Upper Bounds for Hessian Matrices of Positive Solutions to Heat Equations on Kähler Manifolds

Abstract

We prove global and local upper bounds for the Hessian matrices of positive solutions to the heat equation on Kähler manifolds whose bisectional curvature is bounded from below. We also improve a result of Han and Zhang by weakening the curvature assumptions in their Hessian estimates on Riemannian manifolds. More precisely, we extend their global and local upper bounds, originally obtained under two-sided curvature bounds, to Riemannian manifolds with sectional curvature bounded from below.