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A characterization of Weingarten surfaces in hyperbolic 3-space

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Abstract

We study 2-dimensional submanifolds of the space \({\mathbb{L}}({\mathbb{H}}^{3})\) of oriented geodesics of hyperbolic 3-space, endowed with the canonical neutral Kähler structure. Such a surface is Lagrangian iff there exists a surface in ℍ3 orthogonal to the geodesics of Σ.

We prove that the induced metric on a Lagrangian surface in \({\mathbb{L}}({\mathbb{H}}^{3})\) has zero Gauss curvature iff the orthogonal surfaces in ℍ3 are Weingarten: the eigenvalues of the second fundamental form are functionally related. We then classify the totally null surfaces in \({\mathbb{L}}({\mathbb{H}}^{3})\) and recover the well-known holomorphic constructions of flat and CMC 1 surfaces in ℍ3.

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Authors and Affiliations

  1. Department of Computing and Mathematics, Institute of Technology, Tralee, Clash Tralee, Co. Kerry, Ireland

    Nikos Georgiou & Brendan Guilfoyle

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  1. Nikos Georgiou
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  2. Brendan Guilfoyle
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Correspondence to Nikos Georgiou.

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Communicated by V. Cortés.

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Georgiou, N., Guilfoyle, B. A characterization of Weingarten surfaces in hyperbolic 3-space. Abh. Math. Semin. Univ. Hambg. 80, 233–253 (2010). https://doi.org/10.1007/s12188-010-0039-7

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  • DOI: https://doi.org/10.1007/s12188-010-0039-7

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