1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

190 22. TOOLS USED IN PROOF OF PSEUDOLOCALITY

Suppose that there exists ti > l - ~aQ-^1 which is the first time, going

backwards in time from l, such that the closed ball Bg(f)(x, l 0 AQ-~) inter-

sects the boundary of Bg(ti) (xo, dg(f) (x, xo) + 190 AQ-~); we shall show that

this leads to a contradiction. Let x* be such a point of intersection, so that

9 - 1

(22.23) dg(t 1 )(x*,x 0 ) = dg(f)(x,xo) +

10

AQ-2.

Note that since x* E Bg(f)(x, 1 ~AQ-~), we have

1 - 1

(22.24) dg(f)(x*, x 0 ) :::; dg(f)(x, xo) +

10

AQ-2.

By the choice of ti and x* we have dg(t) (x*, xo) :::; dg(f) (x, xo) + { 0 AQ-^1!^2

for any t E [ti, l]. Hence if dg(t) (x, x*) :::; l 0 AQ-^112 , then

so that

(22.25)

(22.26)

dg(t) (x, xo) :S dg(t) (x, x*) + dg(t) (x*, xo)

9 - 1 1 - 1

:::; dg(f)(x,x 0 ) +

10

AQ-2 +

10

AQ-2,

Bg(t) (x*, 1 ~AQ-~) c Bg(t)(xo,dg(f)(x,xo) + AQ-~),

Bg(t) (xo,

1

1

0

AQ-~) c Bg(t)(xo,dg(f)(x,xo) + AQ-~)

for t E [t1, l] (with the second inclusion being obvious).
Note that by combining (22.9) and (22.21) to cover the cases where

(x, t) E Ma and (x, t) tj. Ma, respectively, we have

I Rm I (x, t) :S 4Q in Bg(t) (xo, dg(t) (x, xo) + AQ-~) x [t -~Q-^1 , t].

In particular, by (22.25) and (22.26),

Rc(x, t):::; 4nQ for x E Bg(t) ( x*, 1

1

0

AQ-~) U Bg(t) ( xo,

1

1

0

AQ-~)

and t E [ti, l]. Hence we may apply Theorem 18.7(2) on how distances
change under the Ricci fl.ow (with K = 4nQ and r 0 = lOfoQ-^112 )^5 to
conclude that
d 104 -1
dt dg(t) (x*, xo) ~ -5 (n - 1) vfnQ2 fort E [t 1 , ~.

(^5) This implies
-2(n-l)(Kra+r 01 )

= -2 (n - 1) ( 4nQ

10

~Q-^1 /^2 + 10ynQ1/2)

= _ 1~4 (n _ l) ynQ1;2_