1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

410 8. APPLICATIONS OF THE REDUCED DISTANCE

For any c > 0, Lemma 8.35, after parabolic rescaling g ( T) by Ti, yields

the curvature bound 5-l = 8(n,E,A)-^1 for gri(()) on Bgri(l) (qri,JrI) x

[A-^1 ,AJ. Applying Lemma 8.35 with c = 1andA=2, we obtain

IRmgri (q, 1)1:::=;5-l = 8(n,1, 2)-^1 for q E Bgri(l) (qTi' 1).

Since g( B) is ~-noncollapsed on all scales, we have gri (B) is ~-noncollapsed on

Bgri(l) (qri' 1) and the injectivity radius estimate inj gri(l) (qrJ 2:: 81 (n, ~).

Now we can apply Hamilton's Cheeger-Gromov-type Compactness Theorem

3.10 to the sequence of solutions grJB) to the backward Ricci flow to get

(Mn,gri(B),qrJ---> (M~,goo(B),qoo) for() E [A-^1 ,A].

The limit is a complete solution to the backward Ricci flow. Since each

gri ( B) satisfies the trace Harnack estimate, the limit g 00 ( B) satisfies the trace

Harnack estimate. Note that g 00 (B) is ~-noncollapsed on all scales, has

nonnegative curvature operator, and satisfies inj g 00 (l) ( q 00 ) 2:: 81 ( n, ~).

To finish the proof of Theorem 8.32, we need to show that for each (),

g 00 ( ()) is a nonflat shrinking gradient Ricci soliton.

4.3. The limit of reduced length f,i (q, 0). Let f,i (q, 0) ~ gg'Ti (q, B),

() E [ A-^1 , A J , denote the reduced distance of the solution gri ( B) with respect

to the basepoint Po.

LEMMA 8.36 (Limit of the reduced distance).

(i) The limit

_lim fi ( q, 0) -:-foo ( q, B)

i-->oo

exists in the Cheeger-Gromov sense on M 00 x [A-1,A].^5

(ii) The limiU 00 (q, B) is a locally Lipschitz function on Moo x [A-1, A]

and \lg 00 (e)foo(q,()) and^0 b 0 (q,()) exist a.e. on M 00 x [A-^1 ,A].

PROOF. (i) Suppose g?i : ui c Moo--+ Vi~ g?i (Ui) c M are diffeomor-

phisms which yield the convergence (Mn, gri ( B), qrJ ---> ( M~, goo ( 0), qoo)

in the sense of Definition 3.6. Given any c > 0, by Lemma 8.35, (8.43),

(7.100), and the scaling property (8.40), we obtain for all i, () E [A-^1 ,A],

and q E Bg'Ti(l)(qri' Jrl) c M, the estimates

(8.46) o:::; ei(q, e) :::; 5-^1 ,

l\lg'Ti(e)fi(q,B)l;,.i(e):::; 38-^1 ,

l

aei( 8() q, ()) I :::; 28 -1.

Hence for i large enough, f,i (g?i (·), ·) is a sequence of uniformly Lipschitz

functions on Bg 00 (1) ( qoo, { 0 JrI) C M 00 • By the Arzela-Ascoli theorem

we get fi( g?i (q), B) --+ f 00 (q, 0) on the closed set Bgoo(l) ( q 00 , g.JrI) x

(^5) See the proof below for what we mean by convergence in the Oheeger-Gromov sense.