1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

1. A POINT PICKING METHOD 189

for all (x, t) such that

(22.19) t-~Q-^1 ::; t::; t and dg(f)(x,x)::; ~Q-~,

where Q ~I Rm l(x,[).

REMARK 22.7. The following proof of the claim also easily implies (ex-
ercise) that
I Rm l(x, t) :S 41 Rm l(x, t)

for all (x, t) such that t -~Q-^1 :St :St and dg(t) (x, x) :S A(J-^1 /^2 (where we
use dg(t) instead of dg(f))·

We shall use the following in the proof of the claim.

LEMMA 22.8. If (x, t) satisfies (22.19) and (x, t) E Mau then

(22.20)

Bg(f> (x, ~ 1 ) x [r -
2

a.Q_, rl c LJ Bg(t) (xo, dg(f> (x, x 0 ) + ~ 1 ). x { t}.

lOQ 2 J tE(O,t] Q 2

PROOF OF CLAIM 2. By definition, (x, t) E Ma implies Q > a.[-l and
hence t - ~Q-^1 > !f. Suppose that (x, t) satisfies (22.19).

(1) If (x, t) tf_ Ma, then

(22.21) I Rm l(x, t) :S a.r^1 :Sa (t - ~CJ-^1 )-

1

< 2a.r^1 < 21 Rm l(x, t),

with the second inequality following from t-~Q-^1 :St and the last inequal-
ity following from (x, t) E Ma. Thus we have proved that Claim 2 holds
for those points (x, t) tf_ Ma·

(2) Now suppose that (x, t) E Ma. By Lemma 22.8, we have that if

(x, t) E Man (ag(f) ( x, ~ (J-^1 /^2 ) x [t -~Q-^1 ,t]) ,

then (x, t) satisfies condition (22.9) and we may apply Claim 1 to conclude

I Rm l(x, t) :S 41 Rm l(x,[).

This completes the proof ·Of Claim 2 modulo establishing the lemma. D

PROOF OF LEMMA 22.8. Suppose that t E [t - ~Q-^1 , t] and x E M

with dg(f) (x, x) ::; l 0 AQ-^112 '. We need to show that

• 1
  (22.22) dg(t) (x, x 0 ) :S dg(f) (x, xo) + AQ-2.

By the triangle inequality, we have

1 - 1

dg(f)(x,x 0 ) :S dg(f)(x,xo) +

10

AQ-2.

Therefore at t = t we have the compact containment

Bg(f) - ( x, 10 1 AQ-2 - 1) c Bg(f) ( x 0 , dg(f) (x, x 0 ) + 10 9 AQ-2 - 1).