1547845447-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_IV__Chow_

136 30. TYPE I SINGULARITIES AND ANCIENT SOLUTIONS

leads to the well-definedness and monotonicity of the reduced volume at the singular

time.

LEMMA 30.4 (Uniform upper bound for L i on compact sets). Under the Cheeger-

Gromov convergence set-up in the previous subsection, assume the Type A condition

that there exists M < oo and r E [1, ~) such that

(30.8)

M

IRmg,I (x, t) :::; ( r

Wi - t

for all (x, t) E Mi x [a 00 , wi) and i.^1 Fix to E (a 00 , w 00 ).

(1) !fr> 1, then for all (q,t) E M 00 x [to, w 00 ) we have

(30.9) Li (q, t) :::; 2 exp ( ~':_"[ ( w"'2-f )-(r-l)) · Jw 00 - a 00 d;=~~ ~':

00

)

n^2 M ( _);i,- r

+ -3-- Woo - t 2

2-r

for i sufficiently large.

(2) !fr= 1, then for all (q,l) E M 00 x [to,w 00 ) we have

(30.10) L-·( - ) 2 (2(woo-to))

2

i q, t :::; nM. / V Woo _ aoo d;= (to)(q-~~~-,poo)

Woo - t W 00 - t

+2n^2 M(w 00 -l)~.

A simple consequence is

COROLLARY 30 .5. Under the assumptions of Lemma 30.4, for any compact set

K C Moo, E: E ( 0, Woo -a 00 ], and to E ( a 00 , w 00 - c), there exists C < oo depending

on K , r::, and to such that for any q EK., and any l E [to, w 00 - r::J, we have

(30.11)

for all i suffici ently large.

PROOF OF LEMMA 30.4. We sh all prove the upper bound for L i by considering
elementary "test paths" for the £-length of gi (t). Let q E M 00 and l E [t 0 , w 00 )
and define E: ::;:: w 00 - l. Throughout this proof, we shall assume that i is sufficiently
large and that

c -

Wi > Woo - 2 > t = W 00 - c, and q E Ui·

STEP 1. Bounds for the ev olving m etric. Let do denote the Riemannian dista nce
with respect to g 00 (t 0 ). By assumption (30.8), we have

(30.12)

nM

1Rc 9 , I (x, t) :::; ( r

Wi -t

(^1) 0bserve that (30.8) implies that

IRm 9 = I (x, t) ~ ( Woo M - t r