On graphic Bernstein type results in higher codimension

Abstract

Let \Sigma be a minimal submanifold of \R^{n+m} that can be represented as the graph of a smooth map f:\R^n-->\R^m. We apply a formula 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 $df$.