Complete minimal submanifolds with nullity in the hyperbolic space

TL;DR

This paper classifies complete minimal submanifolds with nullity in hyperbolic space, showing they are either totally geodesic or cones over minimal surfaces in equidistant submanifolds, under scalar curvature bounds.

Contribution

It extends the classification of minimal submanifolds with nullity to hyperbolic space, complementing previous results in Euclidean and spherical spaces.

Findings

Submanifolds are either totally geodesic or cones over minimal surfaces.

Scalar curvature bounds lead to this classification.

Results generalize known classifications to hyperbolic space.

Abstract

We investigate complete minimal submanifolds $f\colon M^3\to\Hy^n$ 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 \cite{dksv1} and \cite{dksv2}, 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 $\Hy^n$.