A Matrix Li-yau-hamilton Estimate for Kähler-ricci Flow

In this paper we prove a new matrix Li-Yau-Hamilton (LYH) estimate for Kähler-Ricci flow on manifolds with nonnegative bisectional curvature. The form of this new LYH estimate is obtained by the interpolation consideration originated in [Ch] by Chow. This new inequality is shown to be connected with Perelman’s entropy formula through a family of differential equalities. In the rest of the paper, We show several applications of this new estimate and its corresponding estimate for linear heat equation. These include a sharp heat kernel comparison theorem, generalizing the earlier result of Li and Tian [LT], a manifold version of Stoll’s theorem [St] on the characterization of ‘algebraic divisors’, and a localized monotonicity formula for analytic subvarieties, which sharpens the Bishop volume comparison theorem. Motivated by the connection between the heat kernel estimate and the reduced volume monotonicity of Perelman [P], we prove a sharp lower bound heat kernel estimate for the time-dependent heat equation, which is, in a certain sense, dual to Perelman’s monotonicity of the reduced volume. As an application of this new monotonicity formula, we show that the blow-down limit of a certain type long-time solution is a gradient expanding soliton. We also illustrate the connection between the new LYH estimate and the Hessian comparison theorem of [FIN] on the forward reduced distance. Local monotonicity formulae on entropy and forward reduced volume are also derived.