Complete Minimal Submanifolds with Nullity in the Hyperbolic Space

Abstract

We investigate complete minimal submanifolds $$f: M^3\rightarrow \mathbb {H}^n$$ in hyperbolic space with index of relative nullity at least one at any point. The case when the ambient space is either the Euclidean space or the round sphere was already studied in Dajczer et al. (Math Z 287: 481–491, 2017 and Comment Math Helv, to appear, 2017), respectively. If the scalar curvature is bounded from below we conclude that the submanifold has to be either totally geodesic or a generalized cone over a complete minimal surface lying in an equidistant submanifold of $$\mathbb {H}^n$$ .