1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

8. KAHLER-RICCI FLOW WITH NONNEGATIVE BISECTIONAL CURVATURE 105

This implies

(2.103)

where we used (2.102). Since (2.103) is tensorial, it holds in any holomorphic
coordinate system. We wish to compare the above formula with

1

f:lRaJh8 ~ 2 (\7 μ \7 P, + \7 P, 'V μ) Rai3'Y8·

To this end we compute (apply the second Bianchi identity (2.9) and
commute covariant derivatives (Exercise 2.22))

and

'V'Y\78Rai3 = 'V'Y\78Rai3μp, = 'V'Y'VμRai3μ8

= \7 P, \7 μRai3'Y8 - ~P,aDRvi3μ8 + R'Yp,vi3RaDμ8

• R'Yp,μDRai3v8 + R"(P,v8Rai3μD

'V p, 'V μRai3"f8 = 'V μ 'V p,Rai3'Y8 + Rμp,avRvi3'Y8 - Rμp,vi3Ra;;'Y8

+ RμP"fDRai3v8 - Rμp,v8Rai3"fD

= 'V μ 'V p,Rai3"f8 + RavRvi3'Y8 - Rvi3RaD"(8

+ R'Y;;Rai3v8 - Rv8Rai3'Yv·

Combining the formulas above yields

8

8t Rai3'Y8 = 'V p, 'V μRai3'Y8 - R'Yp,avRvi3μ8 + R'Yp,vi3Ravμ8

• ~;;Rai3v8 + R'Yp,v8Rai3μv - RaxR'Y8>-.i3
  = flRai3'Y8 - R"(P,aDRvi3μ8 + R'Yp,vi3RaDμ8 + R'Yp,v8Rai3μD
  1

- 2 ( RavRvi3'Y8 + Rvi3RaiJ"(8 + ~DRai3v8 + Rv8Rai3"fD) '

and the proposition follows. D

As a consequence of the proposition we have the following evolution
equations for the bisectional curvature and the Ricci tensor.

COROLLARY 2.82 (Bisectional curvature evolution).

(:t -fl) Raa'Y'? = t (1Rap,v;y[

2

• [Rap,'Y;;[

2

+ Raavp,RμD"f'?)

μ,v=l

n

(2.104) -I: Re ( Rap,Rμa'Y'? + R'Yp,Raaμ"f).

μ=1

Here Re(A) =~(A+ A) denotes the real part of a complex number A.