Data/poincare astrolabe tex

projects/poincare-conjecture/.astrolabe/atoms/002f3496c54c.md

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+---
+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
+
+$$
+d_{t}(x,y)\le d_0(x,y)+16\sqrt{\frac{Q}{3}}T.
+$$

projects/poincare-conjecture/.astrolabe/atoms/00499f9fd446.md

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+---
+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.

projects/poincare-conjecture/.astrolabe/atoms/006e42ef5bb5.md

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+---
+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 $N$.

projects/poincare-conjecture/.astrolabe/atoms/00e5410e6aa1.md

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+---
+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 $(U,g(t)),\ 0\le t<T<\infty$, is a Ricci flow. Suppose
+that $|\mathit{Rm}(x,t)|$ is bounded independent of $(x,t)\in U\times
+[0,T)$. Then for any open subset $V\subset U$ with compact closure
+in $U$, there is an extension of the Ricci flow $(V,g(t)|_V)$ past
+time $T$.

projects/poincare-conjecture/.astrolabe/atoms/01028c93347d.md

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+---
+chapter: 9
+generator: tools/poincare_tex_extract.py
+label: compactcase
+labels:
+- compactcase
+mtref: '9.44'
+ref:
+- 01028c93347d
+sort: corollary
+source: tex
+src: morgan-tian
+tex_file: temp2kappa
+---
+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.

projects/poincare-conjecture/.astrolabe/atoms/012eefbbc44a.md

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+---
+chapter: 18
+generator: tools/poincare_tex_extract.py
+label: xicomp
+labels:
+- xicomp
+mtref: '18.57'
+ref:
+- 012eefbbc44a
+sort: remark
+source: tex
+src: morgan-tian
+tex_file: energy1
+---
+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
+$e^{-C_2(t_1-t_0)}\zeta/3$.

projects/poincare-conjecture/.astrolabe/atoms/016cf36686b2.md

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+---
+chapter: 4
+generator: tools/poincare_tex_extract.py
+label: firststrongmax
+labels:
+- firststrongmax
+mtref: '4.18'
+ref:
+- 016cf36686b2
+sort: theorem
+source: tex
+src: morgan-tian
+tex_file: maxprin
+---
+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 $t\in [0,T]$.

projects/poincare-conjecture/.astrolabe/atoms/01a829e22cf6.md

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+---
+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
+
+$$
+d_a(x_0,x_1)\le d_b(x_0,x_1)+4(n-1)\int_a^b\sqrt{\frac{2K(t)}{3}}dt.
+$$

projects/poincare-conjecture/.astrolabe/atoms/01f0d3baa28f.md

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+---
+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 $0\le t<T<\infty$ satisfying
+Assumptions 11.18. Fix $0<\rho<r_0$. Let
+$\Omega^0(\rho)$ be the union of all components of $\Omega$
+containing points of $\Omega_\rho$. Then $\Omega^0(\rho)$ has
+finitely many components and is a union of a compact set and
+finitely many strong $2\epsilon$-horns each of which is disjoint
+from $\Omega_\rho$ and has its boundary contained in
+$\Omega_{\rho/2C}$.

projects/poincare-conjecture/.astrolabe/atoms/028d3b5182b5.md

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+---
+chapter: 18
+generator: tools/poincare_tex_extract.py
+label: uniformc
+labels:
+- uniformc
+mtref: '18.48'
+ref:
+- 028d3b5182b5
+sort: claim
+source: tex
+src: morgan-tian
+tex_file: energy1
+---
+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
+$n\ge N$.

projects/poincare-conjecture/.astrolabe/atoms/033786a89c6d.md

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+---
+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.)

projects/poincare-conjecture/.astrolabe/atoms/035b5e6ca144.md

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+---
+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.

projects/poincare-conjecture/.astrolabe/atoms/03957890f08f.md

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+---
+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.

projects/poincare-conjecture/.astrolabe/atoms/03c26a36f415.md

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+---
+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 $U_\ell$.

projects/poincare-conjecture/.astrolabe/atoms/0425e2f3cfe1.md

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+---
+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.

projects/poincare-conjecture/.astrolabe/atoms/04518fa1ac2f.md

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+---
+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
+$|s_{b'c'}|\ge |s_{bc}|$.