1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

142 20. COMPACTNESS OF THE SPACE OF 11:-SOLUTIONS

Claim. There exists a constant C1 < oo independent of k such that

(20.31) Rgk (y, O) < C for ally E B 9 k(o) (zk, 2rk).

R9k (zk, 0) -

1

To see the claim, we choose Yk E Mk such that

3

d9k(O) (xk, Yk) = 4rk.

Since Bgk(O) (yk, lrk) C Bgk(O) (xk, rk) -B 9 k(O) (xk, !rk), by (20.30) we have

Rgk (y, 0) < 4rk^2

for any

YE Bgk(O) (Yk, lrk) ·
This local curvature bound and the A;-solution assumption (in particular,
being A;-noncollapsed at all scales^16 ) imply that

Volgk(O) Bgk(O) (yk, lrk) :::::: A; arkf.

For any y E Bgk(o) (zk, 2rk), we have

Bgk(o)(Y, 4rk) =:l Bgk(o) (zk, 2rk) =:l Bgk(o) (xk, rk) =:l B9k(o) (Yk, lrk)

and hence^17

(20.32)

Now consider the rescaled metric

hk ~ Rgk (zk, 0) 9k (0) = rk^2 9k (0) ,

which satisfies Rhk (zk) = 1, together with the ball

Bhk (zk, 2) = Bgk(O) (zk, 2rk).

Now if the claimed local curvature bound (20.31) is not true, then there

exists a subsequence such that

sup Rhk (w)-+ oo

wEBhk (zk,2)

ask-+ oo.

Applying the point picking Lemma 18.15 to this sequence of metrics and

balls, we obtain sequences of points and radii

Wk E Bgk(O) (zk, 2rk) and Sk E (0, 3rk)

such that

(^16) Note that since Rm ::'.". 0, we have IRml :::; R.
(^17) To wit, by 'radii picking' (the Zk determine the radii d 9 k(o)(Xk,Zk)) we have cur-
vature control in balls (centered at Yk), which by K-noncollapsing imply volume (lower)
bounds for those balls, which in turn by containment imply volume bounds for larger balls
with centers not too far away (for which we do not yet have curvature control).