  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.
