Lectures on the Ricci FlowRicci 流讲义
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作者： Peter Topping 著
出 版 社：
出版时间： 2006-11-1字数：版次： 1页数： 113印刷时间： 2006/11/01开本： 16开印次： 1纸张： 胶版纸I S B N ： 9780521689472包装： 平装内容简介
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.
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.
作者简介
Peter Topping is a Senior Lecturer in Mathematics at the University of Warwick.
目录
Preface
1Introduction
1.1 Ricci flow: what is it, and from where did it come?
1.2 Examples and special solutions
1.2.1 Einstein manifolds
1.2.2 Ricci solitons
1.2.3 Parabolic rescaling of Ricci flows
1.3 Getting a feel for Ricci flow
1.3.1 Two dimensions
1.3.2 Three dimensions
1.4 The topology and geometry of manifolds in low dimensions
1.5 Using Ricci flow to prove topological and geometric results
2Riemannian geometry background
2.1 Notation and conventions
2.2 Einstein metrics
2.3 Deformation of geometric quantities as the Riemannian metric is deformed
2.3.1 The formulae
2.3.2 The calculations
2.4 Laplacian of the curvature tensor
2.5 Evolution of curvature and geometric quantities under Ricci flow
3 The maximum principle
3.1 Statement of the maximum principle
3.2 Basic control on the evolution of curvature
3.3 Global curvature derivative estimates
4Comments on existence theory for parabolic PDE
4.1 Linear scalar PDE
4.2 The principal symbol
4.3 Generalisation to Vector Bundles
4.4 Properties of parabolic equations
5Existence theory for the Ricci flow
5.1 Ricci flow is not parabolic
5.2 Short-time existence and uniqueness: The DeTurck trick
5.3 Curvature blow-up at finite-time singularities
6Ricci flow as a gradient flow
6.1 Gradient of total scalar curvature and related functionals
6.2 The 5-functional
6.3 The heat operator and its conjugate
6.4 A gradient flow formulation
6.5 The classical entropy
6.6 The zeroth eigenvalue of-4A + R
7Compactness of Riemannian manifolds and flows
7.1 Convergence and compactness of manifolds
7.2 Convergence and compactness of flows
7.3 Blowing up at singularities I
8Perelman's W entropy functional
8.1 Definition, motivation and basic properties
8.2 Monotonicity of W
8.3 No local volume collapse where curvature is controlled
8.4 Volume ratio bounds imply injectivity radius bounds
8.5 Blowing up at singularities II
9Curvature pinching and preserved curvature properties under Ricci flow
9.1 Overview
9.2 The Einstein Tensor, E
9.3 Evolution of E under the Ricci flow
9.4 The Uhlenbeck Trick
9.5 Formulae for parallel functions on vector bundles
9.6 An ODE-PDE theorem
9.7 Applications of the ODE-PDE theorem
10 Three-manifolds with positive Ricci curvature, and beyond
10.1 Hamilton's theorem
10.2 Beyond the case of positive Ricci curvature
A Connected sum
References
Index
