Mathematische Zeitschrift

, Volume 284, Issue 3–4, pp 1089–1110 | Cite as

Holomorphic differentials and Laguerre deformation of surfaces

  • Emilio Musso
  • Lorenzo NicolodiEmail author


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.


Laguerre surface geometry Laguerre minimal surfaces Laguerre Gauss map L-isothermic surfaces Constant mean curvature surfaces Minimal surfaces Maximal surfaces Isotropic geometry Lawson correspondence 

Mathematics Subject Classification

53A35 53C42 


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Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Dipartimento di Scienze MatematichePolitecnico di TorinoTorinoItaly
  2. 2.Dipartimento di Matematica e InformaticaUniversità degli Studi di ParmaParmaItaly

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