1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

140 20. COMPACTNESS OF THE SPACE OF 11;-SOLUTIONS

Hence, by Theorem 20.1, which says ASCR (gk (0)) = oo if Mk is noncom-

pact, we have that the manifold Mk is compact for all k.

We claim there exists a constant Co < oo independent of k such that

(20.23)

that is, we have a uniform curvature bound for all of the rescaled metrics

9k (0).

If not, then there would exist a subsequence {(Mk,9k (t))} and a corre-

sponding sequence of points {Yk E Mk} such that

(20.24)

as k --+ oo. Note that since A k ::::; 1, this implies

d~k(o)(xk,Yk)--+ 0

as k --+ oo. (Already the reader may sense that this is counterintuitive; we

shall confirm this by rescaling and taking a limit to obtain a contradiction.)

We rescale about the maximum curvature points so as to define

(20.25)

Then R9k (z, 0) :S R9k (yk, 0) = 1 for all z E Mk and k EN. So by the trace

Harnack estimate, which implies that fftR9k 2 0, we have

R 9 k (z, t) ::::; 1

for all z E Mk, t::::; 0, and k EN.

On the other hand, it follows from (20.22), i.e., R 9 k(xk,O)

(20.24) that

R9k (xki 0) --+ 0.

1, and

Since we are assuming Case (a), we have R-tik (yk, O)d~k(o) (xk, Yk) :S 1 and

hence by (20.25) we have the uniform distance bound

(20.26) d~k(O) (xki Yk) :'S 1.

For each k, since (Mh;,gk (t)) has nonnegative curvature operator and
is A;-noncollapsed at all scales, we have .that (Mh;,gk (t)) has uniformly
bounded (independent of k, by our rescaling) nonnegative curvature operator
and is A;-noncollapsed at all scales. Hence we may apply Hamilton's Cheeger-
Gromov-type compactness theorem to get a subsequence {(Mh;, 9k(t), Yk)}
which converges to a complete limit solution

(M~, 900 (t), Yoo)

with bounded nonnegative curvature operator and such that Xk --+ x 00 for
some x 00 E M 00 • Here we used (20.26) and passed to a subsequence.