1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

282 24. HEAT KERNEL FOR EVOLVING METRICS

of a C^2 function f. Using giJ = <\j +o (r (x)^2 ), we compute using (23.113)

that

Therefore

Aj - ij 8

2

~ - g [)xi[)xJ f - ~R 3 kqX q [)xk [) f +^0 ( ( r x )2).

Now applying this formula to the Laplacian of (24.49), we have^4

(~x,r'l/Jo)(x,y,T) = 2giJ .. ( 1^1 )

12

Rij + 4Rij (y,T)

+ 2~ijx~ + ljx~ + lix~)(

2

1

4

\Jk~j + 1

1

2 \JkRij) (y, T)

• 3 ~ Rk q xq a'l/Jo axk + 0 (r r (x)^2 )

1 1

(24.50) = 5R(y,T) + 2,R(y,T)

i

+ x; (\JiR + 2 (divR)i +\Jin) (y, T) + 0 (rr (x)^2 ) ,

where we used Rkqxq~~Z = 0 (rr (x)^2 ) and

( 2 ~ \J kRij + 1

1

2 \J k Rij) gPq [):~xi ( x~x~xt)

= (gijxk r +gkjxi_ 1 +gkixj) r (]_\JkR-· + !\JkR··)

12 iJ 6 iJ

i

= x; (\JiR + 2 (divR)i + \JiR)

(here we applied the contracted second Bianchi identity).

To warm up, we first consider 1/J1 along the diagonal. Evaluating (24.16)

for k = 1 at x = y, we have

1/J1 (y,y,T) = -Lx,r (1/Jo) (x,y,T)lx=y.

(^4) Compare with (23.124) in the static metric case.