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