Language: English
Pages: 1047
Introduction......Page 1
1 Ricci flow as a gradient flow......Page 5
2 No breathers theorem I......Page 6
3 No breathers theorem II......Page 8
4 No local collapsing theorem I......Page 10
5 A statistical analogy......Page 11
6 Riemannian formalism in potentially infinite dimensions......Page 12
7 A comparison geometry approach to the Ricci flow......Page 14
8 No local collapsing theorem II......Page 20
9 Differential Harnack inequality for solutions of the conjugate heat equation......Page 22
10 Pseudolocality theorem......Page 23
11 Ancient solutions with nonnegative curvature operator and bounded entropy......Page 27
12 Almost nonnegative curvature in dimension three......Page 32
13 The global picture of the Ricci flow in dimension three......Page 36
References......Page 38
Ricci flow with surgery on three-manifolds......Page 40
Ancient solutions with bounded entropy......Page 41
The standard solution......Page 43
The structure of solutions at the first singular time......Page 45
Ricci flow with cutoff......Page 46
Justification of the a priori assumption......Page 49
Long time behavior I......Page 52
Long time behavior II......Page 56
On the first eigenvalue of the operator -4+R......Page 59
Finite time extinction......Page 62
Preliminaries on the curve shortening flow......Page 64
Proof of lemma 1.2......Page 66
A complete proof of the Poincaré and geometrization conjectures - Application of the Hamilton-Perelman Theory of the Ricci flow......Page 69
Introduction......Page 71
1.1. The Ricci Flow.......Page 76
1.2. Short-time Existence and Uniqueness.......Page 81
1.3. Evolution of Curvatures.......Page 87
1.4. Derivative Estimates.......Page 94
1.5. Variational Structure and Dynamic Property.......Page 103
2.1. Preserving Positive Curvature.......Page 114
2.2. Strong Maximum Principle.......Page 117
2.3. Advanced Maximum Principle for Tensors.......Page 121
2.4. Hamilton-Ivey Curvature Pinching Estimate.......Page 127
2.5. Li-Yau-Hamilton Estimates.......Page 130
2.6. Perelman’s Estimate for Conjugate Heat Equations.......Page 138
3.1. Riemannian Formalism in Potentially In nite Dimensions.......Page 143
3.2. Comparison Theorems for Perelman’s Reduced Volume.......Page 147
3.3. No Local Collapsing Theorem I.......Page 159
3.4. No Local Collapsing Theorem II.......Page 165
4.1. Cheeger Type Compactness.......Page 171
4.2. Injectivity Radius Estimates.......Page 190
4.3. Limiting Singularity Models.......Page 195
4.4. Ricci Solitons.......Page 206
5. Long Time Behaviors.......Page 211
5.1. The Ricci Flow on Two-manifolds.......Page 212
5.2. Di erentiable Sphere Theorems in 3-D and 4-D.......Page 225
5.3. Nonsingular Solutions on Three-manifolds.......Page 240
6.1. Preliminaries.......Page 261
6.2. Asymptotic Shrinking Solitons.......Page 268
6.3. Curvature Estimates via Volume Growth.......Page 277
6.4. Ancient......Page 288
7.1. Canonical Neighborhood Structures.......Page 302
7.2. Curvature Estimates for Smooth Solutions.......Page 309
7.3. Ricci Flow with Surgery.......Page 317
7.4. Justi cation of the Canonical Neighborhood Assumptions.......Page 336
7.5. Curvature Estimates for Surgically Modi ed Solutions.......Page 356
7.6. Long Time Behavior.......Page 372
7.7. Geometrization of Three-manifolds.......Page 385
References
......Page 390
Notes on Perelman's papers
......Page 397
1. Introduction......Page 610
2. Estimates and results needed......Page 611
3. Synthesis......Page 616
4. Improving Li-Yau-Hamilton estimates via monotonicity......Page 619
5. Local monotonicity formulae......Page 625
References......Page 627
Ancient solutions to Kähler-Ricci flow......Page 629
Gaussian densities and stability for some Ricci solitons......Page 648
Second Variation of the Entropy......Page 649
Second Variation of the Shrinker Entropy......Page 651
The Central Density of a Shrinker......Page 655
Table of Values......Page 658
Introduction......Page 663
Preliminaries......Page 665
Convergence toward the solutions of the Ricci flow......Page 666
Continuity of the minimizers for W......Page 667
Further estimates on the minimizers......Page 671
Ricci soliton in the limit......Page 680
Some properties of the limit solitons......Page 684
References......Page 693
Ricci Flow with Surgery on Four-manifolds with Positive Isotropic Curvature......Page 695
