref: +- ecd62fc2ea2b +- 7464980c4085 +rel: uses +reviewStatus: candidate +scope: same-chapter-term-match +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- definition-term:pinched toward positive +--- +Morgan--Tian same-chapter definition term match. This is a low-confidence candidate edge for review. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/ed44e20975f9.md b/projects/poincare-conjecture/.astrolabe/edges/ed44e20975f9.md new file mode 100644 index 00000000..cf136a21 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/ed44e20975f9.md @@ -0,0 +1,103 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: C1C4delta + proofSpan: + bodyEnd: 136538 + bodyStart: 123471 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 136549 + file: energy1.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 123458 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 129449 + file: energy1.tex + sourceTransform: tex2mdx.strip_comments + start: 129434 + via: proof + - context: proof + label: C1C4delta + proofSpan: + bodyEnd: 136538 + bodyStart: 123471 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 136549 + file: energy1.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 123458 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 129537 + file: energy1.tex + sourceTransform: tex2mdx.strip_comments + start: 129522 + via: proof + - context: proof + label: C1C4delta + proofSpan: + bodyEnd: 136538 + bodyStart: 123471 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 136549 + file: energy1.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 123458 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 135847 + file: energy1.tex + sourceTransform: tex2mdx.strip_comments + start: 135832 + via: proof + sourceStatement: + chapter: 18 + file: energy1.tex + label: xistronger + mtref: '18.53' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 123454 + sourceTransform: tex2mdx.strip_comments + start: 122771 + targetStatement: + chapter: 18 + file: energy1.tex + label: C1C4delta + mtref: '18.54' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 124884 + sourceTransform: tex2mdx.strip_comments + start: 124119 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 9df99ae6f4cb +- 7e0b4b1362c0 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/ed8b605a231a.md b/projects/poincare-conjecture/.astrolabe/edges/ed8b605a231a.md new file mode 100644 index 00000000..129b85b7 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/ed8b605a231a.md @@ -0,0 +1,87 @@ +--- +confidence: 0.35 +edgeClass: prose +evidence: + mentionTriggers: + - label: vclaim + refSpan: + coordinateSpace: comment-stripped-tex + end: 62848 + file: energy1.tex + sourceTransform: tex2mdx.strip_comments + start: 62836 + sentence: 'Then by Claim~\ref{vclaim} and Corollary~\ref{shlnsharea}, there is + $0<\zeta<\eta/2$ such that the following two conditions hold: \begin{enumerate} + \item[(a)] Any loop in ${\mathcal X}(t_1)$ of length less than $\zeta$ bounds + a disk of area less than $\eta$. \item[(b)] For every $a\in [0,W_\xi(t_0)+2\zeta]$ + the solution $w_{a}$ satisfies $w_{a}(t_1)0$, the injectivity + set $\widetilde{\mathcal U}_{x}(\tau)\subset T_pM$ consists of all $Z\in T_pM$ + for which (i) the ${\mathcal L}$-geodesic $\gamma_Z|_{[0,\tau]}$ is the unique + minimizing ${\mathcal L}$-geodesic from $x$ to its endpoint, (ii) the differential + of ${\mathcal L}{\rm exp}_{x}^\tau$ is an isomorphism at $Z$, and (iii) for + all $Z'$ sufficiently close to $Z$ the ${\mathcal L}$-geodesic $\gamma_{Z'}|_{[0,\tau]}$ + is the unique minimizing ${\mathcal L}$-geodesic to its endpoint + sentenceSpan: + coordinateSpace: comment-stripped-tex + end: 7270 + file: newcomp2.tex + sourceTransform: tex2mdx.strip_comments + start: 6718 + sourceStrategy: nearest-previous-statement + via: prose + sourceStatement: + chapter: 7 + file: newcomp2.tex + label: '' + mtref: '7.3' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 5342 + sourceTransform: tex2mdx.strip_comments + start: 5096 + targetStatement: + chapter: 6 + file: newcompar.tex + label: injdefn + mtref: '6.25' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 35883 + sourceTransform: tex2mdx.strip_comments + start: 34564 + type: prose-mention-nearest-previous-statement + via: + - prose +generator: tools/poincare_tex_extract.py +inference: weak +kind: mention +ref: +- 44032fc43e5d +- 90fe97584bdd +rel: mentions +reviewStatus: candidate +scope: prose +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- prose +--- +Morgan--Tian prose mention. The source is the nearest preceding statement in the same chapter segment. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/ee7b71351d60.md b/projects/poincare-conjecture/.astrolabe/edges/ee7b71351d60.md new file mode 100644 index 00000000..3f159792 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/ee7b71351d60.md @@ -0,0 +1,62 @@ +--- +confidence: 0.9 +edgeClass: structural +evidence: + containmentDepth: 1 + proofSpan: + bodyEnd: 35374 + bodyStart: 16867 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 35385 + file: singlimit2.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 16854 + sourceStatement: + chapter: 11 + file: singlimit2.tex + label: '' + mtref: '11.9' + sort: remark + span: + coordinateSpace: comment-stripped-tex + end: 16852 + sourceTransform: tex2mdx.strip_comments + start: 16134 + targetContainedSpan: + coordinateSpace: comment-stripped-tex + end: 17643 + file: singlimit2.tex + sourceTransform: tex2mdx.strip_comments + start: 17183 + targetStatement: + chapter: 11 + file: singlimit2.tex + label: maxT + mtref: '11.10' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 17643 + sourceTransform: tex2mdx.strip_comments + start: 17183 + type: proof-containment + via: + - proof-contains +generator: tools/poincare_tex_extract.py +inference: structural +kind: containment +ref: +- 5e06d2593bfa +- d6919616448d +rel: contains +reviewStatus: accepted +scope: proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof-contains +--- +Morgan--Tian proof containment. The source is the statement whose proof contains the target statement. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/eec44ce43494.md b/projects/poincare-conjecture/.astrolabe/edges/eec44ce43494.md new file mode 100644 index 00000000..556c623d --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/eec44ce43494.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 6 + file: newcompar.tex + label: Jlimit + mtref: '6.78' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 114106 + sourceTransform: tex2mdx.strip_comments + start: 113458 + targetStatement: + chapter: 6 + file: newcompar.tex + label: Jlimit1 + mtref: '6.79' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 118112 + sourceTransform: tex2mdx.strip_comments + start: 117801 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 4e6330b2aa46 +- f1123507a2ea +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/eed2f79b34a7.md b/projects/poincare-conjecture/.astrolabe/edges/eed2f79b34a7.md new file mode 100644 index 00000000..f116db64 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/eed2f79b34a7.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: nonembeasycase + proofSpan: + bodyEnd: 136538 + bodyStart: 123471 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 136549 + file: energy1.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 123458 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 134150 + file: energy1.tex + sourceTransform: tex2mdx.strip_comments + start: 134130 + via: proof + sourceStatement: + chapter: 18 + file: energy1.tex + label: xistronger + mtref: '18.53' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 123454 + sourceTransform: tex2mdx.strip_comments + start: 122771 + targetStatement: + chapter: 18 + file: energy1.tex + label: nonembeasycase + mtref: '18.32' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 74095 + sourceTransform: tex2mdx.strip_comments + start: 73662 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 9df99ae6f4cb +- 27b452d6cb48 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/ef42c9e48f66.md b/projects/poincare-conjecture/.astrolabe/edges/ef42c9e48f66.md new file mode 100644 index 00000000..901689b7 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/ef42c9e48f66.md @@ -0,0 +1,82 @@ +--- +confidence: 0.35 +edgeClass: prose +evidence: + mentionTriggers: + - label: BARMU + refSpan: + coordinateSpace: comment-stripped-tex + end: 145883 + file: energy1.tex + sourceTransform: tex2mdx.strip_comments + start: 145872 + sentence: Before stating that proposition, we show that its hypotheses hold in + the situation that arises in Lemma~\ref{BARMU}. Let us recall the situation + of Lemma~\ref{BARMU} + sentenceSpan: + coordinateSpace: comment-stripped-tex + end: 145934 + file: energy1.tex + sourceTransform: tex2mdx.strip_comments + start: 145767 + sourceStrategy: nearest-previous-statement + via: prose + - label: BARMU + refSpan: + coordinateSpace: comment-stripped-tex + end: 145934 + file: energy1.tex + sourceTransform: tex2mdx.strip_comments + start: 145923 + sentence: Before stating that proposition, we show that its hypotheses hold in + the situation that arises in Lemma~\ref{BARMU}. Let us recall the situation + of Lemma~\ref{BARMU} + sentenceSpan: + coordinateSpace: comment-stripped-tex + end: 145934 + file: energy1.tex + sourceTransform: tex2mdx.strip_comments + start: 145767 + sourceStrategy: nearest-previous-statement + via: prose + sourceStatement: + chapter: 18 + file: energy1.tex + label: mulength + mtref: '18.60' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 141140 + sourceTransform: tex2mdx.strip_comments + start: 140651 + targetStatement: + chapter: 18 + file: energy1.tex + label: BARMU + mtref: '18.59' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 140535 + sourceTransform: tex2mdx.strip_comments + start: 139920 + type: prose-mention-nearest-previous-statement + via: + - prose +generator: tools/poincare_tex_extract.py +inference: weak +kind: mention +ref: +- 52f36e456355 +- 8e814eab5ddf +rel: mentions +reviewStatus: candidate +scope: prose +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- prose +--- +Morgan--Tian prose mention. The source is the nearest preceding statement in the same chapter segment. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/eff441b99f1a.md b/projects/poincare-conjecture/.astrolabe/edges/eff441b99f1a.md new file mode 100644 index 00000000..de5e6e1b --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/eff441b99f1a.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: min-max + proofSpan: + bodyEnd: 92724 + bodyStart: 91854 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 92735 + file: newcompar.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 91841 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 92722 + file: newcompar.tex + sourceTransform: tex2mdx.strip_comments + start: 92709 + via: proof + sourceStatement: + chapter: 6 + file: newcompar.tex + label: minmax + mtref: '6.61' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 91839 + sourceTransform: tex2mdx.strip_comments + start: 91251 + targetStatement: + chapter: 6 + file: newcompar.tex + label: min-max + mtref: '6.60' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 90048 + sourceTransform: tex2mdx.strip_comments + start: 89598 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- a7c2ebee07a3 +- 333be2e58780 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/f0096df878d4.md b/projects/poincare-conjecture/.astrolabe/edges/f0096df878d4.md new file mode 100644 index 00000000..617e3c51 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/f0096df878d4.md @@ -0,0 +1,56 @@ +--- +confidence: 0.85 +edgeClass: semantic +evidence: + containmentReason: target theorem-like statement occurs inside source proof + proofSpan: + bodyEnd: 97366 + bodyStart: 94222 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 97377 + file: newcompar.