1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

104 2. KAHLER-RICCI FLOW

PROPOSITION 2. 79 (Evolution equation for the curvature). Under the

Kahler-Ricci flow,

(2.100)

( :t -~) RoJJy8 = Rap,v8Rμ,j3"(D - Rap,'YvRμ,j3v8 + Raj3vp,Rμ,iJ"(8

1

• 2 (Rap,Rμ,j3"(8 + Rμ,j3Rafj,"(8 + R'YμRaj3μ,8 + Rμ,8Raj3"(μ).

REMARK 2.80. The Riemannian analogue of this formula is given by

Lemma 6.15 on p. 179 of [108].

In the proof of the proposition we find it convenient to use a formula re-

lating ordinary derivatives and covariant derivatives at the center of normal

holomorphic coordinates.

LEMMA 2.81 (Relation between ordinary and covariant derivatives). If

T/ is a closed (1, 1)-form, then, at the center of normal holomorphic coordi-

nates, we have

(2.101)

(2.102)

()2

\7 j3\7 a'T/'Y8 = {)za{)zf3 T/'Y8 + T/>..8Rai3'Y5..'

()2

\7 a \7 j3T/'Y8 = {)za{)zf3 T/'Y8 + T/'Y>..Raj3>..8·

PROOF. We compute that at the center of normal holomorphic coordi-

nates,

\7 j3\7 a'T/'Y8 = Oj3\7 a'T/'Y8 - r~o \7 a'T/"(E

= 8i3 ( 8a'T/'Y8 - r~'YT/eJ)

= 8j38a'T/'Y8 - oi3r~'YT/e8

a2 - Re -

0 za 0 zf3 T/'Yo + aiJy T/ i;o'

where we used (2.4) in the last line; this proves (2.101). Note that (2.102)
is just the conjugate of (2.101). D

Now we give the

PROOF OF PROPOSITION 2. 79. We compute the evolution equation for
Rai3'Y8 at any point x and time t using normal holomorphic coordinates

{za} centered at x with respect tog (t). In such coordinates,^8 J;f (x, t) = 0.

Recall from (2.5) that

R ____ 02 9aj3 + pa-ogao-og pj3

a(3"(8 - {)z'Y {)z8 g {)z'Y {)z8.