Complete minimal submanifolds with nullity in Euclidean space

TL;DR

This paper classifies complete minimal submanifolds in Euclidean space with high nullity, showing they are cylinders over minimal surfaces under certain geometric conditions.

Contribution

It provides a classification result for minimal submanifolds with high nullity, extending understanding of their structure under the Omori-Yau maximum principle.

Findings

Submanifolds with nullity at least m-2 are cylinders over minimal surfaces.

The classification holds if the Omori-Yau maximum principle applies.

Properness of the immersion ensures the submanifold's structure.

Abstract

In this paper, we investigate minimal submanifolds in Euclidean space with positive index of relative nullity. Let $M^m$ be a complete Riemannian manifold and let $f\colon M^m\to\R^n$ be a minimal isometric immersion with index of relative nullity at least $m-2$ at any point. We show that if the Omori-Yau maximum principle for the Laplacian holds on $M^m$, for instance, if the scalar curvature of $M^m$ does not decrease to $-\infty$ too fast or if the immersion $f$ is proper, then the submanifold must be a cylinder over a minimal surface.