Surfaces in Laguerre geometry

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-Kähler theorem for exterior differential systems 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.2000 Mathematics Subject Classification. 53A35, 53C42.

Weierstrass-type representations

“…Thus x t is a marginally trapped surface and L t := x t + G t is the envelope of an L-isothermic surface. This is the T-transform of L-isothermic surfaces (see [25,26]). Notice that we obtain a new closed 1-form…”

“…Suppose that x : Σ → R 3,1 is Christoffel dual to g ∈ ΓG. By defining L := x + G, we see that L is Ω-dual to L * := G. Moreover, since x has the same induced conformal structure on T Σ as G, one identifies x as the middle sphere congruence of L, see [4,26]. One deduces from [8] that such L are the envelopes of L-isothermic surfaces, that is, surfaces in Euclidean 3-space that admit curvature line coordinates which are conformal with respect to the third fundamental form.…”

Geometry of Lagrangian Grassmannians and nonlinear PDEs

“…Finally, Musso and Nicolodi's paper [41] provides a lucid introduction to Laguerre geometry with a clean presentation of the fundamental constructions. It contains helpful comparisons to surface theory in other, classical geometries.…”

Polynomial conserved quantities of Lie applicable surfaces

“…L-isothermic surfaces. L-isothermic surfaces were originally discovered by Blaschke [3] and have been the subject of interest recently in for example [20,28,29,30,32,36,39]. They are the surfaces in R 3 that admit curvature line coordinates that are conformal with respect to the third fundamental of the surface, or as Musso and Nicolodi [30] put it, there exists a holomorphic 4 (with respect to the third fundamental form) quadratic differential q that commutes with the second fundamental form, i.e., if we use the complex structure induced on Σ by III to split the second fundamental form into bidegrees,…”