On graphic Bernstein type results in higher codimension

Let Sigma be a minimal submanifold of Rn+m that can be represented as the graph of a smooth map f : R-n bar right arrow 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 a ne subspace. Our result covers all known ones in the general case. The conditions are stated in terms of the singular values of df.