1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

3. SPACE-TIME POINT PICKING WITH RESTRICTIONS 59

Since A:::; Ai (CT) :::; 23 ~ 4 ( C + B(ti - to)-^1 ), by (18.37) we obtain

1 4 1

(18.38) dh(tk) (yk, p) < 4 + V2304 3

Since nonnegative Ricci curvature implies that distances are nonincreasing

in time under Ricci fl.ow and since tk :::; t 1 :::; 0, we have

1

(18.39) dh(O) (yk,P) < 3·

We estimate

k-1

0 :::; ti - tk :::; I: (ti - ti+i)

i=l

k-1 k-1

:::; I: AR-^1 (Yi, ti) :::; I: A ( C + B(t1 - to)-^1 )-^121 -i

i=l i=l

(18.40)

where the third inequality follows from (18.33) and the fourth inequality

follows from (18.36).

Since A:::; Ai(CT):::; ~ (C + B(ti - to)-^1 ) (ti - to), from (18.40) we ob-

tain

that is,

(18.41)

Since R (Yk, tk) -+ oo if a= oo and since (yk, tk) E Bh(O) (p, i) x [to, ti],^11

which is a set on which R (y, t) is bounded, we must have a< oo.

STEP 2. (y*, t*) = (Yai ta) is the desired point of the lemma.

By Step 1 we have that (18.29) holds for (y*, t*) = (Ya, ta) and all

(y, t) E NB,o satisfying (18.30). On the other hand, (18.41) and (18.38)

imply that

ta - to 2 (1 - CT) (ti - to) and Ya E Bh(ta) (p, ~).

Therefore the lemma holds for (y, t) = (Ya, ta)· D

The point (y*, t*) in Lemma 18.24 satisfies the following properties.

PROPOSITION 18.25 (Curvature control in a parabolic cylinder). Let

the hypotheses of Lemma 18.24 hold and let (y*, t*) be a space-time point

satisfying the conclusion of Lemma 18.24, so that

(i) t* E (to, ti],

(ii) t* -to 2 (1-CT) (t1 -to),

(iii) y* E Bh(t*) (p, i),

(^11) By (18.39) we have Yk E Bh(o) (p, ~).