Hostname: page-component-76fb5796d-2lccl Total loading time: 0 Render date: 2024-04-29T23:17:46.597Z Has data issue: false hasContentIssue false
  • English
  • Français

Free elasticae and Willmore tori in warped product spaces

Published online by Cambridge University Press:  18 May 2009

Manuel Barros
Manuel Barros
Affiliation:
Departamento De Geometría Y Topología, Facultad De Ciencias, Universidad De Granada, 18071, Granada, Spain Email: Mbarros@goliat.ugr.es
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We use the principle of symmetric criticality to connect the Willmore variational problem for surfaces in a warped product space with base a circle, and the free elastica variational problem for curves on its fiber. In addition we obtain a rational oneparameter family of closed helices in the anti De Sitter 3-space which are critical points of the total squared curvature functional. This means they are free elasticae. Also they are spacelike; this allows us to construct a corresponding family of spacelike Willmore tori in a certain kind of spacetime close to the Robertson-Walker spaces.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 40 , Issue 2 , May 1998 , pp. 265 - 272
Copyright
Copyright © Glasgow Mathematical Journal Trust 1998

References

1.Barros, M., Willmore tori in non-standard 3-spheres, Math. Proc. Camb. Phil. Soc. 121 (1997), 321–324.Google Scholar
2.Barros, M., Lorentzian Willmore tori in the 3-dimensional anti De Sitter space, preprint (1996).Google Scholar
3.Barros, M. and Chen, B. Y., Stationary 2-type surfaces in a hypersphere, J. Math. Soc. Japan 39 (1987), 627–648.CrossRefGoogle Scholar
4.Barros, M., Ferrandez, A., Lucas, P. and Meroño, M. A., Hopf cylinders and filament equation in the three-dimensional anti-De Sitter space, preprint (1996).Google Scholar
5.Barros, M., Ferrandez, A., Lucas, P. and Meroño, M. A., Willmore tori into pseudo-Riemannian space forms of low dimension, preprint (1996).Google Scholar
6.Barros, M. and Garay, O. J., Hopf submanifolds in S 7 which are Willmore-Chen submanifolds, Math. Z., to appear.Google Scholar
7.Barros, M., Garay, O. J. and Singer, D. A., New examples of Willmore surfaces, preprint (1995).Google Scholar
8.Chen, B. Y., Some conformal invariants of submanifolds and their applications, Boll. Un. Mat. Ital. 10 (1974), 380–385.Google Scholar
9.Ejiri, N., A counter-exmpe for a Weiner open question, Indiana Univ. Math. J. 31 (1982), 209–211.Google Scholar
10.Greub, W., Halperin, S. and Vanstone, R., Connections, Curvature and Cohomology, 3 Vols. (Academic Press, 1972, 1973, 1976).Google Scholar
11.Langer, J. and Singer, D. A., The total squared curvature of closed curves, J. Diff. Geom. 20 (1984), 1–22.Google Scholar
12.Langer, J. and Singer, D. A., Curves in the hyperbolic plane and the mean curvature of tori in 3-space, Bull. London Math. Soc. 16 (1984), 531–534.CrossRefGoogle Scholar
13.O'Neill, B., Semi-Riemannian Geometry (Academic Press, 1983).Google Scholar
14.Palais, R. S., Critical point theory and the minimax principle, Global Analysis, Proc. Sympos. Pure Math. 15 (1970), 185–212.Google Scholar
15.Pinkall, U., Hopf tori in S 3, Invent. Math. 81 (1985), 379–386.Google Scholar