1547845447-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_IV__Chow_

142 30. TYPE I SINGULARITIES AND ANCIENT SOLUTIONS

satisfying the following properties:

(i) ep,T is locally Lipschitz.

(ii) The inequality

(30.32)

holds both in the support sense and in the weak sense.

PROOF OF THEOREM 30.10. (i) There exists a subsequence such that ii con-

verges in Ck,c to a locally Lipschitz function £ 00 : M 00 x ( a 00 , w 00 ) --+ JR. Recall that

the Arzela- Ascoli theorem says that if we have a sequence of Lipschitz functions on

a compact metric space with Lipschitz constant L , then there exists a subsequence

which converges uniformly to a Lipschitz function with Lipschitz constant L. Thus

(i) follows from Lemmas 30.4, 30.7, and 30.8.

(ii) Inequality (30.30) holds in the weak sense. I.e., for any nonnegative C^2

function cp on M 00 x [t1, t2] with support in K x [t1, t2], where a 00 < ti < t2 < w 00

and K is compact, we have

(30.33)

ltJM,xJ\Jeoo, Vcp)+ (-O~~ + IV£ool

2

-R 900 + 2 (w2-t)) cp] dμ 900 dt ~ 0.

Let '!Ji~ <I?igi· By Lemma 7.129(i) in Part I , since the solutions gi (t) have bounded

sectional curvatures (depending on i) and by pulling back quantities on <J?i(Ui) c

Mi to Ui c M 00 , we have

(30.34)

l t

2

JM

00

[\vii, Vcp)

9

, + (-~~i + lviil;, -R 9 , + 2 (w~ _ t)) cp] dμ9,dt ~ 0

provided i is sufficiently large so that [t 1 , t2] c (ai, wi) and Kc Ui· Thus (30.33)

shall follow from the convergence of each term on the LHS of (30.34) as i--+ oo. We

achieve this in several steps.

STEP 1. Convergence of ftt,^2 J Moo ( -R9, + 2 (w7-t)) cp dμ9, dt. By the Cheeger-

Gromov convergence of gi (t) to g 00 (t), we clearly have

(30.35) lt2 !Moo (-R9, + 2 (w~ - t)) cpdμ9,dt

--+ lt2 !Moo (-Rgoo + 2 (w2 -t)) cp dμgoodt.

STEP 2. Convergence of J/ 12 JM 00 ((vii, Vcp) §, - !?Jt-cp) dμ9, dt. By Lemmas

30.7 and 30.8, there exists C < oo such that lliillwi,oo(Kx[t,,t 2 ]) ~ C, and hence

(30.36)