Laguerre minimal surfaces, isotropic geometry and linear elasticity

Abstract

Laguerre minimal (L-minimal) surfaces are the minimizers of the energy (H 2 − K)/KdA. They are a Laguerre geometric counterpart of Willmore surfaces, the minimizers of (H 2 − K)dA, 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.2000 Mathematics Subject Classification: 68U05, 53A40, 52C99, 51B15.Keywords differential geometry, Laguerre geometry, Laguerre minimal surface, isotropic geometry, linear elasticity, Airy stress function, biharmonic function, thin plate spline, geometrical optics.

Rotational surfaces in isotropic spaces satisfying weingarten conditions

“…The work of surfaces with certain properties in the isotropic 3-space I 3 has important applications in several applied sciences, e.g., computer science, image processing, architectural design and microeconomics, see [3,4,6,8,[30][31][32]. Differential geometry of isotropic spaces has been introduced by K. Strubecker [38], H. Sachs [33][34][35], D. Palman [28] and others.…”

Ruled Laguerre minimal surfaces

“…This is the third in a series of papers [19,18] where we develop and study a novel approach to the Laguerre differential geometry of immersed Legendre surfaces in R 3 . Laguerre geometry is the Euclidean geometry of oriented planes and spheres.…”

“…Our approach is based on a recent result [19] which shows that Laguerre minimal surfaces appear as graphs of biharmonic functions in the isotropic model of Laguerre geometry. This result has various corollaries on Laguerre minimal surfaces, geometric optics and linear elasticity.…”

“…The aim is to study (discrete, see [18], and continuous, see [19]) minimizers of geometric energies which are invariant under Laguerre transformations. The simplest energy of this type has been introduced by Blaschke [2,3,4].…”

Classification of Some Special Types Ruled Surfaces in Simply Isotropic 3-Space

“…Baikoussis and Verstraelen ( [3]) studied the helicoidal surfaces in E 3 . Yoon ( [19,20]) classified the surfaces of revolution and the translation surfaces in the 3-dimensional Galilean space and pseudo-Galilean 3-space under the condition ∆x i = λ i x i and ∆r i = λ i r i ,where λ i ∈ R. Karacan and Yoon ( [10,11]) classified translation surfaces and helicoidal surfaces in the threedimensional simply isotropic space I Kamenarović ( [9]) studied the natural geometry of ruled surfaces and defined equations for the three types ruled surfaces in simply isotropic space I 1 3 . Sipus and Divjak ( [15]) studied some mappings of skew ruled surfaces in simply isotropic space which preserve the generators.…”