296 24. HEAT KERNEL FOR EVOLVING METRICS
On the other hand, we shall obtain a better estimate for the normal deriv-
ative of ii on 8M.
Let a be any number in the interval ( ~, 1).
(1) k = 1. We have for 0:::; O' < T and xo, z E 8M with xo i= z(24.93a)
(24.93b)
(24.93c)8HJM1J (xo,T;z,O') = 2 -
8
- (xo,T;z,O')
Vz,u
n d;.(x 0 ,z)
:::; C (T _ 0')-2 e - 6(r-a):::; C# (T - O')-a d°;n+2a (xo, z)'
where C# < oo, where we used Lemma 24.35 below to obtain (24.93a), and
where the fourth line is true since f (x) ~ x>-e-x, ,\ 2:: 0, is bounded for
x 2:: 0.
Note that for the model case of Euclidean space ffi.n with heat kernel
H (x, t) = ( 47rtr~ e-
1
~~2
centered at the origin and a hyperplane P perpen-
dicular to a vector v and passing through the origin, we have
aH (x, v)
~ (x,t) = ---H(x,t) = 0 for x E P, t > 0.
uV 2tThis is essentially the reason for the improvement in the estimate (24.93a)
for la~:al on BM as compared to j\7g(u)fij. Formally, we may summarize
the above as follows.
LEMMA 24.35. Let g (T), TE [O,T], be a smooth family of Riemannian
metrics on a closed Riemannian manifold Mn with heat kernel H (x, T; y, O')
for the operator Lx,r· Let Nn-I C M be a compact smooth hypersurface.
Then there exists a constant C < oo such that
I
--[)H I (x,T;y,O'):s;C(T-O'r2 n (d^2 r (x ' y) +C ) e-5(r-a) d;.(x,y)
OVy,u T - O'
n d;.(x,y):::; C ( T _ O' )-2 e -6(r-a)
for any x, y E N, 0 :::; O' < T :::; T, and unit normal Vy,u to N at y with
respect to g ( O").
EXERCISE 24.36. Prove this lemma using the facts thata ( d; (x, y)) = d.,-(x, y) /a, \7d.,-(y))
ovy,r 4(T-O') 2(T-O') \ovy,r
< 0 d; (x,y)
- T-0'