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Annals of Global Analysis and Geometry

, Volume 48, Issue 3, pp 233–254 | Cite as

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

  • Antonio MartínezEmail author
  • Pedro Roitman
  • Keti Tenenblat
Article

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.

Keywords

Minimal surfaces Flat fronts Ribaucour transformations Hyperbolic space 

Mathematics Subject Classification

53A35 53C42 

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Copyright information

© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

  • Antonio Martínez
    • 1
    Email author
  • Pedro Roitman
    • 2
  • Keti Tenenblat
    • 2
  1. 1.Departamento de Geometría y TopologíaUniversidad de GranadaGranadaSpain
  2. 2.Departamento de MatemáticaUniversidade de BrasíliaBrasíliaBrazil

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