1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

9. MATRIX DIFFERENTIAL HARNACK ESTIMATE 115

where Wa = 9f3a Wf3, then (2.119) implies

( :t -~) (zai3WaW,e)

= (Rai3'YJZ^0 1) WaW,e + (PaJ'YPei3 1 - PaJ;yPoi3'Y) WaW,e

1 2

• 2 (Ra1Z'Yi3 + Za1R'Yi3) WaWf3 - tzaiJWaW,e

= Rai3'YJZo;yWaWf3 + IM'YJl^2 - IM 10 j^2.

Here

and

Now we use the facts that Zai3 ~ 0 (at least on all of M x [O, to]) and Wis

a null vector of Zai3· An algebraic fact, similar to Lemma 2.86 and using a
second variation computation similar to that in the proof of Theorem 2. 78,
shows that

Rai3'YJZo;yWaWf3 ~ IM'YJl^2 - IM'Yol

2

.

Thus at a point where W satisfies (2.120)~(2.121), we have

Hence, by the maximum principle, on a closed manifold we have Zai3 ~ 0 on
all of space and time. In the complete noncompact case, one can adapt the
proof in Part II of this volume of Hamilton's matrix Harnack estimate for
complete solutions to the Ricci fl.ow with nonnegative curvature operator to
this Kahler setting without significant modifications.
Now we give the two lemmas which are needed to complete the proof of
Proposition 2.93.

LEMMA 2.94.

( :t -~) ( ~Rai3 + Rai3'YJR10)

1 1

= 2~RapRpi3 + 2Rap~Rpi] + 2Rap,"'fRpi3,1

1

+ 2Ra1 ('\Ji\l'YRai3 + '\J'Y'\JJRai3) - ~ (Ra1R'Yi3)

1

+ 2 ('\J J'\J 'YRai]+ '\J 'Y '\JJRai3-Rap"'fJRpij-Ra-pRPi3'YJ) R^01

(2.122)

a

+ 2Rai3'YJRa-pRp;y + RahJ at Ra1·