CHAPTER 30
Type I Singularities and Ancient Solutions
It' s a tad bit late.- From "Regulate" by Warren G featuring Nate Dogg
The classification of 3-dimensional singularity models is used to understand the
high-curvature regions of singular solutions on closed 3 -manifolds. In this chapter
we mainly study various properties of Type I singular solutions and Type I ancient
solutions in higher dimensions, where much less is known.
In §1 we consider a condition for singular solutions that is weaker than Type
I. We develop the tool of the reduced distance based at the singular time.
In §2 we prove the monotonicity of the reduced volume based at the singular
time for Type I singular solutions.
In §3, by using the aforementioned monotonicity, we show that for any Type I
singular solution, (1) there exists a nonflat shrinker singularity model and (2) the
solution must have unbounded scalar curvature.
In §4 we discuss some results about K:-noncollapsed Type I ancient solutions
using the ideas developed in the previous sections.
1. Reduced distance of Type A solutions
In the study of singular solutions it is helpful to impose restrictions on the
growth rates of their curvatures. In this section, under the Type I assumption,
and more generally under the so-called Type A assumption defined below, we shall
estimate the reduced distance function. This enables us to define the reduced
distance based at the singular time (as a limit) and to use it as a tool in singularity
analysis. In the next section we consider the reduced volume based at the singular
time.
Up to this date, there are more results about Type I solutions than Type II
solutions. Since Type II solutions lack curvature growth control, results about
them are generally more difficult to prove; they are also expected to be nongeneric
in some sense.
1.1. Derivative estimates for Type A singular solutions.
Generalizing the notion of a Type I singular solution, we have the following
condition, which is geared toward the study of the reduced distance function based
at the singular time.
DEFINITION 30 .1 (Type A singular solution). We say that a singular solutionof the Ricci flow (Mn, g (t)), t E (a, w), where -oo < a < w < oo, is Type A if
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