374 26. BOUNDS FOR THE HEAT KERNEL FOR EVOLVING METRICS
where ii is the outward unit normal to 8D and where d& is the volume
element of 8D, both with respect tog.
Recall that Er= {(y,r): H(xo,y,r) 2: r-n}. We define
D~ ~ {(y,r(t)) E Er: t < s}
and two portions of its boundary
Pf~ {(y, T (t)) : H(xo, y, T (t)) = r-n and t < s}
and
P2, ~ {(y,r (t)) ED~: t = s}.
Applying (26.116) to D~, we have
0 = ks ( ~~ -b.u) H dμdt
r= Ls uH dμ + Ls ( uH :t + u \7 H - H\7 u, ii) d&
2 1= f uHdμ+r-n f lu
8
8
-\Ju,v) d&+ f u(\JH,il)d&.
}p,s 2 }ps 1 \ t }ps 1
Lettings--+ 0, we then have
Summing together the above equalities, we have the following mean value
property for solutions to the heat equation.
THEOREM 26.42 (MVP using heat spheres). Let (Mn, g) be a Riemann-ian manifold such that the heat kernel H(xo, y, r) satisfies (26.104). If
u : M x [T1, O] --+ JR is a solution to the heat equation) then
(26.117)
i
l\7Hl2 -
u(xo, 0) = u (y, t) (xo, y, T (t)) dO" (y, t),
8Er Vl\7 Hl2+Iaa1i12
where we note that 8Er = PE.
EXERCISE 26.43. Give another proof of Theorem 26.41 by integrating
(26.117) and applying the co-area formula as in [60].