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Stability of Hypersurfaces with Constant \(r\)-Mean Curvature

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Abstract

We deal with compact hypersurfaces immersed in space forms with constant \(r\) -mean curvature. They are critical points for a variational problem. We show they are stable if and only if they are geodesic spheres, generalizing results on constant curvature hypersurfaces.

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References

  1. Alencar, H., do Carmo, M. and Colares, A. G.: Stable Hypersurfaces with Constant Scalar Curvature, Math. Z. 213 (1993), 117–131.

    Google Scholar 

  2. Alencar, H., do Carmo, M. and Rosenberg, H.: On the first eigenvalue of the linearized operator of the rth mean curvature of a hypersurface, Ann. Global Anal. Geom. 11 (1993), 387–395.

    Google Scholar 

  3. Barbosa J. L. and do Carmo, M.: Stability of hypersurfaces with constant mean curvature, Math. Z. 185 (1984), 339–353.

    Google Scholar 

  4. Barbosa, J. L., do Carmo, M. and Eschenburg, J.: Stability of hypersurfaces of constant mean curvature in Riemannian manifolds, Math. Z. 197 (1988), 123–138.

    Google Scholar 

  5. Caffarelli, Nirenberg and Spruck: The Dirichlet problem for nonlinear second order elliptic equations. III: Functions of the eigenvalues of the Hessian, Acta Math. 155 (1985), 262–301.

    Google Scholar 

  6. Cheng, S. Y. and Yau, S. T.: Hypersurfaces with constant scalar curvature, Math. Ann. 225 (1977), 195–204.

    Google Scholar 

  7. Hardy, G., Littlewood, J. E. and Pólya, G.: Inequalities, Cambridge University Press, Cambridge, 1991.

    Google Scholar 

  8. Korevaar, N.: Sphere theorems via Alexandrov for constant Weingarten curvature hypersurfaces, Appendix to a Note of A. Ros, J. Diff. Geom. 27 (1988), 221–223.

    Google Scholar 

  9. Reilly, R.: Variational properties of functions of the mean curvatures for hypersurfaces in space forms, J. Diff. Geom. 8 (1973), 465–477.

    Google Scholar 

  10. Rosenberg, H.: Hypersurfaces of constant curvature in space forms, Bull. Sc. Math., 2 e Série 117 (1993), 211–239.

    Google Scholar 

  11. Voss, K.: Einige Differentialgeometrische Kongruenzsätze für geschlossene Flächen und Hyperflächen, Math. Ann. 131 (1956), 180–218.

    Google Scholar 

  12. Voss, K.: Variation of curvature integrals, Results in Mathematics 20 (1991), 789–796.

    Google Scholar 

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Authors and Affiliations

  1. Departamento de Matemática, Universidade Federal do Ceará, Campus do Pici, 60455-760, Fortaleza-Ce, Brazil

    João Lucas Marques Barbosa & Antônio Gervasio Colares

Authors
  1. João Lucas Marques Barbosa
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  2. Antônio Gervasio Colares
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Marques Barbosa, J.L., Colares, A.G. Stability of Hypersurfaces with Constant \(r\)-Mean Curvature. Annals of Global Analysis and Geometry 15, 277–297 (1997). https://doi.org/10.1023/A:1006514303828

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