1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

108 2. KAHLER-RICCI FLOW

where we used l::~=l Re (Rap,Rμaaii + Rap,Raaμa) = 0 at (xo, to). On the

other hand, by (2.109) with a="(, we have

n n

L IRaavp,1

2

2:: L IRap,avl

2
·
μ,v=l μ,v=l
Hence we conclude
Raavp, = Rap,aD = 0
at (xo, to) for allμ, v. This in turn implies Raa = 0, which is a contradiction.
(1) continued. We now verify (2.109). Consider the following Hermitian
symmetric form defined by the bisectional curvatures:

Q(X,Y,s) - ~Rmc (8 ~ +sX, ~^8 +sX,~^8 +sY,~^8 +sY ) 2:: 0

uza uzOt uz'Y uz'Y.

for X, Y E T^1 ,o M and s E R At a point where the bisectional curvatures
are nonnegative and Raii'Yi = 0, we have Q(X, Y, s) 2:: 0 and Q(X, Y, 0) = 0
for all X, YE T^1 ,o M ands ER Therefore the second variation at s = 0 is
nonnegative:

d2 I -

0 :S ds2 s=O Q(X, Y, s)

(

=Rmc X,X,~,~ -88) +Rmc (88-) ~,~,Y,Y

uz'Y uz'Y · uza uza

• 2Re ( Rmc ( X, 8 ~a' Y, 8 ~ 7 ) + Rmc ( X, 8 ~a' 8 ~ 7 , Y)).

In terms of a unitary (1,0)-frame {ei}f= 1 , we may write this as

Ri]"(;yXi Xj + 2 Re ( Rwfixiyj + Ria 7 3Xiyj) + Raai] yiyj 2:: 0,

where X ~ 2::~ 1 Xiei and Y ~ l::j= 1 Yjej. By Lemma 2.86 below, we have

n n

L Ri]7;yRaaj'i 2:: L (1Riafil

2

-1Riii'YJl

2
) ·
i,j=l i,j=I
The claimed inequality (2.109) follows and this completes the proof of
Theorem 2.78. D

LEMMA 2.86. Let Q(X, Y) be a Hermitian symmetric quadratic form

defined by

Q(X, Y) =Ai] Xi xj + 2 Re ( Bijxiyj + Di]xiyj) + ci]yiyj.

If Q is semi-positive definite, then

n n

L Ai]Cj"i 2:: L (IBijl

2

• IDi3l

2

).

i,j=l i,j=l