Complete flat fronts as hypersurfaces in Euclidean space

2017/09/07 by Atsufumi Honda, Honda, Atsufumi
Engineering · Mathematics · #53C42 #57R45 #Advanced Numerical Analysis Techniques #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Point processes and geometric inequalities

paper · pdf · doi:10.48550/arxiv.1709.02178

openalex publication_date 2017/09/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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≥ 3).

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