Complete minimal submanifolds with nullity in Euclidean space

摘要: 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 \(f:M^m\rightarrow \mathbb {R}^n\) isometric immersion nullity at least \(m-2\) any point. We show that if the Omori–Yau maximum principle for Laplacian holds on \(M^m\), instance, scalar curvature does not decrease to \(-\infty \) too fast or f is proper, then submanifold must cylinder over surface.