Elliptic Gradient estimates for the heat equation on a weighted manifold with time-dependent metrics and potentials

Abstract

In this paper, we get some elliptic type gradient estimates on positive solutions to the heat equation on a weighted Riemannian manifold with time dependent metrics and potentials. The
geometry of the space in terms of curvature bounds play crucial role in determining the estimates. The gradient estimates derived are useful in proving the classical Harnack inequalities, Liouville type theorems, heat kernel bounds, e.t.c. As an example, we discuss Liouville principle on bounded positive solution. Indeed, each gradient estimate obtained is equivalent to
saying bounded weighted harmonic function is a constant.