The half-space property and entire positive minimal graphs in $M \times \mathbb{R}$.

Abstract

We show that a properly immersed minimal hypersurface in $M \times \mathbb{R}^+$ equals some $M \times\{c\}$ when $M$ is a complete, recurrent $n$-dimensional Riemannian manifold with bounded curvature. If on the other hand, $M$ is not necessarily recurrent but has nonnegative Ricci curvature with curvature bounded below, the same result holds for any positive entire minimal graph over $M$.