Bernstein and Half-Space Properties for Minimal Graphs Under Ricci Lower Bounds
Giulio Colombo, Marco Magliaro, Luciano Mari, Marco Rigoli
Abstract
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.
Topics & Concepts
MathematicsRicci curvatureBounded functionSectional curvatureCurvatureManifold (fluid mechanics)Duality (order theory)Space (punctuation)Constant (computer programming)Curvature of Riemannian manifoldsPure mathematicsScalar curvatureMathematical analysisGeometryProgramming languageComputer scienceLinguisticsMechanical engineeringPhilosophyEngineeringGeometric Analysis and Curvature FlowsNonlinear Partial Differential EquationsGeometry and complex manifolds