Calculus of Variations and Geometric Measure Theory

D. Mucci

On generalized nonparametric minimal hyperfurfaces in high dimension

created by mucci on 28 Dec 2022
modified on 28 Sep 2023


Published Paper

Inserted: 28 dec 2022
Last Updated: 28 sep 2023

Journal: J. Geom. Anal.
Volume: 33
Number: 356
Pages: 16
Year: 2023
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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.

Keywords: Generalized surfaces, Canonical forms, Parameterizations