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 94209 + sourceStatement: + chapter: 6 + file: newcompar.tex + label: upperbd + mtref: '6.62' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 94207 + sourceTransform: tex2mdx.strip_comments + start: 92964 + targetStatement: + chapter: 6 + file: newcompar.tex + label: '6.66' + mtref: '6.63' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 95387 + sourceTransform: tex2mdx.strip_comments + start: 95213 + type: proof-contained-theorem-like-statement + via: + - proof-contained-statement +generator: tools/poincare_tex_extract.py +inference: inferred +kind: dependency +ref: +- 7d130fba9693 +- 409f236af149 +rel: depends +reviewStatus: accepted +scope: proof-containment +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof-contained-statement +--- +Morgan--Tian proof-contained statement dependency. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/f041b2d5b08d.md b/projects/poincare-conjecture/.astrolabe/edges/f041b2d5b08d.md new file mode 100644 index 00000000..adaa8513 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/f041b2d5b08d.md @@ -0,0 +1,62 @@ +--- +confidence: 0.35 +edgeClass: prose +evidence: + mentionTriggers: + - label: smlmtflow + refSpan: + coordinateSpace: comment-stripped-tex + end: 6936 + file: singlimit2.tex + sourceTransform: tex2mdx.strip_comments + start: 6921 + sentence: Now we begin the proof of Theorem~\ref{smlmtflow}. + sentenceSpan: + coordinateSpace: comment-stripped-tex + end: 6937 + file: singlimit2.tex + sourceTransform: tex2mdx.strip_comments + start: 6886 + sourceStrategy: nearest-previous-statement + via: prose + sourceStatement: + chapter: 11 + file: singlimit2.tex + label: controlnbhd + mtref: '11.2' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 4360 + sourceTransform: tex2mdx.strip_comments + start: 3597 + targetStatement: + chapter: 11 + file: singlimit2.tex + label: smlmtflow + mtref: '11.1' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 3459 + sourceTransform: tex2mdx.strip_comments + start: 1498 + type: prose-mention-nearest-previous-statement + via: + - prose +generator: tools/poincare_tex_extract.py +inference: weak +kind: mention +ref: +- 0ddfe578961e +- d51aab8a8065 +rel: mentions +reviewStatus: candidate +scope: prose +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- prose +--- +Morgan--Tian prose mention. The source is the nearest preceding statement in the same chapter segment. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/f0546005c214.md b/projects/poincare-conjecture/.astrolabe/edges/f0546005c214.md new file mode 100644 index 00000000..f621f8d9 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/f0546005c214.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: Revol + proofSpan: + bodyEnd: 136142 + bodyStart: 134948 + coordinateSpace: comment-stripped-tex + depth: 2 + end: 136153 + file: temp2kappa.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 134935 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 135757 + file: temp2kappa.tex + sourceTransform: tex2mdx.strip_comments + start: 135746 + via: proof-aux-label-proxy + sourceStatement: + chapter: 9 + file: temp2kappa.tex + label: zkbound + mtref: '9.68' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 134931 + sourceTransform: tex2mdx.strip_comments + start: 134780 + targetStatement: + chapter: 3 + file: flowbasics.tex + label: thmRmevol + mtref: '3.13' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 16962 + sourceTransform: tex2mdx.strip_comments + start: 16120 + type: tex-reference + via: + - proof-aux-label-proxy +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- f61f72020102 +- 1515151790b2 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof-aux-label-proxy +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/f05b0d8899cb.md b/projects/poincare-conjecture/.astrolabe/edges/f05b0d8899cb.md new file mode 100644 index 00000000..1871b0bf --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/f05b0d8899cb.md @@ -0,0 +1,56 @@ +--- +confidence: 0.85 +edgeClass: semantic +evidence: + containmentReason: target theorem-like statement occurs inside source proof + proofSpan: + bodyEnd: 47393 + bodyStart: 39592 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 47404 + file: maxprin.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 39579 + sourceStatement: + chapter: 4 + file: maxprin.tex + label: '' + mtref: '4.27' + sort: remark + span: + coordinateSpace: comment-stripped-tex + end: 39436 + sourceTransform: tex2mdx.strip_comments + start: 39187 + targetStatement: + chapter: 4 + file: maxprin.tex + label: '' + mtref: '4.30' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 43335 + sourceTransform: tex2mdx.strip_comments + start: 43182 + type: proof-contained-theorem-like-statement + via: + - proof-contained-statement +generator: tools/poincare_tex_extract.py +inference: inferred +kind: dependency +ref: +- 1160f53cc783 +- d3f49c0de6a0 +rel: depends +reviewStatus: accepted +scope: proof-containment +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof-contained-statement +--- +Morgan--Tian proof-contained statement dependency. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/f0b9155e45da.md b/projects/poincare-conjecture/.astrolabe/edges/f0b9155e45da.md new file mode 100644 index 00000000..c1409109 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/f0b9155e45da.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 13 + file: surgery.tex + label: delta0 + mtref: '13.5' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 12487 + sourceTransform: tex2mdx.strip_comments + start: 12049 + targetStatement: + chapter: 13 + file: surgery.tex + label: Rcompar + mtref: '13.6' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 12879 + sourceTransform: tex2mdx.strip_comments + start: 12743 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 33f705095319 +- 7f0b4a692c5e +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/f0ec2cfdd42b.md b/projects/poincare-conjecture/.astrolabe/edges/f0ec2cfdd42b.md new file mode 100644 index 00000000..02806c33 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/f0ec2cfdd42b.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: rm>0 + proofSpan: + bodyEnd: 51714 + bodyStart: 50457 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 51725 + file: maxprin.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 50444 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 51114 + file: maxprin.tex + sourceTransform: tex2mdx.strip_comments + start: 51104 + via: proof + sourceStatement: + chapter: 4 + file: maxprin.tex + label: '' + mtref: '4.35' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 50442 + sourceTransform: tex2mdx.strip_comments + start: 50109 + targetStatement: + chapter: 4 + file: maxprin.tex + label: rm>0 + mtref: '4.14' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 16848 + sourceTransform: tex2mdx.strip_comments + start: 16607 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- c7dbaa8c2ca5 +- 0db415314964 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/f123f69af538.md b/projects/poincare-conjecture/.astrolabe/edges/f123f69af538.md new file mode 100644 index 00000000..99759a54 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/f123f69af538.md @@ -0,0 +1,103 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: Dxacross + proofSpan: + bodyEnd: 188096 + bodyStart: 187577 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 188107 + file: energy1.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 187564 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 187634 + file: energy1.tex + sourceTransform: tex2mdx.strip_comments + start: 187620 + via: proof + - context: proof + label: Dxacross + proofSpan: + bodyEnd: 188096 + bodyStart: 187577 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 188107 + file: energy1.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 187564 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 187825 + file: energy1.tex + sourceTransform: tex2mdx.strip_comments + start: 187811 + via: proof + - context: proof + label: Dxacross + proofSpan: + bodyEnd: 188096 + bodyStart: 187577 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 188107 + file: energy1.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 187564 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 188094 + file: energy1.tex + sourceTransform: tex2mdx.strip_comments + start: 188080 + via: proof + sourceStatement: + chapter: 18 + file: energy1.tex + label: '' + mtref: '18.82' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 187562 + sourceTransform: tex2mdx.strip_comments + start: 187208 + targetStatement: + chapter: 18 + file: energy1.tex + label: Dxacross + mtref: '18.74' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 173839 + sourceTransform: tex2mdx.strip_comments + start: 173679 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 9e3a849a6a4c +- 0d6f74e37efb +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/f162b7cea6e6.md b/projects/poincare-conjecture/.astrolabe/edges/f162b7cea6e6.md new file mode 100644 index 00000000..cb0f6ead --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/f162b7cea6e6.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: 2ndvari + proofSpan: + bodyEnd: 22715 + bodyStart: 21316 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 22726 + file: newcompar.tex + ownerLabel: 2ndvarj + sourceTransform: tex2mdx.strip_comments + start: 21303 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 21388 + file: newcompar.tex + sourceTransform: tex2mdx.strip_comments + start: 21375 + via: proof + sourceStatement: + chapter: 6 + file: newcompar.tex + label: 2ndvarj + mtref: '6.13' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 16190 + sourceTransform: tex2mdx.strip_comments + start: 15621 + targetStatement: + chapter: 6 + file: newcompar.tex + label: 2ndvari + mtref: '6.14' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 17350 + sourceTransform: tex2mdx.strip_comments + start: 16270 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 26b7f69e803e +- f48558c60ca2 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/f163bff3c9b9.md b/projects/poincare-conjecture/.astrolabe/edges/f163bff3c9b9.md new file mode 100644 index 00000000..af1b5546 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/f163bff3c9b9.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 6 + file: newcompar.tex + label: diffeo + mtref: '6.28' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 38046 + sourceTransform: tex2mdx.strip_comments + start: 37549 + targetStatement: + chapter: 6 + file: newcompar.tex + label: DL + mtref: '6.29' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 39591 + sourceTransform: tex2mdx.strip_comments + start: 39266 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 17254b50b27a +- 033786a89c6d +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/f1662df03116.md b/projects/poincare-conjecture/.astrolabe/edges/f1662df03116.md new file mode 100644 index 00000000..cbd8b19d --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/f1662df03116.md @@ -0,0 +1,65 @@ +--- +confidence: 0.45 +edgeClass: semantic +evidence: + endpointSemanticDegreeBeforeTermBackfill: + source: 0 + target: 3 + selectionReason: semantic-connectivity-backfill + sourceMatchSpan: + coordinateSpace: comment-stripped-tex + end: 57256 + file: surgery.tex + sourceTransform: tex2mdx.strip_comments + start: 57233 + sourceStatement: + chapter: 14 + file: surgery.tex + label: '' + mtref: '14.18' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 58570 + sourceTransform: tex2mdx.strip_comments + start: 57157 + targetStatement: + chapter: 14 + file: surgery.tex + label: '' + mtref: '14.8' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 47114 + sourceTransform: tex2mdx.strip_comments + start: 46506 + targetTermSpan: + coordinateSpace: comment-stripped-tex + end: 46550 + file: surgery.tex + sourceTransform: tex2mdx.strip_comments + start: 46527 + term: ricci flow with surgery + termAmbiguity: unique-in-chapter + termSource: italic + termSpecificity: no-longer-defined-term + type: same-chapter-definition-term-match + via: + - definition-term:ricci flow with surgery +generator: tools/poincare_tex_extract.py +inference: weak +kind: definition-use +ref: +- 399dd63f056b +- 842f5a7c856b +rel: uses +reviewStatus: candidate +scope: same-chapter-term-match +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- definition-term:ricci flow with surgery +--- +Morgan--Tian same-chapter definition term match. This is a low-confidence candidate edge for review. