Submanifolds with parallel Gaussian mean curvature vector in Euclidean spaces

Abstract

In the present paper, we prove a rigidity theorem for complete submanifolds with parallel Gaussian mean curvature vector in the Euclidean space $${\mathbb {R}}^{n+p}$$ under an integral curvature pinching condition, which is a unified generalization of some rigidity results for self-shrinkers and the $$\lambda $$-hypersurfaces in Euclidean spaces.