1547845447-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_IV__Chow_

4. SOME RESULTS ON TYPE I ANCIENT SOLUTIONS 159

same reasoning used to obtain (30.96) (using Perelman's pseudolocality theorem),

we have

1Rm 9 ; 0 I (x, t) ~ (era)-^2 in B 9 ; 0 (-(e:ro)2) (p, era) x [-(era)^2 , 0),

which is equivalent to

1Rm 91 (x, t) ~ >..^10 2 in B ( _«ral2) (p, e:u 2 ) x [w - (e:;^0 l

2

,w).

(era) g w >-; 0 >-;o^10

This implies that Bg(w->.jo'(e:rol2) (p, Xj;/^12 era) c M - L:, which in turn implies

that p E M - L:. D

As a consequence, for Type I solutions, we have the following characterization

of whether a point is singular or nonsingular.

COROLLARY 30.26 (Distinguishing between singular points and nonsingular

points). Let (Mn,g(t)), t E [0,w), be a Type I singular solution on a closed mani-

fold. For each p E M, either

(1) liminft-.w(w-t)R(p,t) > 0 or

(2) there exists C < oo such that for every ti ---+ w and Pi ---+ p,

limsup IRml (pi, ti) ~ C.

i-Too

Another direct consequence of Theorem 30.25 is the following strengthening of

Theorem 30.22.

THEOREM 30.27 (Limits of Type I solutions with basepoints in L: are nonflat

shrinkers). Let (Mn,g(t)), t E [O,w), be a Type I singular solution on a closed

man if old. Suppose that p E M is such that there exist ti ---+ w and Pi ---+ p with

I Rm I (Pi, ti) ---+ oo (such a point always exists). Then for any ti ---+ w, the sequence

(M, gi (t) ,p) subconverges to a non-Ricci fiat shrinking GRS, where

gi (t) ~ IRml (p, ti) g (ti+ IRml-

1

(p, ti) t).

PROBLEM 30.28. For any Type A singular solution on a closed manifold, does

there exist a corresponding singularity model which is a nonflat shrinking GRS?

We are not aware of an example of a singular solution on a closed manifold which

does not have a nonflat shrinking GRS singularity model.

4. Some results on Type I ancient solutions

The general classification of K-noncollapsed ancient solutions in high dimensions

is a difficult problem. In this section we discuss some qualitative properties of Type

I ancient solutions, including shrinkers.

4.1. Limits of K-noncollapsed Type I ancient solutions.

Let (Mn,g(t)), t E (-oo,O), be an ancient solution to the Ricci flow. Let
Tt ')i 0 and Ti-/" oo be sequences of positive numbers. Define the corresponding
forward ( +) and backward ( - ) rescaled solutions of the Ricci flow by

(30.101) gi ± (t) =;:::. ±^1 g ( Ti ± ) t , t E (-oo, 0),

Ti

where g"f denotes either g°t or gi (and similarly for Ti±). We shall prove a result
regarding the limits of g"f (t).