1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

(^148) 20. COMPACTNESS .OF THE SPACE OF 11:-SOLUTIONS

. First we prove a claim by contradiction. (This claim corresponds to
(20.41) in our first proof of Theorem 20.9.)
CLAIM 20.16 (Volume noncollapsing only assuming curvature bounded

at center). There exists a constant co = co ( K,, n) > 0 such that

Vol,qk(o) B,qk(o)(xk, 1) ~co

for all k ~ 1.

Proof of Claim 20.16. Suppose that the claim is false. Then, for some

subsequence,

(20.43)

as k --+ 00. o: ,::Hnce l' lm 8 _,o+ Vol9k(o)B9k(o)(xk,s) 8 n = Wn, W h ere Wn. h lS t e VO^1 ume

of the unit Euclidean n-ball, for k large enough we can find 6k E (0, 1) such

that

Vol,qk(o) B,qk(o)(xk, 6k) 1

(20.44) Jn = 2Wn.

k

Since Vol,qk(o) B,qk(o) (xk, 1) ~ Vol,qk(O) B,qk(o) (xk, 6k), it follows from (20.43)

and (20.44) that

ask--+ oo.

Consider the blown-up sequence of solutions

(20.45) §k (t) ~ 6"k^2 9k (6~t).

Then (Mk, 9k ( t)) is a K,-solution with Harnack with

1

Vol,qk(o) B,qk(o)(xk, 1) = 2Wn.

For any r > 1 we have

Vol,qk(o) B,qk(O) (xk, 4r) ~ Vol,qk(O) B,qk(o) (xk, 1)

1 Wn ( )n

= 2Wn = 2 ( 4r r 4r.

Applying Proposition 20.6(b), with w = 2 (~;r and r 0 = 4r, to the ball

B,qk(o) (xk, 4r), we get for z E B,qk(O) (xk, r)

c(~) B(~)

(20.46) R_gk (z, 0) < 2(4r)n + 2(4rr

• 16r2 16r2To (~)
  2(4r)n

for some B = B(w) < oo, C = C(w) < oo, and To(w) > 0. Note that the

bound on the RHS of (20.46) depends only on rand n.
We can now apply Hamilton's Cheeger-Gromov-type compactness the-
orem (Corollary 3.18 in Part I) to get a convergent subsequence:

(B,qk(O) (xk,r) ,gk (t) ,xk)--+ (M~,§oo (t) ,x 00 )