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 L-Gauss maps, the T-transforms of L-minimal isothermic surfaces have constant mean curvature H= r in some translate of hyperbolic 3-space H3(-r2)⊂R14, de Sitter 3-space S13(r2)⊂R14, or have mean curvature H= 0 in some translate of a time-oriented lightcone in R14. 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.