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, $M$ has nonnegative Ricci curvature with curvature bounded below, the same result holds for any positive entire minimal graph over $M$.
This is joint work with H. Rosenberg and J. Spruck.