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.
Holomorphic differentials and Laguerre deformation of surfaces / Musso, Emilio; Nicolodi, Lorenzo. - In: MATHEMATISCHE ZEITSCHRIFT. - ISSN 0025-5874. - ELETTRONICO. - 284:4(2016), pp. 1-22. [10.1007/s00209-016-1689-7]
Holomorphic differentials and Laguerre deformation of surfaces
MUSSO, EMILIO;
2016
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 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.File | Dimensione | Formato | |
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