Skip to main content
SpringerLink
Log in

Stable constant mean curvature tori and the isoperimetric problem in three space forms

  • Published:
Commentarii Mathematici Helvetici

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

References

  1. F. J. Almgren, “Existence and regularity almost everywhere of solutions to elliptic variational problems with constrains”, Mem A.M.S.,4, 165, 1976.

    MathSciNet  Google Scholar 

  2. J. L. Barbosa andM. Do Carmo, “Stability of hypersurfaces with constant mean curvature”, Math. Z.,185, 339–353, 1984.

    Article  MathSciNet  Google Scholar 

  3. J. L. Barbosa andM. Do Carmo andJ. Eschenburg “Stability of hypersurfaces with constant mean curvature in Riemannian manifolds”, Math. Z.197, 123–138, 1988.

    Article  MathSciNet  Google Scholar 

  4. M. Do Carmo, “Hypersurfaces with constant mean curvature”, Proc. 3rd Symp. Diff. Geom., Peñíscola, Lecture Notes in Mathematics 1410, 128–144, 1988.

    MathSciNet  Google Scholar 

  5. B. Y. Chen, Geometry of submanifolds, Marcel Dekker, New York, 1973.

    MATH  Google Scholar 

  6. S. Y. Cheng, “Eigenfunctions and nodal sets”, Comm. Math. Helv.,51, 43–55, 1976.

    Google Scholar 

  7. L. P. Eisenhart, Riemannian Geometry, Princeton University Press, Princeton, N.J., 1925.

    Google Scholar 

  8. A. El Soufi andS. Ilias, “Majoration de la seconde valeur propre d'un operateur de Schrödinger sur una variete compacte et applications”, (to appear in Jour. of Func. Anal.).

  9. D. Fischer-Colbrie, “On complete minimal surfaces with finite Morse index in three-manifolds”, Invent. Math.,82, 121–132, 1985.

    Article  MathSciNet  Google Scholar 

  10. T. Frankel, “On the fundamental group of a compact minimal submanifold”, Ann. of Math.,83, 68–73, 1966.

    Article  MathSciNet  Google Scholar 

  11. K. Frensel, “Stable complete surfaces with constant mean curvature”, Anais Acad. Bras. Cien.,60, 115–117, 1988.

    MathSciNet  Google Scholar 

  12. P. Griffiths andJ. Harris, Principles of Algebraic Geometry, Pure and Applied Math., Wiley-Interscience series, 1978.

  13. E. Heintze, “Extrinsic upper bounds for λ1”, Math. Ann.280, 389–402, 1988.

    Article  MathSciNet  Google Scholar 

  14. H. Hopf, Differential Geometry in the Large, Lecture Notes on Mathematics 1000, 2nd edition, Springer-Verlag, 1983.

  15. H. Karcher, “The triply periodic minimal surfaces of Alan Schoen and their constant mean curvature companions”, Manusc. Math.,64, 291–357, 1989.

    Article  MathSciNet  Google Scholar 

  16. N. J. Korevaar, R. Kusner andB. Solomon, “The structure of complete embedded surfaces with constant mean curvature”, Jour. of Diff. Geom.,30, 465–503, 1989.

    MathSciNet  Google Scholar 

  17. R. Kusner, “Conformal geometry and complete minimal surfaces”, Bull. A.M.S.,17, 291–295. 1987.

    MathSciNet  Google Scholar 

  18. H. B. Lawson, “Complete minimal surfaces in S 3”, Ann. of Math.,92, 335–374, 1970.

    Article  MathSciNet  Google Scholar 

  19. P. Li andS. T. Yau, “A new conformal invariant and its applications to the Willmore conjecture and the first eigenvalue of compact surfaces”, Invent. Math.,69, 269–291, 1982.

    Article  MathSciNet  Google Scholar 

  20. F. J. López andA. Ros, “Complete minimal surfaces with index one and stable constant mean curvature surfaces”, Comm. Math. Helv.,64, 34–43, 1989.

    Google Scholar 

  21. S. Montiel andA. Ros, “Schrödinger operators associated to a holomorphic map”, Proceedings Conference on Global Analysis and Global Differential Geometry, Berlin, 1990 (to appear).

  22. B. Palmer, PhD Thesis, Standford, 1986.

  23. R. Pedrosa, “On the uniqueness of isoperimetric regions in cylindrical spaces”, PhD Thesis, Berkeley, 1988.

  24. M. Ross, “Stability properties of complete two-dimensional minimal surfaces in Euclidean space”, PhD Thesis, Berkeley, 1989.

  25. A. Da Silveira, “Stability of complete noncompact surfaces with constant mean curvature”, Math. Ann.,277, 629–638, 1987.

    Article  MathSciNet  Google Scholar 

  26. E. Schmidt, “Breweis der isoperimetrischen Eigenschaft der Kugel in hyperbolischen und sphärischen Raum jeder Dimensionszahl”, Math. Z.,49, 1–109, 1943.

    Article  MathSciNet  Google Scholar 

  27. J. E. Taylor, “The structure of singularities in solutions to ellipsoidal problems with constrains in ℝ3”, Ann. of Math.,103, 541–546, 1976.

    Article  MathSciNet  Google Scholar 

  28. S. T. Yau, “Nonlinear analysis in geometry”, L'enseignement Math.,33, 109–158, 1987.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Departamento de Matemáticas, Universidad de Extremadura, 06071, Badajoz, (Spain)

    Manuel Ritoré & Antonio Ros

  2. Departamento de Geometría y Topología, Universidad de Granada, 18071, Granada, (Spain)

    Manuel Ritoré & Antonio Ros

Authors
  1. Manuel Ritoré
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Antonio Ros
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Supported by DGICYT grant PG89-08823714.

Partially supported by DGICYT grant PS87-0115-C03-02.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Ritoré, M., Ros, A. Stable constant mean curvature tori and the isoperimetric problem in three space forms. Commentarii Mathematici Helvetici 67, 293–305 (1992). https://doi.org/10.1007/BF02566501

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02566501

Keywords

Search

Navigation