- CHARACTERIZING RICCI FLOW BY ASYMPTOTICS OF HEAT KERNEL 285
LEMMA 24.25 (Asymptotic expansion for '1/J1).'1/J1 (x, y, T)(24.57) = (~R + ~R-Q) (y, T)
1.
+
12(''ViR + 2 (divR)i + \liR-6\liQ) (y, T) · x~ (x)
- 0 (rT (x)^2 ).
4. Characterizing Ricci flow by the asymptotics of the heat kernel
In this section, by modifying the discussion regarding the static metric
case in subsection 4.2 of Chapter 23, we consider calculations related to the
asymptotic formula in §9.6 of Perelman [152]. There he wrote:
"Ricci fl.ow can be characterized among all other evolution
equations by the infinitesimal behavior of the fundamental
solutions of the conjugate heat equation. Namely, suppose
we have a Riemannian metric 9ij (t) evolving with time ac-cording to an equation (9ij)t = Aj(t). Then we have the heat
operator D = gt - 6 and its conjugate D* = -gt - 6 -! A,
so that jt J uv = J ((Du) v - u (D*v)). (Here A = gij Aij)
Consider the fundamental solution u = ( -47rt)-~ e-f for D*,
starting as c5-function at some point (p, 0). Then for general
Aij the function (DJ+ {)(q, t), where J = f -J Ju, is of theorder 0(1) for (q, t) near (p, 0). The Ricci fl.ow Aij = -2Rij
is characterized by the condition (DJ+ {)(q, t) = o(l); in
fact, it is O(lpql^2 + ltl) in this case."Let g (T), TE [O, T], be a smooth family of metrics on a closed manifold
M^11 satisfying (24.1), i.e.,
0
OT 9ij = 2Rij'
where Rij is some time-dependent symmetric 2-tensor. Given y EM, let
H: M x (O,T]-+ (O,oo)
be the corresponding fundamental solution to the adjoint heat equation:
oH.
(24.58) D* H ~ OT - /:).. 9 ( 7 )H + R ( ·, T) H = 0,(24.59) lim H ( · , T) = by.
T-+0Define f: M x (0, T]-+ IR by
(24.60)