Higher dimensional Scherk's hypersurfaces

2001/09/19 by Frank Pacard, Pacard, Frank
Mathematics · #53A07 #53A10 #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Point processes and geometric inequalities #math.AP #math.DG #msc:53A07 #msc:53A10

paper · pdf · doi:10.48550/arxiv.math/0109131

22 pages. Improved version

openalex publication_date 2001/09/19 · arxiv created 2001/11/15 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 \Rⁿ⁺¹, for n ≥ 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.

Related

• Riemann minimal surfaces in higher dimensions
• Higher dimensional Schwarz's surfaces and Scherk's surfaces
• The embedded singly periodic Scherk-Costa surfaces
• The space of doubly periodic minimal tori with parallel ends: Standard Examples
• The classification of doubly periodic minimal tori with parallel ends