1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

120 2. KAHLER-RICCI FLOW

LEMMA 2.99. We have the following equations and their complex conju-

gates:

(1)

(2.132) ( :t - A) div (h)a = RμD 'V vhaμ + hμv 'V aRμv - ~Rav div (h)v,

. (2)

(2.133) ( :t - A) (gai3'V ,a div (h )a) = Rμa 'V μdiv (h)a + 'V aRμv 'V vhaμ

+ 'V aRμv 'V ahμv + RμJ 'Va 'V vhaμ

+ hμv 'Va 'V aRμD·

PROOF. Using (2.128) and (2.131), we compute

( :t - A) V' 'Yha,6 = 'V 'Y ( :t - A) hai3 + ('V 'YA - A 'V 'Y) ha,6 - ( :t r~a) h 0 i3

= 'V 'Y ( Rai3e8hoe - ~ ( Raehr;,a + Rr;,ahae))

1 1

• 2 'V 'YRa8ho,a + 2 'V 'YRo,aha8 - R'Yf!a8'V 'T/ho,a
  1

• R'Yf!oi3 'V 'T/ha8 - 2 R'Y8'V ohai3 + 'V 'YRa8ho,a,
  since

2 ('V ,,A - A 'V 1') ha,6 = 'V 1' ('V 'T/ 'V fj + 'V fj 'V 'T/) ha,6 - ('V 'T/ 'V fj + 'V fj 'V 'T/) 'V /'ha,6

= 'V 'T/ ('V 1' 'V fj - 'V fj 'V 1') ha,6 + ('V 'Y 'V fj - 'V fj 'V 1') 'V 'T/ha,6

= 'V 'T/ (-R,,fja8ho,a + R,,f}o,6ha8)

Simplifying, we have

• R'Yf!'T/8 'V 8 h a,6 - R"(fja8 'V 'T/ h 8 i3 + R,,fjo i3 'V 'T/ h a8
  = -'V 'YRa8ho,a + 'V 'YRo,aha8


• 2R,,f!a8'V 'T/h 8 ,a + 2R,,f! 8 ,a'V 'T/ha8 - R,,8'V oha,6·

( :t - A) V' 'Yha,6 = 'V 'YRa,6e8hoe - R'Yfia8 'V 'T/ho,a + R'Yf!oi3 'V 'T/ha8

1

+ Ra,6e8'V 'Yhoe - 2 (Rae'V 'Yhe,6 + Re,a'V ,,hae + R,,8'V ohai3).

Since div (h)a = gl',6\J ,,ha,a, taking the trace and cancelling terms, we have

( :t - A) div (h) a = g 'Y,6 ( :t - A) 'V 1'hai3 + R(J;y 'V /'ha,6

= 'V aRr;8hoe + Rof!\l 'T/ha8 - ~Rae'V (Jhr;,a, ·

which is (2.132).