1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

4. CHARACTERIZING RICCI FLOW BY ASYMPTOTICS OF HEAT KERNEL 289

Applying (24.72) and (24.73) to (24.71) yields

(24.74) W = -n + (R;j - Rij) (y, T) x~x~ + 21/Ji + (~ + Llx,T) 1/Jo

2T 47 1/Jo

REMARK 24.27 (About the 0 (rr~)

3

) term). If r 7 (x)^2 + T --+ 0 and

r.,-(x)z < C then r.,-(x)3 --+ 0.

T - ' T

By applying to (24.74) the formulas 1/Jo = 1 + 0 (r 7 (x)^2 ), (24.57)

(24.50), and (24.55), we calculate that

w = 2T -n + (1 3 )

2

R +

2

n - 2Q (y, T)

xi

+ ; (\i'iR + 2 (divR)i + \i'in-3\i'iQ) (y, T)

i j (Rij - nij) (y, T)

XTXT 4T

• O (r, ~)') + 0 (n r, (x)').

In particular, if Q = R, then

xi

+ ; (\i'iR + 2 (divR)i - 2\i'iR)(y, T)

i j (Rij - nij) (y, T)

XTXT 4T

+o (r, ~)') + 0 (T+rr (x)').

Further specializing, we have the following.

LEMMA 24.28. If g ( T) evolves by the backward Ricci flow (i.e., Rij =
Rij) and if Q = n, then the approximate adjoint heat kernel HN in (24.8)
satisfies (see (24.66) for the definition of W = W1 + W2)

W = -

2

: + 0 (r, ~x)') + 0 (T+rr (x)')

PROBLEM 24.29. Prove Claim 24.26.