# Higher dimensional Scherk's hypersurfaces

Abstract : In 3-dimensional Euclidean space, Scherk second surfaces are singly periodic embedded minimal surfaces with four planar ends. In this paper, we obtain a natural generalization of these minimal surfaces in any higher dimensional Euclidean space ${\R}^{n+1}$, for $n \geq 3$. More precisely, we show that there exist $(n-1)$-periodic embedded minimal hypersurfaces with four hyperplanar ends. The moduli space of these hypersurfaces forms a 1-dimensional fibration over the moduli space of flat tori in ${\R}^{n-1}$. A partial description of the boundary of this moduli space is also given.
Document type :
Journal articles
Domain :

https://hal.archives-ouvertes.fr/hal-00127006
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Submitted on : Saturday, January 27, 2007 - 3:39:09 PM
Last modification on : Thursday, March 19, 2020 - 12:26:02 PM

### Citation

Frank Pacard. Higher dimensional Scherk's hypersurfaces. Journal de Mathématiques Pures et Appliquées, Elsevier, 2002, 81, pp.241-258. ⟨hal-00127006⟩

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