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.
Holomorphic differentials and Laguerre deformation of surfaces / Musso, Emilio; Nicolodi, Lorenzo. - In: MATHEMATISCHE ZEITSCHRIFT. - ISSN 0025-5874. - 284:3-4(2016), pp. 1089-1110. [10.1007/s00209-016-1689-7]