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The Gauss Map of Algebraic Complete Minimal Surfaces Omits Hypersurfaces in Subgeneral Position

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Abstract

The first aim of this paper is to show a second main theorem for algebraic curves into the n-dimensional projective space sharing hypersurfaces in subgeneral position. We then use it to study the value distribution of the generalized Gauss map of the complete (regular) minimal surfaces in \(\mathbb {R}^{m}\) with finite total curvature, as well as the unicity problem, sharing hypersurfaces in subgeneral position. Our results generalize and complete previous results in this area, especially the works of Chern and Osserman (J. Anal. Math. 19, 15–34, 1967), Jin and Ru (Differ. Geom. Appl. 25: 701–712, 2007).

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Acknowledgements

The research of the authors is supported by a NAFOSTED grant of Vietnam (Grant No. 101.04-2017.317).

This work was done during a stay of the authors at the Vietnam Institute for Advanced Study in Mathematics (VIASM). We would like to thank VIASM for partial support, and the staff of VIASM for their hospitality.

We also would like to express our gratitude to the referees. His/her valuable comments made on the first version of this paper led to improvements in the paper.

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Authors and Affiliations

  1. Department of Mathematics, Hanoi National University of Education, 136 Xuan Thuy str., Hanoi, Vietnam

    Do Duc Thai

  2. Department of Mathematics, National University of Civil Engineering, 55 Giai Phong str., Hanoi, Vietnam

    Pham Duc Thoan

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  1. Do Duc Thai
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  2. Pham Duc Thoan
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Correspondence to Do Duc Thai.

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Thai, D.D., Thoan, P.D. The Gauss Map of Algebraic Complete Minimal Surfaces Omits Hypersurfaces in Subgeneral Position. Vietnam J. Math. 46, 579–591 (2018). https://doi.org/10.1007/s10013-017-0259-6

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  • DOI: https://doi.org/10.1007/s10013-017-0259-6

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