Uniqueness of the Ricci Flow on Complete Noncompact Manifolds......Page 800
Introduction......Page 801
Harmonic map flow coupled with the Ricci flow......Page 803
Expanding base and target metrics at infinity......Page 804
Short time existence of the modified harmonic map flows......Page 809
Proof of theorem 2.6 and Theorem 2.1......Page 822
Preliminary estimates for the Ricci-De Turck flow......Page 824
Ricci-De Turck flow......Page 827
Proof of the main theorem......Page 830
Perspectives on Geometric Analysis......Page 833
1.1. Founding fathers of the subject......Page 837
1.2. Modern Contributors......Page 839
2.1. Polynomials from ambient space.......Page 840
2.2. Geometric construction of functions......Page 842
2.3. Functions and tensors defined by linear differential equations......Page 845
3.1. Embedding......Page 858
3.2. Rigidity of harmonic maps with negative curvature......Page 860
3.3. Holomorphic maps......Page 861
3.4. Harmonic maps from two dimensional surfaces and pseudoholomorphic curves......Page 862
3.6. Wave maps......Page 863
3.8. Regularity theory......Page 864
4.2. Classical minimal surfaces in Euclidean space......Page 865
4.4. Surfaces related to classical relativity......Page 866
4.5. Higher dimensional minimal subvarieties......Page 867
4.6. Geometric flows......Page 869
5.1. Geometric structures with noncompact holonomy group......Page 871
5.2. Uniformization for three manifolds......Page 872
5.4. Special connections on bundles......Page 875
5.5. Symplectic structures......Page 877
5.6. Kähler structure......Page 878
5.7. Manifolds with special holonomy group......Page 882
5.9. Obstruction for existence of Einstein metrics on general manifolds......Page 883
5.10. Metric Cobordism......Page 884
References......Page 885
Geometrization of 3-Manifolds via the Ricci Flow......Page 908
Structures of Three Manifolds......Page 918
Conformal geometry......Page 919
Hamilton's equation on Surfaces......Page 921
Topological Surgery......Page 923
Structure of Three Dimensional Spaces......Page 925
Einstein metrics......Page 927
The dynamics of Einstein equation......Page 929
Singularity......Page 930
Hamilton's Program......Page 931
Uniqueness of standard solutions in the work of Perelman......Page 942
Bounding scalar curvature and diameter along the Kaehler-Ricci flow (after Perelman) and some applications......Page 962
1. introduction......Page 981
2. Upper bounds for the rate of change of area of minimal 2--spheres......Page 983
3. Extinction in finite time......Page 984
4. Remarks on the reducible case......Page 986
Appendix A. Lower bounds for the rate of change of area of minimal 2--spheres......Page 987
References......Page 988
1. Introduction......Page 990
2. Strain radii and geometry of Alexandrov surfaces......Page 992
3. Local structure......Page 994
4. Decomposition......Page 999
5. Gluing......Page 1002
6. Proof of Lemma ??......Page 1008
7. Thick-thin decomposition......Page 1010
8. Appendix: Collapsing under a local lower curvature bound......Page 1011
References......Page 1013
0. Introduction......Page 1014
1. Width and finite extinction......Page 1015
1.1. Width......Page 1016
1.2. Finite extinction......Page 1017
1.4. Existence of good sweepouts......Page 1018
1.5. Upper bounds for the rate of change of width......Page 1019
2. The energy decreasing map and its consequences......Page 1021
2.2. Constructing good sweepouts from the energy decreasing map on......Page 1022
3.1. Harmonic replacement......Page 1023
3.2. Continuity of harmonic replacement on C0 (B1) W1,2(B1)......Page 1024
3.3. Uniform continuity of energy improvement on W1,2......Page 1025
3.4. Constructing the map from to......Page 1029
A.1. Bubble convergence and the topology on......Page 1031
A.3. Bubble convergence implies varifold convergence......Page 1032
Appendix B. The proof of Proposition 2.2......Page 1033
B.1. Harmonic maps on cylinders......Page 1034
B.2. Weak compactness of almost harmonic maps......Page 1036
B.4. The compactness argument......Page 1038
B.5. The proof of Proposition B.29......Page 1039
B.6. Bubble compactness......Page 1040
C.1. An application of the Wente lemma......Page 1041
C.2. An application to harmonic maps......Page 1042
C.3. The proof of Theorem 3.1......Page 1043
D.1. Density of smooth mappings......Page 1044
D.2. Equivalence of energy and area......Page 1045
References......Page 1046
