1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

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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)
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