Skip to main content
Log in

Existence an regularity of constant mean curvature hypersurfaces in hyperbolic space

  • Published:
Mathematische Zeitschrift Aims and scope Submit manuscript

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

Access this article

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

Instant access to the full article PDF.

References

  1. Anderson, M.,Complete minimal varieties in hyperbolic space, Invent Math.69, 477–494 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  2. Anderson, M.,Complete minimal hypersurfaces in hyperbolic n-manifolds, Comment. Math. Helv.58, 264–290 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  3. Anderson, M.,The Bernstein problem in complete Riemannian manifolds, PhD Thesis, University of California, Berkeley, 1981.

    Google Scholar 

  4. Allard, W.K.,On the first variation of a varifold, Ann. Math.,95, 417–491 (1972)

    Article  MathSciNet  Google Scholar 

  5. Alencar, H., Rosenberg, H.,Some remarks on the existence of hypersurfaces of constant mean curvature with a given boundary, or asymptotic boundary, in hyperbolic space, Preprint.

  6. Baouendi, M.S., Goulaouic, C.,Régularité et théorie spectrale pour une classe d’opérateurs elliptiques dégénérés, Arch. Rat. Mech. Anal.34 361–379 (1969)

    Article  MATH  MathSciNet  Google Scholar 

  7. Baouendi, M.S., Goulaouic, C.,Etude de l’analyticité et de la régularité Gevrey pour une classe d’opérateurs elliptiques dégénérés, Ann. Sc. E. N. S., tome 4, fasc. 1 (1971)

  8. Bombieri, E.,Regularity theory for almost minimal currents, Arch. Rat. Mech. Anal.78 99–130 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  9. Bryant, R.,Surfaces of mean curvature one in hyperbolic space, Astérisque 154–155, Soc. Math. de France, 321–347 (1987)

  10. Cheng, S.-Y., Yau, S.-T.,On the existence of a complete Kähler metric on noncompact complex manifolds and the regularity of Fefferman’s equation, Comm. Pure Appl. Math.33 507–544 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  11. Federer, H.,Geometric Measure Theory, Berlin- Heidelberg-New York 1969

  12. Fefferman, C.,Monge-Ampère equations, the Bergman kernel, and geometry of pseudoconvex domains, Ann. of Math.103 395–416 (1976)

    Article  MathSciNet  Google Scholar 

  13. Graham, H.,The Dirichlet problem for the Bergman Laplacian, Comm. P.D.E.8 433–476, 563-641 (1983)

    MATH  Google Scholar 

  14. Gidas, B., Ni, W.M., Nirenberg, L.,Symmetry and related properties via the maximum principle, Commun. Math. Phys.68 209–243 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  15. Gilberg, D., Trudinger, N.,Elliptic partial differential equations of second order, Springer-Verlag, 1977.

  16. Hardt, R., Lin, F.-H.,Regularity at infinity for area-minimizing hypersurfaces in hyperbolic space, Invent. Math.88 217–224 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  17. Korevaar, N.J., Kusner, R., Meeks III, W.H., Solomon, B.,Constant mean curvature surfaces in hyperbolic space, Amer. J. Math.114 1–43 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  18. Kohn, J.J., Nirenberg, L.,Degenerated elliptic-parabolic equations of second order, Comm. Pure Appl. Math.20 797–872 (1967)

    Article  MATH  MathSciNet  Google Scholar 

  19. Lin, F.-H.,On the Dirichlet problem for minimal graphs in hyperbolic space, Invent. Math.96 593–612 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  20. Lin, F.-H.,Asymptotic behavior of area-minimizing currents in hyperbolic space, Comm. Pure Appl. Math.42 229–242 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  21. Li, P., Tam, L.-F.,Uniqueness and regularity of proper harmonic maps, Ann. of Math.137 167–201 (1993)

    Article  MathSciNet  Google Scholar 

  22. Li, P., Tam, L.-F.,Uniqueness and regularity of proper harmonic maps II, Indiana Math. J.42 591–635 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  23. Nelli, B., Spruck, J.,On existence and uniqueness of mean curvature hypersurfaces in hyperbolic space, Preprint.

  24. Simon, L.,Lectures on geometric measure theory, Proc. Center for Mathematical Analysis. Australian Nat. Univ. 3 (1983)

  25. Umehara, M., Yamada, K.,Complete surfaces of constant mean curvature-1 in the hyperbolic 3-space, Ann. Math.137 611–638 (1993)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Additional information

This article was processed by the author using the IATEX style filepljourIm from Springer-Verlag.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Tonegawa, Y. Existence an regularity of constant mean curvature hypersurfaces in hyperbolic space. Math Z 221, 591–615 (1996). https://doi.org/10.1007/BF02622135

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02622135

Keywords

Navigation