Original Paper

Bulletin of the Brazilian Mathematical Society

, Volume 35, Issue 2, pp 165-175

First online:

A characterization of Clifford tori with constant scalar curvature one by the first stability eigenvalue

  • Luis J. AlíasAffiliated withDepartamento de Matemáticas, Universidad de Murcia Campus de Espinardo Email author 
  • , Aldir BrasilJrAffiliated withDepartamento de Matemática, Universidade Federal do Ceará, Campus do Pici
  • , Luiz A. M. SousaJr.Affiliated withDepartamento de Matemática e Estatística, UNIRIO

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Abstract.

Let M be a compact hypersurface with constant scalar curvature one immersed into the unit Euclidean sphere \( \mathbb{S}^{{n + 1}} \). As is well-known, such hypersurfaces can be characterized variationally as critical points of the integral M Hdv. In this paper we derive a sharp upper bound for the first eigenvalue of the corresponding Jacobi operator in terms of the mean curvature of the hypersurface. Moreover, we prove that this bound is achieved only for the Clifford tori in \( \mathbb{S}^{{n + 1}} \) with scalar curvature one.

Keywords:

constant scalar curvature Clifford torus Jacobi operator first eigenvalue

Mathematical subject classification:

Primary 53C42 Secondary 53A10