Surfaces in Laguerre Geometry

TL;DR

This paper introduces the theory of surfaces in Laguerre geometry, focusing on L-isothermic, L-minimal, and generalized L-minimal surfaces, using the quadric model and moving frames, and applying the Cartan-Kaehler theorem to study related differential systems.

Contribution

The paper surveys recent results on special classes of surfaces in Laguerre geometry, emphasizing the application of the Cartan-Kaehler theorem and the quadric model for Lie sphere geometry.

Findings

Analysis of L-minimal surfaces using Cartan-Kaehler theorem

Survey of L-isothermic and generalized L-minimal surfaces

Application of moving frames in Laguerre surface theory

Abstract

This exposition gives an introduction to the theory of surfaces in Laguerre geometry and surveys some results, mostly obtained by the authors, about three important classes of surfaces in Laguerre geometry, namely L-isothermic, L-minimal, and generalized L-minimal surfaces. The quadric model of Lie sphere geometry is adopted for Laguerre geometry and the method of moving frames is used throughout. As an example, the Cartan-Kaehler theorem is applied to study the Cauchy problem for the Pfaffian differential system of L-minimal surfaces. This is an elaboration of the talks given by the authors at IMPAN, Warsaw, in September 2016. The objective was to illustrate, by the subject of Laguerre surface geometry, some of the topics presented in a series of lectures held at IMPAN by G. R. Jensen on Lie sphere geometry and by B. McKay on exterior differential systems.