Contents Preface ix What this book is about ix Highlights of Part I xi Acknowledgments xiii Contents of Part I of Volume Two xvii Chapter 1. Ricci Solitons 1 1. General solitons and their canonical forms 2 2. Differentiating the soliton equation local and global analysis 6 3. Warped products and 2-dimensional solitons 11 4. Constructing the Bryant steady soliton 17 5. Rotationally symmetric expanding solitons 26 6. Homogeneous expanding solitons 32 7. When breathers and solitons are Einstein 41 8. Perelman's energy and entropy in relation to Ricci solitons 44 9. Buscher duality transformation of warped product solitons 46 10. Summary of results and open problems on Ricci solitons 50 11. Notes and commentary 52 Chapter 2. Kahler-Ricci Flow and Kahler-Ricci Solitons 55 1. Introduction to Kahler manifolds 55 2. Connection, curvature, and covariant differentiation 62 3. Existence of Kahler-Einstein metrics 70 4. Introduction to the Kahler-Ricci flow 74 5. Existence and convergence of the Kahler-Ricci flow 80 6. Survey of some results for the Kahler-Ricci flow 95 7. Examples of Kahler-Ricci solitons 97 8. Kahler-Ricci flow with nonnegative bisectional curvature 103 9. Matrix differential Harnack estimate for the Kahler-Ricci flow 109 10. Linear and interpolated differential Harnack estimates 118 11. Notes and commentary 124 Chapter 3. The Compactness Theorem for Ricci Flow 127 1. Introduction and statements of the compactness theorems 127 2. Convergence at all times from convergence at one time 132 3. Extensions of Hamilton's compactness theorem 138
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