Calculus of Variations and Partial Differential Equations

, Volume 47, Issue 3, pp 711-737

First online:

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

  • J. JostAffiliated withMax Planck Institute for Mathematics in the Sciences Email author 
  • , Y. L. XinAffiliated withInstitute of Mathematics, Fudan University
  • , Ling YangAffiliated withMax Planck Institute for Mathematics in the SciencesInstitute of Mathematics, Fudan University


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