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Ruled Laguerre minimal surfaces

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Abstract

A Laguerre minimal surface is an immersed surface in \({\mathbb{R}^3}\) being an extremal of the functional \({\int (H^2/K-1)dA}\). In the present paper, we prove that the only ruled Laguerre minimal surfaces are up to isometry the surfaces \({\mathbf{R}(\varphi,\lambda) = ( A\varphi,\, B\varphi,\, C\varphi + D\cos 2\varphi\, ) + \lambda\left(\sin \varphi,\, \cos \varphi,\, 0\,\right)}\), where \({A,B,C,D\in \mathbb{R}}\) are fixed. To achieve invariance under Laguerre transformations, we also derive all Laguerre minimal surfaces that are enveloped by a family of cones. The methodology is based on the isotropic model of Laguerre geometry. In this model a Laguerre minimal surface enveloped by a family of cones corresponds to a graph of a biharmonic function carrying a family of isotropic circles. We classify such functions by showing that the top view of the family of circles is a pencil.

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Author notes

  1. Mikhail Skopenkov

    Present address: King Abdullah University of Science and Technology, 4700, P. O. Box 2187, Thuwal, 23955-6900, Kingdom of Saudi Arabia

Authors and Affiliations

  1. Institute for Information Transmission Problems of the Russian Academy of Sciences, Bolshoy Karetny per. 19, bld. 1, Moscow, 127994, Russian Federation

    Mikhail Skopenkov

  2. King Abdullah University of Science and Technology, 4700, Thuwal, 23955-6900, Kingdom of Saudi Arabia

    Helmut Pottmann

  3. Seminar for Applied Mathematics, ETH Zentrum, Rämistrasse 101, 8092, Zurich, Switzerland

    Philipp Grohs

Authors
  1. Mikhail Skopenkov
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  2. Helmut Pottmann
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  3. Philipp Grohs
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Correspondence to Mikhail Skopenkov.

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Skopenkov, M., Pottmann, H. & Grohs, P. Ruled Laguerre minimal surfaces. Math. Z. 272, 645–674 (2012). https://doi.org/10.1007/s00209-011-0953-0

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