1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

250 23. HEAT KERNEL FOR STATIC METRICS

where t* E (t, t + h) also depends on x, y, s. We claim that by taking the

limit as h '\i 0, we have the fact that the right derivative is

a

a+ (PN G) (x, y, t) = lim JN (x, y, t, t + h) + rt aa JN (x, y, s, t) ds

t h '\,.O Jo t

(23.110) = lim r PN (x, z, t + h - t*) G (z, y, t*) dμ (z)

h'\,.O} M

+lat JM :tPN(x,z,t-s)G(z,y,s)dμ(z)ds

= G (x,y, t)

+lat JM gt PN (x, z, t - s) G (z, y, s) dμ (z) ds,

where. t* E ( t, t + h). (Exercise: Prove that we have the same formula for

a~-(PN * G) (x, y, t).)

By (23.79) we have

(23.111)

lat gt JN (x, y, s, t) ds =lat JM gt PN (x, z, t - s) G (z, y, s) dμ (z) ds,

which yields the second equality in (23.110). Since PN satisfies (23.34a), we

also have the third equality in (23.110). Thus we only need to justify the

first equality in (23.110), i.e.,

(23.112) lim rt

8

8

JNI (x, y, s, t) ds = rt

8

8

JN (x, y, s, t) ds

h'\i.O ) 0 t t=t* } 0 t

converges absolutely. First, fftJN (x, y, s, t) is uniformly continuous for s

away from t. We also have that for every c: > 0 there exists te < t such that

1: I :t JN ( x, y, s, t) ds I < c:.

Second, recall by (23.44) that we have

Thus

1gtJN(x,y,s,t)I = IJM :tPN(x,z,t-s)G(z,y,s)dμ(z)I

::; IJM ~xPN (x, z, t-s) G (z, y, s) dμ (z)I

+ C IJM (t-s)N-~ exp ( :~;~ :~) G (z, y, s) dμ (z)I

::;C(t-s)-a+c,