Bernstein and Half-Space Properties for Minimal Graphs Under Ricci Lower Bounds

Abstract

In this paper, we prove a new gradient estimate for minimal graphs defined on domains of a complete manifold $M$ with Ricci curvature bounded from below. This enables us to 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.

Liouville-type theorems for minimal graphs over manifolds

“…Here, the recurrence is equivalent to the parabolicity (see Grigor'yan [19]). Recently, Colombo-Magliaro-Mari-Rigoli [9] proved that a complete parabolic manifold with Ricci curvature bounded below has the half-space property.…”