Plurisubharmonic functions and the Kähler-Ricci flow

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.