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Hamilton's Ricci flow has attracted considerable attention since its introduction in 1982, owing partly to its promise in addressing the Poincaré conjecture and Thurston's geometrization conjecture. This book gives a concise introduction to the subject with the hindsight of Perelman's breakthroughs from 2002/2003. After describing the basic properties of, and intuition behind the Ricci flow, core elements of the theory are discussed such as consequences of various forms of maximum principle, issues related to existence theory, and basic properties of singularities in the flow. A detailed exposition of Perelman's entropy functionals is combined with a description of Cheeger-Gromov-Hamilton compactness of manifolds and flows to show how a 'tangent' flow can be extracted from a singular Ricci flow. Finally, all these threads are pulled together to give a modern proof of Hamilton's theorem that a closed three-dimensional manifold which carries a metric of positive Ricci curvature is a spherical space form.Read more
- Presents the state of the art view of the subject
- Includes results not available in book form anywhere else
- Written in a style which makes it ideal for use in a graduate level course
Reviews & endorsements
"... The freedom to skip some of the proofs, and the lucid presentation, this small book is pleasant to read."
Peng Lu, Mathematical Reviews
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- Date Published: November 2006
- format: Paperback
- isbn: 9780521689472
- length: 124 pages
- dimensions: 230 x 153 x 7 mm
- weight: 0.189kg
- contains: 22 b/w illus.
- availability: Available
Table of Contents
2. Riemannian geometry background
3. The maximum principle
4. Comments on existence theory for parabolic PDE
5. Existence theory for the Ricci flow
6. Ricci flow as a gradient flow
7. Compactness of Riemannian manifolds and flows
8. Perelman's W entropy functional
9. Curvature pinching and preserved curvature properties under Ricci flow
10. Three-manifolds with positive Ricci curvature and beyond.
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