A Bernstein-type theorem for $xi$ -submanifolds with flat normal bundle in the Euclidean spaces

Abstract

$xi$ -Submanifolds in the Euclidean spaces are a natural extension of self-shrinkers and a generalization of$lambda$-hypersurfaces. Moreover, $xi$ -submanifolds are expected to take the place of submanifolds with parallel mean curvaturevector. In this paper, we establish a Bernstein-type theorem for $xi$ -submanifolds in the Euclidean spaces. More precisely,we prove that an n-dimensional smooth graphic $xi$ -submanifold with flat normal bundle in $mathbb{R}^{n+p}$ is an affine n-plane.