The Geometry of Maximum Principles and a Bernstein Theorem in Codimension 2

Abstract

Moser's Bernstein theorem \cite{moser61} says that an entire minimal graph of codimension 1 with bounded slope must be a hyperplane. An analogous result for arbitrary codimension is not true, by an example of Lawson-Osserman. Here, we show that Moser's theorem nevertheless extends to codimension 2, i.e., a minimal $p$-submanifold in $R^{p+2}$, which is the graph of a smooth function defined on the entire $R^p$ with bounded slope, must be a $p$-plane. Our method depends on convexity properties of Grassmannians which come into play as the targets of the (harmonic) Gauss maps of our minimal submanifolds. In fact, we develop a general method to construct subsets of complete Riemannian manifolds that cannot contain images of non-constant harmonic maps from compact manifolds. When applied to Grassmannians, for codimension 2, it yields an appropriate domain that cannot contain nontrivial images of such harmonic maps. Our conclusion will then be reached by an application of Allard's theorem to handle the issue that the minimal submanifolds in question are not compact.