Skip to main content
SpringerLink
Log in

Complete and Stable O(p+1)×O(q+1)-Invariant Hypersurfaces with Zero Scalar Curvature in Euclidean Space ℝp+q+2

  • Published:
Annals of Global Analysis and Geometry Aims and scope Submit manuscript

Abstract

We classify the zero scalar curvature O(p+1)×O(q+1)-invariant hypersurfaces in the euclidean space ℝp+q+2, p,q > 1, analyzing whether they are embedded and stable. The Morse index of the complete hypersurfaces show the existence of embedded, complete and globally stable zero scalar curvature O(p+1)×O(q+1)-invariant hypersurfaces in ℝp+q+2, p+q≥ 7, which are not homeomorphic to ℝp+q+1. Such stable examples provide counter-examples to a Bernstein-type conjecture in the stable class, for immersions with zero scalar curvature.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Alencar, H.: Minimal Hypersurfaces in ℝ2m invariant by SO(m) × SO(m), Trans. Amer. Math. Soc. 337(1) (1993), 129–141.

    Article  MATH  MathSciNet  Google Scholar 

  2. Aronszajn, N.: A unique continuation theorem for solutions of elliptic partial differential equations or inequalities of second order, J. Math. Pure and App. 36 (1957), 235–249.

    MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  4. Bombieri, E., de Giorgi, E. and Giusti, E.: Minimal cones and the Berstein problem, Invents. Math. 7 (1969), 243–269.

    Article  MATH  MathSciNet  Google Scholar 

  5. Do Carmo, M. P. and Dajczer, M.: Rotational hypersurfaces in spaces of constant curvature, Trans. Amer. Math. Soc. 277-2 (1983), 685–709.

    Article  MathSciNet  Google Scholar 

  6. Elbert, M. F.: Sobre hipersuperfícies com r-curvatura média constante, Doctor Thesis, IMPA – Brasil 1998.

  7. Fischer-Colbrie, D.: On complete minimal surfaces with finite Morse index in three-manifolds, Invents. Math. 82 (1985), 121–132.

    Article  MATH  MathSciNet  Google Scholar 

  8. Hartaman, P.: Ordinary Differential Equations, Witly, New York 1964.

  9. Hounie, J. and Leite, M.L.: The maximum principle for hypersurfaces with vanishing curvature functions, J. Diff. Geom. 41 (1995), 247–258.

    MATH  MathSciNet  Google Scholar 

  10. Hsiang, W. Y. and Lawson, H. B.: Minimal submanifolds of low cohomogeneity, J. Differential erential Geom. 5 (1971), 1–38.

    MATH  MathSciNet  Google Scholar 

  11. Hsiang, W. Y., Teng, Z. H. and Yu, W. C.: New examples of constant mean curvature immersions of (2k−1)-spheres into Euclidean 2k-space, Ann. Math. 117 (1983), 609–625.

    Article  MathSciNet  Google Scholar 

  12. Hsiang, W. Y.: Generalized rotational hypersurfaces of constant mean curvature in the Euclidean spaces I, Journal Diff. Geom. 17 (1982), 337–356.

    MATH  MathSciNet  Google Scholar 

  13. Leite, M. L.: Rotational hypersurfaces of space forms with constant scalar curvature, Manuscripta Math. 67 (1990), 285–304.

    Article  MATH  MathSciNet  Google Scholar 

  14. Okayasu, T.: O(2) × O(2)-invariant hypersurfaces with constant negative scalar curvature in E 4, Proc. Amer. Math. Soc. 107 (1989), 1045–1050.

    Google Scholar 

  15. Palmas, O.: Complete rotation hypersurfaces with H k constant in space forms, Bol. Soc. Brasileira Mat.(N. S.) 30(2) (1999), 139–161.

    Article  MATH  MathSciNet  Google Scholar 

  16. Palmas, O.: O(2) × O(2)-invariant hypersurfaces with zero scalar curvature, Archiv Math. 74-(3) (2000), 226–233.

    Article  MathSciNet  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  18. Rosenberg, H.: Hypersurfaces of constant curvatures in space forms, Bull. Sci. Math. 117 (1993), 211–239.

    MATH  MathSciNet  Google Scholar 

  19. Sato, J.: Stability of O(p+1) × O(p+1)-invariant hypersurfaces with zero scalar curvature in Euclidena Space, Ann. Global Anal. Geom. 22 (2002), 153–153.

    Article  Google Scholar 

  20. Smale, S.: On the Morse index theorem, J. Math. Mech. 14 (1965), 1049–1056.

    MATH  MathSciNet  Google Scholar 

  21. Sotomayor, J.: Lições de equaçõ es diferenciais ordinárias, Projeto Euclides, IMPA, Brasil.

  22. Traizet, M.: On the stable surfaces of constant Gauss curvature in space forms, Ann. Global Anal. Geom. 13 (1995), 141–148.

    Article  MATH  MathSciNet  Google Scholar 

  23. Uhlenbeck, K.: The Morse theorem in Hilbert space, J. Differential Geom. 8 (1973), 555–564.

    MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Faculdade de Matemática, Universidade Federal de Uberlândia, 38400-902, Uberlândia, Brazil

    Jocelino Sato

  2. Departamento de Matemática, Universidade Católica de Pernambuco, Rua do Principe 526, Recife, Brazil

    Vicente Francisco De Souza Neto

Authors
  1. Jocelino Sato
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Vicente Francisco De Souza Neto
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jocelino Sato.

Additional information

Mathematics Subject Classifications (2000): 53A10, 53C42,49005.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Sato, J., Neto, V.F.D.S. Complete and Stable O(p+1)×O(q+1)-Invariant Hypersurfaces with Zero Scalar Curvature in Euclidean Space ℝp+q+2 . Ann Glob Anal Geom 29, 221–240 (2006). https://doi.org/10.1007/s10455-005-9006-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10455-005-9006-4

Keywords

Search

Navigation