1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

8. KAHLER-RICCI FLOW WITH NONNEGATIVE BISECTIONAL CURVATURE 107

PROOF OF THEOREM 2. 78. (1) We first prove that the nonnegativity of

the bisectional curvature is preserved under the fl.ow. Analogous to Theorem

4.6 on p. 97 of Volume One, by the Kahler version of the maximum principle

for tensors (see Proposition 1 in §4 of [19]), we need to show that the

quadratic on the RHS of (2.104), i.e.,

(2.107)

n

Qodi"f1 ~ L (JRap,v;yJ

2

• JRap,'Y;;J

2

+ Raavp,Rμv"f"'f)

μ,v=l

n

• L Re ( RaμRμa'Y"'f + ~p,Raaμ;y) ,
  μ=1

satisfies the null eigenvector assumption. That is, we assume Raa'Y"'f = 0 for
some a and/ at some point x, and we shall prove that

(2.108)

at x. First observe that since Raa'Y"'f = 0 at x and the bisectional curvatures
are nonnegative, we have at x,

n n

L RaμRμa"f"'f = L ~μRaaμ"'f = 0.

μ=1 μ=1

By (2.107), in order to prove (2.108) at x, it suffices to show that

n n

(2.109) L Raavp,Rμv"f"'f :2: L (1Raji,"(vJ^2 - JRap,v;yJ^2 ) ·

μ,v=l μ,v=l

We shall prove (2.109) below, but first we show how the positivity of the
Ricci tensor and holomorphic sectional curvatures follow from the quasi-
positivity of the Ricci curvature at t = 0.

(2) Recall that the Ricci tensor Ra;a satisfies the Lichnerowicz heat equa-
tion, so that by taking a= f3 in (2.105), we have

(2.110)

{)

f)t Raa = b..Raa + Raa'Y8R5;y - Ra;y~a·

Since the nonnegativity of the bisectional curvature is preserved, by apply-
ing Hamilton's strong maximum principle for tensors to (2.110) (see
Theorem A.53 and also Part II of this volume), the Ricci tensor becomes
positive for all positive time. Now suppose there exists a space-time point

(x 0 , t 0 ), with to > 0, at which some holomorphic sectional curvature Raaaa is

zero. Since the holomorphic sectional curvature is nonnegative everywhere,
by (2.104), we have at (xo, to),

0 :2: (:t -b..) Raaaa = t (2 JRaav,uJ

2

• JRap,avl

2

),

μ,v=l