On graphic Bernstein type results in higher codimension

Abstract

Let $\Sigma$ be a minimal submanifold of $\mathbb {R}^{n+m}$ that can be represented as the graph of a smooth map $f:\mathbb {R}^n\mapsto \mathbb {R}^m$. We apply a formula that we derived in the study of mean curvature flow to obtain conditions under which $\Sigma$ must be an affine subspace. Our result covers all known ones in the general case. The conditions are stated in terms of the singular values of $df$.