364 26. BOUNDS FOR THE HEAT KERNEL FOR EVOLVING METRICS
PROOF. Consider the function of r defined by the RHS of (26.83), which
is the average of u on the ball of radius r,
(26.84) </>o(r) ~ ~ f u(x)dμJE (x).
Wnr JB(O,r)
Note that
lim ¢o(r) = u(O)
r-+0
since u is continuous. We compute
(26.85)
~ </>o(r) = - nn+l { u(x)dμJE (x) + ~ { u(x)do-r (x),
ur Wnr } B(O,r) Wnr J aB(O,r)
where do-r is the induced volume form on 8B(O, r).
On the other hand,^4
2n r u(x)dμJE (x) = { ~ (1x1^2 ) u(x)dμJE (x)
JB(O,r) j B(O,r)= - f V7 (1xl^2 ) · V'u(x)dμJE (x)
jB(O,r)+ { 2lxlu(x)do-r(x)
laB(O,r)(26.86) = { (1xl^2 - r^2 ) ~u(x)dμJE (x)
jB(O,r)+ 2r { u(x)do-r (x).
j 8B(O,r)
Since ~u = 0, by combining (26.85) and (26.86), we conclude that
8(26.87) or </>o(r) = 0.
Since <Po is constant, we have<Po(r) = lim <Po(p) = u(O).
p-+0
DNote that the boundary of the ball that appears in (26.84), i.e., 8B(O, r),
is a level set of the fundamental solution of the Laplace equation centered
at 0. Note that by (26.83) and (26.86) we also have(26.88) u(O) =
1
_ 1 f u(x)do-r (x),
nwnrn laB(O,r)
where the RHS is the average of u on the sphere of radius r.(^4) 0ne may think of this as a Pohozaev-type identity; note that x = ~ 'V lxl (^2) is a
conformal Killing vector field.