Complete flat fronts as hypersurfaces in Euclidean space

Abstract

By Hartman–Nirenberg’s theorem, any complete flat hypersurface in Euclidean space muast be a cylinder over a plane curve. However, if we admit some singularities, there are many non-trivial examples. {\it 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 $(n\geq 3)$.