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)