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

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Abstract

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 

Notes

Acknowledgments

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.

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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

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