Half-Space Theorems for Minimal Surfaces with Bounded Curvature

Abstract

First we prove a version of the Strong Half-Space Theorem for minimal surfaces with bounded curvature in ℝ³. With the techniques developed in our proof we give criteria for deciding if a complete minimal surface is proper. We prove a mixed version of the Strong Half-Space Theorem. Turning to 3-dimensional manifolds of bounded geometry and positive Ricci curvature, we show that complete injectively immersed minimal surfaces with bounded curvature are proper and as a corollary we have a Half-Space Theorem in this setting. Finally we show an application of the maximum principle for nonproper minimal immersions in ℝ³.