1547671870-The_Ricci_Flow__Chow

CHAPTER 9

Type I singularities

Suppose that (M^3 , g (t)) is a solution of the Ricci fl.ow which becomes

singular at some time T < oo. In order to extract geometric and topolog-
ical information about M^3 from the solution g (t), one wants to perform
geometric-topological surgeries on M^3 as the singularity develops in such
a way that the maximum curvature of the solution is reduced sufficiently.
In order to perform such surgeries, one needs a good understanding of the
possible singularities that can arise and of the possible limits of dilations
about them. Ultimately, one then wants to argue that only geometrically
recognizable pieces will remain after finitely many surgeries.
This is the program for proving the Geometrization Conjecture that was
designed by Hamilton and advanced by Perelman [105, 106, 107]. We will
further discuss the details of this program in a successor to this volume. In
this chapter, we will begin to explore some of the reasons why singularities
are expected to be topologically and geometrically tractable in dimension 3,
and why near a 'typical' singularity, one expects to see a neck: a piece of the
manifold which is geometrically close to a quotient of the shrinking round
product cylinder. This heuristic notion will be made precise in Section 4
below. (Recall that we made a rigorous study of neckpinch singularities
under certain symmetry assumptions in Section 5 of Chapter 2.)

1. Intuition

Singularities of the Ricci fl.ow (Mn,g(t)) in high dimensions are ex-

pected to be very complex. In dimension n = 3 however, there are three

observations that lead one to expect singularities to be relatively tractable.

The first observation is the pinching estimate of Theorem 9.4, below.

This estimate says that at any point and time where a sectional curvature

is negative and large in absolute value, one finds a much larger positive

sectional curvature. It implies in particular that any singularity model must

have nonnegative sectional curvature at t = 0.

The second observation which restricts the possible singularity models

one may see in dimension n = 3 is the fact that so (2) is the only proper

nontrivial Lie subalgebra of so (3). This fact says that at the origin of any

singularity model, the eigenvalues of the rescaled curvature operator must

conform to either the signature ( +, +, +) or else (0, 0, + ). The signature

(0 , +, + ) is ruled out by the Lie algebra structure of so (3). The other

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