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A connection between flat fronts in hyperbolic space and minimal surfaces in Euclidean space

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Abstract

A geometric construction is provided that associates to a given flat front in \(\mathbb {H}^3\) a pair of minimal surfaces in \(\mathbb {R}^3\) which are related by a Ribaucour transformation. This construction is generalized associating to a given frontal in \(\mathbb {H}^3\), a pair of frontals in \(\mathbb {R}^3\) that are envelopes of a smooth congruence of spheres. The theory of Ribaucour transformations for minimal surfaces is reformulated in terms of a complex Riccati ordinary differential equation for a holomorphic function. This enables one to simplify and extend the classical theory, that in principle only works for umbilic free and simply connected surfaces, to surfaces with umbilic points and non-trivial topology. Explicit examples are included.

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Authors and Affiliations

  1. Departamento de Geometría y Topología, Universidad de Granada, 18071, Granada, Spain

    Antonio Martínez

  2. Departamento de Matemática, Universidade de Brasília, 70910-900, Brasília, Brazil

    Pedro Roitman & Keti Tenenblat

Authors
  1. Antonio Martínez
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  2. Pedro Roitman
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  3. Keti Tenenblat
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Corresponding author

Correspondence to Antonio Martínez.

Additional information

Research partially supported by Ministerio de Educación Grants No: MTM2013-43970-P, No: PHB2010-0109, Junta de Anadalucía Grants No. FQM325, N0. P06-FQM-01642. Ministério de Ciência e Tecnologia, CNPq Proc. No. 303774/2009-6. Ministério de Educação, CAPES/DGU Proc. No. 23038010833/2010-37.

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Martínez, A., Roitman, P. & Tenenblat, K. A connection between flat fronts in hyperbolic space and minimal surfaces in Euclidean space. Ann Glob Anal Geom 48, 233–254 (2015). https://doi.org/10.1007/s10455-015-9468-y

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  • DOI: https://doi.org/10.1007/s10455-015-9468-y

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