For additional information and updates on this book, visit www.ams.org/bookpages/surv-110 Library of Congress Cataloging-in-Publication Data Chow, Bennett. The Ricci flow: an introduction/Bennett Chow, Dan Knopf. p.cm.-(Mathematical surveys and monographs, ISSN 0076-5376; v. 110) Includes bibliographical references and index. ISBN 0-8218-3515-7 (alk. paper) 1. Global differential geometry. 2. Ricci flow. 3. Riemannian manifolds. I. Knopf, Dan, 1959-II. Title. III. Mathematical surveys and monographs; no. 110. QA670.C46 2004 516.3 / 62-dc22 2004046148 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given.

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.

We show that the analog of Hamilton's Ricci flow in the combinatorial setting produces solutions which converge exponentially fast to Thurston's circle packing on surfaces. As a consequence, a new proof of Thurston's existence of circle packing theorem is obtained. As another consequence, Ricci flow suggests a new algorithm to find circle packings.) exists for all i, and (2) lim t→∞ r i (t) = r i (∞) ∈ R >0 exists for all i. A convergent solution is called convergent exponentially fast if there are positive constants c 1 , c 2 so that for all time t ≥ 0,andGiven any subset I ⊂ V , let F I be the set of all cells in T whose vertices are in I and let the link of I, denoted by Lk(I), be the set of pairs (e, v) of an edge e and a vertex v so that (1) the end points of e are not in I and (2) the vertex v is in I and (3) e and v form a triangle.

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