Bernstein type gradient estimation for weighted local heat equation

Abstract

In this article we derive Bernstein type gradient estimation for local weighted heat equation on static weighted Riemannian manifold and evolving weighted Riemannian manifold along Ricci flow and extended Ricci flow. We showed that along Ricci flow and extended Ricci flow we can derive Bernstein type estimation for weighted heat equation without any assumption on the boud of Bakry-\'Emery Ricci curvature.