2009/03/07 by Oscar Perdomo, Óscar Perdomo, Perdomo, Oscar · 4 citations
Engineering · Mathematics · #3D Shape Modeling and Analysis #53 C 42 #Advanced Numerical Analysis Techniques #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #math.DG #msc:42 #msc:53
paper · pdf · doi:10.48550/arxiv.0903.1321
38 pages, 21 figures
arxiv created 2009/03/07 · openalex publication_date 2009/03/07 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let m>1 and n>1 be any pair of integers. In this paper we prove that if H is between the numbers \cot(\fracπm) and bm,n=((m2-2)√(n-1))/(n√(m2-1)), then, there exists a non isoparametric, compact embedded hypersurface in Sn+1 with constant mean curvature H that admits the group O(n)x Zm in their group of isometries, here O(n) is the set of n x n orthogonal matrices and Zm are the integers mod m. When m=2 and H is close to the boundary value 0, the hypersurfaces look like two very close n-dimensional spheres with two catenoid necks attached, similar to constructions made by Kapouleas. When m>2 and H is close to \cot(\fracπm), the hypersurfaces look like a necklets made out of m spheres with (m+1) catenoid necks attached, similar to constructions made by Butscher and Pacard. In general, when H is close to bm,n the hypersurface is close to an isoparametric hypersurface with the same mean curvature. As a consequence of the expression of these bounds for H, we have that every H different from 0,±(1)/(√(3)) can be realized as the mean curvature of a non isoparametric CMC surface in S3. For hyperbolic spaces we prove that every non negative H can be realized as the mean curvature of an embedded CMC hypersurface in Hn+1, moreover we prove that when H>1 this hypersurface admits the group O(n)× Z in its group of isometries. Here Z are the integer numbers. As a corollary of the properties proven for these hypersurfaces, for any n> 5, we construct non isoparametric compact minimal hypersurfaces in Sn+1 which cone in Rn+2 is stable. Also, we will prove that the stability index of every non isoparametric minimal hypersurface with two principal curvatures in Sn+1 is greater than 2n+5.