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New minimal surfaces in the hyperbolic space

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Abstract

We obtain new complete minimal surfaces in the hyperbolic space ℍ3, by using Ribaucour transformations. Starting with the family of spherical catenoids in ℍ3 found by Mori (1981), we obtain 2- and 3-parameter families of new minimal surfaces in the hyperbolic space, by solving a non trivial integro-differential system. Special choices of the parameters provide minimal surfaces whose parametrizations are defined on connected regions of ℝ2 minus a disjoint union of Jordan curves. Any connected region bounded by such a Jordan curve, generates a complete minimal surface, whose boundary at infinity of ℍ3 is a closed curve. The geometric properties of the surfaces regarding the ends, completeness and symmetries are discussed.

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Acknowledgements

The first author of this work was supported by a Post-Doctoral Fellowship offered by CNPq. The second author was partially supported by CNPq, Ministry of Science and Technology, Brazil (Grant No. 312462/2014-0).

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Authors and Affiliations

  1. School of Mathematics, Southwest Jiaotong University, Chengdu, 610031, China

    NingWei Cui

  2. Department of Mathematics, University of Brasilia, Brasilia, 70910-900, Brazil

    Keti Tenenblat

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  1. NingWei Cui
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  2. Keti Tenenblat
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Correspondence to Keti Tenenblat.

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Cui, N., Tenenblat, K. New minimal surfaces in the hyperbolic space. Sci. China Math. 60, 1679–1704 (2017). https://doi.org/10.1007/s11425-016-0356-1

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  • DOI: https://doi.org/10.1007/s11425-016-0356-1

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