1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

4. ASYMPTOTICS OF THE HEAT KERNEL FOR A STATIC METRIC 251

where a E G, 1) and where we used (23.102) and (23.106). From the above

we conclude (23.112). D

This concludes our discussion of the existence of the heat kernel on a

closed Riemannian manifold.

4. Asymptotics of the heat kernel for a static metric

In this section, we continue to consider the case of a fixed Riemannian

metric g on a closed manifold M. We wish to compute asymptotic expan-

sions along the diagonal for the functions cPk (x, y) in the finite series (23.9)
for HN. We also consider an aspect of the heat kernel asymptotics related
to §9.6 in Perelman [152]. In the next chapter we shall consider the Ricci
flow case.

4.1. The first few terms of the asymptotic expansion of the

heat kernel.

We start with the computation of (the asymptotics of) ¢0 in Minj(g) and

of ¢ 1 along the diagonal of M x M. By (23.24), i.e.,

<Po (x, y) = a-^1 /^2 (x, y),

and the k = 1 case of (23.27), we have

where, in geodesic normal coordinates { xk}~=l centered at y EM, we have

(23.15), i.e., a (x, y) = .Jdetgkc(x).
Again let r (x) ~ d (x, y). Recall that in the geodesic normal coordinates
above, the components of the metric g have the following expansion near y
(see formula (3.4) on p. 211 of [168])

(23.113)

9kR. (x) = likR. - ~RkpqR.XpXq - ~\lrRkpqR.XpXqXr

+ (-

2

1

0

\lr \lsRkpqR. + 4 ~RkpqmRR.rsm) xPxqxrxs + 0 (r (x)

5

),

where, on the RHS, the components of the curvature and its covariant deriva-
tives are evaluated at y. From (23.113) one can show that the determinant