1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

136 20. COMPACTNESS OF THE SPACE OF t;;-SOLUTIONS

It follows from the contradiction assumption lt1I :S To (w) that we have

2 (n - 1) ( C (~;w) ltil^312 + (B (~~nw) + 160) ltil^112 )

< 2 (n - 1) (C (5-nw) lt1l1/2 + (B (5-nw) + 160) lt1l1/2)

- 180 60

< 2 (n -1) (C (5-nw) + B (5-nw) + 160) To (w)l/2

• 180 60
  1
  160
  Hence we have
  1 37 19

d 9 (t 1 )(xo,Yo) :S d 9 (t 2 )(xo,Yo) + l60 = 160 <so·

This contradicts the choice of Yo, i.e., (20.18). We have proved t1 :S -To (w)

when t1 > -1.

STEP 3. Finishing the proof of the proposition.

Since ro (w) < 1, we have proved that

19

dg(t) (y, xo) .::::; 80

for ally E B 9 (o) (xo, ~)and t E [-ro (w) ,OJ. From (20.17) we have

Vol 9 (t) Bg(t)(xo, 1) ~ 5-nw

fort E [-ro (w), O]. Now that we have volume noncollapsing on a time inter-

val instead of at just one time, Proposition 20.6(b) follows from Proposition

20.6(a).

3. The compactness of K--solutions

An important step in understanding the high curvature regions of finite

time singular solutions is to understand the compactness, modulo scaling, of

K--solutions. That is, given a sequence of K--solutions and a sequence of space-

time points in this sequence, one would like to obtain a limit of rescalings

which is a K--solution. Remarkably, in dimension 3, this is possible. That

is, the collection of 3-dimensional K--solutions is precompact modulo scaling

(see Corollary 20.10 below).

3.1. Statement of the main theorem.

We say that a collection of ancient solutions of Ricci flow with positive

scalar curvature is precompact modulo scaling if for any sequence of

solutions (M'k,gk (t)), t :S 0, in the collection, for any points Xk E Mk and

for any times tk.::::; 0 the following is true. A subsequence of (M'k, 9k (t), xk),
t :S 0, where

(20.21)