1547671870-The_Ricci_Flow__Chow

CHAPTER 8

Singularities and the limits of their dilations

In this chapter, we shall survey the standard classification of maximal
solutions to the Ricci flow. From the perspective of this classification, one
regards a solution (Mn, g (t)) that exists up to a maximal time T ::::; oo as
becoming singular at T, because it cannot be extended further in time. So
each maximal solution may in this way be thought of as a singular solution.
We shall then study how singularities may be removed by dilations. In many
cases, one can pass to a limit flow, called a singularity model, whose proper-
ties can yield information about the geometry of the original manifold near
the singularity just prior to its formation. We discussed some examples of
singularity models in Chapter 2. Singularity models always exist on infinite
time intervals. In dimension n = 3, singularity models have other special
properties (in particular nonnegative sectional curvature) that imply that
they are simple topologically, and thus make it reasonable to expect that
one could obtain a classification of such limit solutions that is adequate to
derive geometric and topological conclusions about the underlying manifold
M^3 of the original singular solution g (t). In Chapter 9, we shall see some
of the reasons why singularity models in dimension three are so nice.

1. Classifying maximal solutions
   Consider a solution (Mn, g (t)) of the Ricci flow which exists on a maxi-

mal time interval [O, T), where TE (0, oo]. We call T the singularity time.

This divides solutions into two categories: those where T < oo and those

where T = oo. In both cases, there are further subdivisions into two types

of singularities.

To motivate these subdivisions, we look to the simplest solutions, those

which are the fixed points of the normalized Ricci flow. So let us consider

the evolution (Mn, g ( t)) of an Einstein metric by the Ricci flow. There are

three possibilities: the scalar curvature R may be positive, zero or negative.

If R (x , 0) > 0, it follows from the evolution equation

aaR = 6R+2 1 Rcl^2 = L\R+ ~R^2

t n

and the fact that R is constant in space that T < oo; in fact,

n

R(x,t)=2(T-t)

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