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/f18ab48b9f39.md b/projects/poincare-conjecture/.astrolabe/edges/f18ab48b9f39.md new file mode 100644 index 00000000..3e0afb38 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/f18ab48b9f39.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 16 + file: surgery.tex + label: '' + mtref: '16.16' + sort: remark + span: + coordinateSpace: comment-stripped-tex + end: 138750 + sourceTransform: tex2mdx.strip_comments + start: 138540 + targetStatement: + chapter: 16 + file: surgery.tex + label: '' + mtref: '16.17' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 140633 + sourceTransform: tex2mdx.strip_comments + start: 140555 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 4ee1bc9acc48 +- 9b7f421ab0b9 +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/f1d1662e33ea.md b/projects/poincare-conjecture/.astrolabe/edges/f1d1662e33ea.md new file mode 100644 index 00000000..81af4f0a --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/f1d1662e33ea.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: Revol + proofSpan: + bodyEnd: 25052 + bodyStart: 23931 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 25063 + file: maxprin.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 23918 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 24724 + file: maxprin.tex + sourceTransform: tex2mdx.strip_comments + start: 24713 + via: proof-aux-label-proxy + sourceStatement: + chapter: 4 + file: maxprin.tex + label: firststrongmax + mtref: '4.18' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 23916 + sourceTransform: tex2mdx.strip_comments + start: 23628 + targetStatement: + chapter: 3 + file: flowbasics.tex + label: thmRmevol + mtref: '3.13' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 16962 + sourceTransform: tex2mdx.strip_comments + start: 16120 + type: tex-reference + via: + - proof-aux-label-proxy +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 016cf36686b2 +- 1515151790b2 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof-aux-label-proxy +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/f1e0b8c9c94b.md b/projects/poincare-conjecture/.astrolabe/edges/f1e0b8c9c94b.md new file mode 100644 index 00000000..4d911d2e --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/f1e0b8c9c94b.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: Jacobi + proofSpan: + bodyEnd: 59995 + bodyStart: 59477 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 60006 + file: newcompar.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 59464 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 59814 + file: newcompar.tex + sourceTransform: tex2mdx.strip_comments + start: 59802 + via: proof + sourceStatement: + chapter: 6 + file: newcompar.tex + label: adapjab + mtref: '6.41' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 59462 + sourceTransform: tex2mdx.strip_comments + start: 58890 + targetStatement: + chapter: 6 + file: newcompar.tex + label: Jacobi + mtref: '6.40' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 57503 + sourceTransform: tex2mdx.strip_comments + start: 57209 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 72949e6350d4 +- 44abf07e60ec +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/f262cb781db0.md b/projects/poincare-conjecture/.astrolabe/edges/f262cb781db0.md new file mode 100644 index 00000000..28bd907d --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/f262cb781db0.md @@ -0,0 +1,82 @@ +--- +confidence: 0.35 +edgeClass: prose +evidence: + mentionTriggers: + - label: newkappa0r0t0 + refSpan: + coordinateSpace: comment-stripped-tex + end: 79513 + file: surgery.tex + sourceTransform: tex2mdx.strip_comments + start: 79494 + sentence: Then according to Claim~\ref{newkappa0r0t0} the flow exists on $[0,T_1]$ + and the norm of the Riemann curvature is bounded by $2$ on $[0,T_1]$, so that + by Assumption (7) there are no surgeries in this time interval + sentenceSpan: + coordinateSpace: comment-stripped-tex + end: 79683 + file: surgery.tex + sourceTransform: tex2mdx.strip_comments + start: 79468 + sourceStrategy: nearest-previous-statement + via: prose + - label: newkappa0r0t0 + refSpan: + coordinateSpace: comment-stripped-tex + end: 79719 + file: surgery.tex + sourceTransform: tex2mdx.strip_comments + start: 79700 + sentence: Also, by Claim~\ref{newkappa0r0t0} there is a $\kappa_0>0$ so that ${\rm + Vol}\,B(x,t,r)\le \kappa_0r^3$ for every $t\le T_1$ and $x\in M_t$ and every + $r\le \epsilon$. + sentenceSpan: + coordinateSpace: comment-stripped-tex + end: 79851 + file: surgery.tex + sourceTransform: tex2mdx.strip_comments + start: 79684 + sourceStrategy: nearest-previous-statement + via: prose + sourceStatement: + chapter: 15 + file: surgery.tex + label: '' + mtref: '15.6' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 79252 + sourceTransform: tex2mdx.strip_comments + start: 79163 + targetStatement: + chapter: 15 + file: surgery.tex + label: newkappa0r0t0 + mtref: '15.1' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 60765 + sourceTransform: tex2mdx.strip_comments + start: 60324 + type: prose-mention-nearest-previous-statement + via: + - prose +generator: tools/poincare_tex_extract.py +inference: weak +kind: mention +ref: +- 1f2e2e449b34 +- f5e180ecfd18 +rel: mentions +reviewStatus: candidate +scope: prose +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- prose +--- +Morgan--Tian prose mention. The source is the nearest preceding statement in the same chapter segment. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/f26c6a7ea3df.md b/projects/poincare-conjecture/.astrolabe/edges/f26c6a7ea3df.md new file mode 100644 index 00000000..cd00c247 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/f26c6a7ea3df.md @@ -0,0 +1,56 @@ +--- +confidence: 0.85 +edgeClass: semantic +evidence: + containmentReason: target theorem-like statement occurs inside source proof + proofSpan: + bodyEnd: 54967 + bodyStart: 37263 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 54978 + file: energy1.tex + ownerLabel: W_2 + sourceTransform: tex2mdx.strip_comments + start: 37250 + sourceStatement: + chapter: 18 + file: energy1.tex + label: W_2 + mtref: '18.11' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 36484 + sourceTransform: tex2mdx.strip_comments + start: 36186 + targetStatement: + chapter: 18 + file: energy1.tex + label: '' + mtref: '18.16' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 47275 + sourceTransform: tex2mdx.strip_comments + start: 47013 + type: proof-contained-theorem-like-statement + via: + - proof-contained-statement +generator: tools/poincare_tex_extract.py +inference: inferred +kind: dependency +ref: +- 7bb0fa0c9412 +- 39199318fe64 +rel: depends +reviewStatus: accepted +scope: proof-containment +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof-contained-statement +--- +Morgan--Tian proof-contained statement dependency. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/f27699f8a181.md b/projects/poincare-conjecture/.astrolabe/edges/f27699f8a181.md new file mode 100644 index 00000000..a658a294 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/f27699f8a181.md @@ -0,0 +1,45 @@ +--- +confidence: 0.2 +edgeClass: structural +evidence: + sourceStatement: + chapter: 13 + file: surgery.tex + label: capball + mtref: '13.4' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 7464 + sourceTransform: tex2mdx.strip_comments + start: 6854 + targetStatement: + chapter: 13 + file: surgery.tex + label: delta0 + mtref: '13.5' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 12487 + sourceTransform: tex2mdx.strip_comments + start: 12049 + type: chapter-reading-order-adjacency + via: + - chapter-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 752133843e1e +- 33f705095319 +rel: in-chapter +reviewStatus: accepted +scope: chapter +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- chapter-sequence +--- +Morgan--Tian reading-order adjacency across sections in a chapter. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/f2c5a97c349f.md b/projects/poincare-conjecture/.astrolabe/edges/f2c5a97c349f.md new file mode 100644 index 00000000..8053c9ed --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/f2c5a97c349f.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: weak + proofSpan: + bodyEnd: 35538 + bodyStart: 33593 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 35549 + file: temp2kappa.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 33580 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 34041 + file: temp2kappa.tex + sourceTransform: tex2mdx.strip_comments + start: 34031 + via: proof + sourceStatement: + chapter: 9 + file: temp2kappa.tex + label: '' + mtref: '9.23' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 33578 + sourceTransform: tex2mdx.strip_comments + start: 33087 + targetStatement: + chapter: 7 + file: newcomp2.tex + label: weak + mtref: '7.13' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 23409 + sourceTransform: tex2mdx.strip_comments + start: 22180 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 3776c086ec3b +- 7d5d94efe4e9 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/f2cfafc9f4aa.md b/projects/poincare-conjecture/.astrolabe/edges/f2cfafc9f4aa.md new file mode 100644 index 00000000..f25bcce1 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/f2cfafc9f4aa.md @@ -0,0 +1,120 @@ +--- +confidence: 0.35 +edgeClass: prose +evidence: + mentionTriggers: + - label: asympt + refSpan: + coordinateSpace: comment-stripped-tex + end: 63853 + file: stdsoln.tex + sourceTransform: tex2mdx.strip_comments + start: 63841 + sentence: then it follows from Claim~\ref{asympt} that $G(\tilde{\rho}_{i-1},{\bf + x},t)$ is bounded on $\mathbb{R}^{n+2} \times [0,T]$ + sentenceSpan: + coordinateSpace: comment-stripped-tex + end: 63938 + file: stdsoln.tex + sourceTransform: tex2mdx.strip_comments + start: 63813 + sourceStrategy: nearest-previous-statement + via: prose + - label: asympt + refSpan: + coordinateSpace: comment-stripped-tex + end: 64038 + file: stdsoln.tex + sourceTransform: tex2mdx.strip_comments + start: 64026 + sentence: '\[ |G(\tilde{\rho}_{i-1},x,t)| \leq C_*(C_1,\tilde{h} ), \] and also + because of Claim~\ref{asympt} both $|\nabla B|$ and $|\nabla \tilde h|$ are + bounded on all of $\Ar^{n+2}\times [0,T]$, it follows that $F(x, \tilde{\rho}_{i-1},\nabla\tilde{\rho}_{i-1},t)$ + is bounded: \begin{align*} & |F(x, \tilde{\rho}_{i-1},\nabla\tilde{\rho}_{i-1},t)| + \\ \leq & \left[(n-1)\sup |\nabla \tilde{h}| + \sup |\nabla B| \right]C_2+C_2^2+ + C_*(C_1,\tilde{h} ) = C_3 \end{align*} Hence $\tilde{\rho}_i$ exists.' + sentenceSpan: + coordinateSpace: comment-stripped-tex + end: 64432 + file: stdsoln.tex + sourceTransform: tex2mdx.strip_comments + start: 63939 + sourceStrategy: nearest-previous-statement + via: prose + - label: asympt + refSpan: + coordinateSpace: comment-stripped-tex + end: 70596 + file: stdsoln.tex + sourceTransform: tex2mdx.strip_comments + start: 70584 + sentence: It follows from (\ref{rho_i est}) and Claim~\ref{asympt} that there + is a constant $C_8$ independent of $i$ such that + sentenceSpan: + coordinateSpace: comment-stripped-tex + end: 70656 + file: stdsoln.tex + sourceTransform: tex2mdx.strip_comments + start: 70539 + sourceStrategy: nearest-previous-statement + via: prose + - label: asympt + refSpan: + coordinateSpace: comment-stripped-tex + end: 72629 + file: stdsoln.tex + sourceTransform: tex2mdx.strip_comments + start: 72617 + sentence: From (\ref{rhoHolder}) and Claim~\ref{asympt} we know that $\nabla [ + (n-1)\tilde{h} -B+ \tilde{\rho}_\infty ]$ has $C^{\alpha,\alpha/2}$-H\"older-norm + bounded (this means $\alpha$-H\"older norm in space and the $\alpha/2$-H\"older + norm in time) + sentenceSpan: + coordinateSpace: comment-stripped-tex + end: 72828 + file: stdsoln.tex + sourceTransform: tex2mdx.strip_comments + start: 72583 + sourceStrategy: nearest-previous-statement + via: prose + sourceStatement: + chapter: 12 + file: stdsoln.tex + label: '' + mtref: '12.24' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 62221 + sourceTransform: tex2mdx.strip_comments + start: 61942 + targetStatement: + chapter: 12 + file: stdsoln.tex + label: asympt + mtref: '12.23' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 61284 + sourceTransform: tex2mdx.strip_comments + start: 60562 + type: prose-mention-nearest-previous-statement + via: + - prose +generator: tools/poincare_tex_extract.py +inference: weak +kind: mention +ref: +- 43ea5e1d04cf +- ac4ac4fb06f8 +rel: mentions +reviewStatus: candidate +scope: prose +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- prose +--- +Morgan--Tian prose mention. The source is the nearest preceding statement in the same chapter segment. