Complete flat fronts as hypersurfaces in Euclidean space

TL;DR

This paper proves that in Euclidean spaces of dimension three and higher, complete flat fronts with singularities cannot exist, extending the understanding of flat hypersurfaces beyond classical smooth cases.

Contribution

It establishes a non-existence result for complete flat fronts with singularities in Euclidean spaces of dimension three and above, generalizing previous results in three dimensions.

Findings

Complete flat fronts with singularities do not exist in Euclidean (n+1)-space for n ≥ 3.

The classical cylinder over a plane curve is the only complete flat hypersurface without singularities.

The result contrasts with the three-dimensional case where such fronts do exist.

Abstract

By Hartman--Nirenberg's theorem, any complete flat hypersurface in Euclidean space must be a cylinder over a plane curve. However, if we admit some singularities, there are many non-trivial examples. Flat fronts are flat hypersurfaces with admissible singularities. Murata--Umehara gave a representation formula for complete flat fronts with non-empty singular set in Euclidean $3$-space, and proved the four vertex type theorem. In this paper, we prove that, unlike the case of $n=2$, there do not exist any complete flat fronts with non-empty singular set in Euclidean $(n+1)$-space $(n\geq 3)$.