Holomorphic differentials and Laguerre deformation of surfaces

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 \({\mathbb {H}}^3(-r^2)\subset \mathbb {R}^4_1\), de Sitter 3-space \({\mathbb {S}}^3_1(r^2)\subset \mathbb {R}^4_1\), or have mean curvature \(H=0\) in some translate of a time-oriented lightcone in \(\mathbb {R}^4_1\). 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.