CHAPTER 1

The Ricci flow of special geometries

The Ricci flow

—5 = - 2 Re

or

g (o) = 50

and its cousin the normalized Ricci flow

dt n JMndn

5(0) = 50

are methods of evolving the metric of a Riemannian manifold

{Mn,

go) that

were introduced by Hamilton in [58]. They differ only by a rescaling of space

and time. Hamilton has crafted a well-developed program to use these flows

to resolve Thurston's Geometrization Conjecture for closed 3-manifolds. The

intent of this volume is to provide a comprehensive introduction to the foun-

dations of Hamilton's program. Perelman's recent ground-breaking work

[105, 106, 107] is aimed at completing that program.

Roughly speaking, Thurston's Geometrization Conjecture says that any

closed 3-manifold can be canonically decomposed into pieces in such a way

that each admits a unique homogeneous geometry. (See Section 1 below.)

As we will learn in the chapters that follow, one cannot in general expect

a solution

(Ms,g(t))

of the Ricci flow starting on an arbitrary closed 3-

manifold to converge to a complete locally homogeneous metric. Instead, one

must deduce topological and geometric properties of

M3

from the behavior

of g (t). Hamilton's program outlines a highly promising strategy to do so.

By way of an intuitive introduction to this strategy, this chapter ad-

dresses the following natural question:

If 9o ^ a* complete locally homogeneous metric, how will g(t) evolve?

The observations we collect in examining this question are intended to

help the reader develop a sense and intuition for the properties of the flow

in these special geometries. While knowledge of the Ricci flow's behavior

in homogeneous geometries does not appear necessary for understanding

its topological consequences, such knowledge is valuable for understanding

analytic aspects of the flow, particularly those related to collapse.

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http://dx.doi.org/10.1090/surv/110/01