A Weierstrass representation for linear Weingarten spacelike surfaces of maximal type in the Lorentz-Minkowski space.
type: Item
instance of: scholarly article
title: A Weierstrass representation for linear Weingarten spacelike surfaces of maximal type in the Lorentz-Minkowski space.
zbMATH Open document ID: 1035.53014
DOI: 10.1016/S0022-247X(03)00166-5
author: Juan A. Aledo
author: José A. Gálvez
published in: Journal of Mathematical Analysis and Applications
publication date: 2003-08-25T00:00:00Z
review text: The authors extend the Weierstrass representation for maximal spacelike surfaces in the 3-dimensional Lorentz-Minkowski space \({\mathbf L}^3\) to a conformal representation of the spacelike surfaces whose mean curvature \(H\) is proportional to its Gaussian curvature \(K= -2aH+ bK= 0\), \(a\), \(b\) being two real constants with \(a\neq 0\) (``linear Weingarten surfaces of maximal type'', shortly: LWM-spacelike surfaces; see Theorem 9). This representation is used in order to study the Gaussian curvature and the Gauss map of such surfaces when the immersion is complete (it is shown that for any \(b\neq 0\) there exist nonflat complete LWM-spacelike surfaces) proving that the surface is a plane or the supremum of its Gaussian curvature is a negative constant \((=4a^2/b^2)\) and its Gauss map is a diffeomorphism onto the hyperbolic plane (Theorem 12). The rotation LWM-spacelike surfaces can be described explicitly as follows (Theorem 20): every rotation LWM-spacelike surface in \({\mathbf L}^3\) is isometric to a piece of one of the following: (i) plane; (ii) LWM-catenoid with vertical axis; (iii) LWM-catenoid with horizontal axis; (iv) LWM-Enneper surface.
Mathematics Subject Classification ID: 53A10
Mathematics Subject Classification ID: 53C42
Mathematics Subject Classification ID: 53C50
zbMATH DE Number: 1970505
zbMATH Keywords: Lorentz-Minkowski space
zbMATH Keywords: linear Weingarten surface of maximal type
zbMATH Keywords: conformal representation
zbMATH Keywords: rotation surface
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1970505