E-Book Overview
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.
E-Book Content
lectures on the ricci flow Peter Topping March 9, 2006 1 c Peter Topping 2004, 2005, 2006. 1 Contents 1 Introduction 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 2 Riemannian geometry background 2.1 Notation and conventions . . .