**Graduate Studies in Mathematics**

Volume: 77;
2006;
608 pp;
Hardcover

MSC: Primary 53; 58; 35;

Print ISBN: 978-0-8218-4231-7

Product Code: GSM/77

List Price: $86.00

Individual Member Price: $68.80

**Electronic ISBN: 978-1-4704-2111-3
Product Code: GSM/77.E**

List Price: $86.00

Individual Member Price: $68.80

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#### Supplemental Materials

# Hamilton’s Ricci Flow

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*Bennett Chow; Peng Lu; Lei Ni*

A co-publication of the AMS and Science Press

Ricci flow is a powerful analytic method for studying the geometry and topology of manifolds. This book is an introduction to Ricci flow for graduate students and mathematicians interested in working in the subject. To this end, the first chapter is a review of the relevant basics of Riemannian geometry. For the benefit of the student, the text includes a number of exercises of varying difficulty.

The book also provides brief introductions to some general methods of geometric analysis and other geometric flows. Comparisons are made between the Ricci flow and the linear heat equation, mean curvature flow, and other geometric evolution equations whenever possible.

Several topics of Hamilton's program are covered, such as short time existence, Harnack inequalities, Ricci solitons, Perelman's no local collapsing theorem, singularity analysis, and ancient solutions.

A major direction in Ricci flow, via Hamilton's and Perelman's works, is the use of Ricci flow as an approach to solving the Poincaré conjecture and Thurston's geometrization conjecture.

#### Table of Contents

# Table of Contents

## Hamilton's Ricci Flow

- Cover Cover11 free
- Title iii4 free
- Copyright iv5 free
- Contents v6 free
- Preface xi12 free
- Acknowledgments xvii18 free
- A Detailed Guide for the Reader xxi22 free
- Notation and Symbols xxxv36 free
- Chapter 1. Riemannian Geometry 138 free
- §1. Introduction 138
- §2. Metrics, connections, curvatures and covariant differentiation 239
- §3. Basic formulas and identities in Riemannian geometry 1047
- §4. Exterior differential calculus and Bochner formulas 1451
- §5. Integration and Hodge theory 2057
- §6. Curvature decomposition and locally conformally flat manifolds 2562
- §7. Moving frames and the Gauss-Bonnet formula 3269
- §8. Variation of arc length, energy and area 4178
- §9. Geodesics and the exponential map 5289
- §10. Second fundamental forms of geodesic spheres 5895
- §11. Laplacian, volume and Hessian comparison theorems 67104
- §12. Proof of the comparison theorems 73110
- §13. Manifolds with nonnegative curvature 80117
- §14. Lie groups and left-invariant metrics 87124
- §15. Notes and commentary 89126

- Chapter 2. Fundamentals of the Ricci Flow Equation 95132
- §1. Geometric flows and geometrization 96133
- §2. Ricci flow and the evolution of scalar curvature 98135
- §3. The maximum principle for heat-type equations 100137
- §4. The Einstein-Hilbert functional 104141
- §5. Evolution of geometric quantities 108145
- §6. DeTurck's trick and short time existence 113150
- §7. Reaction-diffusion equation for the curvature tensor 119156
- §8. Notes and commentary 123160

- Chapter 3. Closed 3-manifolds with Positive Ricci Curvature 127164
- §1. Hamilton's 3-manifolds with positive Ricci curvature theorem 127164
- §2. The maximum principle for tensors 128165
- §3. Curvature pinching estimates 131168
- §4. Gradient bounds for the scalar curvature 136173
- §5. Curvature tends to constant 140177
- §6. Exponential convergence of the normalized flow 142179
- §7. Notes and commentary 149186

- Chapter 4. Ricci Solitons and Special Solutions 153190
- Chapter 5. Isoperimetric Estimates and No Local Collapsing 181218
- §1. Sobolev and logarithmic Sobolev inequalities 181218
- §2. Evolution of the length of a geodesic 186223
- §3. Isoperimetric estimate for surfaces 188225
- §4. Perelman's no local collapsing theorem 190227
- §5. Geometric applications of no local collapsing 198235
- §6. 3-manifolds with positive Ricci curvature revisited 206243
- §7. Isoperimetric estimate for 3-dimensional Type I solutions 208245
- §8. Notes and commentary 211248

