1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

4. INTRODUCTION TO THE KAHLER-RICCI FLOW 77

Differentiating (2.3), we find that for both the Kahler-Ricci ft.ow and

the normalized Kahler-Ricci flow, the Christoffel symbols evolve by

(2. 33) at^8 I'"Y af3 = -g 7J \7 a R f3J.

The volume form and scalar curvature evolve according to the following.

LEMMA 2.38 (Evolution of dμ and R for normalized ft.ow). Under the

normalized Kahler-Ricci flow ( 2 .29),

8

at dμ = (r - R) dμ

and

(2.34) -8R at = ~R + I R a 13 - 12 - n -R. r

In particular, since JM (r - R) dμ = 0, the normalized Kahler-Ricci flow

preserves the volume.

PROOF. We first compute, using (2.29), that

8 58

(2.35) at log det g 75 = g^7 atg 75 = r - R.

Hence

8

atdμ = (r - R) dμ.

The evolution of the Ricci tensor is

(2.36)

From this and

at^8 Rai3 = -8a8f3 -(8 at log det g ) -

75 = 8a8f3~·

8R - ai3 ~ R -- ~ -. R -

at - g at a(3 8t^9 af3 af3

we easily derive (2.34). D

EXERCISE 2.39 (Evolution of R for unnormalized flow). Show that under
the Kahler-Ricci flow gt9ai3 = -Rai3' we have gtdμ = -Rdμ, gtRa!3 =
8aB(3R, and

(2.37)

EXERCISE 2.40 (Total and average scalar curvature evolution). Show

that if Mn is closed, then under the Kahler-Ricci ft.ow gt9ai3 = -Rai3'

! JM Rdμ= JM (IRa13l2 -R2) dμ,

and hence we have

~: = (JM dμ )-

1

JM (IRa13j

2

• R

2

) dμ + r

2

.