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 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.

23 pages. Final version, corrected misquotation at p.5

References in corpus (3)

• The half-space property and entire positive minimal graphs in M x R
• Liouville type theorems for minimal graphs over manifolds
• Survey on the asymptotic Dirichlet problem for the minimal surface equation

Cited by in corpus (8)

• Detecting the completeness of a Finsler manifold via potential theory for its infinity Laplacian
• Non-negative Ricci curvature and Minimal graphs with linear growth
• Recent rigidity results for graphs with prescribed mean curvature
• A splitting theorem for capillary graphs under Ricci lower bounds
• On splitting complete manifolds via infinity harmonic functions
• Stochastic half-space theorems for minimal surfaces and $H$-surfaces of ${\mathbb{R}}^3$
• Half-space Liouville-type theorems for minimal graphs with capillary boundary
• Liouville theorem for minimal graphs over manifolds of nonnegative Ricci curvature