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 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.Comment: 25 page

Weierstrass-type representations

“…Now, by setting c : Hence, from Subsection 4.1 we know that x m is a zero mean curvature surface in an affine hyperplane of R 3,1 . Therefore, we recover the result of [25] that the Ttransform of L-isothermic surfaces perturbs the cases (1), (2), (3) of Subsection 4.2 into the cases (1), (2), (3) of Subsection 4.1, respectively. This generalises the Umehara-Yamada perturbation [31] and gives a Laguerre geometric analogue of its interpretation in [21].…”

“…Thus x t is a marginally trapped surface and L t := x t + G t is the envelope of an L-isothermic surface. This is the T-transform of L-isothermic surfaces (see [25,26]). Notice that we obtain a new closed 1-form…”

“…Remark 4.1. In [25] it was shown that surfaces in Euclidean space that are simultaneously L-minimal and L-isothermic are those surfaces whose middle sphere congruence is one of the three cases above. On the other hand, a Weierstrass-type representation was developed for such surfaces in [30].…”

Surfaces in Laguerre geometry

“…This feature indicates that L-isothermic surfaces constitute an integrable system. For more details on the aspects of L-isothermic surfaces related to the theory of integrable system and for the study of their rich transformation theory, including the analogues of the T -transformation and of the Darboux transformation in Möbius geometry, we refer the reader to [45,44,46,47,49,53,60,62].…”

“…As an application of these results (cf. [53]), various instances of the Lawson isometric correspondence [35] between certain isometric constant mean curvature (CMC) surfaces in different hyperbolic 3-spaces and of the generalizations of Lawson's correspondence in the Lorentzian [55] and the (degenerate) isotropic situations [1,2,34,36], can be viewed as special cases of the T -transformation of L-isothermic surfaces with holomorphic quartic differential.…”