On Generalized Nonparametric Minimal Hyperfurfaces in High Dimension

Nonparametric g-surfaces in Euclidean space have recently been characterized by Bildhauer-Fuchs in terms of closure of a 1-form associated to the so called asymptotic normal. This 1-form can be written by means of the pull-back of a canonical vector-valued 1-form through a suitable map depending on the asymptotic normal, that in the minimal surfaces case agrees with the Gauss graph map. We show that a similar characterization holds true for g-hypersurfaces of any high dimension N, but this time in terms of a canonical vector valued form of degree N - 1. In the minimal hypersurfaces case, we finally discuss the lack of a relationship between the previous result and existence of good parameterizations, when N is greater than two.