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/f2dd77d90830.md b/projects/poincare-conjecture/.astrolabe/edges/f2dd77d90830.md new file mode 100644 index 00000000..740d99c2 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/f2dd77d90830.md @@ -0,0 +1,64 @@ +--- +confidence: 0.35 +edgeClass: prose +evidence: + mentionTriggers: + - label: hatM + refSpan: + coordinateSpace: comment-stripped-tex + end: 66986 + file: surgery.tex + sourceTransform: tex2mdx.strip_comments + start: 66976 + sentence: Let ${\mathcal M}_{(-\infty,t)}$ be ${\bf t}^{-1}((-\infty,t))$ and + let $(\widehat {\mathcal M}_{(-\infty,t)},\widehat G)$ be the maximal extension + of $({\mathcal M}_{(-\infty,t)},G)$ to time $t$, as given in Definition~\ref{hatM}. + sentenceSpan: + coordinateSpace: comment-stripped-tex + end: 66987 + file: surgery.tex + sourceTransform: tex2mdx.strip_comments + start: 66755 + sourceStrategy: nearest-previous-statement + via: prose + sourceStatement: + chapter: 15 + file: surgery.tex + label: neckglue + mtref: '15.2' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 61465 + sourceTransform: tex2mdx.strip_comments + start: 60812 + targetStatement: + chapter: 11 + file: singlimit2.tex + label: hatM + mtref: '11.22' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 42568 + sourceTransform: tex2mdx.strip_comments + start: 41534 + type: prose-mention-nearest-previous-statement + via: + - prose +generator: tools/poincare_tex_extract.py +inference: weak +kind: mention +ref: +- a992d6a89479 +- 8651cf20bd2d +rel: mentions +reviewStatus: candidate +scope: prose +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- prose +--- +Morgan--Tian prose mention. The source is the nearest preceding statement in the same chapter segment. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/f329f293eb18.md b/projects/poincare-conjecture/.astrolabe/edges/f329f293eb18.md new file mode 100644 index 00000000..27dc9052 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/f329f293eb18.md @@ -0,0 +1,55 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: statement + label: statement + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 52847 + file: bddcurvbdddist.tex + sourceTransform: tex2mdx.strip_comments + start: 52832 + via: statement + sourceStatement: + chapter: 10 + file: bddcurvbdddist.tex + label: secondstatment + mtref: '10.29' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 53510 + sourceTransform: tex2mdx.strip_comments + start: 52640 + targetStatement: + chapter: 10 + file: bddcurvbdddist.tex + label: statement + mtref: '10.12' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 36191 + sourceTransform: tex2mdx.strip_comments + start: 35406 + type: tex-reference + via: + - statement +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- b34d555cd7b5 +- 9cf6ad839dbb +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- statement +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/f32b0e60af24.md b/projects/poincare-conjecture/.astrolabe/edges/f32b0e60af24.md new file mode 100644 index 00000000..c1cf02da --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/f32b0e60af24.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 19 + file: canonnbhd.tex + label: coreint + mtref: '19.8' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 11926 + sourceTransform: tex2mdx.strip_comments + start: 11192 + targetStatement: + chapter: 19 + file: canonnbhd.tex + label: neckdistcomp + mtref: '19.9' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 14663 + sourceTransform: tex2mdx.strip_comments + start: 13813 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 5ef5906bb370 +- 6cf1cbc39ada +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/f33b9167f4df.md b/projects/poincare-conjecture/.astrolabe/edges/f33b9167f4df.md new file mode 100644 index 00000000..ea25cc3c --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/f33b9167f4df.md @@ -0,0 +1,55 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: statement + label: surgeryparams + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 96556 + file: surgery.tex + sourceTransform: tex2mdx.strip_comments + start: 96537 + via: statement + sourceStatement: + chapter: 15 + file: surgery.tex + label: surgerydist + mtref: '15.12' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 97780 + sourceTransform: tex2mdx.strip_comments + start: 96249 + targetStatement: + chapter: 15 + file: surgery.tex + label: surgeryparams + mtref: '15.7' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 80864 + sourceTransform: tex2mdx.strip_comments + start: 79853 + type: tex-reference + via: + - statement +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 9021a4eca03f +- 22c40a5664ac +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- statement +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/f35ecd8324ab.md b/projects/poincare-conjecture/.astrolabe/edges/f35ecd8324ab.md new file mode 100644 index 00000000..123520d5 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/f35ecd8324ab.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 15 + file: surgery.tex + label: surgerytoptype + mtref: '15.3' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 69964 + sourceTransform: tex2mdx.strip_comments + start: 69392 + targetStatement: + chapter: 15 + file: surgery.tex + label: Tgood0good + mtref: '15.4' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 74045 + sourceTransform: tex2mdx.strip_comments + start: 73002 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 7946423e82a4 +- 7822d06083e0 +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/f360f4ccd72e.md b/projects/poincare-conjecture/.astrolabe/edges/f360f4ccd72e.md new file mode 100644 index 00000000..3e0e3fc6 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/f360f4ccd72e.md @@ -0,0 +1,79 @@ +--- +confidence: 0.35 +edgeClass: prose +evidence: + mentionTriggers: + - label: pi2 + refSpan: + coordinateSpace: comment-stripped-tex + end: 36542 + file: energy1.tex + sourceTransform: tex2mdx.strip_comments + start: 36533 + sentence: Let us show how this lemma implies Proposition~\ref{pi2} + sentenceSpan: + coordinateSpace: comment-stripped-tex + end: 36542 + file: energy1.tex + sourceTransform: tex2mdx.strip_comments + start: 36485 + sourceStrategy: nearest-previous-statement + via: prose + - label: pi2 + refSpan: + coordinateSpace: comment-stripped-tex + end: 37207 + file: energy1.tex + sourceTransform: tex2mdx.strip_comments + start: 37198 + sentence: This shows that to complete the proof of Proposition~\ref{pi2} we need + only establish Lemma~\ref{W_2}. + sentenceSpan: + coordinateSpace: comment-stripped-tex + end: 37247 + file: energy1.tex + sourceTransform: tex2mdx.strip_comments + start: 37144 + sourceStrategy: nearest-previous-statement + via: prose + sourceStatement: + chapter: 18 + file: energy1.tex + label: W_2 + mtref: '18.11' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 36484 + sourceTransform: tex2mdx.strip_comments + start: 36186 + targetStatement: + chapter: 18 + file: energy1.tex + label: pi2 + mtref: '18.9' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 30087 + sourceTransform: tex2mdx.strip_comments + start: 29657 + type: prose-mention-nearest-previous-statement + via: + - prose +generator: tools/poincare_tex_extract.py +inference: weak +kind: mention +ref: +- 7bb0fa0c9412 +- 4415092c2079 +rel: mentions +reviewStatus: candidate +scope: prose +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- prose +--- +Morgan--Tian prose mention. The source is the nearest preceding statement in the same chapter segment. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/f36f2c416f94.md b/projects/poincare-conjecture/.astrolabe/edges/f36f2c416f94.md new file mode 100644 index 00000000..ca84f9b9 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/f36f2c416f94.md @@ -0,0 +1,45 @@ +--- +confidence: 0.2 +edgeClass: structural +evidence: + sourceStatement: + chapter: 9 + file: temp2kappa.tex + label: '' + mtref: '9.82' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 157267 + sourceTransform: tex2mdx.strip_comments + start: 157017 + targetStatement: + chapter: 9 + file: temp2kappa.tex + label: infprod2 + mtref: '9.83' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 157798 + sourceTransform: tex2mdx.strip_comments + start: 157331 + type: chapter-reading-order-adjacency + via: + - chapter-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 1138b028cc69 +- 25e0946e7222 +rel: in-chapter +reviewStatus: accepted +scope: chapter +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- chapter-sequence +--- +Morgan--Tian reading-order adjacency across sections in a chapter. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/f3baa8360405.md b/projects/poincare-conjecture/.astrolabe/edges/f3baa8360405.md new file mode 100644 index 00000000..31809b81 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/f3baa8360405.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: soul + proofSpan: + bodyEnd: 103549 + bodyStart: 101913 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 103560 + file: temp2kappa.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 101900 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 103073 + file: temp2kappa.tex + sourceTransform: tex2mdx.strip_comments + start: 103063 + via: proof + sourceStatement: + chapter: 9 + file: temp2kappa.tex + label: 3DGSSkappa + mtref: '9.53' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 101898 + sourceTransform: tex2mdx.strip_comments + start: 101535 + targetStatement: + chapter: 2 + file: prelim.tex + label: soul + mtref: '2.7' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 57159 + sourceTransform: tex2mdx.strip_comments + start: 56544 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 351e4ac14e4e +- 2b5f18292e62 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/f3d809a193b0.md b/projects/poincare-conjecture/.astrolabe/edges/f3d809a193b0.md new file mode 100644 index 00000000..65c9255a --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/f3d809a193b0.md @@ -0,0 +1,45 @@ +--- +confidence: 0.2 +edgeClass: structural +evidence: + sourceStatement: + chapter: 12 + file: stdsoln.tex + label: '' + mtref: '12.30' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 88901 + sourceTransform: tex2mdx.strip_comments + start: 88738 + targetStatement: + chapter: 12 + file: stdsoln.tex + label: Restim + mtref: '12.31' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 90335 + sourceTransform: tex2mdx.strip_comments + start: 90184 + type: chapter-reading-order-adjacency + via: + - chapter-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- cf6eed7b1da4 +- ffae4504318a +rel: in-chapter +reviewStatus: accepted +scope: chapter +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- chapter-sequence +--- +Morgan--Tian reading-order adjacency across sections in a chapter. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/f4240797c00b.md b/projects/poincare-conjecture/.astrolabe/edges/f4240797c00b.md new file mode 100644 index 00000000..0d4ff7af --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/f4240797c00b.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: compint + proofSpan: + bodyEnd: 13705 + bodyStart: 11943 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 13716 + file: canonnbhd.