- ASYMPTOTICS OF THE HEAT KERNEL FOR A STATIC METRIC 257
More generally, for x near y and t near 0, we have(f) ")(l H n 1 ( 4 )) logHN+~log(4Kt)
fJt + ux og N + 2 og Kt + t_ - (2n - ~Rpq (y) xPxq) + k.Riiq (y) xPxq
4t+ 2 ( R ~Y) + 112 (\7 rR) (y) Xr) + R ~Y) + ~ (\7 rR) (y) Xr+o (r(:)') +o (r(x)') +O(t)
(23.131) = -!!'._ 2t + 2! R ( ) Y + Rpq (y) 4t xP xq + 3! (\7 r R) ( ) Y X r
- O (r (:)') + 0 (r (xJ^2 ) + 0 (t)
Recall that for Euclidean space we haver (4Kt)-n/2 e-1~~2 dx = K-n/2 r e-lxl2 dx = 1,
}~n }~nby the change of variables x = x / .;4t. Differentiating this under the integral
sign, we see that
0 = r (-!!'._ + lxl:) (4Kt)-n/^2 e-
1
~~2
dx.
}~n 2t 4tHence
1
lxl2- ( 4 Kt )-n/2 e -~d 4t X = -. n
~n 4t 2
Moreover, we also deduce from the same change of variables that
(1)
(23.132)and
(2)(23.133) L. 0 c~I') (41rt)-•1^2 e-1¥.'rm: = 0 (t;i').
(3) If A= (Aj) is a symmetric n x n matrix, then
(23.134)
Indeed, after conjugating by a rotation, we may assume that Aei =
Aiei, i = 1, ... , n, where { ei, ... , en} is the standard basis of ffi.n.