A Bernstein-Type Theorem for Minimal Graphs of Higher Codimension via Singular Values

Abstract

Instead on the slope function, we establish a condition on the singular values of the differential of a vector-valued function $f:\mathbb {R}^{n} \to \mathbb {R}^{m}$ which ensures that such f satisfying the minimal surface equations is affine linear. This is a Bernstein type theorem for entire minimal graphs of arbitrary dimension and codimension, improving the results in Jost and Xin (Calc. Var. PDE 9, 277–296, 1999) and Wang (Trans. Amer. Math. Soc. 355, 265–271, 2003). The proof depends on the superharmoncity of an auxiliary function and on conclusions from geometric measure theory.