- CHARACTERIZING RICCI FLOW BY ASYMPTOTICS OF HEAT KERNEL 287
By Theorem 24.21, for T small and for x and y close, HN is a good approx-
imation to the adjoint heat kernel H. For this reason we shall calculate the
asymptotics of the quantity:
where
(24.66a)(24.66b)our calculation culminates in Lemma 24.28 below.
Note that, by (24.44), we have the expansion
where { x~} are the coordinates of x in geodesic normal coordinates with
respect tog (T) centered at y. From this we may deduce
r^2 7 (x) = r 0 2 (x) + 2Rij (y, 0) Xi 7 X~T. + 0 ( Trr (x)^3 + T rr^2 (x) 2).Furthermore, by (23.118), we haveHence
(24.68)
(:T +L1x,r) (r;)(x)=2n-(~Rij-2Rij) (y,T)x~x{+o(r 7 .(x)
3
).The logarithm of (24.64) yields(24.69) log HN + -n log (47rT) = - 4 r2 r +log ( L N '1/JkTk ) •
2 T k=OTaking its time derivative and Laplacian, we have