On the index of minimal hypersurfaces of spheres

2019/02/27 by Óscar Perdomo, Perdomo, Oscar M.
Mathematics · #53A10 #53C42 #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Point processes and geometric inequalities

paper · pdf · doi:10.48550/arxiv.1902.10801

openalex publication_date 2019/02/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let M⊂ Sⁿ⁺¹⊂ℝⁿ⁺² be a compact minimal hypersurface of the n-dimensional Euclidean unit sphere. Let us denote by |A|² the square of the norm of the second fundamental form and J(f)=-Δf-nf-|A|²f the stability operator. It is known that the index (the number of negative eigenvalues of J) is 1 when M is a totally geodesic sphere, and it is n+3 when M is a Clifford minimal hypersurface. It has been conjectured that for any other minimal hypersurface, the index must be greater than n+3. One partial result for this conjecture states that if the index is n+3 and M is not Clifford, then ∫_M |A|²

Citations

• First stability eigenvalue characterization of Clifford hypersurfaces
• Minimal Varieties in Riemannian Manifolds

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