Abstract
In this paper, it is characterized the generating curves of minimal rotational surfaces in the de Sitter space \({\mathbb {S}}_1^3\) as solutions of a variational problem. More exactly, it is proved that these curves are the critical points of a potential energy functional involving the distance to a given plane among all curves of \({\mathbb {S}}_1^2\) with prescribed endpoints and fixed length. This extends the known Euler’s result that asserts that the catenary is the generating curve of the catenoid.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Data Availability Statement
This manuscript has no associate data. No funding was received for conducting this study. The author has no competing interests to declare that are relevant to the content of this article.
References
Akutagawa, K.: On spacelike hypersurfaces with constant mean curvature in the de Sitter space. Math. Z. 196, 13–19 (1987)
Brasil, A., Jr., Colares, A.G., Palmas, O.: Complete spacelike hypersurfaces with constant mean curvature in the de Sitter space: a gap theorem. Illinois J. Math. 47, 847–866 (2003)
da Silva, L. C. B., López, R.: Catenaries and minimal surfaces of revolution in hyperbolic space. arXiv:2211.15297 [math.DG] (2022)
Demirci, B. B.: On rotational surfaces in 3-dimensional de Sitter space with Weingarten condition. Mediterr. J. Math. 19 (2022), Paper No. 138, 17 pp
do Carmo, M., Dajczer, M.: Rotation hypersurfaces in spaces of constant curvature. Trans. Am. Math. Soc. 277, 685–709 (1983)
Euler, L.: Opera omnia. Series prima. Opera mathematica. Vol. XXIV. Methodus inveniendi lineas curvas maximi minimive proprietate gaudentes sive solutio problematis isoperimetrici latissimo sensu accepti, Societas Scientiarum Naturalium Helveticae, Bern (1952)
Liu, H.: Maximal and minimal rotation surfaces in 3-dimensional de Sitter space and their global stability. Atti Sem. Mat. Fis. Univ. Modena XLII, 319–327 (1994)
Liu, H.: Rotational surfaces of finite type in 3-dimensional de Sitter space. Algebra Groups Geom. 13, 359–369 (1996)
Liu, H., Liu, G.: Hyperbolic rotation surfaces of constant mean curvature in 3-de Sitter space. Bull. Belg. Math. Soc. Simon Stevin 7, 455–466 (2000)
Liu, H., Liu, G.: Weingarten rotation surfaces in 3-dimensional de Sitter space. J. Geom. 79, 156–168 (2004)
López, R.: The hanging chain problem in the sphere and in the hyperbolic plane. arXiv:2208.13694 [math.DG] (2022)
Mori, H.: Rotational surfaces in a pseudo-Riemannian \(3\)-sphere. Tohoku Math. J. 38, 29–36 (1986)
Yang, S-D.: Björling formula for mean curvature one surfaces in hyperbolic three-space and in de Sitter three-space. Bull. Korean Math. Soc. 54, 159–175 (2017)
Acknowledgements
Rafael López is a member of the Institute of Mathematics of the University of Granada. This work has been partially supported by the Projects I+D+i PID2020-117868GB-I00, supported by MCIN/ AEI/10.13039/501100011033/, A-FQM-139-UGR18 and P18-FR-4049.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
López, R. A Characterization of Minimal Rotational Surfaces in the de Sitter Space. Mediterr. J. Math. 20, 68 (2023). https://doi.org/10.1007/s00009-023-02275-8
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00009-023-02275-8