1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

306 7. THE REDUCED DISTANCE

Letting O'j ~ 2..Jij and /Ji (O') ~'Yi (0'^2 /4), the above formula says

1

0'2 Id/Ji 12 dO' < c.

0'1 dO' g( 0'2 I 4) -

From standard theory in the calculus of variations, we can conclude that

there exists a subsequence such that 'Yi converges to a path 'Yoo : [r1, r2] -+

M with [, ('"'! 00 ) = inLy [, ( i'). All £-length minimizing paths satisfy the £-

geodesic equation in the weak sense. By standard theory again, we have

that such paths are smooth. D

4. The gradient and time-derivative of the £-distance function

In this section (Mn, g ( r)) , r E [O, T] , shall again denote a complete

solution to the backward Ricci flow satisfying the pointwise curvature bound

max{IRml, IRcJ} ::::; Co < oo on M x [O,T], and p E M shall denote a

basepoint.

4.1. L :is locally Lipschitz. Before we study \7 L and ~~, we prove

that, as a consequence of Lemma 7.13, Lis locally Lipschitz. As is the case

with this and the following one-sided Lipschitz result, we shall prove effective

estimates. When we prove L is locally Lipschitz in the space variables, the

following one-sided Lipschitz property will be used.

LEMMA 7.28 (One-sided locally Lipschitz in time). Given 0 < ro < T,

let c ~ min { rn ' T 1;^0 ' lo } > 0. For any r1 < r2 in ( ro - c' ro + c) ' qo E M'

and q E Bg(O) (qo, c), we have

where

d;h) (p, q)

C1=C(n,T,Co,ro)+C(Co,T).

4r2

PROOF. Let'"'( : [O, r2] -+ M be a minimal £-geodesic from p to q. We
define the piecewise linearly reparametrized path 'I] : [O, r 1 ] -+ M by

where

</> ( r) ~ 2r + r2 - 2r1 2: r

for r E [2r1 - r2, r1]. Although '!] is only piecewise smooth, we still have
L (q, r1) ::::; [, ('!]). (We will use this fact a few times later.) Hence, since