Half-space theorems for recurrent minimal and $H$-surfaces of ${\mathbb{R}}^3$

Abstract

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 ${\mathbb{R}}^3$. 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 $M\times {\mathbb{R}}_{+}$ equals some $M\times \{s\}$ provided $M$ is a complete, recurrent $n$-dimensional Riemannian manifold with ${\text{Ric}}_M\ge 0$ 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.