Skip to main content
Log in

On the Gauss map of complete surfaces of constant mean curvature in R3 and R4

  • Published:
Commentarii Mathematici Helvetici

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. S. Berstein,Sur les surfaces définies au moyen de leur courbure moyenne ou totale, Ann. Ec. Norm. Sup.27 (1910), 233–256.

    Google Scholar 

  2. W. Blaschke,Sulla geometria differenziale delle superficie S 2 nello spazio euclideo S 4, Ann. Mat. Pura Appl. (4)28 (1949), 205–209.

    MathSciNet  MATH  Google Scholar 

  3. C. C. Chen, On the image of the generalized Gauss map of a complete minimal surface inR 4, Pacific J. Math. (to appear).

  4. S.-S. Chern Minimal surfaces in an euclidean space of N dimensions, Differential and Combinatorial Topology, Princeton University Press (1965), 187–198.

  5. S.-S. Chern,On the curvature of a piece of hypersurface in Euclidean space, Abh. Math. Sem. Hamburg29 (1965), 77–91.

    MathSciNet  MATH  Google Scholar 

  6. S.-S. Chern andR. Osserman,Complete minimal surfaces in Euclidean n-space, J. Analyse Math.19 (1967), 15–34.

    Article  MathSciNet  MATH  Google Scholar 

  7. S.-S. Chern andE. Spanier,A theorem on orientable surfaces in four-dimensional space, Comment. Math. Helv.25 (1951), 1–5.

    Article  MathSciNet  Google Scholar 

  8. J. Eells andL. Lemaire,A report on harmonic maps, Bull. London Math. Soc.10 (1978), 1–68.

    MathSciNet  MATH  Google Scholar 

  9. D. Fischer-Colbrie andR. Schoen,The structure of complete stable minimal surfaces in 3-manifolds of nonnegative scalar curvature, Comm. Pure Applied Math.33 (1980), 199–211.

    MathSciNet  MATH  Google Scholar 

  10. Erhard Heinz, Über Flächen mit eineindeutiger Projection auf einer Ebene, deren Krümmungen durch Ungleichugen eingeschränkt sind, Math. Ann.192 (1955), 451–454.

    Article  MathSciNet  Google Scholar 

  11. S. Hildebrandt, J. Jost andK.-O. Widman,Harmonic mappings and minimal submanifolds, Invent. Math.62 (1980), 269–298.

    Article  MathSciNet  MATH  Google Scholar 

  12. D. A. Hoffman,Surfaces of constant mean curvature in manifolds of constant curvature, J. Differential Geometry8 (1973), 161–176.

    MathSciNet  MATH  Google Scholar 

  13. D. A. Hoffman andR. Osserman,The geometry of the generalized Gauss map, Amer. Math. Soc. Memoir No. 236, 1980.

  14. D. A. Hoffman andR. Osserman,The Gauss map of surfaces in R n (preprint).

  15. K. Kenmotsu,Wierstrass formula for surfaces of prescribed mean curvature, Math. Ann.245 (1979), 89–99.

    Article  MathSciNet  MATH  Google Scholar 

  16. R. Osserman,Proof of a conjecture of Nirenberg, Comm. Pure Appl. Math.12 (1959), 229–232.

    MathSciNet  MATH  Google Scholar 

  17. R. Osserman,Global properties of minimal surfaces in E 3 and E n, Ann. of Math.80 (1964), 340–364.

    Article  MathSciNet  Google Scholar 

  18. E. A. Ruh,Asymptotic behavior of non-parametric minimal hypersurfaces, J. Differential Geometry4 (1970), 509–513.

    MathSciNet  MATH  Google Scholar 

  19. E. A. Ruh andJ. Vilms,The tension field of the Gauss map, Trans. Amer. Math. Soc.149 (1970), 569–573.

    Article  MathSciNet  MATH  Google Scholar 

  20. R. Schoen andS.-T. Yau,On univalent harmonic maps between surfaces, Invent. Math.44 (1978), 265–278.

    Article  MathSciNet  MATH  Google Scholar 

  21. J. L. Weiner,The Gauss map for surfaces in 4-space (preprint).

  22. F. Xavier,The Gauss map of a complete non-flat minimal surface cannot omit 7 points on the sphere, Ann. of Math.13 (1981), 211–214.

    Article  MathSciNet  Google Scholar 

  23. S.-T. Yau,Submanifolds with constant mean curvature, II, Amer. J. Math.97 (1975), 76–100.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Additional information

This work was supported in part by NSF grants at the University of Massachusetts, Amherst; Stanford University, Stanford; University of California, Berkeley.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Hoffman, D.A., Osserman, R. & Schoen, R. On the Gauss map of complete surfaces of constant mean curvature in R3 and R4 . Commentarii Mathematici Helvetici 57, 519–531 (1982). https://doi.org/10.1007/BF02565874

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02565874

Keywords

Navigation