1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

60 18. GEOMETRIC TOOLS AND POINT PICKING METHODS

Then j

(18.42) R (y, t) :S 2R(y*, t*)

PROOF. ·we argue by contradiction. Suppose there exists

(18.43) (w,s) EBh(t*) (y*,

1

1

0

A^1 l^2 R-^112 (y,t)) x [t-DR-^1 (y,t),t]

satisfying

(18.44)

Then s satisfies

s - to 2: (t - to) - DR-^1 (y, t*)

D

(^2) (t - to) - C + B (t - to)- 1
t - to

-*--

• 2 '
  where we have used assumption (iv) and, in the last inequality, D :S C(t*-io)+B.
  Hence, using (iv) again, we have

Thus (w, s) E NB,c·

R (w, s) > 2R(y, t)

> 2 ( C + B (t* - to)-^1 )

2: C + B ( s - to )-^1.

It follows from D :S 10 ~~~l) <A that

s E (t -AR-^1 (y, t), t].

If we can show the claim that

(18.45) w E Bh(s) (p,dh(t) (y,p) +A^112 R-^112 (y,t)),

then we can apply Lemma 18.24 to get R (w, s) :S 2R(y*, t*), which is the

desired contradiction to (18.44).

Now we estimate the distance dh(s) (w,p). Let

(t$,t] c [t-DR-^1 (y,t),t*]