Table of contents for Hamilton's Ricci flow / Bennett Chow, Peng Lu, Lei Ni.


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Chapter 1. Riemannian Geometry                                     1
1  Introduction                                                 1
2. Metrics, connections, curvatures and covariant differentiation  2
3. Basic formulas and identities in Riemannian geometry        10
4. Exterior differential calculus and Bochner formulas         14
5. Integration and Hodge theory                                20
6. Curvature decomposition and locally conformally flat manifolds 25
7. Moving frames and the Gauss-Bonnet formula                 32
8. Variation of arc length, energy and area                    41
9. Geodesics and the exponential map                           52
10. Second fundamental forms of geodesic spheres               58
11   Laplacian, volume and Hessian comparison theorems         67
12. Proof of the comparison theorems                           73
13   Manifolds with nonnegative curvature                      80
14. Lie groups and left-invariant metrics                      87
15. Notes and commentary                                       89
Chapter 2. Fundamentals of the Ricci Flow Equation                 95
5 I  Geometric flows and geometrization                         96
2. Ricci flow and the evolution of scalar curvature             98
3. The maximum principle for heat-type equations               00
4. The Einstein-Hilbert functional                             104
5. Evolution of geometric quantities                           1.08
6. DeTurck's trick and short time existence                    113
7. Reaction-diffusion equation for the curvature tensor        119
8. Notes and commentary                                        123
Chapter 3. Closed 3-manifolds with Positive Ricci Curvature       127
1. Hamilton's 3-manifolds with positive Ricci curvature theorem  127
2. The maximum principle for tensors                           128
3. Curvature pinching estimates                               131
4. Gradient bounds for the scalar curvature                   136
S5. Curvature tends to constant                                 140
6. Exponential convergence of the normalized flow             142
7. .Notes and commentary                                       149
Chapter 4. Ricci Solitons and Special Solutions                   153
 1  Gradient Ricci solitons                                    154
2. Gaussian and cylinder solitons                              157
3. Cigar steady soliton                                        159
4. Rosenau solution                                            162
5. An expanding soliton                                        164
6. Bryant soliton                                              167
7. Homogeneous solutions                                       169
8. The isometry group                                          175
9   Notes and commentary                                       176
Chapter 5. Isoperimetric Estimates and No Local Collapsing        181
1   Sobolev and logarithmic Sobolev inequalities               181
2. Evolution of the length of a geodesic                        186
*3.  Isoperimetric estimate for surfaces                        188
4. Perelman's no local collapsing theorem                      190
5. Geometric applications of no local collapsing               198
6. 3-manifolds with positive Ricci curvature revisited         206
7. Isoperimetric estimate for 3-dimensional Type I solutions  208
8. Notes and commentary                                       211
Chapter 6. Preparation for Singularity Analysis                  213
1. Derivative estimates and long time existence              213
2. Proof of Shi's local first and second derivative estimates  218
3. Cheeger-Gromov-type compactness theorem for Ricci flow      233
4. Long time existence of solutions with bounded Ricci curvature 237
5. The Hamilton-Ivey curvature estimate                       240
6. Strong maximum principles and metric splitting             245
7. Rigidity of 3-manifolds with nonnegative curvature         248
8. Notes and commentary                                       250
Chapter 7. High-dimensional and Noncompact Ricci Flow            253
1. Spherical space form theorem of Huisken-Margerin-Nishikawa  254
2. 4-manifolds with positive curvature operator               259
3. Ma.nifolds with nonnegative curvature operator             263
4. The maximum principle on noncompact manifolds              272
5. Complete solutions of the Ricci flow on noncompact manifolds 279
6. Notes and commentary                                      286
Chapter 8. Singularity Analysis                                  291
1. Singularity dilations and types                            292
2. Point picking and types of singularity models             297
3. Geometric invariants of ancient solutions                  307
4. Dimension reduction                                        316
5, Notes and commentary                                       326
Chapter 9. Ancient Solutions                                     327
1. Classification of ancient solutions on surfaces            328
2. Properties of ancient solutions that relate to their type  338
3. Geometry at infinity of gradient Ricci solitons            353
4. Injectivity radius of steady gradient Ricci solitons       364
5. Towards a classification of 3-dimensional ancient solutions  368
6. Classification of 3-dimensional shrinking Ricci solitons   375
7. Summary and open problems                                  388
Chapter 10. Differential Harnack Estimates                       391
1. Harnack estimates for the heat and Laplace equations        392
2. Harnack estimate on surfaces with y > 0                     397
,3. Linear trace and interpolated Harnack estimates on surfaces  401
S4. Hamilton's matrix Harnack estimate for the Ricci flow      405
5. Proof of the matrix Harnack estimate                        410
6   Harnack and pinching estimates for linearized Ricci flow  415
7. Notes and commentary                                        420
Chapter 11. Space-time Geometry                                   425
1. Space-time solution to the Ricei flow for degenerate metrics  426
2. Space-time curvature is the matrix Harnack quadratic       .33
3. P otentially infinite metrics and potentially infinite dimensions 434
4. Renormalizing the space-time length yields the f-length    452
5. Space-time DeTurck's trick and fixing the measure           453
<6. Notes and commentary                                       456



Library of Congress subject headings for this publication: Global differential geometry, Ricci flow, Riemannian manifolds