The half-space property and entire positive minimal graphs in M x R
  We show that a properly immersed minimal hypersurface in M x R_+ equals some M x {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. 
Harold Rosenberg, Felix Schulze, Joel Spruck
math.DG math.AP