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 H3(−r 2) ⊂ R41, de Sitter 3-space S31(r 2) ⊂ R41, or have mean curvature H = 0 in some translate of a time-oriented lightcone in R41. 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 withholomorphic quartic differential.