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

  • Luis J. Alías
  • Aldir BrasilJr
  • Luiz A. M. SousaJr.
Original Paper


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.


constant scalar curvature Clifford torus Jacobi operator first eigenvalue 

Mathematical subject classification:

Primary 53C42 Secondary 53A10 


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Copyright information

© Sociedade Brasileira de Matemática 2004

Authors and Affiliations

  • Luis J. Alías
    • 1
  • Aldir BrasilJr
    • 2
  • Luiz A. M. SousaJr.
    • 3
  1. 1.Departamento de MatemáticasUniversidad de Murcia Campus de EspinardoEspinardo, MurciaSPAIN
  2. 2.Departamento de MatemáticaUniversidade Federal do Ceará, Campus do PiciFortaleza-CeBRAZIL
  3. 3.Departamento de Matemática e EstatísticaUNIRIORio de Janeiro-RJBRAZIL

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