- Chapter 6. Preparation for Singularity Analysis 213250
- §1. Derivative estimates and long time existence 213250
- §2. Proof of Shi's local first and second derivative estimates 218255
- §3. Cheeger-Gromov-type compactness theorem for Ricci flow 233270
- §4. Long time existence of solutions with bounded Ricci curvature 237274
- §5. The Hamilton-Ivey curvature estimate 240277
- §6. Strong maximum principles and metric splitting 245282
- §7. Rigidity of 3-manifolds with nonnegative curvature 248285
- §8. Notes and commentary 250287

- Chapter 7. High-dimensional and Noncompact Ricci Flow 253290
- §1. Spherical space form theorem of Huisken-Margerin-Nishikawa 254291
- §2. 4-manifolds with positive curvature operator 259296
- §3. Manifolds with nonnegative curvature operator 263300
- §4. The maximum principle on noncompact manifolds 272309
- §5. Complete solutions of the Ricci flow on noncompact manifolds 279316
- §6. Notes and commentary 286323

- Chapter 8. Singularity Analysis 291328
- Chapter 9. Ancient Solutions 327364
- §1. Classification of ancient solutions on surfaces 328365
- §2. Properties of ancient solutions that relate to their type 338375
- §3. Geometry at infinity of gradient Ricci solitons 353390
- §4. Injectivity radius of steady gradient Ricci solitons 364401
- §5. Towards a classification of 3-dimensional ancient solutions 368405
- §6. Classification of 3-dimensional shrinking Ricci solitons 375412
- §7. Summary and open problems 388425

- Chapter 10. Differential Harnack Estimates 391428
- §1. Harnack estimates for the heat and Laplace equations 392429
- §2. Harnack estimate on surfaces with X > 0 397434
- §3. Linear trace and interpolated Harnack estimates on surfaces 401438
- §4. Hamilton's matrix Harnack estimate for the Ricci flow 405442
- §5. Proof of the matrix Harnack estimate 410447
- §6. Harnack and pinching estimates for linearized Ricci flow 415452
- §7. Notes and commentary 420457

- Chapter 11. Space-time Geometry 425462
- §1. Space-time solution to the Ricci flow for degenerate metrics 426463
- §2. Space-time curvature is the matrix Harnack quadratic 433470
- §3. Potentially infinite metrics and potentially infinite dimensions 434471
- §4. Renormalizing the space-time length yields the l-length 452489
- §5. Space-time DeTurck's trick and fixing the measure 453490
- §6. Notes and commentary 456493

- Appendix A. Geometric Analysis Related to Ricci Flow 461498
- §1. Compendium of inequalities 461498
- §2. Comparison theory for the heat kernel 463500
- §3. Green's function 465502
- §4. The Liouville theorem revisited 466503
- §5. Eigenvalues and eigenfunctions of the Laplacian 467504
- §6. The determinant of the Laplacian 476513
- §7. Parametrix for the heat equation 485522
- §8. Monotonicity for harmonic functions and maps 492529
- §9. Bieberbach theorem 494531
- §10. Notes and commentary 500537

- Appendix B. Analytic Techniques for Geometric Flows 503540
- Appendix S. Solutions to Selected Exercises 535572
- Bibliography 573610
- Index 603640
- Back Cover Back Cover1648

#### Readership

Graduate students and research mathematicians interested in geometric analysis, the Poincaré conjecture, Thurston's geometrization conjecture, and 3-manifolds.

#### Reviews

The style of the book is very pleasant, including lots of motivations and background material, course outlines and exercises (with solutions), the bibliography is rather comprehensive. This work is certain to become one of the main references in this field of great current interest.

-- M. Kunzinger

This book is a very well written introduction to and resource for study of the Ricci flow. It is quite self-contained, but relevant references are provided at appropriate points. The style of the book renders it accessible to graduate students (suggested course outlines and many relevant further references are provided), while its substance provides an essential resource for background, key concepts and fundamental ideas for further study in the area.

-- James McCoy, Mathematical Reviews