Hamilton’s Ricci Flow

The aim of this project is to introduce the basics of Hamilton’s Ricci Flow. The Ricci flow is a pde for evolving the metric tensor in a Riemannian manifold to make it “rounder”, in the hope that one may draw topological conclusions from the existence of such “round” metrics. Indeed, the Ricci flow has recently been used to prove two very deep theorems in topology, namely the Geometrization and Poincaré Conjectures. We begin with a brief survey of the differential geometry that is needed in the Ricci flow, then proceed to introduce its basic properties and the basic techniques used to understand it, for example, proving existence and uniqueness and bounds on derivatives of curvature under the Ricci flow using the maximum principle. We use these results to prove the “original” Ricci flow theorem – the 1982 theorem of Richard Hamilton that closed 3-manifolds which admit metrics of strictly positive Ricci curvature are diffeomorphic to quotients of the round 3-sphere by finite groups of isometries acting freely. We conclude with a qualitative discussion of the ideas behind the proof of the Geometrization Conjecture using the Ricci flow. Most of the project is based on the book by Chow and Knopf [6], the notes by Peter Topping [28] (which have recently been made into a book, see [29]), the papers of Richard Hamilton (in particular [9]) and the lecture course on Geometric Evolution Equations presented by Ben Andrews at the 2006 ICE-EM Graduate School held at the University of Queensland. We have reformulated and expanded the arguments contained in these references in some places. In particular, the proof of Theorem 7.19 is original, based on a suggestion by Gerhard Huisken. We also diverge from the existing references by emphasising the analogy between the techniques applied to the Ricci flow and those applied to the curve-shortening flow, which we feel helps clarify the important ideas behind the technical details of the Ricci flow. Chapter 6 is based on [6, Chap. 6, 7], but we have significantly reformulated the material and elaborated on the proofs. We feel that our organization is easier to follow than Chow and Knopf’s book. The attempt to motivate the compactness result in Section 8.1 is also original.