1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

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

where a < 1 is a small positive constant to be chosen later. On the other

hand, by (20.34) and fJR-

8 :k (x, t) ;:::^0 we have the curvature bound

(20.36) R-gk (y, t) ::; C1R9k (zk, 0) = C1rJ;^2

in B-gk(o) (zk, 2rk) x [-2r~, OJ. This and Rc-gk ::; R-gk imply

d~ Bk ( -ark 2) (xk, Zk) ::; e^2 a^01 d~ 9 k (O) (xk, Zk) = e^2 aGir~.

Hence by (20.22) and (20.35),

that is,

(20.37)

Recall that Shi's local derivative estimate says the following (see Theo-

rem 14.14 in Part II). For any a, K, r, n, and m EN there exists a constant

C = C (a, m, n) < oo depending only on a, m, and n such that if Mn is
a manifold, p E M, and g (t), t E [O, T], 0 < T ::; a/ K, is a solution to
the Ricci fl.ow on an open neighborhood U of p containing. Bg(O) (p, r) as a
compact subset, and if
IRm (x, t)I ::; K for all x EU and t E [O, T],

then

(20.38) I m ( )I Cmax{K,r-

(^2) }
V' Rm y, t. ::; tm/2

for ally E B 9 (o) (p,min{r,K-^112 } /2) and t E (0,T]. Note that the depen-

dence of the constant on the RHS of (20.38) is more explicit in terms of K
and r than in Theorem 14.14 in Part II; however, this dependence follows
easily from the latter result for K = 1 and r = 1 and from rescaling the
solution (in space and time) by max { K, r-^2 }.
Since by (20.36) we have the bound

IRm-gk I :S C1R9k (zk, 0) = C1rJ;^2

for the curvature of 9k (t) in B-gk(o) (zki 2rk) x [-2r~, OJ, we may apply Shi's
local derivative estimate to obtain^20

/V';,: RmYk/ (y,t)::; C(C1,m,n)rJ;m-^2

in B 9 k(- 2 r~) (zk, rk) x [-r~, OJ. In particular,

tiR-gk (y, t) :SC (C1, n) rJ;^4 = C (C1, n) R 9 k (zk, 0)^2