Matrix Li-Yau-Hamilton Estimates under K\"ahler-Ricci Flow

TL;DR

This paper establishes matrix Li-Yau-Hamilton estimates for heat equations under the Kähler-Ricci flow, leading to a new monotonicity formula that enhances understanding of geometric evolution in complex manifolds.

Contribution

It introduces matrix Li-Yau-Hamilton estimates for heat equations coupled with the Kähler-Ricci flow, providing new tools for analyzing geometric flows in Kähler geometry.

Findings

Proved matrix Li-Yau-Hamilton estimates for heat and conjugate heat equations.

Derived a monotonicity formula from these estimates.

Enhanced understanding of the behavior of solutions under Kähler-Ricci flow.

Abstract

We prove matrix Li-Yau-Hamilton estimates for positive solutions to the heat equation and the backward conjugate heat equation, both coupled with the K\"ahler-Ricci flow. As an application, we obtain a monotonicity formula.