The Ricci Flow: An Introduction (Mathematical Surveys and Monographs)

This is quite simply the best book on the Ricci Flow that I have ever encountered. This is the only book on the Ricci Flow that I have ever encountered. I believe that its value to the development and application of geometric analysis for the study of manifolds is incalculable (no pun intended). I must say, I have never seen nicer functions or such improved metrics. I am particularly impressed by by the talents of the second author, Dr. Dan Knopf, as displayed in this fine publication. His commanding grasp of the material, lucid and thought-provoking presentation, and fine expository style are nothing short of breathtaking. I am pleased to recommend this excellent work. The Ricci Flow Method Is Now Central To Our Understanding Of The Geometry And Topology Of Manifolds. The Book Is An Introduction To That Program And To Its Connection To Thurston's Geometrization Conjecture. The Book Is Suitable For Geometers And Others Who Are Interested In The Use Of Geometric Analysis To Study The Structure Of Manifolds.--book Jacket. Chapter 1. The Ricci Flow Of Special Geometries 1 -- 1. Geometrization Of Three-manifolds 2 -- 2. Model Geometries 4 -- 3. Classifying Three-dimensional Maximal Model Geometries 6 -- 4. Analyzing The Ricci Flow Of Homogeneous Geometries 8 -- 5. The Ricci Flow Of A Geometry With Maximal Isotropy So (3) 11 -- 6. The Ricci Flow Of A Geometry With Isotropy So (2) 15 -- 7. The Ricci Flow Of A Geometry With Trivial Isotropy 17 -- Chapter 2. Special And Limit Solutions 21 -- 1. Generalized Fixed Points 21 -- 2. Eternal Solutions 24 -- 3. Ancient Solutions 28 -- 4. Immortal Solutions 34 -- 5. The Neckpinch 38 -- 6. The Degenerate Neckpinch 62 -- Chapter 3. Short Time Existence 67 -- 1. Variation Formulas 67 -- 2. The Linearization Of The Ricci Tensor And Its Principal Symbol 71 -- 3. The Ricci-deturck Flow And Its Parabolicity 78 -- 4. The Ricci-deturck Flow In Relation To The Harmonic Map Flow 84 -- 5. The Ricci Flow Regarded As A Heat Equation 90 -- Chapter 4. Maximum Principles 93 -- 1. Weak Maximum Principles For Scalar Equations 93 -- 2. Weak Maximum Principles For Tensor Equations 97 -- 3. Advanced Weak Maximum Principles For Systems 100 -- 4. Strong Maximum Principles 102 -- Chapter 5. The Ricci Flow On Surfaces 105 -- 1. The Effect Of A Conformal Change Of Metric 106 -- 2. Evolution Of The Curvature 109 -- 3. How Ricci Solitons Help Us Estimate The Curvature From Above 111 -- 4. Uniqueness Of Ricci Solitons 116 -- 5. Convergence When X (m[superscript 2]) [less Than Sign] 0 120 -- 6. Convergence When X (m[superscript 2]) = 0 123 -- 7. Strategy For The Case That X (m[superscript 2] [greater Than Sign] 0) 128 -- 8. Surface Entropy 133 -- 9. Uniform Upper Bounds For R And [vertical Bar Down Triangle, Open]r[vertical Bar] 137 -- 10. Differential Harnack Estimates Of Lyh Type 143 -- 11. Convergence When R(.,0) [greater Than Sign] 0 148 -- 12. A Lower Bound For The Injectivity Radius 149 -- 13. The Case That R(.,0) Changes Sign 153 -- 14. Monotonicity Of The Isoperimetric Constant 156 -- 15. An Alternative Strategy For The Case X (m[superscript 2] [greater Than Sign] 0) 165 -- Chapter 6. Three-manifolds Of Positive Ricci Curvature 173 -- 1. The Evolution Of Curvature Under The Ricci Flow 174 -- 2. Uhlenbeck's Trick 180 -- 3. The Structure Of The Curvature Evolution Equation 183 -- 4. Reduction To The Associated Ode System 187 -- 5. Local Pinching Estimates 189 -- 6. The Gradient Estimate For The Scalar Curvature 194 -- 7. Higher Derivative Estimates And Long-time Existence 200 -- 8. Finite-time Blowup 209 -- 9. Properties Of The Normalized Ricci Flow 212 -- 10. Exponential Convergence 218 -- Chapter 7. Derivative Estimates 223 -- 1. Global Estimates And Their Consequences 223 -- 2. Proving The Global Estimates 226 -- 3. The Compactness Theorem 231 -- Chapter 8. Singularities And The Limits Of Their Dilations 233 -- 1. Classifying Maximal Solutions 233 -- 2. Singularity Models 235 -- 3. Parabolic Dilations 237 -- 4. Dilations Of Finite-time Singularities 240 -- 5. Dilations Of Infinite-time Singularities 246 -- 6. Taking Limits Backwards In Time 250 -- Chapter 9. Type I Singularities 253 -- 1. Intuition 253 -- 2. Positive Curvature Is Preserved 255 -- 3. Positive Sectional Curvature Dominates 256 -- 4. Necklike Points In Finite-time Singularities 262 -- 5. Necklike Points In Ancient Solutions 271 -- 6. Type I Ancient Solutions On Surfaces 274 -- Appendix A. The Ricci Calculus 279 -- 1. Component Representations Of Tensor Fields 279 -- 2. First-order Differential Operators On Tensors 280 -- 3. First-order Differential Operators On Forms 283 -- 4. Second-order Differential Operators 284 -- 5. Notation For Higher Derivatives 285 -- 6. Commuting Covariant Derivatives 286 -- Appendix B. Some Results In Comparison Geometry 287 -- 1. Some Results In Local Geometry 287 -- 2. Distinguishing Between Local Geometry And Global Geometry 295 -- 3. Busemann Functions 303 -- 4. Estimating Injectivity Radius In Positive Curvature 312. Bennett Chow, Dan Knopf. Includes Bibliographical References (p. 317-322) And Index. The Ricci flow is a powerful technique that integrates geometry, topology, and analysis. Intuitively, the idea is to set up a PDE that evolves a metric according to its Ricci curvature. The resulting equation has much in common with the heat equation, which tends to "flow" a given function to ever nicer functions. By analogy, the Ricci flow evolves an initial metric into improved metrics. Richard Hamilton began the systematic use of the Ricci flow in the early 1980s and applied it in particular to study 3-manifolds. Grisha Perelman has made recent breakthroughs aimed at completing Hamilton's program. The Ricci flow method is now central to our understanding of the geometry and topology of manifolds. This book is an introduction to that program and to its connection to Thurston's geometrization conjecture. The authors also provide a "Guide for the hurried reader", to help readers wishing to develop, as efficiently as possible, a nontechnical appreciation of the Ricci flow program for 3-manifolds, i.e., the so-called "fast track". The Ricci flow is a powerful technique that integrates geometry, topology, and analysis. Intuitively, the idea is to set up a PDE that evolves a metric according to its Ricci curvature. The resulting equation has much in common with the heat equation, which tends to “flow” a given function to ever nicer functions. By analogy, the Ricci flow evolves an initial metric into improved metrics. Richard Hamilton began the systematic use of the Ricci flow in the early 1980s and applied it in particular to study 3-manifolds. Grisha Perelman has made recent breakthroughs aimed at completing Hamilton's program. The Ricci flow method is now central to our understanding of the geometry and topology of manifolds. This book is an introduction to that program and to its connection to Thurston's geometrization conjecture. The book is suitable for geometers and others who are interested in the use of geometric analysis to study the structure of manifolds.