Preface

This book and its planned sequel(s) are intended to compose an introduc-

tion to the Ricci flow in general and in particular to the program originated

by Hamilton to apply the Ricci flow to approach Thurston's Geometrization

Conjecture. The Ricci flow is the geometric evolution equation in which one

starts with a smooth Riemannian manifold

{Mn,go)

and evolves its metric

by the equation

| , = -2Rc,

where Re denotes the Ricci tensor of the metric g. The Ricci flow was

introduced in Hamilton's seminal 1982 paper, "Three-manifolds with pos-

itive Ricci curvature". In this paper, closed 3-manifolds of positive Ricci

curvature are topologically classified as spherical space forms. Until that

time, most results relating the curvature of a 3-manifold to its topology in-

volved the influence of curvature on the fundamental group. For example,

Gromov-Lawson and Schoen-Yau classified 3-manifolds with positive scalar

curvature essentially up to homotopy. Among the many results relating cur-

vature and topology that hold in arbitrary dimensions are Myers' Theorem

and the Cartan-Hadamard Theorem.

A large number of innovations that originated in Hamilton's 1982 and

subsequent papers have had a profound influence on modern geometric anal-

ysis. Here we mention just a few. Hamilton's introduction of a nonlinear

heat-type equation for metrics, the Ricci flow, was motivated by the 1964

harmonic heat flow introduced by Eells and Sampson. This led to the re-

newed study by Huisken, Ecker, and many others of the mean curvature

flow originally studied by Brakke in 1977. One of the techniques that has

dominated Hamilton's work is the use of the maximum principle, both for

functions and for tensors. This technique has been applied to control vari-

ous geometric quantities associated to the metric under the Ricci flow. For

example, the so-called pinching estimate for 3-manifolds with positive cur-

vature shows that the eigenvalues of the Ricci tensor approach each other

as the curvature becomes large. (This result is useful because the Ricci

flow shrinks manifolds with positive Ricci curvature, which tends to make

the curvature larger.) Another curvature estimate, due independently to

Ivey and Hamilton, shows that the singularity models that form in dimen-

sion three necessarily have nonnegative sectional curvature. (A singularity

V l l