1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

278 24. HEAT KERNEL FOR EVOLVING METRICS

for some constants c > 0 and C < oo. Thus, defining

. c~ ( CVol (g (T)) r-l
ck =;= ,
( k - 1)! ( N - ~ + 1) k-l

(24.39)

we obtain
n c2kd~(x,y)

[(Lx,rPN)*k+ll (x,T;y,v) :S Ck+l (T-v)(k+l)(N-2+1)-l e- S(r-v).

By induction we conclude that for all k E N we have (24.37) with Ck defined
by (24.39). The c^0 convergence of the convolution series (24.30) follows
since

< 00.

Compare the above with Lemma 23.20.

Analogous to Lemma 23.24 we have the following.

LEMMA 24.19 (Covariant derivatives of the convolution series). Given

£, m EN U {O} and N > ~ + 2£ + m, the series of covariant m-tensors

00

L af'V;i ( (Lx,rPN )*k)

k=l

converges absolutely and uniformly on M x M x [O, T]. Hence

af 'V;' (t, (L,,,PN )'") ~ t, af'V;' ( (L,,,PN )")

exists and is continuous.

This concludes our discussion of the sketch of a proof of the existence of

heat-type kernels in the time-dependent metric case.

EXERCISE 24.20. Complete the details of the proof of the existence of a
fundamental solution to the heat-type equation Lx,rU = 0 (defined using an
evolving metric).

3. Aspects of the asymptotics of the heat kernel for a

time-dependent metric
Similarly to as in the last chapter, the proof of Theorem 24.2 yields the
following asymptotic expansion.