2. EXISTENCE OF THE HEAT KERNEL FOR A TIME-DEPENDENT METRIC 271
where the infimum is taken over all minimal geodesics joining y to x, with
respect to g ( T). Hence, assuming IRij I :::; C on M x [O, T], we haveI~; (x)I:::; CLg(T) (!') = Cr 7 (x),
which is (24.20). D2. Existence of the heat kernel for a time-dependent metric
In this section we sketch how to proceed analogously to §1 and §2 of the
previous chapter to complete the proof of Theorem 24.2 on the existence of
the heat kernel for the operator Lx, 7 = g 7 - .6.x, 7 + Q. In particular, we
discuss a version of the Levi-Minakshisundaram-Pleijel method to derive a
parametrix for the heat operator with respect to a time-dependent metric
on a closed manifold. We leave it as an exercise for the reader to fill in the
details of the proof (consult [61], [69], [70], and [85]).
2.1. Parametrix for the heat operator for a time-dependent
metric.
For N > ~we may extend Lx, 7 (HN) continuously to LJ 7 E[O,T] Minj(g( 7 )) x
{r} x [O, r] by having it take the value 0 on UTE[O,T] Minj(g( 7 )) x { T} x { T}
(this generalizes extending DxHN in subsection 1.3 of Chapter 23). More-
over, given local coordinates (U, {xi} ~= 1 ), we have for k + 2£ < N - ~
ofo~ (Lx,THN) (x,y,r,v)
= (r - v)N-(n/^2 )-k-^2 £ exp (-d;_ (x, y)) F (x y r v)
4 ( T - V) k,£ ' ' 'in LJ 7 E[O,T] ( (U x M) n Minj(g( 7 ))) x { r} x [O, r], where
Fk,£ E C^00 ( u ( (U x M) n Minj(g(T))) x { T} x [O, T]).
TE[O,Tj
From this we may obtain the following (compare with Lemma 23.10).LEMMA 24.8 (Covariant derivatives of Lx, 7 HN)· For any k,f E NU{O},
(24.22)laevk (L HN)I (x y T v) < c (r - v)N-(n/^2 )-k-^2 £ exp (-d; (x, y))
t X x,T ' ' ' - 4 ( T - V)on LJ 7 E[O,T] Minj(g( 7 )) x { r} x [O, T]. Moreover, if N > ~ + k + 2£, then
(24.23)of\l~ (Lx,THN) (x,y,T,v)= (4"'")-n/2 " (T - v)N-(n/2)-k-2£ exp ( d;_ (x, y)) F- ( )