Contents
Preface vii
A guide for the reader viii
A guide for the hurried reader x
Acknowledgments xi
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
Notes and commentary 19
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
Notes and commentary 66
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
Notes and commentary 92
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
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