- HEAT KERNEL FOR AN EVOLVING METRIC 335
A and B are functions on M x [O, T] which are both C^2 in space and C^1 intime, then for any 0 S r1 < r2 S T we have
(26.11)
1T2
dr r ((Lx,TA) (x, r) B (x, r) - A (x, r) (L~,TB) (x, r)) dμg(T) (x)
Tl JM=JM A (x, r2) B (x, r2) dμg(T 2 ) (x) - JM A (x, r1) B (x, r1) dμg(Tl) (x).
PROOF. The LHS of (26.11) is equal to1:
2
dr JM ( ( ~~) (x, r) B (x, r) +A (x, r) ( ~!) (x, r)) dμg(T) (x)- 1T
2
dr r (-(L:lg(T)A) (x, r) B (x, r) +A (x, r) (L:lg(T)B) (x, r)) dμg(T) (x)
Tl JM- 1T
2
dr r A (x, r) R (x, r) B (x, r) dμg(T) (x)
n JM= 1:
2
d~ (JM A (x, r) B (x, r) dμg(c! (x)) dr=JM A (x, r2) B (x, r2) dμg( 72 ) (x) - JM A (x, r1) B (x, r1) dμg( 71 ) (x),
where we used (26.3) and the divergence theorem, i.e.,
0
As a special case of (26.11), we see that if a and bare C^2 functions with
compact support in M x (0, T), then
(26.12)
So, indeed, the operator L;, 7 is the (formal) adjoint of Lx,T·
EXERCISE 26.2 (Duhamel's principle on a compact manifold with bound-ary). Show that if M is compact and has nonempty boundary 8M, then
(26.11) still holds provided both A= 0 and B = 0 on 8M x [O, T].
Let Hand H* be the heat kernel and adjoint heat kernel in (26.6) and
(26.9), respectively. Given 0 Sp< v ST, for r E (p, v) we take
A (x, r) = H (x, r; z, p) and B (x, r) = H* (x·, r; y, v) ·'