Abstract:
Laguerre minimal (Lminimal) surfaces are theminimizers of the energy ò(H²  K)KdA. They are a Laguerre geometric counterpart ofWillmore surfaces, the minimizers of ò(H²  K)dA, which are known to be an entity of Möbius sphere geometry. The present paper provides a new and simple approach to Lminimal surfaces by showing that they appear as graphs of biharmonic functions in the isotropic model of Laguerre geometry. Therefore, Lminimal surfaces are equivalent to Airy stress surfaces of linear elasticity. In particular, there is a close relation between Lminimal 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 Lminimal surfaces via thin plate splines and numerical solutions of the biharmonic equation. Finally, metric duality in isotropic space is used to derive an isotropic counterpart to Lminimal surfaces and certain Lie transforms of Lminimal surfaces in Euclidean space. The latter surfaces possess an optical interpretation as anticaustics of graph surfaces of biharmonic functions.
Bibtex:
@article{pottmann2009fs,
title = "Laguerre Minimal Surfaces, Isotropic Geometry and Linear Elasticity",
author = "Helmut Pottmann and Philipp Grohs and Niloy J. Mitra",
journal = {Adv. Comp. Math.},
year = 2009,
volume = 31,
pages="391419",
}

