1547845447-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_IV__Chow_

162 30. TYPE I SINGULARITIES AND ANCIENT SOLUTIONS

By Corollary 30.11, the reduced distance functions

±

C(~o,o) : M x (-oo, 0) ---+ JR

for gf (t) based at the singular time 0 exist. Moreover, Proposition 30. 18 implies

±

that each function C(~o,O) ( <I>T° ( ·), ·) is locally bounded and locally Lipschitz, uni-

formly in i. Thus, by the Arzela- Ascoli theorem and (30.70), they subconverge to

a locally Lipschitz continuous function

C~: M~ x (-oo,O)---+ JR

satisfying

d~± (t) (q, x~) ± Cd~± (t) (q, x~)

00 -Ct -C-5:.Coo(q,t)-5:. oo -t +C

on M~ x ( -oo, 0) for some C < oo.

We define the reduced volume corresponding to £~ (based at the singular time

0) by

V~ (t) ~JM~ (-47rt)-n/2 e-£!(t)dμg~(t)

fort E (-oo, 0). As in the proof of Proposition 30.19 (see (30.92)), we have that

(30.108) V~ (t) = .lim ii,(g'{-O) (t) = ,lim ii,(g O) (Ti±t) := c±

t - HX> XQ' t --+OO XQ'

fort E (-oo, 0) and for some constants c± E [O, l]. From this we then obtain that

(30.109) Re 900 ± (t) + v^9 !<tl v^9 !<t)g± 00 (t) + ~2t 900 ± (t) = o· '

i.e., g~ (t) is a gradient shrinker.

Furthermore, by (30.109) and (30.82), we have

(30.110)

(30.111)

which imply

(30.112) Rg~(t) + I !Coo ± (t)^12 + £~ -t-(t) = 0,

(30.113) t::.g~(t)Coo ± (t) - I !Coo ± (t)^12 + c~ (t) -t - ~ = 0.

By (7.94) in Part I, we have

ac~ I ± 12 c~ (t)

28t (t) - \1£ 00 (t) + R 9 ~(t) + -t-= 0.

Combining this with (30.112), we obtain^8 ~f = l\1£~1

2

. By Lemma 30.29, we

conclude that the shrinkers (M~,g~ (t) ,£~ (t), t) are in canonical form. D

Regarding the backwards limit in Theorem 30.31, we have the following general
result for shrinkers. The idea is that we have good control backwards in time of
the 1-parameter family of diffeomorphisms associated to shrinkers.