1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

3. THE COMPACTNESS OF /;;-SOLUTIONS 141

The limit also satisfies R9= (y 00 , 0) = 1 and R9= (x 00 , 0) = 0. However

this contradicts the strong maximum principle (see Corollary 12.43 in Part

II) and hence we have proved

sup R-gk (z, 0) s Co

zEMk

for some Co < oo as claimed.

Since now we have a uniform curvature bound, i.e., (20.23), we may

apply Hamilton's Cheeger-Gromov-type compactness theorem to obtain a

subsequence {(Mk',gk (t) ,xk)} converging to a complete nonfl.at limit solu-

tion (M~, goo(t), x~) with bounded nonnegative curvature operator. We

have that (M 00 ,g 00 (t)) is /'\;-noncollapsed at all scales and satisfies the trace

Harnack estimate since these properties are preserved under C^00 Cheeger-

Gromov convergence. This finishes the proof of Case (a).

REMARK 20.14. To crystallize one of the ideas in Case (a), we note the
following. Suppose that (Mn, g) is a closed Riemannian manifold, x E M
with R (x) = 1, and R (y) d (y, x)^2 :; C for ally EM. Given w EM, define
the rescaled metric g = R ( w) g. Then

R(x) = R(w)-^1 ,

.R(w)=l,

d(w,x) = R(w)^112 d(w,x):; c^112.

Case (b). Ak > 1 for all k.

STEP 1. Point picking.

There exists Zk E Mk such that Zk is a point which is closest to Xk in

the nonempty set of all points z satisfying

(20.27) R-gk (z, O)d~k(O) (xk, z) = 1.

Let

(20.28)

Note that (20.27) implies

(20.29)

for all z E B-g k (o) (xk, rk) = B-gk(o) (xk, v' Rgk _ \ zk,o )). Note also that we have

bounded curvature in an annulus, i.e.,

(20.30)

for z E B-gk(o) (xk, rk) - B-gk(O) (xk, ~rk).
STEP 2. Relative curvature bound in balls centered at Zk and containing

B-gk(o) (xk, rk).