Hardcover ISBN:  9780821842317 
Product Code:  GSM/77 
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Electronic ISBN:  9781470421113 
Product Code:  GSM/77.E 
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Book DetailsGraduate Studies in MathematicsVolume: 77; 2006; 608 ppMSC: Primary 53; 58; 35;
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 solitons, 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.ReadershipGraduate students and research mathematicians interested in geometric analysis, the Poincaré conjecture, Thurston's geometrization conjecture, and 3manifolds.

Table of Contents

Chapters

Chapter 1. Riemannian geometry

Chapter 2. Fundamentals of the Ricci flow equation

Chapter 3. Closed 3manifolds 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. Highdimensional and noncompact Ricci flow

Chapter 8. Singularity analysis

Chapter 9. Ancient solutions

Chapter 10. Differential Harnack estimates

Chapter 11. Spacetime geometry

Appendix A. Geometric analysis related to Ricci flow

Appendix B. Analytic techniques for geometric flows

Appendix S. Solutions to selected exercises


Additional Material

Reviews

The style of the book is very pleasant, including lots of motivations and background material, course outlines and exercises (with solutions), the bibliography is rather comprehensive. This work is certain to become one of the main references in this field of great current interest.
M. Kunzinger 
This book is a very well written introduction to and resource for study of the Ricci flow. It is quite selfcontained, but relevant references are provided at appropriate points. The style of the book renders it accessible to graduate students (suggested course outlines and many relevant further references are provided), while its substance provides an essential resource for background, key concepts and fundamental ideas for further study in the area.
James McCoy, Mathematical Reviews


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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 solitons, 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.
Graduate students and research mathematicians interested in geometric analysis, the Poincaré conjecture, Thurston's geometrization conjecture, and 3manifolds.

Chapters

Chapter 1. Riemannian geometry

Chapter 2. Fundamentals of the Ricci flow equation

Chapter 3. Closed 3manifolds 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. Highdimensional and noncompact Ricci flow

Chapter 8. Singularity analysis

Chapter 9. Ancient solutions

Chapter 10. Differential Harnack estimates

Chapter 11. Spacetime geometry

Appendix A. Geometric analysis related to Ricci flow

Appendix B. Analytic techniques for geometric flows

Appendix S. Solutions to selected exercises

The style of the book is very pleasant, including lots of motivations and background material, course outlines and exercises (with solutions), the bibliography is rather comprehensive. This work is certain to become one of the main references in this field of great current interest.
M. Kunzinger 
This book is a very well written introduction to and resource for study of the Ricci flow. It is quite selfcontained, but relevant references are provided at appropriate points. The style of the book renders it accessible to graduate students (suggested course outlines and many relevant further references are provided), while its substance provides an essential resource for background, key concepts and fundamental ideas for further study in the area.
James McCoy, Mathematical Reviews