1. HEAT KERNEL FOR A TIME-DEPENDENT METRIC 267
THEOREM 24.2 (Existence of the heat kernel on closed (Mn,g(r))).
In the setup above, assume that M is closed. Then there exists a uniquefundamental solution H (x, r; y, v) for t,-~x, 7 +Q. Moreover, His positive
and C^00 , and hence H is the heat kernel.1.2. A good approximation to the heat kernel for a time-depen-
dent metric.
In order to prove Theorem 24.2, we first construct a good approximation
to the heat kernel on a closed manifold Mn with respect to a time-dependent
metric g (r), r E [O, T].
Define the transplanted heat kernel E : M x M x IR~ --+ (0, oo) by(24.7) E -( x,y,r,v )-'--( --,-- 47r ( r-v ))-n/2 exp ( - d';.(x,y))
4 (r-v) ,
where
dT ; M x M --+ [O, 00)denotes the distance function with respect tog (r) for r E [O, T]. Recall that
Minj(g(T)) ~ {(x, y) EM X M : d 7 (x, y) < inj (g (r))}.
For each NE N, with N > n/2, we shall construct a function
of the form
(24.8)
HN: LJ Minj(g(T)) X { r} X [O, r)--+ IR
TE(O,T]N
HN (x, y, r,v) ~ E (x, y, r, v) L '¢k (x,y, r, v) (r - v)k,
k=Owhere the functions
'¢k: LJ Minj(g( 7 ))x{r}x[O,r)-+IR
TE(O,T]are to be defined. Here, as in (23.8),
Minj(g( 7 )) ~ {(x, y) EM X M: d 7 (x, y) < inj(g (r))}.
Note that, unlike in our previous discussions of the heat kernel parametrix,
'¢k depends on ( r, v). The reason for why this is assumed is that the recursive
OD Es we shall use to define the '¢k depend on ( r, v). Contrast this with the
time independence of the ODE (23.23a)-(23.23b).
Given r E [O, T] and y EM, let
rT (x) ~ dT (x,y).