Ruled Laguerre minimal surfaces

TL;DR

This paper classifies all ruled Laguerre minimal surfaces in Euclidean space, showing they are essentially a specific family of surfaces, and explores their invariance under Laguerre transformations using isotropic models.

Contribution

It provides a complete classification of ruled Laguerre minimal surfaces and characterizes their invariance properties via isotropic Laguerre geometry.

Findings

Only a specific family of ruled Laguerre minimal surfaces exist up to isometry.

Laguerre minimal surfaces enveloped by cones correspond to biharmonic functions with isotropic circle families.

The top view of isotropic circles forms a pencil, aiding classification.

Abstract

A Laguerre minimal surface is an immersed surface in the Euclidean space 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 R(u,v) = (Au, Bu, Cu + D cos 2u) + v (sin u, cos u, 0), where A, B, C, D are fixed real numbers. 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.