1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

(^280) 24. HEAT KERNEL FOR EVOLVING METRICS
LEMMA 24.22 (Expansion for °;;). Let x~ = x~ (x), so that r;. (x)
2:7= 1 ( x~)
2

. Then

(24.44)

Now we compute the first few terms of the expansion for 'I/Jo. By (24.15),

we have

Since 'I/Jo (y, y, T) = 1, we have

(24.45)

(

1 r,,.(x) or )

'I/Jo (x, y, T) = a_;=-^1 /^2 (x, y) exp 2 lo O; (/ (s)) ds ,

where / is the unique unit speed minimal geodesic joining x to y with respect

tog (T) and where d 7 (x, y) < inj (g (T)). From (23.116), we have

(24.46)

(a;^1!^2 )(x,y) = 1+: 2 Rpq(y,T)x~x~+

2

~ \7rRpq(y,T)x~x~x~+o(r 7 (x)^4 ).

Moreover, from (24.44) we have

where xi = r,,.(x) x~ and where { x~} ~=l are the coordinates of x. Making the
change of variable u =i= r,,. (x) , we have (note that r 7 (r ( s)) = ur 7 ( x))