On graphic Bernstein type results in higher codimension

Source abstract

Let Σ \Sigma be a minimal submanifold of R n + m \mathbb {R}^{n+m} that can be represented as the graph of a smooth map f : R n ↦ R m 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 Σ \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 d f df .