1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

INCOMPRESSIBLE SURFACES AND GEOMETRIZATION CONJECTURE 437 and Shalen [224] and Johannson [225] says the following. (For simpli ...

438 9. BASIC TOPOLOGY OF 3-MANIFOLDS assume that M is irreducible. Since M has a trivial fundamental group, it is geometrically ...

DECOMPOSITION THEOREMS AND THE RICCI FLOW 439 In particular, this implies that the boundary of M is a possibly empty col- lect ...

440 9. BASIC TOPOLOGY OF 3-MANIFOLDS sectional curvature. By the strong maximum principle, the universal covers of the ancient s ...

DECOMPOSITION THEOREMS AND THE RICCI FLOW 441 regions with curvature comparable to their spatial maximum are geomet- rically c ...

442 9. BASIC TOPOLOGY OF 3-MANIFOLDS After the surgery, one continues the Ricci fl.ow on the resulting (possi- bly disconnected) ...

NOTES AND COMMENTARY 443 [267] and Morgan's [272] are two excellent survey papers introducing Ricci fl.ow on 3-manifolds and g ...

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Appendix A. Basic Ricci Flow Theory Heat, like gravity, penetrates every substance of the universe, its rays occupy all parts of ...

446 A. BASIC RICCI FLOW THEORY • trg denotes the trace with respect tog (e.g., of a symmetric (2, 0)- tensor). sn usually denot ...

RIEMANNIAN GEOMETRY 447 (3) Contracted second Bianchi identity: . 1 (Vl-3.13) }J Rij = 2 "\liR, which is obtained from (Vl-3.1 ...

448 A. BASIC RICCI FLOW THEORY LEMMA A.1 (Metric variation formulas). Suppose that g (s) is a smooth 1-parameter family of metri ...

If () is a 1-form, then (Vl-p. 286b) RIEMANNIAN GEOMETRY More generally, if A is any (p, q)-tensor field, one has the commutat ...

450 A. BASIC RICCI FLOW THEORY where the dot denotes the metric inner product, i.e., X · Y = (X, Y) = gijxiyj. If gtgij = -2Rij ...

RIEMANNIAN GEOMETRY 451 As a consequence, we have the following characterization of Euclidean space. COROLLARY A.5 (Volume cha ...

452 A. BASIC RICCI FLOW THEORY then inj (p) ;:::: lo. 1.11. Laplacian and Hessian comparison theorems. Given KE JR. and r > 0 ...

RIEMANNIAN GEOMETRY 453 1.12. Li-Yau differential Harnack estimate. Let u : Mn x [O, oo) --+ ~ be a positive solution to the h ...

454 A. BASIC RICCI FLOW THEORY so that Cut (p) = expP (8Cp). The cut locus is a closed set with measure zero. We have expPjintCp ...

RIEMANNIAN GEOMETRY 455 (the above calculation holds for any C^2 function¢). That is, if x tJ. Cut (p), then by (A.17) we have ...

456 A. BASIC RICCI FLOW THEORY (xo, to). However, since d(·, q) is 02 at xo, we may apply the maximum principle to the equation ...