- NOTES AND COMMENTARY 263
By the fundamental theorem of calculus, we haved r d 1[
dt JM f (x, f) dμ (x) = dt °' cp (t) dt= cp (f)
r^01
=JM &t (x, f) dμ (x).06. Notes and commentary
First we make some general comments regarding the history and litera-
ture for the heat kernel on manifolds. The parametrix method employed to
prove the existence of a fundamental solution to a linear elliptic or parabolic
partial differential equation of second order is due to E. E. Levi. For a thor-
ough discussion of the parabolic case, see Chapter 1 of Friedman [61]. For
the heat kernel on a closed Riemannian manifold, this method was adapted
by Minakshisundaram and Pleijel [131], [130], which also leads to a geomet-
ric understanding of the asymptotics of the heat kernel. We closely follow
the presentations in [61], Berger, Gauduchon, and Mazet [13], and Chavel
[27]. See also the forthcoming book by Li [118].
For some additional references besides those mentioned above on the
parametrix and the existence and asymptotics of the heat kernel, see Berline,
Getzler, and Vergne [14] and Gilkey [73].
Second we mention some source material for this chapter.
§1. For Lemma 23.9 seep. 209 of Berger, Gauduchon, and Mazet [13]
or p. 150 of Chavel [27]. Proposition 23.12 is Lemma E.III.3 on pp. 210-211
of [13].
§2. Equation (23.54) is (4.1) on p. 14 of Friedman [61]. Definition(23.55) is (4.4)-(4.5) on p. 14 of [61]. We refer the reader to§ 4 in Chapter
1 of [61] for a proof of the convergence of the series for fundamental solutions
of variable coefficient second-order parabolic equations.
§3. We follow Friedman [61] in presenting the properties of differenti-
ating a convolution with the parametrix. For Lemma 23.26 see Theorem 3
on pp. 8-9 of [61]. For Lemma 23.27 see Theorem 4 on p. 9 of [61].
§4. Regarding the formulas for the expansion of
see Branson, Gilkey, and Vassilevich [17] for much more general formulas
for the heat kernel asymptotic coefficients for 'Laplace-type' operators.