ricci flows and poincare conjecture

474页 英文 3.82M PDF

John W. Morgan and Gang Tian July 25, 2006

庞加莱猜想的证明被认为是06年最伟大的科学进展,这本书是Tiangang和J Morgan为使更多人了解过程而写的较perelman证明更清晰版本

　　　　　　　　　　　　　　　　　　　Contents
0.1 Overview of Perelman’s argument . . . . . . . . . . . . . . . . . . . . 9
0.2 Background material from riemannian geometry . . . . . . . . . . . 12
0.2.1 Volume and injectivity radius . . . . . . . . . . . . . . . . . . 12
0.2.2 Manifolds of non-negative curvature . . . . . . . . . . . . . . 13
0.2.3 Canonical Neighborhoods . . . . . . . . . . . . . . . . . . . . 13
0.3 Background Material from Ricci Flow . . . . . . . . . . . . . . . . . 15
0.3.1 First results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
0.3.2 Gradient Shrinking Solitons . . . . . . . . . . . . . . . . . . . 16
0.3.3 Controlling higher derivatives of curvature . . . . . . . . . . . 16
0.3.4 Generalized Ricci flows . . . . . . . . . . . . . . . . . . . . . 16
0.3.5 The maximum principle . . . . . . . . . . . . . . . . . . . . . 18
0.3.6 Geometric Limits . . . . . . . . . . . . . . . . . . . . . . . . . 19
0.4 Perelman’s advances . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
0.4.1 The reduced length function . . . . . . . . . . . . . . . . . . . 20
0.4.2 Application to non-collapsing results . . . . . . . . . . . . . . 21
0.4.3 Application to ancient κ-non-collapsed solutions . . . . . . . 22
0.4.4 Bounded Curvature at Bounded Distance . . . . . . . . . . . 24
0.5 The standard solution and the surgery process . . . . . . . . . . . . 25
0.5.1 The standard solution . . . . . . . . . . . . . . . . . . . . . . 25
0.5.2 Ricci Flows with surgery . . . . . . . . . . . . . . . . . . . . . 26
0.5.3 The Inductive Conditions Necessary for doing Surgery . . . . 27
0.5.4 Surgery . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
0.5.5 Topological effect of surgery . . . . . . . . . . . . . . . . . . . 28
0.6 Surgery and canonical neighborhoods for generalized Ricci flows . . . 29
0.7 Finite Extinction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
0.8 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
0.9 List of related papers . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
1 Preliminaries from Riemannian Geometry 36
1.1 Riemannian metrics and the Levi-Civit´a connection . . . . . . . . . 36
1.2 Curvature of a riemannian manifold . . . . . . . . . . . . . . . . . . 38
1.2.1 Consequences of the Bianchi identities . . . . . . . . . . . . . 40
1.2.2 First Examples . . . . . . . . . . . . . . . . . . . . . . . . . . 41
1.3 Cones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
1.4 Geodesics and the exponential map . . . . . . . . . . . . . . . . . . . 43
1.4.1 Geodesics and the energy functional . . . . . . . . . . . . . . 43
1.4.2 Families of geodesics and Jacobi fields . . . . . . . . . . . . . 44
1.4.3 Minimal geodesics . . . . . . . . . . . . . . . . . . . . . . . . 45
1.4.4 The exponential mapping . . . . . . . . . . . . . . . . . . . . 47
1.5 Computations in Gaussian normal coordinates . . . . . . . . . . . . 48
1.6 Basic Curvature Comparison Results . . . . . . . . . . . . . . . . . . 50
1.7 Local Volume and the injectivity radius . . . . . . . . . . . . . . . . 52

。。。。。

18.6 Proof of the first inequality in Lemma 18.42 . . . . . . . . . . . . . . 443
18.6.1 A bound for R kds . . . . . . . . . . . . . . . . . . . . . . . . 444
18.6.2 Writing the curve flow as a graph . . . . . . . . . . . . . . . . 447
18.6.3 t3 = t2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 449
18.6.4 t2 = t′ + ǫr2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 451
19 Appendix: Canonical Neighborhoods 453
19.1 Shortening Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453
19.2 The geometry of an ǫ-neck . . . . . . . . . . . . . . . . . . . . . . . . 453
19.3 Overlapping ǫ-necks . . . . . . . . . . . . . . . . . . . . . . . . . . . 458
19.4 Regions covered by ǫ-necks and (C, ǫ)-caps . . . . . . . . . . . . . . . 460
19.4.1 Chains of ǫ-necks . . . . . . . . . . . . . . . . . . . . . . . . . 460
19.5 Subsets of the union of cores of (C, ǫ)-caps and ǫ-necks. . . . . . . . 463