Matrix Li-Yau-Hamilton estimates for nonlinear heat equations

In this paper we are concerned with the matrix Li-Yau-Hamilton estimates for nonlinear heat equations. Firstly, we derive such an estimate on a Kahler manifold with a fixed Kahler metric. Then we consider the estimate on Kahler manifolds with Kahler metrics evolving under the rescaled Kahler-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 Kahler manifolds.