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

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.