The unboundedness of certain minimal submanifolds of positively curved Riemannian spaces

Abstract

We prove that a minimal immersion of a complete Riemannian manifold M into another complete noncompact Riemannian manifold N of positive curvature must have an unbounded image provided that M has scalar curvature bounded away from −∞. This extends the unboundedness theorems of Gromoll and Meyer for complete geodesics and of Galloway and Rodriguez for parabolic minimal surfaces. Furthermore, we prove that in case M is of codimension 1, only the Ricci curvature and not necessarily the full sectional curvature of the ambient space N need be positive in order for the same conclusion to hold.