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
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