1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

4. ASYMPTOTICS OF THE HEAT KERNEL FOR A STATIC METRIC 255

Now we compute the expansion for </>1 up to first order. From (23.26)

we have

</>1 (x, y) = a-^1 /^2 (x, y) fo

1

( a^112 b.x<Po) (expy (pr (x). V), y) dp,

where V E TyM is the unit vector tangent to the unique minimal geodesic

from y to x. Let {xi} be geodesic normal coordinates centered at y and let

xi also denote the i-th coordinate of x. We have xi (expy (pr (x) V)) = pxi.

By (23.124) and since a^112 = 1+0 (r^2 ), we have

( a^112 b.xo) (expy (pr (x) V), y) = R ~y) + ~ ('\JrR) (y) xr + 0 (p^2 r (x)^2 ).

Hence, using a-^112 (x, y) = 1+0 (r (x)^2 ), we have

From this we immediately obtain

LEMMA 23.33.

(23.125)

REMARK 23.34. We have

1 12 1 2 1 2

(23.126) </>2 (y, y) =

30

b.R + 72 R - 180 IRcl + 180 IRml ,

where 1Rcl^2 ~ gikgj.e.RijRke and 1Rml^2 ~ gipgjqgkris~jk£Rpqrs and the RHS
is evaluated at y (see Gilkey [72]).

time-dependent metric 3. Aspects of the asymptotics of the heat kernel for a

manifold.

Motivated by §9.6 in Perelman [152], we consider an aspect of the heat

kernel asymptotics on a fixed Riemannian manifold related to entropy.

Differentiating the logarithm of (23.9), i.e., differentiating the equation

(23.127) 2( ) ( N )

n d x,y k

logHN=--log(41T't)-

4

+log L<f>kt ,

2 t k=O

in both time and space, we have

.§.__lo H - _!!_ d2 (x, y) I:f=1 k<f>ktk-1

at g N - 2t + 4t^2 + °'\:"'N A-tk

L..k=O '+'k

= _!!_ d2 (x, y) </>1 0 (t)

2t + 4t^2 + <Po +