Beitr\ EMIS ELibM Electronic Journals Beiträge zur Algebra und Geometrie
Contributions to Algebra and Geometry
Vol. 45, No. 1, pp. 191-208 (2004)

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On a conjecture about the Gauss map of complete spacelike surfaces with constant mean curvature in the Lorentz-Minkowski space

Rosa M. B. Chaves and Claudia Cueva C\^ andido

Departamento de Matemática, Instituto de Matemática e Estatística, Universidade de S\ ao Paulo, Caixa Postal 66281, CEP 05315--970, SP Brazil, e-mail: e-mail:

Abstract: The Gauss map of complete helicoidal (consequently rotational) surfaces with non-zero constant mean curvature in the Euclidean 3-space contains a maximal circle of the sphere. Observing the Gauss map image for complete spacelike surfaces in the Lorentz-Minkowski 3-space ${\mathbb L}^3$, we propose the following conjecture: ``Given a complete spacelike surface in ${\mathbb L}^3$, with non-zero constant mean curvature, its Gauss map image contains an arbitrary maximal geodesic of the hyperboloid contained in ${\mathbb L}^3$''. We answer the conjecture for the special class of spacelike rotational surfaces in ${\mathbb L}^3$ and obtain that, in this case, the conjecture is also true, as in the Euclidean space ${\mathbb R}^3$.

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Electronic version published on: 5 Mar 2004. This page was last modified: 4 May 2006.

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