On the hypersufaces of the Euclidean space which are simultaneously minimal and maximal

TL;DR

This paper investigates hypersurfaces in higher-dimensional Euclidean spaces that are both minimal and maximal, revealing that their level curves are minimal hypersurfaces, extending classical results from surfaces to higher dimensions.

Contribution

It generalizes the classification of hypersurfaces that are both minimal and maximal from surfaces to higher-dimensional Euclidean spaces.

Findings

Level curves of such hypersurfaces are minimal hypersurfaces.

The work extends classical surface results to higher dimensions.

Provides a new perspective on the geometry of hypersurfaces.

Abstract

It is well known that the only surfaces that are simultaneously minimal in $\mathbb{R}^3$ and maximal in $\mathbb{L}^3$ are open pieces of helicoids (in the region in which they are spacelike) and of spacelike planes (O. Kobayashi 1983). The proof of this result consists in showing that the level curves of those surfaces are lines, and so the surfaces are ruled. And it finishes comparing the classification of minimal ruled surfaces to that of maximal ruled surfaces. In this manuscript we consider the general case of spacelike hypersurfaces in the $(n+1)$-dimensional Euclidean space which are simultaneously maximal and minimal. We show that its level curves are minimal hypersurfaces in the $n$-dimensional Euclidean space.