1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

3. HEAT KERNEL ASYMPTOTICS FOR A TIME-DEPENDENT METRIC 283

Recall that by assumption (24.11), we have

o'l/Jo

OT (y,y,r) = 0.

Hence, by (24.50), along the diagonal of M x M x (0, T],

'l/Ji(y, y, r) = (~x,T'l/Jo) (x, y, r)lx=y - Q (y, r)

1 1

(24.51) =

6

R (y, r) + -{R, (y, r) - Q (y, r).

Now we compute the first few terms of the expansion for 'l/J1 (x, y, r) for

x near y and T near 0. First note that from (24.46) we may easily deduce

(24.52) (a,;1^2 ) (1' (s), y) = 1 -

1

1

2

82

2 Rpq (y, r) x~x~ +^0 (s

(^3) ).
rT (x)
Moreover, by (24.17), we have
'l/J1 (x, y, r) = -rT (x)-1 a-;1/2 (x, y) e~ J;'T<x) a;; ('y(s))ds
rr'T (x) 1 s l'!!::r. - -
x lo a,;1^2 (1' (s), y) e-2 fo a'T ('y(s))ds Lx,T ('I/Jo) (1' (s), y, r) ds,
where Lx,T = ff 7 - ~x,T + Q. Therefore, by applying (24.52) and (24.48),
we have
1 r'T(x) 1/2 1 rs ar'T ( (-))d-
'l/J1 (x, y, r) = -rT(x )-lo a 7 ('y(s),y)e-2Joar'Y^8 8
x Lx,T ('I/Jo) (1' (s), y, r) ds

• 0 (rT (x)
  2
  )

(24.53)

Observe that by differentiating (24.45) we immediately get

(24.54) ---^1 O'l/JO =-log^0 ( a-1/2) + --^1 0 (1r'T(x) --OTT (1' (s)) ds ).

'I/Jo OT OT^7 2 OT 0 OT

Now we expand the factor Lx,T ('I/Jo) (1' (s), y, T) on the RHS of the equation

above. Since by (24.46) we easily deduce

:

7

log ( a-;^112 ) ('y(s) ,y) = 0 (s^2 )