Atsufumi Honda
Published 2017-09-07Version 1
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 $(n\geq 3)$.
Soap bubbles and isoperimetric regions in the product of a closed manifold with Euclidean space
A pinching theorem for the first eigenvalue of the laplacian on hypersurface of the euclidean space
Isothermic submanifolds of Euclidean space
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