1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

4. GRADIENT AND TIME-DERIVATIVE OF THE L-DISTANCE FUNCTION 309

is a piecewise smooth path from p to q2. We estimate, using I~~ (T)l~(T) :S

e2CoT I da dT (T) (^12) g(O) ~ e2CoT ' that

L (q2, Tr+ do (q1, q2))

:::; ,C ('Y) + 1T1+do(q1,q2) VT (R (a (T) 'T) +I ~a (T)l2 ) dT

Tl T g(T)

:SL(q1,T1)+ 2 ( nCo + e^200 T) ( 3/2 3/2)

3

(T1+do(q1,q2)) -T 1

:SL (q1, Tr)+ C1do (qi, q2).

By Lemma 7.28 we have

L (q2, Tr) :SL (q2, Tr+ do (q1, q2)) + C1do (q1, q2)

:::; L (q1, Tr)+ C1do (q1, q2)

:SL (qi, Tr)+ C1dT 1 (q1, q2),

where we have used do (q1, q2) :S e^00 T dT 1 (q1, q2). By the symmetry between

qi and q2 we get

IL (q2, Tr) - L (q1, Tr) I :S C1dT 1 (q1, q2).

( 2) L ( q, ·) is locally Lipschitz in the time variable uniformly in q E

Bg(o)(qo,c:). For any Tr< T2 in (To-c:,To+c:), let')': [O,T1] -----t M be a

minimal £-geodesic from p to q and let {3 : [T1, T2] -----t M be the constant

path f3 ( T) = q. Then 'Y '-' {3 : [O, T2] -----t M is a piecewise smooth path from

p to q. Hence

L (q, T2) :S £ ('Y) + £ ((3) ~ £ ('Y) + 1T

2

JTR (q, T) dT

Tl

< - L ( q,Tl ) + -3-2nCo ( T2 3/2 -Tl 3/2)

:::; L ( q' Tr) + C1 ( T2 - Tl) '

where C 1 depends only on Co and T. Combining this with Lemma 7.28, we

obtain

where

(7.53)

D

COROLLARY 7.31. L is differentiable almost everywhere on M x (0, T)

and LE Vl!i~:i (M x (0, T)).

PROOF. See Lemmas 7.110 and 7.111. D