Surfaces in Laguerre geometry
This exposition gives an introduction to the theory of surfaces in Laguerre geometry and surveys some significant results concerning 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. The Cartan–Kähler theorem for exterior differential systems is applied to study the Cauchy problem for the Pfaffian differential system of $L$-minimal surfaces. This paper is an elaboration of our talks at the IMPAN Workshop in Warsaw. Our objective was to illustrate, by the subject of Laguerre surface geometry, some of the main concepts presented in the lecture series by G. R. Jensen on Lie sphere geometry and by B. McKay on exterior differential systems.