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Book DetailsGraduate Studies in MathematicsVolume: 111; 2010; 176 ppMSC: Primary 53
In 1982, R. Hamilton introduced a nonlinear evolution equation for Riemannian metrics with the aim of finding canonical metrics on manifolds. This evolution equation is known as the Ricci flow, and it has since been used widely and with great success, most notably in Perelman's solution of the Poincaré conjecture. Furthermore, various convergence theorems have been established.
This book provides a concise introduction to the subject as well as a comprehensive account of the convergence theory for the Ricci flow. The proofs rely mostly on maximum principle arguments. Special emphasis is placed on preserved curvature conditions, such as positive isotropic curvature. One of the major consequences of this theory is the Differentiable Sphere Theorem: a compact Riemannian manifold, whose sectional curvatures all lie in the interval (1,4], is diffeomorphic to a spherical space form. This question has a long history, dating back to a seminal paper by H. E. Rauch in 1951, and it was resolved in 2007 by the author and Richard Schoen.
This text originated from graduate courses given at ETH Zürich and Stanford University, and it is directed at graduate students and researchers. The reader is assumed to be familiar with basic Riemannian geometry, but no previous knowledge of Ricci flow is required.
ReadershipGraduate students and research mathematicians interested in differential geometry and topology of manifolds.

Table of Contents

Chapters

Chapter 1. A survey of sphere theorems in geometry

Chapter 2. Hamilton’s Ricci flow

Chapter 3. Interior estimates

Chapter 4. Ricci flow on $S^2$

Chapter 5. Pointwise curvature estimates

Chapter 6. Curvature pinching in dimension 3

Chapter 7. Preserved curvature conditions in higher dimensions

Chapter 8. Convergence results in higher dimensions

Chapter 9. Rigidity results

Appendix A. Convergence of evolving metrics

Appendix B. Results from complex linear algebra

Problems


Additional Material

Reviews

This book is a great selfcontained presentation of one of the most important and exciting developments in differential geometry. It is highly recommended for both researchers and students interested in differential geometry, topology and Ricci flow. As the main technical tool used in the book is the maximum principle, familiar to any undergraduate, this book would make a fine reading course for advanced undergraduates or postgraduates and, in particular, is an excellent introduction to some of the analysis required to study Ricci flow.
Huy The Nyugen, Bulletin of the LMS 
This is an excellent selfcontained account of exciting new developments in mathematics suitable for both researchers and students interested in differential geometry and topology and in some of the analytic techniques used in Ricci flow. I very strongly recommend it.
Jahresbericht Der Deutschen Mathematiker  Vereinigung


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In 1982, R. Hamilton introduced a nonlinear evolution equation for Riemannian metrics with the aim of finding canonical metrics on manifolds. This evolution equation is known as the Ricci flow, and it has since been used widely and with great success, most notably in Perelman's solution of the Poincaré conjecture. Furthermore, various convergence theorems have been established.
This book provides a concise introduction to the subject as well as a comprehensive account of the convergence theory for the Ricci flow. The proofs rely mostly on maximum principle arguments. Special emphasis is placed on preserved curvature conditions, such as positive isotropic curvature. One of the major consequences of this theory is the Differentiable Sphere Theorem: a compact Riemannian manifold, whose sectional curvatures all lie in the interval (1,4], is diffeomorphic to a spherical space form. This question has a long history, dating back to a seminal paper by H. E. Rauch in 1951, and it was resolved in 2007 by the author and Richard Schoen.
This text originated from graduate courses given at ETH Zürich and Stanford University, and it is directed at graduate students and researchers. The reader is assumed to be familiar with basic Riemannian geometry, but no previous knowledge of Ricci flow is required.
Graduate students and research mathematicians interested in differential geometry and topology of manifolds.

Chapters

Chapter 1. A survey of sphere theorems in geometry

Chapter 2. Hamilton’s Ricci flow

Chapter 3. Interior estimates

Chapter 4. Ricci flow on $S^2$

Chapter 5. Pointwise curvature estimates

Chapter 6. Curvature pinching in dimension 3

Chapter 7. Preserved curvature conditions in higher dimensions

Chapter 8. Convergence results in higher dimensions

Chapter 9. Rigidity results

Appendix A. Convergence of evolving metrics

Appendix B. Results from complex linear algebra

Problems

This book is a great selfcontained presentation of one of the most important and exciting developments in differential geometry. It is highly recommended for both researchers and students interested in differential geometry, topology and Ricci flow. As the main technical tool used in the book is the maximum principle, familiar to any undergraduate, this book would make a fine reading course for advanced undergraduates or postgraduates and, in particular, is an excellent introduction to some of the analysis required to study Ricci flow.
Huy The Nyugen, Bulletin of the LMS 
This is an excellent selfcontained account of exciting new developments in mathematics suitable for both researchers and students interested in differential geometry and topology and in some of the analytic techniques used in Ricci flow. I very strongly recommend it.
Jahresbericht Der Deutschen Mathematiker  Vereinigung