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