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PREFACE

model is a solution of the flow that arises as the limit of a sequence of dila-

tions of an original solution approaching a singularity.) Since the underlying

manifolds of such limit solutions are topologically simple, a detailed analy-

sis of the singularities which arise in dimension three is therefore possible.

For example, showing that certain singularity models are shrinking cylin-

ders is one of the cornerstones for enabling geometric-topological surgeries

to be performed on singular solutions to the Ricci flow on 3-manifolds. The

Li-Yau-Hamilton-type differential Harnack quantity is another innovation;

this yields an a priori estimate for a certain expression involving the cur-

vature and its first and second derivatives in space. This estimate allows

one to compare a solution at different points and times. In particular, it

shows that in the presence of a nonnegative curvature operator, the scalar

curvature does not decrease too fast. Another consequence of Hamilton's

differential Harnack estimate is that slowly forming singularities (those in

which the curvature of the original solution grows more quickly than the

parabolically natural rate) lead to singularity models that are stationary

solutions.

The convergence theory of Cheeger and Gromov has had fundamental

consequences in Riemannian geometry. In the setting of the Ricci flow,

Gromov's compactness theorem may be improved to obtain C°° convergence

of a sequence of solutions to a smooth limit solution. For singular solutions,

Perelman's recent No Local Collapsing Theorem allows one to dilate about

sequences of points and times approaching the singularity time in such a way

that one can obtain a limit solution that exists infinitely far back in time.

The analysis of such limit solutions is important in Hamilton's singularity

theory.

A guide for the reader

The reader of this book is assumed to have a basic knowledge of Rie-

mannian geometry. Familiarity with algebraic topology and with nonlinear

second-order partial differential equations would also be helpful but is not

strictly necessary.

In Chapters 1 and 2, we begin the study of the Ricci flow by considering

special solutions which exhibit typical properties of the Ricci flow and which

guide our intuition. In the case of an initial homogeneous metric, the solution

remains homogenous so that its analysis reduces to the study of a system of

ODE. We study examples of such solutions in Chapter 1. Solutions which

exist on long time intervals such as those which exist since time — oo or until

time +oo are very special and can appear as singularity models. In Chapter

2, we both present such solutions explicitly and provide some intuition for

how they arise. An important example of a stationary solution is the so-

called "cigar soliton" on

R2;

intuitive solutions include the neckpinch and

degenerate neckpinch. In Chapter 2, we discuss degenerate neckpinches