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

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• 作者：Antonio Martínez ; Pedro Roitman ; Keti Tenenblat
• 关键词：Minimal surfaces ; Flat fronts ; Ribaucour transformations ; Hyperbolic space ; 53A35 ; 53C42
• 刊名：Annals of Global Analysis and Geometry
• 出版年：2015
• 出版时间：October 2015
• 年：2015
• 卷：48
• 期：3
• 页码：233-254
• 全文大小：631 KB
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• 作者单位：Antonio Martínez (1)
  Pedro Roitman (2)
  Keti Tenenblat (2)
  
  1. Departamento de Geometría y Topología, Universidad de Granada, 18071, Granada, Spain
  2. Departamento de Matemática, Universidade de Brasília, 70910-900, Brasília, Brazil
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Analysis
  Statistics for Business, Economics, Mathematical Finance and Insurance
  Mathematical and Computational Physics
  Group Theory and Generalizations
  Geometry
  
• 出版者：Springer Netherlands
• ISSN：1572-9060

文摘

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