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 11930 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 12626 + file: canonnbhd.tex + sourceTransform: tex2mdx.strip_comments + start: 12613 + via: proof + sourceStatement: + chapter: 19 + file: canonnbhd.tex + label: coreint + mtref: '19.8' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 11926 + sourceTransform: tex2mdx.strip_comments + start: 11192 + targetStatement: + chapter: 19 + file: canonnbhd.tex + label: compint + mtref: '19.6' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 8416 + sourceTransform: tex2mdx.strip_comments + start: 8079 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 5ef5906bb370 +- 82f43e137897 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/f44eafe18526.md b/projects/poincare-conjecture/.astrolabe/edges/f44eafe18526.md new file mode 100644 index 00000000..e3851837 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/f44eafe18526.md @@ -0,0 +1,45 @@ +--- +confidence: 0.2 +edgeClass: structural +evidence: + sourceStatement: + chapter: 1 + file: prelim.tex + label: calabi + mtref: '1.29' + sort: remark + span: + coordinateSpace: comment-stripped-tex + end: 39490 + sourceTransform: tex2mdx.strip_comments + start: 38777 + targetStatement: + chapter: 1 + file: prelim.tex + label: '' + mtref: '1.30' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 40898 + sourceTransform: tex2mdx.strip_comments + start: 40693 + type: chapter-reading-order-adjacency + via: + - chapter-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 9f5db77333af +- b4ef6801b3e2 +rel: in-chapter +reviewStatus: accepted +scope: chapter +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- chapter-sequence +--- +Morgan--Tian reading-order adjacency across sections in a chapter. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/f45859b93aee.md b/projects/poincare-conjecture/.astrolabe/edges/f45859b93aee.md new file mode 100644 index 00000000..9b7d7f7f --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/f45859b93aee.md @@ -0,0 +1,68 @@ +--- +confidence: 0.72 +edgeClass: semantic +evidence: + anaphoraTriggers: + - context: statement + direction: previous + phrase: previous theorem + sort: theorem + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 38714 + file: singlimit2.tex + sourceTransform: tex2mdx.strip_comments + start: 38698 + via: local-anaphora-statement + - context: statement + direction: previous + phrase: previous theorem + sort: theorem + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 38853 + file: singlimit2.tex + sourceTransform: tex2mdx.strip_comments + start: 38837 + via: local-anaphora-statement + sourceStatement: + chapter: 11 + file: singlimit2.tex + label: Rproper + mtref: '11.21' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 39168 + sourceTransform: tex2mdx.strip_comments + start: 38496 + targetStatement: + chapter: 11 + file: singlimit2.tex + label: omega + mtref: '11.19' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 38153 + sourceTransform: tex2mdx.strip_comments + start: 36599 + type: local-anaphora-reference + via: + - local-anaphora-statement +generator: tools/poincare_tex_extract.py +inference: inferred +kind: dependency +ref: +- a99536ac8110 +- af00774dbaa9 +rel: depends +reviewStatus: accepted +scope: local-context +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- local-anaphora-statement +--- +Morgan--Tian local anaphora dependency. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/f4735aaba62d.md b/projects/poincare-conjecture/.astrolabe/edges/f4735aaba62d.md new file mode 100644 index 00000000..504cb0bf --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/f4735aaba62d.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: R_0cannbhd + proofSpan: + bodyEnd: 96054 + bodyStart: 95212 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 96065 + file: stdsoln.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 95199 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 95240 + file: stdsoln.tex + sourceTransform: tex2mdx.strip_comments + start: 95224 + via: proof + sourceStatement: + chapter: 12 + file: stdsoln.tex + label: '' + mtref: '12.33' + sort: remark + span: + coordinateSpace: comment-stripped-tex + end: 95195 + sourceTransform: tex2mdx.strip_comments + start: 94631 + targetStatement: + chapter: 12 + file: stdsoln.tex + label: R_0cannbhd + mtref: '12.28' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 83517 + sourceTransform: tex2mdx.strip_comments + start: 82969 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 246771506ee5 +- 85e3c925177e +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/f4738635fb16.md b/projects/poincare-conjecture/.astrolabe/edges/f4738635fb16.md new file mode 100644 index 00000000..bf30e0ab --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/f4738635fb16.md @@ -0,0 +1,63 @@ +--- +confidence: 0.35 +edgeClass: prose +evidence: + mentionTriggers: + - label: sliceconv + refSpan: + coordinateSpace: comment-stripped-tex + end: 60559 + file: bddcurvbdddist.tex + sourceTransform: tex2mdx.strip_comments + start: 60544 + sentence: Hence, by Theorem~\ref{sliceconv} we can pass to a subsequence and extract + a geometric limit + sentenceSpan: + coordinateSpace: comment-stripped-tex + end: 60618 + file: bddcurvbdddist.tex + sourceTransform: tex2mdx.strip_comments + start: 60525 + sourceStrategy: nearest-previous-statement + via: prose + sourceStatement: + chapter: 10 + file: bddcurvbdddist.tex + label: '' + mtref: '10.30' + sort: remark + span: + coordinateSpace: comment-stripped-tex + end: 58650 + sourceTransform: tex2mdx.strip_comments + start: 58353 + targetStatement: + chapter: 5 + file: converge2.tex + label: sliceconv + mtref: '5.11' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 21468 + sourceTransform: tex2mdx.strip_comments + start: 20456 + type: prose-mention-nearest-previous-statement + via: + - prose +generator: tools/poincare_tex_extract.py +inference: weak +kind: mention +ref: +- c397ced906ac +- d9f3522dec12 +rel: mentions +reviewStatus: candidate +scope: prose +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- prose +--- +Morgan--Tian prose mention. The source is the nearest preceding statement in the same chapter segment. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/f4950deb4bc0.md b/projects/poincare-conjecture/.astrolabe/edges/f4950deb4bc0.md new file mode 100644 index 00000000..54bef89b --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/f4950deb4bc0.md @@ -0,0 +1,56 @@ +--- +confidence: 0.85 +edgeClass: semantic +evidence: + containmentReason: target theorem-like statement occurs inside source proof + proofSpan: + bodyEnd: 129463 + bodyStart: 113176 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 129474 + file: surgery.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 113163 + sourceStatement: + chapter: 16 + file: surgery.tex + label: altern + mtref: '16.5' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 112999 + sourceTransform: tex2mdx.strip_comments + start: 110961 + targetStatement: + chapter: 16 + file: surgery.tex + label: '' + mtref: '16.10' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 125643 + sourceTransform: tex2mdx.strip_comments + start: 125039 + type: proof-contained-theorem-like-statement + via: + - proof-contained-statement +generator: tools/poincare_tex_extract.py +inference: inferred +kind: dependency +ref: +- ad4f67ea75bc +- 683bb06c1059 +rel: depends +reviewStatus: accepted +scope: proof-containment +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof-contained-statement +--- +Morgan--Tian proof-contained statement dependency. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/f4ab4a1e6983.md b/projects/poincare-conjecture/.astrolabe/edges/f4ab4a1e6983.md new file mode 100644 index 00000000..3985d069 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/f4ab4a1e6983.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 16 + file: surgery.tex + label: goodtotheta + mtref: '16.9' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 120111 + sourceTransform: tex2mdx.strip_comments + start: 119688 + targetStatement: + chapter: 16 + file: surgery.tex + label: '' + mtref: '16.10' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 125643 + sourceTransform: tex2mdx.strip_comments + start: 125039 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 712124d7ea02 +- 683bb06c1059 +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/f4dba2e1f402.md b/projects/poincare-conjecture/.astrolabe/edges/f4dba2e1f402.md new file mode 100644 index 00000000..96b83260 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/f4dba2e1f402.md @@ -0,0 +1,62 @@ +--- +confidence: 0.9 +edgeClass: structural +evidence: + containmentDepth: 1 + proofSpan: + bodyEnd: 80533 + bodyStart: 76639 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 80544 + file: prelim.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 76626 + sourceStatement: + chapter: 2 + file: prelim.tex + label: narrows + mtref: '2.19' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 76622 + sourceTransform: tex2mdx.strip_comments + start: 76377 + targetContainedSpan: + coordinateSpace: comment-stripped-tex + end: 77469 + file: prelim.tex + sourceTransform: tex2mdx.strip_comments + start: 76866 + targetStatement: + chapter: 2 + file: prelim.tex + label: neckseparate + mtref: '2.20' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 77469 + sourceTransform: tex2mdx.strip_comments + start: 76866 + type: proof-containment + via: + - proof-contains +generator: tools/poincare_tex_extract.py +inference: structural +kind: containment +ref: +- cb9ca6d6db9f +- 818772530e5a +rel: contains +reviewStatus: accepted +scope: proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof-contains +--- +Morgan--Tian proof containment. The source is the statement whose proof contains the target statement. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/f50686fedb49.md b/projects/poincare-conjecture/.astrolabe/edges/f50686fedb49.md new file mode 100644 index 00000000..a7b252c0 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/f50686fedb49.md @@ -0,0 +1,65 @@ +--- +confidence: 0.35 +edgeClass: prose +evidence: + mentionTriggers: + - label: gradshrink + refSpan: + coordinateSpace: comment-stripped-tex + end: 63982 + file: temp2kappa.tex + sourceTransform: tex2mdx.strip_comments + start: 63966 + sentence: Together with Proposition~\ref{gradshrink}, this completes the proof + of Theorem~\ref{asympGSS}, namely of the fact that $(M_\infty,g_\infty(t)),\ + -\infty0$. + sentenceSpan: + coordinateSpace: comment-stripped-tex + end: 139884 + file: temp2kappa.tex + sourceTransform: tex2mdx.strip_comments + start: 139662 + sourceStrategy: nearest-previous-statement + via: prose + sourceStatement: + chapter: 9 + file: temp2kappa.tex + label: derivcor + mtref: '9.71' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 139489 + sourceTransform: tex2mdx.strip_comments + start: 139133 + targetStatement: + chapter: 9 + file: temp2kappa.tex + label: kappa0 + mtref: '9.58' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 113708 + sourceTransform: tex2mdx.strip_comments + start: 113563 + type: prose-mention-nearest-previous-statement + via: + - prose +generator: tools/poincare_tex_extract.py +inference: weak +kind: mention +ref: +- 8c090fbd21a0 +- 9c54e0d0c7d8 +rel: mentions +reviewStatus: candidate +scope: prose +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- prose +--- +Morgan--Tian prose mention. The source is the nearest preceding statement in the same chapter segment. