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Global rigidity theorems of hypersurfaces

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References

  1. Akutagawa, K., On spacelike hypersurfaces with constant mean curvature in the de Sitter space.Math. Z.196 (1987), 13–19.

    Article  MATH  MathSciNet  Google Scholar 

  2. Alencar, H. andDo Carmo, M., Hypersurfaces with constant mean curvature in spheres,Proc. Amer. Math. Soc.120 (1994), 1223–1229.

    Article  MATH  MathSciNet  Google Scholar 

  3. Chen, B. Y. andOkumura, M., Scalar curvature, inequality and submanifold,Proc. Amer. Math. Soc.38 (1973), 605–608.

    Article  MATH  MathSciNet  Google Scholar 

  4. Cheng, S. Y. andYau, S. T., Maximal spacelike hypersurfaces in the Lorentz-Minkowski spaces,Ann. of Math.104 (1976), 407–419.

    Article  MathSciNet  Google Scholar 

  5. Cheng, S. Y. andYau, S. T., Hypersurfaces with constant scalar curvature,Math. Ann.225 (1977), 195–204.

    Article  MATH  MathSciNet  Google Scholar 

  6. Chern, S. S., Some new characterizations of the Euclidean sphere,Duke Math. J.12 (1945), 279–290.

    Article  MATH  MathSciNet  Google Scholar 

  7. Chern, S. S., Do Carmo, M. andKobayashi, S., Minimal submanifolds of a sphere with second fundamental form of constant length, inFunctional Analysis and Related Fields (F. Brower, ed.), pp. 59–75, Springer-Verlag, Berlin, 1970.

    Google Scholar 

  8. Dajczer, M. andNomizu, K., On the flat surfaces inS 31 andH 31 , inManifolds and Lie Groups (Hano, J., Morimoto, A., Murakami, S., Okamoto, K. and Ozeki, H., eds.), pp. 71–108, Birkhäuser, Boston, Mass., 1981.

    Google Scholar 

  9. Derdziński, A., Some remarks on the local structure of Codazzi tensors, inGlobal Differential Geometry and Global Analysis (Ferus, D., Kühnel, W., Simon, U. and Wegner, B., eds.), Lecture Notes in Math.838, pp. 251–255, Springer-Verlag, Berlin-Heidelberg, 1981.

    Chapter  Google Scholar 

  10. Goddard, A. J., Some remarks on the existence of spacelike hypersurfaces of constant mean curvature,Math. Proc. Cambridge Philos. Soc.82 (1977), 489–495.

    Article  MATH  MathSciNet  Google Scholar 

  11. Hartman, P., On complete hypersurfaces of nonnegative sectional curvature,Trans. Amer. Math. Soc.245 (1978), 363–374.

    Article  MATH  MathSciNet  Google Scholar 

  12. Lawson, H. B., Local rigidity theorems for minimal hypersurfaces,Ann. of Math.89 (1969), 187–197.

    Article  MathSciNet  Google Scholar 

  13. Leung, P. F., An estimate on the Ricci curvature on a submanifold and some applications,Proc. Amer. Math. Soc.114 (1992), 1051–1061.

    Article  MATH  MathSciNet  Google Scholar 

  14. Li, H., Hypersurfaces with constant scalar curvature in space forms,Math. Ann.305 (1966), 665–672.

    Google Scholar 

  15. Montiel, S., An integral inequality for compact spacelike hypersurfaces in de Sitter space and applications to the case of constant mean curvature,Indiana Univ. Math. J.37 (1988), 909–917.

    Article  MATH  MathSciNet  Google Scholar 

  16. Montiel, S. andRos, A., Compact hypersurfaces: The Alexandrov theorem for higher order mean curvatures, inDifferential Geometry, a Symposium in Honor of M. do Carmo (Lawson, H. B. and Tenenblat, K. eds.), Pitman Monographs Surveys Pure Appl. Math.52, pp. 279–296, Longman, Harlow, 1991.

    Google Scholar 

  17. Nomizu, K. andSmyth, B., A formula of Simon's type and hypersurfaces,J. Differential Geom.3 (1969), 367–377.

    MATH  MathSciNet  Google Scholar 

  18. Okumura, M., Hypersurfaces and a pinching problem on the second fundamental tensor,Amer. J. Math.96 (1974), 207–213.

    Article  MATH  MathSciNet  Google Scholar 

  19. Olike, V. I. andSimon, U., Codazzi tensors and equations of Monge-Ampère type on compact manifolds of constant sectional curvature,J. Reine Angew. Math.342 (1983), 35–65.

    MathSciNet  Google Scholar 

  20. Omori, H., Isometric immersions of Riemannian manifolds,J. Math. Soc. Japan19 (1967), 205–214.

    Article  MATH  MathSciNet  Google Scholar 

  21. Ramanathan, J., Complete spacelike hypersurfaces of constant mean curvature in de Sitter space,Indiana Univ. Math. J.36 (1987), 349–359.

    Article  MATH  MathSciNet  Google Scholar 

  22. Ros, A., Compact hypersurfaces with constant scalar curvature and a congruence theorem,J. Differential Geom.27 (1988), 215–220.

    MATH  MathSciNet  Google Scholar 

  23. Simon, U., Codazzi tensors, inGlobal Differential Geometry and Global Analysis (Ferus, D., Kühnel, W., Simon, U. and Wegner, B., eds.), Lecture Notes in Math.838, pp. 289–296, Springer-Verlag, Berlin-Heidelberg, 1981.

    Chapter  Google Scholar 

  24. Simons, J., Minimal varieties in Riemannian manifolds,Ann. of Math.88 (1968), 62–105.

    Article  MathSciNet  Google Scholar 

  25. Spivak, M.,A Comprehensive Introduction to Differential Geometry, Vol. 4, 2nd ed., Publish or Perish, Wilmington, Del., 1979.

    Google Scholar 

  26. Sun, Z. Q., Compact hypersurfaces with constant scalar curvature inS4,Chinese Ann. Math. Ser. A8:3 (1987), 273–286.

    MATH  MathSciNet  Google Scholar 

  27. Yau, S. T., Harmonic functions on complete Riemannian manifolds,Comm. Pure Appl. Math.28 (1975), 201–228.

    Article  MATH  MathSciNet  Google Scholar 

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  1. Department of Applied Mathematics, Tsinghua University, 100084, Beijing, People's Republic of China

    Haizhong Li

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Li, H. Global rigidity theorems of hypersurfaces. Ark. Mat. 35, 327–351 (1997). https://doi.org/10.1007/BF02559973

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  • DOI: https://doi.org/10.1007/BF02559973

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