1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

226 23. HEAT KERNEL FOR STATIC METRICS

(iii) In the intermediate region (Minj(g) - Minj(g)/s) x IRt by (23.38)

and (23.11), we have

IDxPNI (x, t; y, u) :::=; IDxHNI + C IHNI + 2C l\7 HNI

(23.39)

(23.40)

:::; CE (x, t; y, u) (t - u)N +CE (x, t; y, u)

c

+--E(x,t;y,u)

t-u

$ C (4" (t - uW~ exp (-(i:j(;g2 ~)')ct - u)-^1

< ( inj (g)

2

)

_Cexp -257(t-u)

since d (x, y) ~ inj (g) /8 and 257 > 256. The RHS of (23.39), which is

independent of x and y, tends to zero as t \i u.

Thus we see, from considering all three cases (i)-(iii), that we may extend

DxPN continuously to M x M x IRS, so that it takes the value 0 on M x

M x 8 (IRS,).

Similarly, we may show that ( ..6.y + fu) P extends continuously to M x

M x IR$.. We leave this as an exercise.

(2)(a) Let X be a topological space and let f E c^0 (Xx M). We shall

establish (23 .. 34a) for P = PN by showing that

(23.41) lim ( PN(x,t;y,u)f(p,x)dμ(x)=f(p,y),

t"-,,.u}M

where the convergence is uniform in (p, y) on compact subsets of Xx M.

We have

JM PN (x, t; y, u) f (p, x) dμ (x)

N

= I: r.. 'I] ( x, y) ( 47r ( t - u) )-~

k=O j B(y, mJ~g))

x exp (- 4 ~: ~~)) </>k (x, y) (t - u)k f (p, x) dμ (x).

Recall that

(23.42) ( (47r(t-u))-n/^2 exp (-1~-Jll~) dμE(x) = 1

kn 4t-u

for any y E :!Rn and ( t, u) E IRS,. Given 0 :::; k :::; N and factoring out ( t - u )k,
using (23.42), we note that the integral

1

n ~W

'I] (x, y) (47r (t - u))-2 e-4(t-u) </>k (x, y) f (p, x) dμ (x) :::=; C

B(y,inj(g)/2)