1547845447-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_IV__Chow_

l. REDUCED DISTANCE OF TYPE A SOLUTIONS 137

on Mix [a 00 ,wi)· Given any (p,t) E M 00 x [to,w 00 ) and any nonzero tangent

vector VE TpM 00 , by the Ricci flow equation and (30.12) we have

(30.13)

l

ln gi(t) (V, V) I = lln 9i (t) ((<1) V, (<Pi) V) I

gi(to) (V, V) 9i(to) ((<Di)* V, (<Pi)* V)

::::; 2 rt !Reg, I (<Di (p)' s) ds

lto

< r-1 i '

{

(^2) nM (w· - t)-(r-l) if r > 1

- 2nM ln w;-to Wi-t if r = 1.

Since gi(to) --* 900 (to) and since wi --* w 00 , we conclude from (30.13) that for

any point (p, t) in any compact subset K of M 00 x [to, w 00 ) we have

(30.14) K

1

(t) 900 (p, to) ::::; .Yi (p, t) ::::; K (t) 900 (p, to),

where

(30.15)

. { exp ( ~n_A[ (woo - t)-(r-l)) if r > 1,

J{ (t) =;= ( )2nM

~ W oo-t l "f T -1 -.

STEP 2. Test path for the .C-len9th. Recall that (q, l) EM x [to, w 00 ). By the

definition of the L-distance of 9i ( t), we have

(30.16) Li (q, l) =Li (<Pi(q), l)

::::; .c 9 i (I)

= 1Wi VWi - t ( Rg, (I (t)) t) + h'' (t)l~;(t)) dt

for any piecewise smooth path"(: [f,wi]--* Mi with 'Y (wi) =Pi and 'Y(f) = <I?i(q).

Now choose 'Y in (30.16) to be the "test path" uniquely defined so that

'YI [w 00 s 2' w·J 1. = Pi and so that 'YI [f l w 00 £ ] 2 ~ 'f) is a constant speed minimal geodesic

from <I?i(q) to Pi with respect to (<Pi^1 )*9 00 (t 0 ). Using the definition of Cheeger-
Gromov convergence, we see that such a minimal geodesic exists. We then h ave

Li (q, l)::::; 1 w=-~ VWi - t ( Rg, (rJ (t), t) + J{ (t) lrJ' (t)l~;-l)*goo(to)) dt

+ 1 ww, _£ VWi - t R 9 , (pi, t) dt

00 2

by (30.16) and by the upper bound in (30.14).

Since

' ( )I - dcq,;-1)*goo(to) (<I?i(q), pi)

'f) t (<I>;-^1 )*goo(to) - £

2

2dg 00 (to) (q,poo)

Woo -f