Half-space theorems for recurrent minimal and H-surfaces of R³

We prove a version of the strong half-space theorem between the classes of recurrent minimal surfaces and complete minimal surfaces with bounded curvature of R³ . The use of subsolutions in the barrier sense allow us to deal with non-proper minimal surfaces immersed with bounded curvature. We show that any minimal hypersurface immersed with bounded curvature in M× ₊ equals some M× \{s\} provided M is a complete, recurrent n -dimensional Riemannian manifold with {Ric}M ≥ 0 and whose sectional curvatures are bounded from above. Furthermore, we prove a half-space theorem for the class of stochastically complete H -surfaces. We present a maximum principle at infinity assuming M has non-empty boundary. Finally, we present examples of a complete non-proper recurrent minimal surface with unbounded curvature.