Liouville type theorems for minimal graphs over manifolds

TL;DR

This paper proves that on certain complete Riemannian manifolds with specific geometric properties, any positive minimal graph must be constant, extending Liouville type theorems to these settings.

Contribution

It establishes Liouville type theorems for positive minimal graphs over manifolds satisfying volume doubling and Poincaré inequalities, broadening previous results.

Findings

Positive minimal graphic functions are constant on the specified manifolds

The result applies to manifolds with volume doubling and Neumann-Poincaré inequalities

Extends classical Liouville theorems to a broader geometric context

Abstract

Let $\Sigma$ be a complete Riemannian manifold with the volume doubling property and the uniform Neumann-Poincar$\mathrm{\acute{e}}$ inequality. We show that any positive minimal graphic function on $\Sigma$ is a constant.