Kähler-Ricci Flow and the Poincaré-Lelong Equation

has been studied extensively and important applications were given. In [N1] and [N-S-T], the Poincare-Lelong equation has been solved under more general conditions than in [M-S-Y]. The conditions in [N-S-T] are more in line with the conditions in [Sh2-4]. Since a solution of (0.1) is a potential for the Ricci tensor, it is interesting to see if one can apply (0.1) to study solutions of (0.2). In this work, on the one hand we shall study the Kahler-Ricci flows by using solutions of the Poincare-Lelong equation. On the other hand, we will also refine some of the results in [Sh3, C-Z, C-T-Z] and give new applications. The hinge between the equations (0.1) and (0.2) is that by solving (0.1) one can then construct a function u(x, t) which satisfies the time-dependent heat equation ( ∂ ∂t − ∆)u(x, t) = 0 and the time-dependent Poincare-Lelong equation √−1∂∂u = Ricg(t) simultaneously. It then can simplify the study of (0.2) quite a bit. It also suggests some of the refined

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