1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

368 26. BOUNDS FOR THE HEAT KERNEL FOR EVOLVING METRICS

(n - 1)-dimensional measure on 8Er,s(x, t). Since

we have

x-y

"Vy'l/Jr= 2(t-s) and

y-x

v-

- ly-xl'

Ix -yj = -IV'l/Jrl

2(t-s)

on 8Er,s(x, t). Combining this with the fact that u is a solution to the heat

equation, we have

(26.101) u (x, t) = 2_ lt 1 u(y, s)IV'l/Jrl dO's (y) ds.

rn t- 4 r2 71" oEr,s ( x,t )

Note that by (26.91), a normal vector to 8Er(x, t) is ( "V'l/Jn °tt). Thus

dO'(y, s) =

IV'l/Jrl^2 + I 0tt 1

2

IV'l/Jrl dO's (y) ds.

Therefore (26.101) implies

(26.102) u(x,t) = -^1 }A r u(y,s)----=====(y,s)dO'(y,s), IV'l/Jrl^2

rn oEr(x,t) IV'l/Jrl2+1ott12

where dO' is the induced n-dimensional measure of 8Er(x, t) (compare with

( 26 .117) in the more general case of a Riemannian manifold).^5

3.2. Mean value property on Riemannian manifolds.

Let (Mn, g) be a complete Riemannian manifold and let H ( xo, y, T)

denote the heat kernel centered at (xo, 0). Let

T (t) ~to - t,

so that H(xo, y, T (t)) is the adjoint heat kernel centered at (x 0 , to), i.e.,

8

atH(xo,y,T(t)) = -f::.yH(xo,y,T(t)),

lim H(xo, y, T (t)) = Oxa·

t/to

Define the corresponding Riemannian heat ball of radius r based

at (xo, to) by

(26.103) Er(xo, to)~ { (y, t): H(xo, y, T) > r-n} CM x (-oo, to).

(^5) To see the equivalence of (26.102) and (26.117), note that
----;===== J\7HJ^2 = r -n j\7logHJ^2
VIVHJ2+I~1[12 VIV log HJ2+Iin~~H12
since H = r-n on 8Er.