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Remarks on hypersurfaces with constant higher order mean curvature in Euclidean space

  • Luis J. Alías
  • Josué Meléndez
Original Paper

Abstract

In this paper we consider complete oriented hypersurfaces of Euclidean space with constant higher order mean curvature and having two principal curvatures, one of them simple. As an application of the so called principal curvature theorem, a purely geometric result on the principal curvatures of the hypersurface given by Smyth and Xavier (Invent Math 90:443–450, 1987), we characterize those hypersurfaces for which the Gauss–Kronecker curvature does not change sign, extending to the general n-dimensional case a previous result for surfaces due to Klotz and Osserman.

Keywords

Hypersurface Higher order mean curvature Gauss–Kronecker curvature Principal curvature theorem 

Mathematics Subject Classification

53C40 53C42 

Notes

Acknowledgements

The authors would like to thank the referee for reading the manuscript in great detail and giving several valuable suggestions and useful comments which improved the paper. This work was finished while the two authors were visiting the Laboratoire de Mathématiques et Physique Théorique (LMPT) of the Université Francois Rabelais at Tours (France). They would like to thank this institution for its cordial hospitality during this visit.

References

  1. 1.
    Alías, L.J., García-Martínez, S.C.: An estimate for the scalar curvature of constant mean curvature hypersurfaces in space forms. Geom. Dedicata 156, 31–47 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Alías, L.J., Meléndez, J.: Hypersurfaces with constant higher order mean curvature in Euclidean space. Geom. Dedicata 182, 117–131 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Cecil, T., Ryan, P.J.: Geometry of hypersurfaces. Springer Monographs in Mathematics. Springer, New York, (2015). xi+596 pp. ISBN: 978-1-4939-3245-0; 978-1-4939-3246-7Google Scholar
  4. 4.
    do Carmo, M., Dajczer, M.: Rotational hypersurfaces in spaces of constant curvature. Trans. Am. Math. Soc. 277(2), 685–709 (1983)CrossRefzbMATHGoogle Scholar
  5. 5.
    Hartman, P.: On complete hypersurfaces of nonnegative sectional curvatures and constant \(m\)th mean curvature. Trans. Am. Math. Soc. 245, 363–374 (1978)zbMATHGoogle Scholar
  6. 6.
    Klotz, T., Osserman, R.: Complete surfaces in \(E^3\) with constant mean curvature. Comment. Math. Helv. 41, 313–318 (1966/1967)Google Scholar
  7. 7.
    Levi-Civita, T.: Famiglia di superfici isoparametriche nell’ordinario spazio Euclideo. Att. Accad. naz Lincie Rend. Cl. Sci. Fis. Mat. Natur 26, 355–362 (1937)zbMATHGoogle Scholar
  8. 8.
    Núñez, R.A.: On complete hypersurfaces with constant mean and scalar curvatures in Euclidean spaces. Proc. Am. Math. Soc. 145(6), 2677–2688 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Otsuki, T.: Minimal hypersurfaces in a Riemannian manifold of constant curvature. Am. J. Math. 92, 145–173 (1970)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Perelman, G.: Proof of the soul conjecture of Cheeger and Gromoll. J. Differ. Geom. 40, 209–212 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Segre, B.: Famiglie di ipersuperficie isoparametriche negli spazi euclidei ad un qualunque numero di dimensioni. Att. Accad. naz Lincie Rend. Cl. Sci. Fis. Mat. Natur 27, 203–207 (1938)zbMATHGoogle Scholar
  12. 12.
    Smyth, B., Xavier, F.: Efimov’s theorem in dimension greater than two. Invent. Math. 90, 443–450 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Wu, B.-Y.: On hypersurfaces with two distinct principal curvatures in Euclidean space. Houston J. Math. 36, 451–467 (2010)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  1. 1.Departamento de MatemáticasUniversidad de MurciaEspinardo, MurciaSpain
  2. 2.Departamento de MatemáticasUniversidad Autónoma Metropolitana-IztapalapaMexico cityMexico

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