1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

4. GRADIENT AND TIME-DERIVATIVE OF THE £-DISTANCE FUNCTION 307

(7.50)

Recall that (3 (O") is defined by (7.18). By Lemma 7.13(ii), (iii) there

exists T* E (0, T2) such that

(7.51)

Since T2 ~ T - c:, by Shi's derivative estimate we have

\V R (x, T)\ ~ C(n)Co ~ C 2

Jmin {T - T2, C 01 }

for any (x, T) EM x [O, T 2 ]. From equation (7.46), we have

_!!___ I d(J 12 < 3VrCo I d(J 12 + Ci r3/2.

dO" dO" g(u2/ 4 ) - dO" g(u2/ 4 ) 4Co

Integrating the above inequality over [O"*' O"], we get for all O" E [O, 2yrz],

(7.52)

I

dfJ 1

2

~ e3VTColu-u*l I d(J (O"*) 1

2

+ Ci~ e3VTColu-u*I.

dO" g(u2/ 4 ) dO" g(T*) 12C 0

Noting that <P-^1 (T) =^7272 +TI ~ T for TE [2T1 - T2, T2], we can estimate

using (7.52) and (7.51)