In this paper, a sharp linear trace Li-Yau-Hamilton inequality for Kähler-Ricci flow is proved. The new inequality extends the previous trace Harnack inequality obtained by H.-D. Cao. We also establish sharp gradient estimates for the positive solution of the time-dependent heat equation for some cases. Finally, we apply this new linear trace Li-Yau-Hamilton inequality to study the Liouville properties of the plurisubharmonic functions on complete Kähler manifolds with bounded nonnegative holomorphic bisectional curvature.
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Ni et al. (2003) studied this question.
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