- NOTES AND COMMENTARY 303
where 1/ 7 ,i is the outward unit normal to ani with respect to g ( T); here we
used ~~ni r,i ::=; 0.^6 The lemma follows sincelim r Hni (x, T; y, v) dμg(T) (x) = 1.
T\,v Joi
D
With Lemma 24.41, we may apply the parabolic mean value inequality
(i.e., Theorem 25.2, which is a local result) to obtain a uniform upper bound
for Hni on any compact subset of M x M x JR~ (in a bounded subset of
Mand away from v = T). We conclude that HM (x,T;y,v) is finite on all
of M x M x JR~. By the Bernstein local derivative estimates for heat-type
equations, we obtain that HM is C^00 •
To show that HM is a fundamental solution, we may apply the method
in S. Zhang [196]. Finally, HM is the minimal positive fundamental solution
since for any positive fundamental solution H' we have H' 2:: Hni for all i.
6. Notes and commentary
The heat kernel with time-dependent coefficients has been treated in
[14], [61], by Garofalo and Lanconelli [69], [70], and by one of the authors
[85].
There is a work by Molchanov [132] using probabilistic methods regard-
ing the asymptotics of heat kernels with respect to time-dependent metrics
(we would like to thank Peter Topping for bringing this reference to our
attention).
§ 1. The existence of the heat kernel associated to a 1-parameter family
of Riemannian metrics on a closed manifold is due to one of the authors [85]
and is related to [69].
§5. In subsection 5.2 we follow Chapter VII of Chavel [27]. For another
approach to Lemma 24.31, see Theorems 7 and 9 in Chapter 3 of Friedman
[61]. For Lemma 24.33, see §5.2 of Friedman [61] or, for a special case,
see Theorem 1 on pp. 159-160 of Chavel [27]. To solve Exercise 24.34, one
needs to adapt the arguments in [61].
(^6) In fact, ~Hn: < O by the Hopf boundary point lemma.
VT,'/,