1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

106 2. KAHLER-RICCI FLOW

PROOF. Indeed, substituting fJ =a and J = 1 in (2.100), we have

(! -.6.) Raa'Y"f = Rap,v:yRμa"fiJ - Rap,'YvRμav"f + Raavp,RμD"f"f

1

• 2 ( Rap,Rμa"f"f + RμaRaP"f"f + Ryp,Raaμ"f + Rμ:yRaa'Yp,).
  D
  COROLLARY 2.83 (Ricci tensor evolution). The Ricci tensor satisfies the
  Lichnerowicz heat equation:
  8
  (2.105) ot Ra'{J = .6..Ra'/l + Ra'fi"fJRo:y - Ra:yR'Y°!J = .6..LRa°/3·

REMARK 2.84. More generally, we say that a real (1, 1)-tensor ha°/3 sat-
isfies the Lichnerowicz heat equation if

8 1 1

ot ha°/3 = .6..Lha°/3 ~ .6..ha°/3 + Ra°!J'YJho:y - 2Ra:yh'Y°!J - 2R'Y°!Jha1·

See also (2.22).

PROOF. Summing (2.100) over"'(= 5 from 1 ton, we have

( :t -.6..) Ra°/3 = t ( :t -.6..) Ra°!J'Y'Y + Ro:yRa°!J'YJ

k=l

= Rap,v;yRμ°!J"fiJ - Ra'fi,"(vRμ°/Jv;y + Ra°/Jvp,Rμv + Ra°!J'YJRo:y

1

• 2 ( Rap,Rμ°/J + Rμ'{JRap, + RyμRa'{Jμ;y + Rμ:yRa'fi'Yμ)

= Ra°!J'YJRo:y - Rap,Rμ'/J'

after cancelling terms to get the last equality. D

REMARK 2.85. Equation (2.105) may also be derived from (2.36), (2.10)

and commuting covariant derivatives. In particular,

1

.6..Ra°/3 = 2 (\7 'Y Y' ;y + Y' ;y \7 'Y) Ra°/3

1 1

= 2 \7 'Y \7 °!JRa;y + 2 V' ;y \7 aR'Y°!J

1 1

= \7 a \7 '{JR - 2R'Y'fiaJRo:y + 2R 8 °/3RaJ

1 1

+ 2RaJRo°/3 - 2R1a'/JoR'YJ

(2.106) = \7 a \7 '{JR - Ra°!J'YJRo:y + RaJRo°/3·

That is,

\7 a \7 °!JR= .6..LRa°/3·
Based on the evolution equation (2.104) and Hamilton's maximum prin-
ciple for tensors (see Chapter 4 of Volume One or Part II of this volume) we
present the following.