Holomorphic differentials and Laguerre deformation of surfaces

Abstract

A Laguerre geometric local characterization is given of L-minimal surfaces and Laguerre deformations (T-transforms) of L-minimal isothermic surfaces in terms of the holomorphicity of a quartic and a quadratic differential. This is used to prove that, via their Laguerre Gauss maps, the T-transforms of L-minimal isothermic surfaces have constant mean curvature in some translate of hyperbolic 3-space or de Sitter 3-space in Minkowski 4-space, or have mean curvature zero in some translate of a time-oriented lightcone in Minkowski 4-space. As an application, we show that various instances of the Lawson isometric correspondence can be viewed as special cases of the T-transformation of L-isothermic surfaces with holomorphic quartic differential.