Ricci flow is a powerful analytic method for studying the geometry and topology of manifolds. This book is an introduction to Ricci flow for graduate students and mathematicians interested in working in the subject. To this end, the first chapter is a review of the relevant basics of Riemannian geometry. For the benefit of the student, the text includes a number of exercises of varying difficulty.The book also provides brief introductions to some general methods of geometric analysis and other geometric flows. Comparisons are made between the Ricci flow and the linear heat equation, mean curvature flow, and other geometric evolution equations whenever possible.Several topics of Hamilton’s program are covered, such as short time existence, Harnack inequalities, Ricci solutions, Perelman’s no local collapsing theorem, singularity analysis, and ancient solutions.A major direction in Ricci flow, via Hamilton’s and Perelman’s works, is the use of Ricci flow as an approach to solving the Poincaré conjecture and Thurston’s geometrization conjecture.
Bennett Chow, University of California, San Diego, La Jolla, California, USA. Peng Lu, University of Oregon, Eugene, OR, and Lei Ni, University of California, San Diego, La Jolla, California, USA.
Preface Acknowledgments A Detailed Guide for the Reader Notation and Symbols Chapter 1. Riemannian Geometry Chapter 2. Fundamentals of the Ricci Flow Equation Chapter 3. Closed 3-manifolds with Positive Ricci Curvature Chapter 4. Ricci Solitons and Special Solutions Chapter 5. Isoperimetric Estimates and No Local Collapsing Chapter 6. Preparation for Singularity Analysis Chapter 7. High-dimensional and Noncompact Ricci Flow Chapter 8. Singularity Analysis Chapter 9. Ancient Solutions Chapter 10. Differential Harnack Estimates Chapter 11. Space-time Geometry Appendix A. Geometric Analysis Related to Ricci Flow Appendix B. Analytic Techniques for Geometric Flows Appendix C. Solutions to Selected Exercises Bibliography Index