Complete minimal submanifolds with nullity in Euclidean space

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 $f:M^m\to {\mathbb{R}}^n$ 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.

References in corpus (1)

• Associative Submanifolds of the 7-Sphere