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

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.