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Mathematische Zeitschrift

, Volume 272, Issue 1–2, pp 645–674 | Cite as

Ruled Laguerre minimal surfaces

  • Mikhail SkopenkovEmail author
  • Helmut Pottmann
  • Philipp Grohs
Article

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.

Keywords

Laguerre geometry Laguerre minimal surface Ruled surface Biharmonic function 

Mathematics Subject Classification (2000)

53A40 49Q10 31A30 

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Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  • Mikhail Skopenkov
    • 1
    • 2
    Email author
  • Helmut Pottmann
    • 3
  • Philipp Grohs
    • 4
  1. 1.Institute for Information Transmission Problems of the Russian Academy of SciencesMoscowRussian Federation
  2. 2.King Abdullah University of Science and Technology, 4700ThuwalKingdom of Saudi Arabia
  3. 3.King Abdullah University of Science and Technology, 4700ThuwalKingdom of Saudi Arabia
  4. 4.Seminar for Applied MathematicsETH ZentrumZurichSwitzerland

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