x PREFACE derivative estimates established in this chapter enable one to prove the long- time existence theorem for the flow, which states that a unique solution to the Ricci flow exists as long as its curvature remains bounded. In Chapter 8, we begin the analysis of singularity models by discussing the general procedures for obtaining limits of dilations about a singular- ity. This involves taking a sequence of points and times approaching the singularity, dilating in space and time, and translating in time to obtain a sequence of solutions. Depending on the type of singularity, slightly different methods must be employed to find suitable sequences of points and times so that one obtains a singularity model which can yield information about the geometry of the original solution near the singularity just prior to its formation. In Chapter 9, we consider Type I singularities, those where the curvature is bounded proportionally to the inverse of the time remaining until the sin- gularity time. This is the parabolically natural rate of singularity formation for the flow. In this case, one sees the R x 5 2 cylinder (or its quotients) as singularity models. The results in this chapter provide additional rigor to the intuition behind neckpinches. In the two appendices, we provide elementary background for tensor calculus and some comparison geometry. REMARK. At present (December 2003) there has been sustained ex- citement in the mathematical community over Perelman's recent ground- breaking progress on Hamilton's Ricci flow program intended to resolve Thurston's Geometrization Conjecture. Perelman's results allow one to view some of the material in Chapters 8 and 9 in a new light. We have retained that material in this volume for its independent interest, and hope to present Perelman's work in a subsequent volume. A guide for the hurried reader The reader wishing to develop a nontechnical appreciation of the Ricci flow program for 3-manifolds as efficiently as possible is advised to follow the fast track outlined below. In Chapter 1, read Section 1 for a brief introduction to the Geometriza- tion Conjecture. In Chapter 2, the most important examples are the cigar, the neckpinch and the degenerate neckpinch. Read the discussion of the cigar soliton in Section 2. Read the statements of the main results on neckpinches derived in Section 5. And read the heuristic discussion of degenerate neckpinches in Section 6. In Chapter 3, review the variation formulas derived in Section 1. In Chapter 4, read the proof of the first scalar maximum principle (The- orem 4.2) and at least the statements of the maximum principles that follow. In Chapter 5, review the entropy estimates derived in Section 8 and the differential Harnack estimates derived in Section 10. The entropy estimates
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