Bernstein and half-space properties for minimal graphs under Ricci lower bounds

TL;DR

This paper establishes new gradient estimates for minimal graphs on manifolds with Ricci curvature bounds, proving constancy of positive entire minimal graphs and the half-space property under certain conditions, using nonlinear potential theory techniques.

Contribution

It introduces a novel gradient estimate for minimal graphs on manifolds with Ricci lower bounds and applies it to prove constancy and half-space properties without sectional curvature assumptions.

Findings

Positive, entire minimal graphs on non-negative Ricci curvature manifolds are constant.

Complete, parabolic manifolds with Ricci lower bounds have the half-space property.

Avoids sectional curvature bounds by using nonlinear potential theory.

Abstract

In this paper, we prove a new gradient estimate for minimal graphs defined on domains of a complete manifold with Ricci curvature bounded from below. In particular, we show that positive, entire minimal graphs on manifolds with non-negative Ricci curvature are constant, and that complete, parabolic manifolds with Ricci curvature bounded from below have the half-space property. We avoid the need of sectional curvature bounds on $M$ by exploiting a form of the Ahlfors-Khas'minskii duality in nonlinear potential theory.