1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

80 2. KAHLER-RICCI FLOW

Here is an interesting equation due to Hamilton.

LEMMA 2.47 (Ricci soliton vanisher evolution equation). For the poten-

tial function f in Lemma 2.42, we have

(2.49) :t IV' a Y' ,af 12 = .6. l\7a\7,Bf1

2

• IY'-y \7aY',af1

2

• IV' i \7a\7,Bf1

2

• 2Rai3-y8 \7 a \7 if \7 ,6 \7 of.

PROOF. By (2.42) and the commutator formulas, we have

:t (\7 a \7 ,6 f) = \7 a \7 ,6 ( ~~) - ( :t r~,6) \7 1' f

= \7 a Y' ,a ( .6.f + fJ) + \7 aR,B:y Y'-yj.

On the other hand, for any function f,

Y'a\7,a.6.f = Y'aY',aY'-yY';yf = Y'-yY'aY':y\7,af

= Y'-yY';yY'aY',af-Y'-y (Ra:y,68\i'of)

1

= 2 (Y'-y V' i + V' i Y'-y) \7 a \7 ,sf - \7 aR,a5\7 of - Ra:y,68\7-y \7 of

1

• 2 (Ra:y Y'-y \7 ,Bf+ R,a:y \7 a Y'-yf)

since

Hence

a r

at (\7 a \7 ,sf) = .6. V' a \7 ,sf+-:;;, \7 a Y' ,af - Ra:y,a8Y'-y \7 of

1

• 2 (Ra:y Y'-y \7,af + R,a:y \7 a Y'-yf)

and one easily derives (2.49) from this. D

PROBLEM 2.48. Find geometric applications of (2.49) in the study of

the Kahler-Ricci flow.

EXERCISE 2.49. Show that when dime M = 1,

IY'aY',afl

2

= t IY'iY'jf-t.6.fgijl

2

Show also that (2.49) generalizes (2.40).

and

5. Existence and convergence of the Kahler-Ricci flow

In this section we present some of the proofs of the basic global existence

and convergence results for the Kahler-Ricci flow due to H.-D. Cao [46].