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Laguerre minimal surfaces, isotropic geometry and linear elasticity

  • Helmut PottmannEmail author
  • Philipp Grohs
  • Niloy J. Mitra
Article

Abstract

Laguerre minimal (L-minimal) surfaces are the minimizers of the energy \(\int (H^2-K)/K d\!A\). They are a Laguerre geometric counterpart of Willmore surfaces, the minimizers of \(\int (H^2-K)d\!A\), which are known to be an entity of Möbius sphere geometry. The present paper provides a new and simple approach to L-minimal surfaces by showing that they appear as graphs of biharmonic functions in the isotropic model of Laguerre geometry. Therefore, L-minimal surfaces are equivalent to Airy stress surfaces of linear elasticity. In particular, there is a close relation between L-minimal surfaces of the spherical type, isotropic minimal surfaces (graphs of harmonic functions), and Euclidean minimal surfaces. This relation exhibits connections to geometrical optics. In this paper we also address and illustrate the computation of L-minimal surfaces via thin plate splines and numerical solutions of biharmonic equations. Finally, metric duality in isotropic space is used to derive an isotropic counterpart to L-minimal surfaces and certain Lie transforms of L-minimal surfaces in Euclidean space. The latter surfaces possess an optical interpretation as anticaustics of graph surfaces of biharmonic functions.

Keywords

Differential geometry Laguerre geometry Laguerre minimal surface Isotropic geometry Linear elasticity Airy stress function Biharmonic function Thin plate spline Geometrical optics 

Mathematics Subject Classifications (2000)

68U05 53A40 52C99 51B15 

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Copyright information

© Springer Science+Business Media, LLC. 2008

Authors and Affiliations

  • Helmut Pottmann
    • 1
    Email author
  • Philipp Grohs
    • 1
  • Niloy J. Mitra
    • 1
  1. 1.Institut für Diskrete Mathematik und GeometrieTechnische Universität WienWienAustria

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