- MATRIX DIFFERENTIAL HARNACK ESTIMATE 109
For the ·proof of this lemma, which is elementary in nature, we refer the
re~der to Mok [269].9. Matrix differential Harnack estimate for the Kahler-Ricci flow
In this and the next section we discuss various differential Harnack es-
timates for the Kahler-Ricci fl.ow and their geometric applications. Differ-
ential Harnack estimates for the Riemannian Ricci fl.ow will be discussed in
Part II. For Kahler-Ricci fl.ow, a fundamental result is H.-D. Cao's differen-
tial Harnack estimate for solutions with nonnegative bisectional curvature
(see [46]). Define
(2.111)
. 8. -. J Ra -
Z ( X) a)3 ~ a Ra'/3 + Ra;yR'Y'fJ + \7 'YRa'/JX'Y + \7 1 Ra'/3X'Y + Ra'/J'YJX'Y X + _p
' t ' t
for any (1, 0)-vector X = X'Y 8 ~ 7 and where X'Y ~ X'Y.
THEOREM 2.87 (Kahler matrix Harnack estimate). If (Mn,g(t)) is a
complete solution to the Kahler-Ricci flow with bounded nonnegative bisec-
tional curvature, then ·..(2.112)for any (1, 0)-vector X.
This result may be considered as the space-time·.analogue of Theorem
2.78. We shall also see a similar analogy for the Riemannian Ricci fl.ow in
Part II, where Hamilton's matrix differential Harnack estimate will appear
as the space-time analogue of the result that nonnegative curvature operator
is preserved under the Ricci fl.ow.
9.1. Trace differential Harnack estimate for the Kahler-Ricci
flow. Taking the trace of the estimate (2.112) leads to the so-called trace
differential Harnack estimate, after applying the second Bianchi iden-
tity.
COROLLARY 2.88 (The Kahler trace differential Harnack estimate). Let
(Mn,g(t)) be a complete solution to the Kahler-Ricci flow with bounded
nonnegative bisectional curvature. Then(2.113)PROOF. By (2.112) we have- -8 2 J R
0::::: ga(J za'/3 = ga(J at Ra'/3 + IRa'/31 + \7 'YRX'Y + \7;yRX^1 + R'Y3X'Y x + t'
and (2.113) follows from (2.37). D