1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

(^202) 22. TOOLS USED IN PROOF OF PSEUDOLOCALITY
by (22.67). Since e-x ::;::: 1 - x, this implies
(22.69) JM v h dμg(O) ::S -(31 ( 1 - l~~
2
) = -(31 ( 1 - l0~ 2 ) ,
which is slightly stronger than (22.63).
Now we establish the inclusion (22.68), i.e.,
(22.70) d 9 (i) (x, xo) + ~::::; b - 200nVt,

where t E [t - ~Q-^1 , t] satisfies (22.62). To prove (22.70), we shall apply

the changing distances estimate on the time interval [t, t] to the inequality

(22.71) dg(t) (x, xo) ::::; (2A + l)c:o.

It follows from (22.60) that if (x, t) E M x (0, t] is such that dg(t) (x, xo) ::::;

• 1
  dg(t) (x, x 0 ) + AQ-2, then

I Rm I ( x, t) ::::; max { ~, 4Q}.

Moreover, if t E [t, t], then by (22.61) we have t::;::: t - ~Q-^1 > ~Q-1; that

is,

a -

t <2Q.

Hence, if (x, t) EM x [t, t] is such that

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

then

1 -

n-l Rc(x,t)::::; IRml(x,t)::::; 4Q.

Now by the changing distances Theorem 18.7(2), if Rc(x, t)::::; 4(n-l)Q

for all x E Bg(t) ( xo, Q-~) U Bg(t) ( x, Q-~), then

8 -1

(22.72) otdg(t)(x,x 0 ) 2::-lO(n-l)Q2.

Hence, as long as

(22.73)

so that Bg(t) ( xo, Q-~) u Bg(t) ( x, Q-~) c Bg(t) ( xo, dg(t) (x, x 0 ) + AQ-~),

we have (22.72).
On the other hand, suppose t' E (t, t] is such that (22.73) holds on the
interval [t', t]. Then by integrating (22.72) in time, we have

(22.74) d 9 (t')(x,xo)::::; dg(t)(x,xo) + lO(n-l)Q~ (t-t').