| Preface | p. vii |
| A guide for the reader | p. viii |
| A guide for the hurried reader | p. x |
| Acknowledgments | p. xi |
| The Ricci flow of special geometries | p. 1 |
| Geometrization of three-manifolds | p. 2 |
| Model geometries | p. 4 |
| Classifying three-dimensional maximal model geometries | p. 6 |
| Analyzing the Ricci flow of homogeneous geometries | p. 8 |
| The Ricci flow of a geometry with maximal isotropy SO (3) | p. 11 |
| The Ricci flow of a geometry with isotropy SO (2) | p. 15 |
| The Ricci flow of a geometry with trivial isotropy | p. 17 |
| Notes and commentary | p. 19 |
| Special and limit solutions | p. 21 |
| Generalized fixed points | p. 21 |
| Eternal solutions | p. 24 |
| Ancient solutions | p. 28 |
| Immortal solutions | p. 34 |
| The neckpinch | p. 38 |
| The degenerate neckpinch | p. 62 |
| Notes and commentary | p. 66 |
| Short time existence | p. 67 |
| Variation formulas | p. 67 |
| The linearization of the Ricci tensor and its principal symbol | p. 71 |
| The Ricci-DeTurck flow and its parabolicity | p. 78 |
| The Ricci-DeTurck flow in relation to the harmonic map flow | p. 84 |
| The Ricci flow regarded as a heat equation | p. 90 |
| Notes and commentary | p. 92 |
| Maximum principles | p. 93 |
| Weak maximum principles for scalar equations | p. 93 |
| Weak maximum principles for tensor equations | p. 97 |
| Advanced weak maximum principles for systems | p. 100 |
| Strong maximum principles | p. 102 |
| Notes and commentary | p. 103 |
| The Ricci flow on surfaces | p. 105 |
| The effect of a conformal change of metric | p. 106 |
| Evolution of the curvature | p. 109 |
| How Ricci solitons help us estimate the curvature from above | p. 111 |
| Uniqueness of Ricci solitons | p. 116 |
| Convergence when X (M[superscript 2]) [less than sign] 0 | p. 120 |
| Convergence when X (M[superscript 2]) = 0 | p. 123 |
| Strategy for the case that X (M[superscript 2] [greater than sign] 0) | p. 128 |
| Surface entropy | p. 133 |
| Uniform upper bounds for R and [vertical bar down triangle, open]R[vertical bar] | p. 137 |
| Differential Harnack estimates of LYH type | p. 143 |
| Convergence when R(.,0) [greater than sign] 0 | p. 148 |
| A lower bound for the injectivity radius | p. 149 |
| The case that R(.,0) changes sign | p. 153 |
| Monotonicity of the isoperimetric constant | p. 156 |
| An alternative strategy for the case X (M[superscript 2] [greater than sign] 0) | p. 165 |
| Notes and commentary | p. 171 |
| Three-manifolds of positive Ricci curvature | p. 173 |
| The evolution of curvature under the Ricci flow | p. 174 |
| Uhlenbeck's trick | p. 180 |
| The structure of the curvature evolution equation | p. 183 |
| Reduction to the associated ODE system | p. 187 |
| Local pinching estimates | p. 189 |
| The gradient estimate for the scalar curvature | p. 194 |
| Higher derivative estimates and long-time existence | p. 200 |
| Finite-time blowup | p. 209 |
| Properties of the normalized Ricci flow | p. 212 |
| Exponential convergence | p. 218 |
| Notes and commentary | p. 221 |
| Derivative estimates | p. 223 |
| Global estimates and their consequences | p. 223 |
| Proving the global estimates | p. 226 |
| The Compactness Theorem | p. 231 |
| Notes and commentary | p. 232 |
| Singularities and the limits of their dilations | p. 233 |
| Classifying maximal solutions | p. 233 |
| Singularity models | p. 235 |
| Parabolic dilations | p. 237 |
| Dilations of finite-time singularities | p. 240 |
| Dilations of infinite-time singularities | p. 246 |
| Taking limits backwards in time | p. 250 |
| Notes and commentary | p. 251 |
| Type I singularities | p. 253 |
| Intuition | p. 253 |
| Positive curvature is preserved | p. 255 |
| Positive sectional curvature dominates | p. 256 |
| Necklike points in finite-time singularities | p. 262 |
| Necklike points in ancient solutions | p. 271 |
| Type I ancient solutions on surfaces | p. 274 |
| Notes and commentary | p. 277 |
| The Ricci calculus | p. 279 |
| Component representations of tensor fields | p. 279 |
| First-order differential operators on tensors | p. 280 |
| First-order differential operators on forms | p. 283 |
| Second-order differential operators | p. 284 |
| Notation for higher derivatives | p. 285 |
| Commuting covariant derivatives | p. 286 |
| Notes and commentary | p. 286 |
| Some results in comparison geometry | p. 287 |
| Some results in local geometry | p. 287 |
| Distinguishing between local geometry and global geometry | p. 295 |
| Busemann functions | p. 303 |
| Estimating injectivity radius in positive curvature | p. 312 |
| Notes and commentary | p. 315 |
| Bibliography | p. 317 |
| Index | p. 323 |
| Table of Contents provided by Rittenhouse. All Rights Reserved. |