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/fc091832958f.md b/projects/poincare-conjecture/.astrolabe/edges/fc091832958f.md new file mode 100644 index 00000000..d0be27ff --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/fc091832958f.md @@ -0,0 +1,65 @@ +--- +confidence: 0.35 +edgeClass: prose +evidence: + mentionTriggers: + - label: BishopGromov + refSpan: + coordinateSpace: comment-stripped-tex + end: 42570 + file: temp2kappa.tex + sourceTransform: tex2mdx.strip_comments + start: 42552 + sentence: Since the time slices of all the flows in question have non-negative + curvature, by Theorem~\ref{BishopGromov} the volume of the ball of radius $s$ + is at most $\omega s^{n}$ where $\omega$ is the volume of the ball of radius + one in $\Ar^n$ + sentenceSpan: + coordinateSpace: comment-stripped-tex + end: 42699 + file: temp2kappa.tex + sourceTransform: tex2mdx.strip_comments + start: 42460 + sourceStrategy: nearest-previous-statement + via: prose + sourceStatement: + chapter: 9 + file: temp2kappa.tex + label: '' + mtref: '9.26' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 41863 + sourceTransform: tex2mdx.strip_comments + start: 41621 + targetStatement: + chapter: 1 + file: prelim.tex + label: BishopGromov + mtref: '1.34' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 43978 + sourceTransform: tex2mdx.strip_comments + start: 43265 + type: prose-mention-nearest-previous-statement + via: + - prose +generator: tools/poincare_tex_extract.py +inference: weak +kind: mention +ref: +- d16f43d74c54 +- 1a1b2c4fdc9b +rel: mentions +reviewStatus: candidate +scope: prose +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- prose +--- +Morgan--Tian prose mention. The source is the nearest preceding statement in the same chapter segment. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/fc0a568c8179.md b/projects/poincare-conjecture/.astrolabe/edges/fc0a568c8179.md new file mode 100644 index 00000000..d8dbeb4d --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/fc0a568c8179.md @@ -0,0 +1,62 @@ +--- +confidence: 0.9 +edgeClass: structural +evidence: + containmentDepth: 2 + proofSpan: + bodyEnd: 49745 + bodyStart: 44181 + coordinateSpace: comment-stripped-tex + depth: 2 + end: 49756 + file: canonnbhd.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 44168 + sourceStatement: + chapter: 19 + file: canonnbhd.tex + label: cccomponent + mtref: '19.23' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 44166 + sourceTransform: tex2mdx.strip_comments + start: 43962 + targetContainedSpan: + coordinateSpace: comment-stripped-tex + end: 45026 + file: canonnbhd.tex + sourceTransform: tex2mdx.strip_comments + start: 44680 + targetStatement: + chapter: 19 + file: canonnbhd.tex + label: sinside + mtref: '19.24' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 45026 + sourceTransform: tex2mdx.strip_comments + start: 44680 + type: proof-containment + via: + - proof-contains +generator: tools/poincare_tex_extract.py +inference: structural +kind: containment +ref: +- 327067f8535b +- 1b6c3ac94c37 +rel: contains +reviewStatus: accepted +scope: proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof-contains +--- +Morgan--Tian proof containment. The source is the statement whose proof contains the target statement. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/fc402320d497.md b/projects/poincare-conjecture/.astrolabe/edges/fc402320d497.md new file mode 100644 index 00000000..dbb74cbd --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/fc402320d497.md @@ -0,0 +1,45 @@ +--- +confidence: 0.2 +edgeClass: structural +evidence: + sourceStatement: + chapter: 15 + file: surgery.tex + label: '' + mtref: '15.8' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 89238 + sourceTransform: tex2mdx.strip_comments + start: 89057 + targetStatement: + chapter: 15 + file: surgery.tex + label: MAIN + mtref: '15.9' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 91186 + sourceTransform: tex2mdx.strip_comments + start: 90004 + type: chapter-reading-order-adjacency + via: + - chapter-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 6c167a310932 +- 703e4cfeece7 +rel: in-chapter +reviewStatus: accepted +scope: chapter +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- chapter-sequence +--- +Morgan--Tian reading-order adjacency across sections in a chapter. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/fc86f9a43e4b.md b/projects/poincare-conjecture/.astrolabe/edges/fc86f9a43e4b.md new file mode 100644 index 00000000..eef337e1 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/fc86f9a43e4b.md @@ -0,0 +1,62 @@ +--- +confidence: 0.9 +edgeClass: structural +evidence: + containmentDepth: 1 + proofSpan: + bodyEnd: 34517 + bodyStart: 26920 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 34528 + file: bddcurvbdddist.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 26907 + sourceStatement: + chapter: 10 + file: bddcurvbdddist.tex + label: Uinfty + mtref: '10.7' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 26826 + sourceTransform: tex2mdx.strip_comments + start: 25406 + targetContainedSpan: + coordinateSpace: comment-stripped-tex + end: 33981 + file: bddcurvbdddist.tex + sourceTransform: tex2mdx.strip_comments + start: 33869 + targetStatement: + chapter: 10 + file: bddcurvbdddist.tex + label: '' + mtref: '10.11' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 33981 + sourceTransform: tex2mdx.strip_comments + start: 33869 + type: proof-containment + via: + - proof-contains +generator: tools/poincare_tex_extract.py +inference: structural +kind: containment +ref: +- e2e64cdb0b75 +- 2d6c9dcad762 +rel: contains +reviewStatus: accepted +scope: proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof-contains +--- +Morgan--Tian proof containment. The source is the statement whose proof contains the target statement. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/fcb12312c0e8.md b/projects/poincare-conjecture/.astrolabe/edges/fcb12312c0e8.md new file mode 100644 index 00000000..24b6e44b --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/fcb12312c0e8.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: item5 + proofSpan: + bodyEnd: 96054 + bodyStart: 95212 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 96065 + file: stdsoln.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 95199 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 95644 + file: stdsoln.tex + sourceTransform: tex2mdx.strip_comments + start: 95633 + via: proof + sourceStatement: + chapter: 12 + file: stdsoln.tex + label: '' + mtref: '12.33' + sort: remark + span: + coordinateSpace: comment-stripped-tex + end: 95195 + sourceTransform: tex2mdx.strip_comments + start: 94631 + targetStatement: + chapter: 12 + file: stdsoln.tex + label: item5 + mtref: '12.7' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 18988 + sourceTransform: tex2mdx.strip_comments + start: 18657 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 246771506ee5 +- 77ca4669ffb3 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/fcd595493ce9.md b/projects/poincare-conjecture/.astrolabe/edges/fcd595493ce9.md new file mode 100644 index 00000000..86415f16 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/fcd595493ce9.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: Restim + proofSpan: + bodyEnd: 136429 + bodyStart: 132110 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 136440 + file: surgery.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 132097 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 133009 + file: surgery.tex + sourceTransform: tex2mdx.strip_comments + start: 132997 + via: proof + sourceStatement: + chapter: 16 + file: surgery.tex + label: bigell + mtref: '16.13' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 131955 + sourceTransform: tex2mdx.strip_comments + start: 130235 + targetStatement: + chapter: 12 + file: stdsoln.tex + label: Restim + mtref: '12.31' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 90335 + sourceTransform: tex2mdx.strip_comments + start: 90184 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 0a4d6f17810d +- ffae4504318a +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/fce7e5797e5b.md b/projects/poincare-conjecture/.astrolabe/edges/fce7e5797e5b.md new file mode 100644 index 00000000..3b254d42 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/fce7e5797e5b.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: COR + proofSpan: + bodyEnd: 117032 + bodyStart: 113723 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 117043 + file: temp2kappa.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 113710 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 115758 + file: temp2kappa.tex + sourceTransform: tex2mdx.strip_comments + start: 115749 + via: proof + sourceStatement: + chapter: 9 + file: temp2kappa.tex + label: kappa0 + mtref: '9.58' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 113708 + sourceTransform: tex2mdx.strip_comments + start: 113563 + targetStatement: + chapter: 6 + file: newcompar.tex + label: COR + mtref: '6.74' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 110602 + sourceTransform: tex2mdx.strip_comments + start: 110260 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 9c54e0d0c7d8 +- 541f71387f84 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/fd0e2e5dacd7.md b/projects/poincare-conjecture/.astrolabe/edges/fd0e2e5dacd7.md new file mode 100644 index 00000000..4f758e6d --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/fd0e2e5dacd7.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: embcase + proofSpan: + bodyEnd: 77920 + bodyStart: 74111 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 77931 + file: energy1.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 74098 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 77109 + file: energy1.tex + sourceTransform: tex2mdx.strip_comments + start: 77096 + via: proof + sourceStatement: + chapter: 18 + file: energy1.tex + label: nonembeasycase + mtref: '18.32' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 74095 + sourceTransform: tex2mdx.strip_comments + start: 73662 + targetStatement: + chapter: 18 + file: energy1.tex + label: embcase + mtref: '18.30' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 69934 + sourceTransform: tex2mdx.strip_comments + start: 69415 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 27b452d6cb48 +- f98ccf7ea70d +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/fd3a8e045252.md b/projects/poincare-conjecture/.astrolabe/edges/fd3a8e045252.md new file mode 100644 index 00000000..9dd0b97e --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/fd3a8e045252.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 9 + file: temp2kappa.tex + label: bdnearxk + mtref: '9.69' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 136482 + sourceTransform: tex2mdx.strip_comments + start: 136184 + targetStatement: + chapter: 9 + file: temp2kappa.tex + label: rho + mtref: '9.70' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 137332 + sourceTransform: tex2mdx.strip_comments + start: 137165 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 7c6182738b78 +- 4bff7f1e7bc4 +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/fd4e96fb5a69.md b/projects/poincare-conjecture/.astrolabe/edges/fd4e96fb5a69.md new file mode 100644 index 00000000..6763fe94 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/fd4e96fb5a69.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: defnepsround + proofSpan: + bodyEnd: 204939 + bodyStart: 186733 + coordinateSpace: comment-stripped-tex + depth: 2 + end: 204950 + file: surgery.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 186720 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 189597 + file: surgery.tex + sourceTransform: tex2mdx.strip_comments + start: 189579 + via: proof + sourceStatement: + chapter: 17 + file: surgery.tex + label: '16.2' + mtref: '17.2' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 186718 + sourceTransform: tex2mdx.strip_comments + start: 186200 + targetStatement: + chapter: 9 + file: temp2kappa.tex + label: defnepsround + mtref: '9.76' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 144692 + sourceTransform: tex2mdx.strip_comments + start: 144236 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 778d690c4e7c +- 14ff8384814f +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/fd4f4a3aa8b4.md b/projects/poincare-conjecture/.astrolabe/edges/fd4f4a3aa8b4.md new file mode 100644 index 00000000..a406c25f --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/fd4f4a3aa8b4.