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Stability of area-preserving variations in space forms

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Abstract

In this article, we deal with compact hypersurfaces without boundary immersed in space forms with \(\frac{S_{r+1}}{S_1} = {\rm constant}\) . They are critical points for an area-preserving variational problem. We show that they are r-stable if and only if they are totally umbilical hypersurfaces.

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References

  1. Alencar H., do Carmo M. and Colares A.G. (1993). Stable hypersurfaces with constant scalar curvature. Math. Z. 213: 117–131

    Article  MATH  MathSciNet  Google Scholar 

  2. Alencar H., do Carmo M. and Elbert M.F. (2003). Stability of hypersurfaces with vanishing r-mean curvatures in Euclidean spaces. J. Reine Angew. Math. 554: 201–216

    MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  4. Alencar H., Rosenberg H. and Santos W. (2004). On the Gauss map of hypersurfaces with constant scalar curvature in spheres. Proc. Am. Math. Soc. 132: 3731–3739

    Article  MATH  MathSciNet  Google Scholar 

  5. Barbosa J.L. and Colares A.G. (1997). Stability of hypersurfaces with constant r-mean curvature. Ann. Global Anal. Geom. 15: 277–297

    Article  MATH  MathSciNet  Google Scholar 

  6. Barbosa J.L.M. and do Carmo M. (2005). On stability of cones in \({\mathbb{R}}^{n+1}\) with zero scalar curvature. Ann. Global Anal. Geom. 28: 107–127

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  9. Cao L.F. and Li H. (2007). r-Minimal submanifolds in space forms. Ann. Global Anal. Geom. 32: 311–341

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  11. Hardy G.H., Littlewood J.E. and Polya G. (1934). Inequalities. Cambridge University Press, London

    Google Scholar 

  12. He, Y.J., Li, H.: A new variational characterization of the Wulff shape. Diff. Geom. App. (to appear in)

  13. Li H. (1996). Hypersurfaces with constant scalar curvature in space forms. Math. Ann. 305: 665–672

    Article  MATH  MathSciNet  Google Scholar 

  14. Li H. (1997). Global rigidity theorems of hypersurfaces. Ark. Mat. 35: 327–351

    Article  MATH  MathSciNet  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  16. Rosenberg H. (1993). Hypersurfaces of constant curvature in space forms. Bull. Soc. Math., 26 Série 117: 211–239

    MATH  Google Scholar 

  17. Yano K. (1970). Integral Formulas in Riemannian Geometry. Marcel Dekker, NY

    MATH  Google Scholar 

  18. Voss K. (1991). Variation of curvature integral. Results Math. 20: 789–796

    MathSciNet  MATH  Google Scholar 

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Correspondence to Haizhong Li.

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He, Y., Li, H. Stability of area-preserving variations in space forms. Ann Glob Anal Geom 34, 55–68 (2008). https://doi.org/10.1007/s10455-007-9095-3

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  • DOI: https://doi.org/10.1007/s10455-007-9095-3

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