1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

3. HEAT KERNEL ASYMPTOTICS FOR A TIME-DEPENDENT METRIC 279

THEOREM 24.21 (Heat kernel expansion for evolving metrics). On a

closed manifold there exists a sequence of functions Uj E C^00 (M x M x lRf),

with

(24.40) uk(x, y, r, v) = 'l/Jk (x, y, r, v)

for all x, y EM such that dT (x, y) ::; p = ~ minTE[O,T] inj (g (r)) and for all

(r, v) E lRf, such that the function

(24.41)

wN(x,y,r,v) ~H(x,y,r,v)

• (41r(T - v))-1' exp (-:r;x~ ~)) t,(T - v)'uk(x, y, r, v)

satisfies

WN (x, y, r, v) = 0 ((r -v)N+l-~)

as r '\iv, uniformly for all x, y EM. Moreover, for any k, f. ~ 0,

(24.42) lo;\i'~wNI (x, y, r, v) = O ( (r - v)N+I-~-k-^2 £)

as r '\iv, uniformly for all x, y EM.

We now discuss, based on the above theorem and the recursive formula

(24.17) for 'l/Jk, aspects of the heat kernel asymptotic expansion.

Assume for simplicity that v = 0. With this we abbreviate (x, y, r, 0) as

(x, y, r).

We begin by deriving the first few terms of the expansion for c;; (x) for

x near y. Let x be a point near y with x -I y. As in (24.14), let

"( = "fT,V : [O, rT (x)] -+ M

be the unique unit speed minimal geodesic joining y to x at time r, with

unit tangent vector V E TyM. In geodesic normal coordinates { x~} ~=l with

respect to g ( r) centered at y, we have

.. s.
(24.43) 'Y (sY ~ x~ ("! (s)) = -(-) x~
rT X

for s E [O, rT (x)], where for the last term x~ ~ x~ (x). By (24.21) and
expanding the 2-tensor RT along"(, we have

or rr(x)

a: ( x) = J 0 RT ('Y ( s) ' 'Y ( s)) ds

1

= G~( Rij(y,r)+s-(T) ~ (\7kRij)(y,r)+O(s^2 ) ) _(T ~ )_(T ~ )ds.

0 rT x rT x rT x

Evaluating this integral, we obtain the following.