Matrix Li-Yau-Hamilton Estimates for Nonlinear Heat Equations

TL;DR

This paper develops matrix Li-Yau-Hamilton estimates for nonlinear heat equations on Kähler and Riemannian manifolds, including evolving metrics and constrained cases, extending classical results to broader geometric contexts.

Contribution

It introduces new matrix estimates for nonlinear heat equations on Kähler manifolds with fixed and evolving metrics, generalizing previous scalar estimates and covering more general equations.

Findings

Derived estimates on fixed Kähler manifolds.

Extended estimates to evolving Kähler-Ricci flow.

Generalized results to broader nonlinear heat equations.

Abstract

In this paper we are concerned with the matrix Li-Yau-Hamilton estimates for nonlinear heat equations. Firstly, we derive such estimate on a K\"{a}hler manifold with a fixed K\"{a}hler metric. Then we consider the estimate on K\"{a}hler manifolds with K\"{a}hler metrics evolving under the rescaled K\"{a}hler-Ricci flow. Both of the estimates are generalized to constrained cases. Finally, we extend the estimtes to more general nonlinear heat equations on both Riemannian manifolds and K\"{a}hler manifolds.