The Ricci Flow: An Introduction
نویسندگان
چکیده
Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. @ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. 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 iv CONTENTS Notes and commentary 103 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 2) < 0 120 6. Convergence when \ (M 2) = 0 123 7. Strategy for the case that \ (M 2 > 0) 128 8. Surface entropy 133 9. …
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