Lectures on the Ricci Flow

Peter Topping

doi:10.1017/CBO9780511721465

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 whichcarries a metric of positive Ricci curvature is a spherical space form.

"... 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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Lectures on the Ricci Flow

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Contents

Frontmatter
pp i-iv

Contents
pp v-viii

Preface
pp ix-x

1 - Introduction
pp 1-15

2 - Riemannian geometry background
pp 16-34

3 - The maximum principle
pp 35-42

4 - Comments on existence theory for parabolic PDE
pp 43-47

5 - Existence theory for the Ricci flow
pp 48-54

6 - Ricci flow as a gradient flow
pp 55-64

7 - Compactness of Riemannian manifolds and flows
pp 65-70

8 - Perelman's W entropy functional
pp 71-87

9 - Curvature pinching and preserved curvature properties under Ricci flow
pp 88-104

10 - Three–manifolds with positive Ricci curvature, and beyond
pp 105-107

A - Connected sum
pp 108-108

References
pp 109-111

Index
pp 112-113

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