Liouville theorem for minimal graphs over manifolds of nonnegative Ricci curvature

Abstract

Let $\Sigma$ be a complete Riemannian manifold of nonnegative Ricci curvature. We prove a Liouville-type theorem: every smooth solution $u$ to minimal hypersurface equation on $\Sigma$ is a constant provided $u$ has sublinear growth for its negative part. Here, the sublinear growth condition is sharp. Our proof relies on a gradient estimate for minimal graphs over $\Sigma$ with small linear growth of the negative parts of graphic functions via iteration.

14 pages

References in corpus (1)

• Non-negative Ricci curvature and Minimal graphs with linear growth