1. HEAT KERNEL FOR A TIME-DEPENDENT METRIC 269
(2) If the { 'lfJk}1;= 0 satisfy (24.15)-(24.16), then HN = E L:1:=o 'l/JkTk is
a solution to (24.12), i.e.,REMARK 24.5. Similarly to Remark 23.8, we have that 'lfJk is independent
of N 2:: k.
PROOF. (1) From (24.15)-(24.16) we havewhere rn an 'l/Jk, and Lx,r ('l/Jk-1) are all evaluated at (lr,V (s), y) (in partic-
ular rr = rr (rr,V (s), y) = s). Thus, integrating over the interval [O, rr (x)],
we have the recursive formula
(24.17)
'l/Jk (x,y,T,v)
= -rr (x)-k a.;:-1/2 (x, y) e~ J;r(x) a;; ('-y(s))dsrrr(x) 1 s lirr - -
X Jo a;l^2 (r (s), y) sk-le-2 fa a;r('Y(s))ds Lx,r ('l/Jk-1)(1 (s), y, T, v)ds,where r ~ rr,V is the unit speed minimal geodesic emanating from y to x
(compare with (23.25)). Part (1) easily follows.
(2) Applying the heat-type operator Lx,r to (24.8) and using (24.12) and
(24.10) yields
N -
(T - v) ELx,r ('I/JN)= Lx,r (Et 'lfJk (T-v)k)
k=O=( Tr ologar_ Tr~ )Et'ljJk(T-v)k
2 ( T - V) Orr 2 ( T - V) k=O
N- 2 L (T-v)k (\lxE, \1'1/Jk)
~o gW
N N
- ELLx,r ('lfJk) (T - v)k +EL k'l/Jk (T - v)k-l.
k=O k=O
Factoring out E, collecting like powers of T - v, cancelling the two ( T - v) N
terms in the above expression, and using \1 x log E = - 2 (;:_v) a~r, we then