Bulletin of the Brazilian Mathematical Society

, Volume 35, Issue 2, pp 165–175

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

Authors

    • Departamento de MatemáticasUniversidad de Murcia Campus de Espinardo
  • Aldir BrasilJr
    • Departamento de MatemáticaUniversidade Federal do Ceará, Campus do Pici
  • Luiz A. M. SousaJr.
    • Departamento de Matemática e EstatísticaUNIRIO
Original Paper

DOI: 10.1007/s00574-004-0009-8

Cite this article as:
Alías, L.J., Brasil, A. & Sousa, L.A.M. Bull Braz Math Soc, New Series (2004) 35: 165. doi:10.1007/s00574-004-0009-8

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 MHdv. 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 curvatureClifford torusJacobi operatorfirst eigenvalue

Mathematical subject classification:

Primary 53C42Secondary 53A10

Copyright information

© Sociedade Brasileira de Matemática 2004