1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

4. BACKWARD LIMIT OF ANCIENT i.;-SOLUTION IS A SHRINKER 411

[A-^1 , A] for some subsequence. Since c: is arbitrary, (i) then follows from

a diagonalization argument. The convergence of the pulled-back reduced

distance functions .ei(~i (q), B) is what we mean by .ei(q, B) -+ .e 00 (q, B) on

M 00 x [ A-^1 , A J in the Cheeger-Gromov sense.

(ii) It is clear that .e 00 (q, B) is a locally Lipschitz function on M 00 x
[A-^1 , A]. By Rademacher's Theorem (Lemma 7.110) for locally Lipschitz
functions, we know \7 g 00 (e)f 00 (q, B) and abe (q, B) exist a.e. on M 00 x [A-^1 , A].
D

In the next lemma we show that the equality 2g; = R - l\7fl^2 - ~ is

preserved under the limit.

LEMMA 8.37 (Properties of the limit of the reduced distance functions).

(i) We have

(8.47)

(ii) Foranysmoothcompactlysupportedcp(q,B) 2:: 0 onM 00 x(A-^1 ,A),

we have

(8.48) {A { ( ab'f +\7g;:foo·\7g=I./) )e-.e=(q,fJ)cpdμgoo(e)(q)dB 2:: 0.

JA-^1 }Moo -Rgoo+w

PROOF. (i) Equation (7.94) tells us that 2g; + 1Vfl^2 - R + ~ = 0. By
scaling, we have

a.ei

1

2 .ei

2 ae + IV g,,.i (e).ei - Rg,,.i (e) + e = o.

It suffices to prove that.

a.ei ( ( ) ) a.eoo ( )

ae ~ i q ' e -+ ae q' e

a.e. on Moo x (A-^1 ,A). Applying Lemma 7.63 (with T = oo) to .ei (q,B) =
.e (q, TiB) and using the scale-invariance of (Hess(q, 7 ) .e) (Y, Y), we get for

e E [A-^1 , A] that

(

nfi(q, B) 1) 2

(HesS(q,l'J) .ei) (Y (B) 'y (B)) :::; e + 2B IY (B)lg,,.i(e)

:::; A ( no-^1 + ~) IY (B)l~,,.i(e).

Since gTi(~i (q), B)-+ g 00 (q, B) in the C^00 -norm on any compact subset

KC M 00 x (A-^1 , A), the Hessian Hessg 00 (e) .ei (~i (q), B) on K is uniformly

bounded. From the discussion in Section 9 of Chapter 7, for any q E Moo,