Matrix Li-Yau-Hamilton Estimates for 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.