Advertisement

ANNALI DELL'UNIVERSITA' DI FERRARA

, Volume 64, Issue 2, pp 427–436 | Cite as

Stable hypersurfaces via the first eigenvalue of the anisotropic Laplacian operator

  • Jonatan Floriano da Silva
  • Henrique Fernandes de Lima
  • Marco Antonio Lázaro Velásquez
Article
  • 16 Downloads

Abstract

In this paper we provide a characterization for stable hypersurfaces with constant anisotropic mean curvature immersed in the Euclidean space \(\mathbb {R}^{n+1}\) through the analysis of the first eigenvalue of the anisotropic Laplacian operator.

Keywords

Euclidean space Wulff shape Anisotropic mean curvatures \((r, s, F)\)-linear Weingarten surfaces Stable closed hypersurfaces Jacobi operator 

Mathematics Subject Classification

Primary 53C42 Secondary 53B25 

Notes

Acknowledgements

The authors would like to thank the referee for giving valuable suggestions and useful comments which improved the paper.

References

  1. 1.
    Barbosa, J.L.M., Colares, A.G.: Stability of hypersurfaces with constant \(r\)-mean curvature. Ann. Global Anal. Geom. 15, 277–297 (1997)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Clarenz, U.: The Wulff-shape minimizes an anisotropic Willmore functional. Interfaces Free Bound. 6, 351–359 (2004)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Colares, A.G., da Silva, J.F.: Stable hypersurfaces as minima of the integral of an anisotropic mean curvature preserving a linear combination of area and volume. Math. Z. 275, 595–623 (2013)MathSciNetCrossRefGoogle Scholar
  4. 4.
    de Lima, H.F., Velásquez, M.A.L.: A new characterization of \(r\)-stable hypersurfaces in space forms. Arch. Math. 47, 119–131 (2011)MathSciNetzbMATHGoogle Scholar
  5. 5.
    He, Y., Li, H.: A new variational characterization of the Wulff shape. Differ. Geom. Appl. 26, 377–390 (2008)MathSciNetCrossRefGoogle Scholar
  6. 6.
    He, Y., Li, H.: Integral formulae of Minkowski type and new characterization of the Wulff shape. Acta Math. Sin. 24, 697–704 (2008)MathSciNetCrossRefGoogle Scholar
  7. 7.
    He, Y., Li, H.: Stability of hypersurfaces with constant \((r+1)\)-th anisotropic mean curvature. IlI. J. Math. 52, 1301–1314 (2008)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Koiso, M., Palmer, B.: Geometry and stability of surfaces with constant anisotropic mean curvature. Indiana Univ. Math. J. 54, 1817–1852 (2005)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Li, H.: Hypersurfaces with constant scalar curvature in space forms. Math. Ann. 305, 665–672 (1996)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Li, H.: Global rigidity theorems of hypersurfaces. Ark. Math. 35, 327–351 (1997)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Palmer, B.: Stability of the Wulff shape. Proc. Am. Math. Soc. 126, 3661–3667 (1998)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Rosenberg, H.: Hypersurfaces of constant curvature in space forms. Bull. Sci. Math. 117, 217–239 (1993)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Taylor, J.: Crystalline variational problems. Bull. Am. Math. Soc. 84, 568–588 (1978)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Winklmann, S.: A note on the stability of the Wulff shape. Arch. Math. 87, 272–279 (2006)MathSciNetCrossRefGoogle Scholar

Copyright information

© Università degli Studi di Ferrara 2018

Authors and Affiliations

  • Jonatan Floriano da Silva
    • 1
  • Henrique Fernandes de Lima
    • 2
  • Marco Antonio Lázaro Velásquez
    • 2
  1. 1.Departamento de MatemáticaUniversidade Federal do CearáFortalezaBrazil
  2. 2.Departamento de MatemáticaUniversidade Federal de Campina GrandeCampina GrandeBrazil

Personalised recommendations