1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

284 24. HEAT KERNEL FOR EVOLVING METRICS

and by (24.47) we obtain

:T (las Z; (r (s)) ds) = O (s^2 ),

it follows from (24.54) that

(24.55) °!

0

(r(s),y,T)=O(s^2 )

for s E [O, r 7 (x)]. Furthermore, similar to (24.50), we have

(24.56)

1 1

(f:lx/l/Jo) (r(s) ,y,T) =

6

R(y,T) + 2,R,(y,T)

i

+ rT (x) x; ("\7iR + 2 (divR)i + V'iR)(y, T) + O(s^2 ).

Using (24.55), (24.56), and

Q (r ( S) , T) = Q (y, T) + -(-) s Xi^2

rT X^7 \i' iQ (y, T) + Q ( S ) ,

we compute

Lx,T ('I/Jo) (r (s), y, T)

= ( °!

0

• !:lx,T'l/JO + Q'l/Jo) (r (s), y, T)

1 1 s x~.

= -6R (y, T) - 2 n (y, T) - rT (x) 6 (V'iR + 2 (d1vR)i + Y'iR) (y, T)

+Q(y,T)+-(-)xTV'iQ(y,T)+O(s) s i^2

rT X

= (-~R-tn+ Q) (y,T)

i

• rT (x) x; (V'iR + 2 (divR)i + V'iR - 6\i'iQ) (y, T) + O (s^2 ).

We conclude from (24.53) that

'l/J1 (x, y, T)

rr(x)

= -rT (x)-^1 lo Lx,T ('I/Jo) (r (s), y, T) ds + 0 (r 7 (x)^2 )

= r 7 (x)-^1 forr(x) ( (~R + tn-Q) (y, T) + 0 (s^2 )) ds

rrr(x) s xi

+ rT (x)-

1

lo rT (x) ; (V'iR + 2 (divR)i + V'iR-6\i'iQ) (y, T) ds

• 0 (rT (x)^2 ).

Thus we obtain the following.