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 13 + file: surgery.tex + label: Rhatmatrix + mtref: '13.9' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 17454 + sourceTransform: tex2mdx.strip_comments + start: 16730 + targetStatement: + chapter: 13 + file: surgery.tex + label: '13.10' + mtref: '13.10' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 18442 + sourceTransform: tex2mdx.strip_comments + start: 17697 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- 0a41b5c61dd2 +- 347896f0385b +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/fd60b2c40f4c.md b/projects/poincare-conjecture/.astrolabe/edges/fd60b2c40f4c.md new file mode 100644 index 00000000..38f0c6ae --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/fd60b2c40f4c.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: shiw/deriv + proofSpan: + bodyEnd: 122171 + bodyStart: 120127 + coordinateSpace: comment-stripped-tex + depth: 2 + end: 122182 + file: surgery.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 120114 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 121504 + file: surgery.tex + sourceTransform: tex2mdx.strip_comments + start: 121488 + via: proof + sourceStatement: + chapter: 16 + file: surgery.tex + label: goodtotheta + mtref: '16.9' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 120111 + sourceTransform: tex2mdx.strip_comments + start: 119688 + targetStatement: + chapter: 3 + file: flowbasics.tex + label: shiw/deriv + mtref: '3.29' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 42464 + sourceTransform: tex2mdx.strip_comments + start: 41312 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 712124d7ea02 +- e0b6a1f646a7 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/fd8531c5632c.md b/projects/poincare-conjecture/.astrolabe/edges/fd8531c5632c.md new file mode 100644 index 00000000..fcfed0ce --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/fd8531c5632c.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 2 + file: prelim.tex + label: epsneck + mtref: '2.18' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 76303 + sourceTransform: tex2mdx.strip_comments + start: 73778 + targetStatement: + chapter: 2 + file: prelim.tex + label: narrows + mtref: '2.19' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 76622 + sourceTransform: tex2mdx.strip_comments + start: 76377 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- d91d54ddbd1c +- cb9ca6d6db9f +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/fda6d179aeeb.md b/projects/poincare-conjecture/.astrolabe/edges/fda6d179aeeb.md new file mode 100644 index 00000000..77e0da56 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/fda6d179aeeb.md @@ -0,0 +1,63 @@ +--- +confidence: 0.35 +edgeClass: prose +evidence: + mentionTriggers: + - label: lipaty + refSpan: + coordinateSpace: comment-stripped-tex + end: 97615 + file: newcompar.tex + sourceTransform: tex2mdx.strip_comments + start: 97603 + sentence: In order to complete the proof of Proposition~\ref{lipaty} we must establish + inequalities in the opposite direction + sentenceSpan: + coordinateSpace: comment-stripped-tex + end: 97672 + file: newcompar.tex + sourceTransform: tex2mdx.strip_comments + start: 97556 + sourceStrategy: nearest-previous-statement + via: prose + sourceStatement: + chapter: 6 + file: newcompar.tex + label: '' + mtref: '6.64' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 96436 + sourceTransform: tex2mdx.strip_comments + start: 96257 + targetStatement: + chapter: 6 + file: newcompar.tex + label: lipaty + mtref: '6.59' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 87473 + sourceTransform: tex2mdx.strip_comments + start: 86282 + type: prose-mention-nearest-previous-statement + via: + - prose +generator: tools/poincare_tex_extract.py +inference: weak +kind: mention +ref: +- 30414a5a14a9 +- 96ced8c484df +rel: mentions +reviewStatus: candidate +scope: prose +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- prose +--- +Morgan--Tian prose mention. The source is the nearest preceding statement in the same chapter segment. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/fdd6aa7e8902.md b/projects/poincare-conjecture/.astrolabe/edges/fdd6aa7e8902.md new file mode 100644 index 00000000..8d283db4 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/fdd6aa7e8902.md @@ -0,0 +1,56 @@ +--- +confidence: 0.85 +edgeClass: semantic +evidence: + containmentReason: target theorem-like statement occurs inside source proof + proofSpan: + bodyEnd: 80533 + bodyStart: 76639 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 80544 + file: prelim.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 76626 + sourceStatement: + chapter: 2 + file: prelim.tex + label: narrows + mtref: '2.19' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 76622 + sourceTransform: tex2mdx.strip_comments + start: 76377 + targetStatement: + chapter: 2 + file: prelim.tex + label: neckseparate + mtref: '2.20' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 77469 + sourceTransform: tex2mdx.strip_comments + start: 76866 + type: proof-contained-theorem-like-statement + via: + - proof-contained-statement +generator: tools/poincare_tex_extract.py +inference: inferred +kind: dependency +ref: +- cb9ca6d6db9f +- 818772530e5a +rel: depends +reviewStatus: accepted +scope: proof-containment +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof-contained-statement +--- +Morgan--Tian proof-contained statement dependency. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/fdef039e72e4.md b/projects/poincare-conjecture/.astrolabe/edges/fdef039e72e4.md new file mode 100644 index 00000000..857a706d --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/fdef039e72e4.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: lipcomplete + proofSpan: + bodyEnd: 20911 + bodyStart: 18068 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 20922 + file: noncoll.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 18055 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 20599 + file: noncoll.tex + sourceTransform: tex2mdx.strip_comments + start: 20582 + via: proof + sourceStatement: + chapter: 8 + file: noncoll.tex + label: '' + mtref: '8.10' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 18052 + sourceTransform: tex2mdx.strip_comments + start: 17502 + targetStatement: + chapter: 7 + file: newcomp2.tex + label: lipcomplete + mtref: '7.5' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 9437 + sourceTransform: tex2mdx.strip_comments + start: 8692 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 198e7475ad2f +- 16fbe263a7c8 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/fe1357de8b82.md b/projects/poincare-conjecture/.astrolabe/edges/fe1357de8b82.md new file mode 100644 index 00000000..ba1aef1a --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/fe1357de8b82.md @@ -0,0 +1,84 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: xistronger + proofSpan: + bodyEnd: 145345 + bodyStart: 144076 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 145356 + file: energy1.tex + ownerLabel: XI + sourceTransform: tex2mdx.strip_comments + start: 144063 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 144317 + file: energy1.tex + sourceTransform: tex2mdx.strip_comments + start: 144301 + via: proof + - context: proof + label: xistronger + proofSpan: + bodyEnd: 145345 + bodyStart: 144076 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 145356 + file: energy1.tex + ownerLabel: XI + sourceTransform: tex2mdx.strip_comments + start: 144063 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 144851 + file: energy1.tex + sourceTransform: tex2mdx.strip_comments + start: 144835 + via: proof + sourceStatement: + chapter: 18 + file: energy1.tex + label: XI + mtref: '18.24' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 58810 + sourceTransform: tex2mdx.strip_comments + start: 58030 + targetStatement: + chapter: 18 + file: energy1.tex + label: xistronger + mtref: '18.53' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 123454 + sourceTransform: tex2mdx.strip_comments + start: 122771 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- fd85732525c1 +- 9df99ae6f4cb +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/fe32b72a6efa.md b/projects/poincare-conjecture/.astrolabe/edges/fe32b72a6efa.md new file mode 100644 index 00000000..727d4d0f --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/fe32b72a6efa.md @@ -0,0 +1,71 @@ +--- +confidence: 0.35 +edgeClass: prose +evidence: + mentionTriggers: + - label: divRic + refSpan: + coordinateSpace: comment-stripped-tex + end: 83370 + file: temp2kappa.tex + sourceTransform: tex2mdx.strip_comments + start: 83358 + sentence: Taking the trace of the gradient shrinking soliton equation yields $$R+\triangle + f-\frac{n}{2}=0,$$ and consequently that $$ dR+d(\triangle f)=0.$$ Using Lemma~\ref{lapformula} + we rewrite this equation as \begin{equation}\label{Rfeqn} dR+\triangle(df)-{\rm + Ric}(\nabla f,\cdot)=0.\end{equation} On the other hand, taking the divergence + of the gradient shrinking soliton equation and using the fact that $\nabla^* + g=0$ gives $$\nabla^*{\rm Ric}+\nabla^*{\rm Hess}(f)=0.$$ Of course, $$\nabla^*{\rm + Hess}(f)=\nabla^*(\nabla\nabla f)=(\nabla^*\nabla)\nabla f=\triangle (df),$$ + so that $$\triangle(df)=-\nabla^*{\rm Ric}.$$ Plugging this into Equation~\ref{Rfeqn} + gives $$dR-\nabla^*{\rm Ric}-{\rm Ric}(\nabla f,\cdot)=0.$$ Now invoking Lemma~\ref{divRic} + we have \begin{equation}\label{dR} dR=2{\rm Ric}(\nabla f,\cdot).\end{equation} + sentenceSpan: + coordinateSpace: comment-stripped-tex + end: 83451 + file: temp2kappa.tex + sourceTransform: tex2mdx.strip_comments + start: 82616 + sourceStrategy: nearest-previous-statement + via: prose + sourceStatement: + chapter: 9 + file: temp2kappa.tex + label: noncompsol + mtref: '9.46' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 82429 + sourceTransform: tex2mdx.strip_comments + start: 82233 + targetStatement: + chapter: 1 + file: prelim.tex + label: divRic + mtref: '1.9' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 12596 + sourceTransform: tex2mdx.strip_comments + start: 12512 + type: prose-mention-nearest-previous-statement + via: + - prose +generator: tools/poincare_tex_extract.py +inference: weak +kind: mention +ref: +- 03957890f08f +- 9c83686198e8 +rel: mentions +reviewStatus: candidate +scope: prose +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- prose +--- +Morgan--Tian prose mention. The source is the nearest preceding statement in the same chapter segment. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/fe563389e321.md b/projects/poincare-conjecture/.astrolabe/edges/fe563389e321.md new file mode 100644 index 00000000..86ffb4df --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/fe563389e321.md @@ -0,0 +1,45 @@ +--- +confidence: 0.25 +edgeClass: structural +evidence: + sourceStatement: + chapter: 10 + file: bddcurvbdddist.tex + label: '' + mtref: '10.16' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 41257 + sourceTransform: tex2mdx.strip_comments + start: 41090 + targetStatement: + chapter: 10 + file: bddcurvbdddist.tex + label: '' + mtref: '10.17' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 42249 + sourceTransform: tex2mdx.strip_comments + start: 41852 + type: section-reading-order-adjacency + via: + - section-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- f83f945f80e7 +- e98dee3d7485 +rel: in-section +reviewStatus: accepted +scope: section +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- section-sequence +--- +Morgan--Tian reading-order adjacency within a section. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/fea45beab1b8.md b/projects/poincare-conjecture/.astrolabe/edges/fea45beab1b8.md new file mode 100644 index 00000000..35020735 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/fea45beab1b8.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: fordiffmax + proofSpan: + bodyEnd: 33957 + bodyStart: 32670 + coordinateSpace: comment-stripped-tex + depth: 2 + end: 33968 + file: stdsoln.