Yau and Souplet-Zhang type gradient estimates on Riemannian manifolds with boundary under Dirichlet boundary condition

TL;DR

This paper develops gradient estimates and Liouville theorems for harmonic functions and ancient heat solutions on Riemannian manifolds with boundary, extending classical results to boundary-including settings.

Contribution

It introduces Yau and Souplet-Zhang type gradient estimates and Liouville theorems specifically for manifolds with boundary under Dirichlet conditions, a novel extension.

Findings

Established Yau type gradient estimate for harmonic functions.

Formulated Souplet-Zhang type gradient estimate for ancient heat solutions.

Proved Liouville theorems under Dirichlet boundary conditions.

Abstract

In this paper, on Riemannian manifolds with boundary, we establish a Yau type gradient estimate and Liouville theorem for harmonic functions under Dirichlet boundary condition. Under a similar setting, we also formulate a Souplet-Zhang type gradient estimate and Liouville theorem for ancient solutions to the heat equation.