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

We develop a general method to construct subsets of complete Riemannian manifolds that cannot contain images of non-constant harmonic maps from compact manifolds. We apply our method to the special case where the harmonic map is the Gauss map of a minimal submanifold and the complete manifold is a Grassmannian. With the help of a result by Allard [All72], we can study the graph case and have an approach to prove Bernstein-type theorems. This enables us to extend Moser’s Bernstein theorem [Mos61] to codimension two, i.e., a minimal p-submanifold in R, which is the graph of a smooth function defined on the entire R with bounded slope, must be a p-plane.