Lectures on the Ricci Flow
1st Edition
Published on 4. April 2011
978-0-511-89375-9 (ISBN)
Description
Hamilton's Ricci flow has attracted considerable attention since its introduction in 1982, owing partly to its promise in addressing the Poincare 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 whichcarries a metric of positive Ricci curvature is a spherical space form.
Reviews / Votes
"... The freedom to skip some of the proofs, and the lucid presentation, this small book is pleasant to read."Peng Lu, Mathematical Reviews
More details
Series
Language
English
Place of publication
Cambridge
United Kingdom
Publishing group
Cambridge University Press
Target group
Professional and scholarly
Illustrations
Worked examples or Exercises; 22 Line drawings, unspecified
File size
1,88 MB
ISBN-13
978-0-511-89375-9 (9780511893759)
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Schweitzer Classification
Other editions
Additional editions
Peter Topping
Lectures on the Ricci Flow
Book
10/2006
Cambridge University Press
€73.60
Shipment within 15-20 days
Person
Peter Topping is a Senior Lecturer in Mathematics at the University of Warwick.
Content
1. Introduction; 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.
