Stability index jump for cmc hypersurfaces of spheres

2012/02/09 by Óscar Perdomo, Oscar M. Perdomo, Aldir Brasil +2
Mathematics · #53C40 53C42 #Algebraic Geometry and Number Theory #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #math.DG #msc:53C40 #msc:53C42

paper · pdf · doi:10.48550/arxiv.1202.2050

4 pages

arxiv created 2012/02/09 · openalex publication_date 2012/02/09 · arxiv updated 2012/02/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

It is known that the totally umbilical hypersurfaces in the (n+1)-dimensional spheres are characterized as the only hypersurfaces with weak stability index 0. That is, a compact hypersurface with constant mean curvature, cmc, in Sⁿ⁺¹, different from an Euclidean sphere, must have stability index greater than or equal to 1. In this paper we prove that the weak stability index of any non-totally umbilical compact hypersurface M⊂ Sⁿ⁺¹ with cmc cannot take the values 1,2,3... n.

Citations

• Minimal Varieties in Riemannian Manifolds
• Bifurcation of constant mean curvature tori in Euclidean spheres

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