A spectral characterization of the H(r)-torus by the first stability eigenvalue

2004/09/16 by Luis J. Alías, Luis J. Alı́as, Abdênago Barros +1 · 6 citations
Mathematics · #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Geometric and Algebraic Topology

paper · pdf · doi:10.1090/s0002-9939-04-07559-8

Abstract

Let M be a compact hypersurface with constant mean curvature immersed into the unit Euclidean sphere \mathbb Sⁿ⁺¹. In this paper we derive a sharp upper bound for the first eigenvalue of the stability operator of M in terms of the mean curvature and the length of the total umbilicity tensor of the hypersurface. Moreover, we prove that this bound is achieved only for the so-called H(r)-tori in \mathbb Sⁿ⁺¹, with r²≤ (n-1)/n. This extends to the case of constant mean curvature hypersurfaces previous results given by Wu (1993) and Perdomo (2002) for minimal hypersurfaces.

Citations

• First stability eigenvalue characterization of Clifford hypersurfaces
• Local Rigidity Theorems for Minimal Hypersurfaces
• Minimal Varieties in Riemannian Manifolds

Cited by

• First eigenvalue of a Jacobi operator of hypersurfaces with a constant scalar curvature
• A lower bound for 𝐿₂ length of second fundamental form on minimal hypersurfaces
• First stability eigenvalue of singular hypersurfaces with constant mean curvature in the unit sphere
• First Eigenvalue of Jacobi operator and Rigidity Results for Constant Mean Curvature Hypersurfaces
• On the first stability eigenvalue of constant mean curvature surfaces\n into homogeneous 3-manifolds
• A lower bound for L₂ length of second fundamental form on minimal hypersurfaces

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