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 32657 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 33689 + file: stdsoln.tex + sourceTransform: tex2mdx.strip_comments + start: 33673 + via: proof + sourceStatement: + chapter: 12 + file: stdsoln.tex + label: '' + mtref: '12.14' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 32654 + sourceTransform: tex2mdx.strip_comments + start: 32551 + targetStatement: + chapter: 2 + file: prelim.tex + label: fordiffmax + mtref: '2.23' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 83307 + sourceTransform: tex2mdx.strip_comments + start: 82254 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- f26ee81ded19 +- 37d7d059a526 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/fecc55e7f396.md b/projects/poincare-conjecture/.astrolabe/edges/fecc55e7f396.md new file mode 100644 index 00000000..09fcf8cf --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/fecc55e7f396.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: tildelK + proofSpan: + bodyEnd: 125200 + bodyStart: 122543 + coordinateSpace: comment-stripped-tex + depth: 2 + end: 125211 + file: newcompar.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 122530 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 124818 + file: newcompar.tex + sourceTransform: tex2mdx.strip_comments + start: 124805 + via: proof + sourceStatement: + chapter: 6 + file: newcompar.tex + label: fmonotone + mtref: '6.81' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 122525 + sourceTransform: tex2mdx.strip_comments + start: 122141 + targetStatement: + chapter: 6 + file: newcompar.tex + label: tildelK + mtref: '6.54' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 82952 + sourceTransform: tex2mdx.strip_comments + start: 82766 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 4dc40f37cd6d +- 451b4cc590c1 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/fefdd6b4eaca.md b/projects/poincare-conjecture/.astrolabe/edges/fefdd6b4eaca.md new file mode 100644 index 00000000..aaf92794 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/fefdd6b4eaca.md @@ -0,0 +1,56 @@ +--- +confidence: 0.85 +edgeClass: semantic +evidence: + containmentReason: target theorem-like statement occurs inside source proof + proofSpan: + bodyEnd: 137790 + bodyStart: 127828 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 137801 + file: temp2kappa.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 127815 + sourceStatement: + chapter: 9 + file: temp2kappa.tex + label: proposition + mtref: '9.65' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 127812 + sourceTransform: tex2mdx.strip_comments + start: 127549 + targetStatement: + chapter: 9 + file: temp2kappa.tex + label: 2Rbound + mtref: '9.66' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 128756 + sourceTransform: tex2mdx.strip_comments + start: 128570 + type: proof-contained-theorem-like-statement + via: + - proof-contained-statement +generator: tools/poincare_tex_extract.py +inference: inferred +kind: dependency +ref: +- ee314aacd207 +- ea9b285e40cb +rel: depends +reviewStatus: accepted +scope: proof-containment +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof-contained-statement +--- +Morgan--Tian proof-contained statement dependency. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/ff0d65ea6653.md b/projects/poincare-conjecture/.astrolabe/edges/ff0d65ea6653.md new file mode 100644 index 00000000..1b917383 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/ff0d65ea6653.md @@ -0,0 +1,63 @@ +--- +confidence: 0.35 +edgeClass: prose +evidence: + mentionTriggers: + - label: splitting + refSpan: + coordinateSpace: comment-stripped-tex + end: 18052 + file: stdsoln.tex + sourceTransform: tex2mdx.strip_comments + start: 18037 + sentence: Since the time-slices of this flow have two ends, it follows from Theorem~\ref{splitting} + that every manifold in the flow is a product of a compact surface with $\Ar$ + sentenceSpan: + coordinateSpace: comment-stripped-tex + end: 18129 + file: stdsoln.tex + sourceTransform: tex2mdx.strip_comments + start: 17962 + sourceStrategy: nearest-previous-statement + via: prose + sourceStatement: + chapter: 12 + file: stdsoln.tex + label: third + mtref: '12.6' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 12060 + sourceTransform: tex2mdx.strip_comments + start: 11922 + targetStatement: + chapter: 2 + file: prelim.tex + label: splitting + mtref: '2.13' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 67619 + sourceTransform: tex2mdx.strip_comments + start: 67365 + type: prose-mention-nearest-previous-statement + via: + - prose +generator: tools/poincare_tex_extract.py +inference: weak +kind: mention +ref: +- 2e8c17743da6 +- e5b196773169 +rel: mentions +reviewStatus: candidate +scope: prose +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- prose +--- +Morgan--Tian prose mention. The source is the nearest preceding statement in the same chapter segment. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/ff51163d9510.md b/projects/poincare-conjecture/.astrolabe/edges/ff51163d9510.md new file mode 100644 index 00000000..3f83550c --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/ff51163d9510.md @@ -0,0 +1,45 @@ +--- +confidence: 0.2 +edgeClass: structural +evidence: + sourceStatement: + chapter: 9 + file: temp2kappa.tex + label: '' + mtref: '9.8' + sort: example + span: + coordinateSpace: comment-stripped-tex + end: 6486 + sourceTransform: tex2mdx.strip_comments + start: 5910 + targetStatement: + chapter: 9 + file: temp2kappa.tex + label: uniformlip + mtref: '9.9' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 7575 + sourceTransform: tex2mdx.strip_comments + start: 6851 + type: chapter-reading-order-adjacency + via: + - chapter-sequence +generator: tools/poincare_tex_extract.py +inference: navigational +kind: sequence +ref: +- d9035559d0a4 +- 9387b1b04824 +rel: in-chapter +reviewStatus: accepted +scope: chapter +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- chapter-sequence +--- +Morgan--Tian reading-order adjacency across sections in a chapter. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/ff578c6a5d22.md b/projects/poincare-conjecture/.astrolabe/edges/ff578c6a5d22.md new file mode 100644 index 00000000..4294574f --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/ff578c6a5d22.md @@ -0,0 +1,55 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: statement + label: LOCALSURGERY + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 78054 + file: surgery.tex + sourceTransform: tex2mdx.strip_comments + start: 78036 + via: statement + sourceStatement: + chapter: 15 + file: surgery.tex + label: hrestricted + mtref: '15.5' + sort: definition + span: + coordinateSpace: comment-stripped-tex + end: 78345 + sourceTransform: tex2mdx.strip_comments + start: 77435 + targetStatement: + chapter: 13 + file: surgery.tex + label: LOCALSURGERY + mtref: '13.2' + sort: theorem + span: + coordinateSpace: comment-stripped-tex + end: 6442 + sourceTransform: tex2mdx.strip_comments + start: 4875 + type: tex-reference + via: + - statement +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 91876608df01 +- cf61b731c6f2 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- statement +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/ff649155c213.md b/projects/poincare-conjecture/.astrolabe/edges/ff649155c213.md new file mode 100644 index 00000000..08d9efd1 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/ff649155c213.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: addaneck + proofSpan: + bodyEnd: 32963 + bodyStart: 31019 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 32974 + file: canonnbhd.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 31006 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 31053 + file: canonnbhd.tex + sourceTransform: tex2mdx.strip_comments + start: 31039 + via: proof + sourceStatement: + chapter: 19 + file: canonnbhd.tex + label: epschains + mtref: '19.19' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 31004 + sourceTransform: tex2mdx.strip_comments + start: 30360 + targetStatement: + chapter: 19 + file: canonnbhd.tex + label: addaneck + mtref: '19.18' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 29774 + sourceTransform: tex2mdx.strip_comments + start: 29005 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- ba681e15e1a6 +- 6dc87ad116fc +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/ff92d49ef581.md b/projects/poincare-conjecture/.astrolabe/edges/ff92d49ef581.md new file mode 100644 index 00000000..bd34e362 --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/ff92d49ef581.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: tausmooth + proofSpan: + bodyEnd: 15504 + bodyStart: 13148 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 15515 + file: newcomp2.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 13135 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 13983 + file: newcomp2.tex + sourceTransform: tex2mdx.strip_comments + start: 13968 + via: proof + sourceStatement: + chapter: 7 + file: newcomp2.tex + label: '' + mtref: '7.9' + sort: remark + span: + coordinateSpace: comment-stripped-tex + end: 13133 + sourceTransform: tex2mdx.strip_comments + start: 12880 + targetStatement: + chapter: 6 + file: newcompar.tex + label: tausmooth + mtref: '6.31' + sort: proposition + span: + coordinateSpace: comment-stripped-tex + end: 44056 + sourceTransform: tex2mdx.strip_comments + start: 43590 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 6262ca70063e +- 4b9af3660539 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/ffe39eb07c39.md b/projects/poincare-conjecture/.astrolabe/edges/ffe39eb07c39.md new file mode 100644 index 00000000..fff8af8e --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/ffe39eb07c39.md @@ -0,0 +1,65 @@ +--- +confidence: 1.0 +edgeClass: semantic +evidence: + refTriggers: + - context: proof + label: COR + proofSpan: + bodyEnd: 18321 + bodyStart: 17794 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 18332 + file: temp2kappa.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 17781 + sourceSpan: + coordinateSpace: comment-stripped-tex + end: 17917 + file: temp2kappa.tex + sourceTransform: tex2mdx.strip_comments + start: 17908 + via: proof + sourceStatement: + chapter: 9 + file: temp2kappa.tex + label: tausqtau + mtref: '9.15' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 17779 + sourceTransform: tex2mdx.strip_comments + start: 17653 + targetStatement: + chapter: 6 + file: newcompar.tex + label: COR + mtref: '6.74' + sort: corollary + span: + coordinateSpace: comment-stripped-tex + end: 110602 + sourceTransform: tex2mdx.strip_comments + start: 110260 + type: tex-reference + via: + - proof +generator: tools/poincare_tex_extract.py +inference: explicit +kind: reference +ref: +- 88571a22a2bc +- 541f71387f84 +rel: references +reviewStatus: accepted +scope: statement-or-proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof +--- +Morgan--Tian cross-reference. \ No newline at end of file diff --git a/projects/poincare-conjecture/.astrolabe/edges/fffd833792e7.md b/projects/poincare-conjecture/.astrolabe/edges/fffd833792e7.md new file mode 100644 index 00000000..101d2ddf --- /dev/null +++ b/projects/poincare-conjecture/.astrolabe/edges/fffd833792e7.md @@ -0,0 +1,62 @@ +--- +confidence: 0.9 +edgeClass: structural +evidence: + containmentDepth: 1 + proofSpan: + bodyEnd: 171471 + bodyStart: 161354 + coordinateSpace: comment-stripped-tex + depth: 1 + end: 171482 + file: energy1.tex + ownerLabel: '' + sourceTransform: tex2mdx.strip_comments + start: 161341 + sourceStatement: + chapter: 18 + file: energy1.tex + label: laest + mtref: '18.66' + sort: lemma + span: + coordinateSpace: comment-stripped-tex + end: 161338 + sourceTransform: tex2mdx.strip_comments + start: 161080 + targetContainedSpan: + coordinateSpace: comment-stripped-tex + end: 165064 + file: energy1.tex + sourceTransform: tex2mdx.strip_comments + start: 164946 + targetStatement: + chapter: 18 + file: energy1.tex + label: '' + mtref: '18.69' + sort: claim + span: + coordinateSpace: comment-stripped-tex + end: 165064 + sourceTransform: tex2mdx.strip_comments + start: 164946 + type: proof-containment + via: + - proof-contains +generator: tools/poincare_tex_extract.py +inference: structural +kind: containment +ref: +- 45c783d384c1 +- a9617f70db9f +rel: contains +reviewStatus: accepted +scope: proof +sort: (morgan-tian, morgan-tian) +source: tex +src: morgan-tian +via: +- proof-contains +--- +Morgan--Tian proof containment. The source is the statement whose proof contains the target statement. \ No newline at end of file