The Gauss image of entire graphs of higher codimension and Bernstein type theorems

TL;DR

This paper develops new conditions on the Gauss map of complete submanifolds in Euclidean space, leading to improved Bernstein-type theorems and regularity results for higher codimension minimal submanifolds.

Contribution

It introduces more general conditions on the Gauss map, constructs a strongly subharmonic function, and derives new a-priori estimates that enhance previous regularity and Bernstein results.

Findings

Improved Bernstein-type theorems for higher codimension minimal submanifolds.

Enhanced regularity results for submanifolds with parallel mean curvature.

Construction of a strongly subharmonic function under broader conditions.

Abstract

Under suitable conditions on the range of the Gauss map of a complete submanifold of Euclidean space with parallel mean curvature, we construct a strongly subharmonic function and derive a-priori estimates for the harmonic Gauss map. The required conditions here are more general than in previous work and they therefore enable us to improve substantially previous results for the Lawson-Osseman problem concerning the regularity of minimal submanifolds in higher codimension and to derive Bernstein type results.