1547671870-The_Ricci_Flow__Chow

CHAPTER 6

Three-manifolds of positive Ricci curvature

The topic of this chapter is Hamilton's application of the Ricci flow to the
classification of closed 3-manifolds of positive Ricci curvature, in particular
his landmark result that any such manifold is diffeomorphic to a spherical
space form. Let (Mn, g) be a closed Riemannian n-manifold with positive
Ricci ~vature. By Myers' Theorem, Mn is compact. Since its universal
cover Mn also has positive Ricci curvature, it too is compact; and hence the
fundamental group of Mn is finite. When n = 3, we have the well-known

CONJECTURE 6.1 (Poincare Conjecture). Any simply connected closed

smooth 3-manifold is diffeomorphic to 53.

A successful resolution of the Poincare Conjecture implies that M3 is

diffeomorphic to 53. Furthermore, there is also the following

CONJECTURE 6.2 (Spherical Space Form Conjecture). Any finite group

of diffeomorphisms acting freely on 53 is conjugate to a group of isometries

of the standard sphere.

In the presence of a complete proof of the Spherical Space Form Conjec-

ture, the diffeomorphism .M3 ;:.::::; 53 would imply that M^3 is diffeomorphic to

53 /r, where r is a finite subgroup of 0 ( 4). In other words, one could con-

clude that M^3 is diffeomorphic to a space form, all of which are classified.

[126] As a consequence, M^3 would admit a metric of constant positive sec-

tional curvature. It is this last statement - the existence of a constant pos-

itive sectional curvature metric on any closed 3-manifold with positive Ricci

curvature - that Hamilton proved in 1982 using the then newly-introduced

Ricci flow. Our goal in this chapter is to present his result:

THEOREM 6.3 (Hamilton). Let (M^3 ,go) be a closed Riemannian 3-

manifold of positive Ricci curvature. Then a unique solution g (t) of the

normalized Ricci flow with g (0) = go exists for all positive time; and as

t-----+ oo, the metrics g(t) converge exponentially fast in every cm-norm to a

metric g 00 of constant positive sectional curvature.

Although the result is stated for the normalized flow, it will be more

convenient to work with the unnormalized flow for as long as possible. The

two equations differ only by rescaling space and time. (See Subsection 9.1.)

In fact, the theorem may be interpreted as a statement of the convergence

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