Riemann minimal surfaces in higher dimensions

2006/03/28 by Saida Kaabachi, S. Kaabachi, Frank Pacard +3
Mathematics · #53A07 #53A10 #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Mathematical Dynamics and Fractals #math.DG #msc:53A07 #msc:53A10

paper · pdf · doi:10.48550/arxiv.math/0603662

arxiv created 2006/03/28 · openalex publication_date 2006/03/28 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We prove the existence of a one parameter family of minimal embedded hypersurfaces in Rⁿ⁺¹, for n ≥ 3, which generalize the well known 2 dimensional "Riemann minimal surfaces". The hypersurfaces we obtain are complete, embedded, simply periodic hypersurfaces which have infinitely many parallel hyperplanar ends. By opposition with the 2-dimensional case, they are not foliated by spheres.

Related

• Higher dimensional Scherk's hypersurfaces
• Existence of infinitely many minimal hypersurfaces in low dimensions, after F.C. Marques, A.A. Neves et A. Song (Bourbaki Seminar)
• The Riemann minimal examples
• Complete embedded minimal surfaces of finite total curvature
• New examples of higher-dimensional minimal hypersurfaces