The Gauss image of entire graphs of higher codimension and Bernstein type theorems

  • J. JostEmail author
  • Y. L. Xin
  • Ling Yang
Open Access


Under suitable conditions on the range of the Gauss map of a complete submanifold of Euclidean space with parallel mean curvature, we construct a strongly subharmonic function and derive a-priori estimates for the harmonic Gauss map. The required conditions here are more general than in previous work and they therefore enable us to improve substantially previous results for the Lawson–Osseman problem concerning the regularity of minimal submanifolds in higher codimension and to derive Bernstein type results.

Mathematics Subject Classification (1991)

58E20 53A10 



Y. L. Xin is grateful to the Max Planck Institute for Mathematics in the Sciences in Leipzig for its hospitality and continuous support. He is also partially supported by NSFC and SFMEC.

Open Access

This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.


  1. 1.
    Allard W.: On the first variation of a varifold. Ann. Math. 95, 417–491 (1972)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Barbosa J.L.M.: An extrinsic rigidity theorem for minimal immersion from S 2 into S n. J. Differ. Geom. 14(3), 355–368 (1980)MathSciNetGoogle Scholar
  3. 3.
    Chern S.S., Osserman R.: Complete minimal surfaces in Euclidean n-space. J. d’Anal. Math. 19, 15–34 (1967)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Ecker K., Huisken G.: A Bernstein result for minimal graphs of controlled growth. J. Differ. Geom. 31, 337–400 (1990)MathSciNetGoogle Scholar
  5. 5.
    Fischer-Colbrie D.: Some rigidity theorems for minimal submanifolds of the sphere. Acta Math. 145, 29–46 (1980)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Giaquinta M., Giusti E.: On the regularity of the minima of variational integrals. Acta Math. 148, 31–46 (1982)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Giaquinta M., Hildebrandt S.: A priori estimates for harmonic mappings. J. Reine Angew. Math. 336, 124–164 (1982)MathSciNetGoogle Scholar
  8. 8.
    Grüter M., Widman K.: The Green function for uniformly elliptic equations. Manuscr. Math. 37, 303–342 (1982)zbMATHCrossRefGoogle Scholar
  9. 9.
    Gulliver R., Jost J.: Harmonic maps which solve a free-boundary problem. J. Reine Angew. Math. 381, 61–89 (1987)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Hildebrandt S., Jost J., Widman K.: Harmonic mappings and minimal submanifolds. Invent. Math. 62, 269–298 (1980)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Jost J.: Generalized Dirichlet forms and harmonic maps. Calc. Var. PDE 5, 1–19 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Jost J., Xin Y.L.: Bernstein type theorems for higher codimension. Calc. Var. 9, 277–296 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Jost J., Xin Y.L., Yang L.: The regularity of harmonic maps into spheres and application to Bernstein problems. J. Differ. Geom. 90, 131–176 (2012)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Lawson H.B., Osserman R.: Non-existence, non-uniqueness and irregularity of solutions to the minimal surface system. Acta Math. 139, 1–17 (1977)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Moser J.: On Harnack’s theorem for elliptic differential equations. Commun. Pure Appl. Math. 14, 577–591 (1961)zbMATHCrossRefGoogle Scholar
  16. 16.
    Ruh E.A., Vilms J.: The tension field of Gauss maps. Trans. AMS 149, 569–573 (1970)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Schoen R., Simon L., Yau S.T.: Curvature estimates for minimal hypersurfaces. Acta Math. 134, 275–288 (1974)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Wang M.-T.: On graphic Bernstein type results in higher codimension. Trans. AMS 355(1), 265–271 (2003)zbMATHCrossRefGoogle Scholar
  19. 19.
    Wong Y.-C.: Differential geometry of Grassmann manifolds. Proc. Natl. Acad. Sci. 57, 589–594 (1967)zbMATHCrossRefGoogle Scholar
  20. 20.
    Xin Y.: Minimal Submanifolds and Related Topics. World Scientific Publications, Singapore (2003)zbMATHGoogle Scholar
  21. 21.
    Xin Y.L., Yang L.: Convex functions on Grassmannian manifolds and Lawson–Osserman problem. Adv. Math. 219(4), 1298–1326 (2008)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© The Author(s) 2012

Authors and Affiliations

  1. 1.Max Planck Institute for Mathematics in the SciencesLeipzigGermany
  2. 2.Institute of MathematicsFudan UniversityShanghaiChina

Personalised recommendations