Complete minimal submanifolds with nullity in Euclidean space

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 → 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 - ∞ too fast or if the immersion f is proper, then the submanifold must be a cylinder over a minimal surface.