1547845447-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_IV__Chow_

3. TYPE I SOLUTIONS HAVE SHRINKER SINGULARITY MODELS 155

PROPOSITION 30.19 (Limits of Type I solutions with fixed basepoints are (pos-

sibly flat) shrinkers). L et (Mn,g (t)), t E [0,w), be a Type I singular solution

on a closed manifold. Then for any p E M and for any Ai --+ oo, the limit

(M~, g= (t) ,p 00 ) in (30.85) with Pi= pis a complete shrinking GRS with bounded

curvature.

PROOF. First let Pi EM be any sequence. By Corollary 30.11, for each i there

exists a reduced distance function e~,,w and the reduced volume

Vl,,w (t) ~JM (47r (w - t))-n/2 e-e~,,w(x,t)dμg(t) (x)

based at the singular time for the original solution g ( t), t E [O, w). By Corollary

30.14(1)- (2), we have that

(30.87)

d - -

dt Vl,,w (t) ;::: 0 and Vl,,w (t) ::::; l.

The reduced distance function £i ~ e~:,o and the reduced volume i/i ~ v::,o

based at the singular time for the rescaled solution gi (t), t E [-.Aiw, 0), are related

to e~,,w and iit,,w by

(30.88)

and

(30.89)

respectively.

Vi (t) = Vl,,w (w + >-i^1 t),

Now Proposition 30.18 implies that each £i is locally bounded and locally Lip-
schitz, uniformly in i. Hence, from the Arzela- Ascoli theorem, by passing to a
subsequence we have that

exists and is locally Lipschitz on M 00 x (-oo, 0), where the <I>i are as in (30.86).

By (30.88) and (30.70), we h ave that

d~(w+.Xilt) (<T>i (q) ,pi) Cd~(w+..Xi't) (<I>i (q) , pi)

(30.90) -CX:- 1 t - C::::; £i(i(q), t)::::; ->.:-it + C.

i i

d~ (t) (q,poo) C 0 ( ) Cd~oo(t) (q,poo) C

(^30. 91)^00 -Ct - < - {, 00 q ' t -< -t + '

and for i large enough,

1 ;::: Vi (t) = Vi_w (w + .>.;^1 t)

is defined and nondecreasing in i. Define the reduced volume corresponding to £ 00

by

v= (t) ~ { (-47rt)-n; 2 e-e=<x,t)dμgoo(t) (x)

}Moo

for t E (-oo, 0). Similarly to the proof of Claim 1 in the proof of Theorem

30.13(2), we have on compact time intervals that the reduced volume integrands