1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

288 24. HEAT KERNEL FOR EVOLVING METRICS

and

respectively. Summing the above two formulas yields the following expres-

sion for (24.66a):

w

1

= r;. (-fr+ ~x,T) (r;.) l\7 x,T1);ol

2

4r^2 4r 1);5

• 1);1 +(-fr :a ~x,T) 1);o + 0 (r).

By (24.69), for (24.66b) we have

(24.70) W 2 = _ r;. + log1);o + 1);1 + 0 (r).

4r^2 T 1);o

Summing these two formulas yields

(24.71) W= -(-fr +~x,T) (r;.) +4log1);o _ l\7x,^7 1);ol

2

4T 1);5

+ 21);1 + ( 1r1);: ~X,T) 1);o + Q ( T).

On the other hand, we obtain from (24.68) and (24.49) that

• ( :T + ~x,T) (r;.) + 4log1);o

(

=-2n+^2 3Rij-2Rij ) (y,r)x~x~..

+ (~~j + Rij) (y, r) x~xt + 0 (r 7 (x)^3 )

(24.72) = -2n + (~j - Rij) (y, r) x~xt + 0 (r 7 (x)^3 ).

We also have from (24.49) that

(24.73)