## Proceedings of the Japan Academy, Series A, Mathematical Sciences

### Complete flat fronts as hypersurfaces in Euclidean space

Atsufumi Honda

#### 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 $(n\geq 3)$.

#### Article information

Source
Proc. Japan Acad. Ser. A Math. Sci. Volume 94, Number 3 (2018), 25-30.

Dates
First available in Project Euclid: 28 February 2018

https://projecteuclid.org/euclid.pja/1519808414

Digital Object Identifier
doi:10.3792/pjaa.94.25

#### Citation

Honda, Atsufumi. Complete flat fronts as hypersurfaces in Euclidean space. Proc. Japan Acad. Ser. A Math. Sci. 94 (2018), no. 3, 25--30. doi:10.3792/pjaa.94.25. https://projecteuclid.org/euclid.pja/1519808414

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