286 24. HEAT KERNEL FOR EVOLVING METRlCS
From (24.58) we derive
8f 2 n
(24.61) ar -l:lg(T)f +IV' fl - R + 27 = 0.Define J: M x (0, T] ---t JR. by
f :!;= f - JM fH dμg( 7 )·
Let D :!;= - gT - flg(T). We have
of+ - -J = --a J J d (l )^1 l
T^8 T - l:lg(T)f + - + TT -d M J Hdμg(T) - -TM JHdμg(T)·
The following is a special case of Perelman's result.^5
CLAIM 24.26 (Heat kernel asymptotics characterization of Ricci flow).
Let H : M >< (0, T] -1 (0, oo) be the adjoint heat kernel centered at (y, 0) and
let f: M x (O,T] ---t JR. be as in (24.60). IfRij = Rij, i.e., g(r) evolves by
backward Ricci flow, then(24.62) Of -(x,r) + J (x, r) (^2 ( ) )
7= 0 dg(O) x,y +r.
In particular,(24.63) Df (x,r) + f (x,r) = o(l)
T
for x near y and T near 0, i.e.,
lim (of (x,r) + f (x,r)) = 0.
d~(O) (x,y)+T---+0 T
Assume, as in the previous section, that v = 0 and abbreviate (x, y, r, 0)
as (x, y, r). A discussion of aspects of the heat kernel asymptotic expansion
related to Claim 24.26 shall occupy the rest of this section. Recall from
(24.8), corresponding to (Mn, g (r)), TE [O, T], we have
N
(24.64) HN (x, y, r) = E -(x, y, r) L../Pk ~ (x, y, r) T k ,
k=O
where Eis given by (24.7), i.e.,
(24.65) E -(x, y, r) :!;= (47rr)-n /2 exp ( - d2(T) g (x, y))
47
,
and where the 1/Jk are defined by (24.15)-(24.16).
By (24.60), we have
DJ - [_ = (~ + /:l) (logH +~log (47rr)) + logH +~log (47rr).
T 8T 2 T(^5) More generally, one may wish to study the asymptotic expansions of fundamental
solutions of heat-type equations coupled to the Ricci flow.