1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

232 23. HEAT KERNEL FOR STATIC METRICS

STEP 3. H is the fundamental solution. Recall from Proposition 23.12
that we have for any f E c^0 (M)

lim ( PN(x,t;y,u)f(x)dμ(x)=f(y).

t\,u}M

By (23.59), that His the fundamental solution follows from the claim that

(23.60) lim ( (PN f (DxPN )k) (x, t; y, u) f (x) dμ (x) = 0.

t\,u} M k=l

Now we prove (23.60). For simplicity we take u = 0. By (23.69) and

(23.70) below, we have

(23.61)

~I L.J (DxPN) *kl (x, y, t) :::; Ct N -2 n exp (CVol N (M) _ I! tN-~+1) + d^2 (x,y)

1

e--5t-.
k=l 2
Therefore
(23.62)

(PN * t, (DxPN)•k) (x, y, t)

00

PN (x, z, s) L (DzPN)*k (z, y, t - s) dμ (z) ds

k=l

CVol(M)(t-s) N -"2" n+l d2(z,y)

PN (x, z, s) C (t - s)N-~ e N-~+^1 e -^5 (t-s) dμ (z) ds,

which tends to zero as t "';; 0 uniformly in ( x, y) since N > ~. Thus we

obtain (23.60) for u = 0. The theorem now follows from Lemma 26.4. D

2.3. The heat kernel expansion.

We have the following.

THEOREM 23.17 (Heat kernel expansion). Ast-+ 0, we have

(23.63) H(x, y, t) ~ ( 4?rt)-n/^2 exp ( - d2 ~; y)) ~ti u; (x, y).

More precisely, equation (23.63) means that there exists a sequence of func-

tions Uj E C^00 (M x M) such that

(23.64) H (x, y, t)-( 4?rt)-n/Z exp ( - d2~; y)) t, t•uk( x, y) "c WN(x, y, t ),

with
uo(y, y) = 1
and

WN(x, y, t) = 0 (tN+l-~)