1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

194 22. TOOLS USED IN PROOF OF PSEUDOLOCALITY

for T E ( 0, (.U]. By the pointed Cheeger---Gromov convergence of 9i ( T) to

g 00 (T), given any R > 0, there exists Co (R) < oo such that

Co (R)---^1 9i (x, 0) S 9i (x, T) S Co (R) 9i (x, 0)

for all x E B 9 i(o) (xi, R), TE (0, w], and i EN. Since

dμ 9 i(r) :2'. Co (R)---n/^2 dμ 9 i(O)

in B 9 i(O) (xi, R), we have by (22.35)

r Hi (x, T) dμgi(O) (x) s Co (Rt^12 r Hi (x, T) dμgi(r) (x)

h~~~~ h~~~~

(22.36) s Co (Rt/^2

for T E (0, w]. We may now obtain (22.33) from either (i) the mean value
inequality (Theorem 25.2) or (ii) the Li-Yau estimate.
We illustrate the proof using method (ii). By (22.36) we have that, for
each T E (0, w], there exists Xr E Bgi(o) (xi, 1) such that

Co (lt^12

Hi(x^7 , T) S ( )

Vol 9 ,(o) B 9 ,(o) Xi, 1

By the Li-Yau estimate (i.e., Corollary 25.13), for any

(x,T) E B 9 i(o) (xi,0-^1 ) x [o,w-o]

we have

c 8 (T + o) 2.:'3c: (Co (o-^1 ) d;i(O) (x,Xr+o))
Hi(x, T) s Hi(Xr+o, T + o)e^12 -T- exp 4(1 - s) 0

S Co (lt/ eC12822-ac: exp '

(^2) n (Co (0- (^1) ) (0- (^1) +1)^2 )
Vol 9 i(o) B 9 i(o) (xi, 1) 4(1-s) o
where C12 depends only n, o, s, and the local bounds on jRc 9 i(r) j, jV' R 9 i(r) j,

and b.Rgi(r) (and we may choose s = 1/2 for example); hence we obtain

(22.33). Regarding the use of Corollary 25.13 (and hence the use of Theorem
25.9), note the following:

(1) Let I : [T, T + o] --+ Mi be a minimal geodesic joining x to Xr+o)

with respect to 9i ( 0). Then / ( T^1 ) E B 9 i (O) (Xi, 5---^1 + 2) for each

T^1 E [T, T + o]. Hence 'Y ( T^1 ) E Bgi(r') (xi,~) for each T^1 E [T, T + o],

(

where Ri ~ C- ) 1/2

0 (0-^1 +2) (o-^1 +2).

(2) Since 9i ( T) is a solution to the backward Ricci flow on [O, w], by

Shi's local derivative estimate, IV' R 9 i(r) I and b.R 9 i(r) are bounded

in B 9 i(o) (xi, 2 Ri) x [O, w - o] only in terms of o and bounds on the

curvatures jRm 9 i(r) I in B 9 i(o) (xi, 2 Ri +1) x [O, w] (in particular, the

bounds are independent of i).

(3) Vol 9 i(O) Bgi(O) (xi, 1) ---+ Vol 900 (0) B 900 (0) (x 00 , 1) > 0 as i--+ oo.