**Mathematical Surveys and Monographs**

Volume: 144;
2008;
458 pp;
Hardcover

MSC: Primary 53; 58; 35;

Print ISBN: 978-0-8218-4429-8

Product Code: SURV/144

List Price: $112.00

Individual Member Price: $89.60

**Electronic ISBN: 978-1-4704-1371-2
Product Code: SURV/144.E**

List Price: $112.00

Individual Member Price: $89.60

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#### Supplemental Materials

# The Ricci Flow: Techniques and Applications: Part II: Analytic Aspects

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*Bennett Chow; Sun-Chin Chu; David Glickenstein; Christine Guenther; James Isenberg; Tom Ivey; Dan Knopf; Peng Lu; Feng Luo; Lei Ni*

Geometric analysis has become one of the most important tools in geometry and topology. In their books on the Ricci flow, the authors reveal the depth and breadth of this flow method for understanding the structure of manifolds. With the present book, the authors focus on the analytic aspects of Ricci flow.

Some highlights of the presentation are weak and strong maximum principles for scalar heat-type equations and systems on manifolds, the classification by Böhm and Wilking of closed manifolds with 2-positive curvature operator, Bando's result that solutions to the Ricci flow are real analytic in the space variables, Shi's local derivative of curvature estimates and some variants, and differential Harnack estimates of Li–Yau-type including Hamilton's matrix estimate for the Ricci flow and Perelman's estimate for fundamental solutions of the adjoint heat equation coupled to the Ricci flow.

The authors have tried to make some advanced material accessible to graduate students and nonexperts. The book gives a rigorous introduction to some of Perelman's work and explains some technical aspects of Ricci flow useful for singularity analysis. They have also attempted to give the appropriate references so that the reader may further pursue the statements and proofs of the various results.

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#### Table of Contents

# Table of Contents

## The Ricci Flow: Techniques and Applications: Part II: Analytic Aspects

- Contents v6 free
- Preface ix10 free
- Acknowledgments xiii14 free
- Contents of Part II of Volume Two xvii18 free
- Notation and Symbols xxiii24 free
- Chapter 10. Weak Maximum Principles for Scalars, Tensors, and Systems 128 free
- 1. Weak maximum principles for scalars and symmetric 2-tensors 229
- 2. Vector bundle formulation of the weak maximum principle for systems 936
- 3. Spatial maximum function and its Dini derivatives 2451
- 4. Convex sets, support functions, ODEs preserving convex sets 3259
- 5. Proof of the WMP for systems: time-dependent sets and avoidance sets 4370
- 6. Maximum principles for weak solutions of heat equations 4774
- 7. Variants of maximum principles 5683
- 8. Notes and commentary 6592

- Chapter 11. Closed Manifolds with Positive Curvature 6794
- 1. Multilinear algebra related to the curvature operator 6996
- 2. Algebraic curvature operators and Rm 77104
- 3. A family of linear transformations and their effect on R[sup(2)] + R[sup(#)] 89116
- 4. Proof of the main formula for D[sub(a,b)](R) 94121
- 5. The convex cone of 2-nonnegative algebraic curvature operators 105132
- 6. A pinching family of convex cones in the space of algebraic curvature operators 116143
- 7. Obtaining a generalized pinching set from a pinching family and the proof of Theorem 11.2 126153
- 8. Summary of the proof of the convergence of Ricci flow 134161
- 9. Notes and commentary 136163

- Chapter 12. Weak and Strong Maximum Principles on Noncompact Manifolds 139166
- 1. Weak maximum principles for scalar heat-type equations 140167
- 2. Mollifying distance functions on Riemannian manifolds 158185
- 3. Weak maximum principle for parabolic systems 170197
- 4. Strong maximum principle for parabolic systems 180207
- 5. Applications to the curvature operator under the Ricci flow 192219
- 6. Notes and commentary 195222

- Chapter 13. Qualitative Behavior of Classes of Solutions 197224
- Chapter 14. Local Derivative of Curvature Estimates 227254
- 1. Introduction—fine versus coarse estimates 228255
- 2. A quick review of the global derivative estimates 232259
- 3. Shi's local derivative estimates 235262
- 4. Modified Shi's local derivative estimates assuming bounds on some derivatives of curvatures of the initial metrics 244271
- 5. Some applications of the local derivative estimates 251278
- 6. Local heat equation and local Ricci flow 253280
- 7. Notes and commentary 258285

- Chapter 15. Differential Harnack Estimates of LYH-type 259286
- 1. Deriving the Harnack expression using Ricci solitons 259286
- 2. Statement of the matrix Harnack estimate 263290
- 3. Proofs: getting started with surfaces 265292
- 4. Proof of the matrix Harnack estimate 268295
- 5. A variant on Hamilton's proof of the matrix Harnack estimate 287314
- 6. Ricci solitons and ancient solutions attaining R[sub(max)] 293320
- 7. Applications of Harnack estimates 300327
- 8. Notes and commentary 303330

- Chapter 16. Perelman's Differential Harnack Estimate 305332
- 1. Entropy and differential Harnack estimates for the heat equation 306333
- 2. Properties of the heat kernel and linear entropy formula on complete manifolds 314341
- 3. Differential Harnack estimate and characterizing R[sup(n)] by linear entropy 325352
- 4. Perelman's differential Harnack estimate 335362
- 5. Notes and commentary 355382

- Appendix D. An Overview of Aspects of Ricci Flow 357384
- 1. Existence, uniqueness, convergence, and curvature evolution 357384
- 2. The rotationally symmetric neckpinch 360387
- 3. Curvature pinching, derivative, and Harnack estimates 365392
- 4. Perelman's energy, entropy, and associated invariants 369396
- 5. Compactness, no local collapsing, and singularity models 373400

- Appendix E. Aspects of Geometric Analysis Related to Ricci Flow 379406
- Appendix F. Tensor Calculus on the Frame Bundle 411438
- 1. Introduction 411438
- 2. Tensors as vector-valued functions on the frame bundle 412439
- 3. Local coordinates on the frame bundle 414441
- 4. The metric on the frame bundle 415442
- 5. A natural frame field on FM 416443
- 6. Covariant differentiation 420447
- 7. Curvature and commuting covariant derivatives 422449
- 8. Reduction to the orthonormal frame bundle 423450
- 9. Time dependent orthonormal frame bundle for solutions to the Ricci flow 426453
- 10. The time vector field and its action on tensors 427454
- 11. The heat operator and commutation formulas 429456
- 12. Notes and commentary 431458

- Bibliography 433460
- Index 455482

#### Readership

Graduate students and research mathematicians interested in geometic analysis; geometry and topology.

#### Reviews

Just like in the first part this book is meant to be a text book as well as a reference book and it includes exercises as well as the solutions for some of them. The authors took a great care in making this a self contained book which compiles a great amount of data related to Ricci flow, organizing the information in a very clear way giving very often information on the content of each chapter and relating it to other parts of the book. An example is the sumary given in the preface, another more detailed sumary given in a special kind of chapter zero, also in the beginning of each chapter and sometimes introducing each section. Besides all this, each chapter includes a special section in the end with notes and commentaries. This approach gives the reader permanently a global view on the subject while the details are being explained. The book also includes a large number of exercises very often with suggestions for its resolutions.

-- Zentralblatt MATH

This book and this series is necessarily quite technical in parts, but it does reveal the full beauty and technical details of the Ricci flow. The authors should be congratulated on presenting such material in a very readable style, creating an excellent resource with thorough discussion of many aspects of Ricci flow for the benefit of mathematical researchers today and in the future.

-- Mathematical Reviews