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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

A criterion for reduction of variables in the Willmore-Chen variational problem and its applications


Authors: Manuel Barros, Angel Ferrández, Pascual Lucas and Miguel A. Meroño
Journal: Trans. Amer. Math. Soc. 352 (2000), 3015-3027
MSC (2000): Primary 53C40, 53A30, 58E30
Published electronically: February 24, 2000
MathSciNet review: 1621713
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Abstract: We exhibit a criterion for a reduction of variables for Willmore-Chen submanifolds in conformal classes associated with generalized Kaluza-Klein metrics on flat principal fibre bundles. Our method relates the variational problem of Willmore-Chen with an elasticity functional defined for closed curves in the base space. The main ideas involve the extrinsic conformal invariance of the Willmore-Chen functional, the large symmetry group of generalized Kaluza-Klein metrics and the principle of symmetric criticality. We also obtain interesting families of elasticae in both lens spaces and surfaces of revolution (Riemannian and Lorentzian). We give applications to the construction of explicit examples of isolated Willmore-Chen submanifolds, discrete families of Willmore-Chen submanifolds and foliations whose leaves are Willmore-Chen submanifolds.


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Additional Information

Manuel Barros
Affiliation: Departamento de Geometría y Topología, Universidad de Granada, 18071 Granada, Spain
Email: mbarros@ugr.es

Angel Ferrández
Affiliation: Departamento de Matemáticas, Universidad de Murcia, 30100 Espinardo, Murcia, Spain
Email: aferr@um.es

Pascual Lucas
Affiliation: Departamento de Matemáticas, Universidad de Murcia, 30100 Espinardo, Murcia, Spain
Email: plucas@um.es

Miguel A. Meroño
Affiliation: Departamento de Matemáticas, Universidad de Murcia, 30100 Espinardo, Murcia, Spain
Email: mamb@um.es

DOI: http://dx.doi.org/10.1090/S0002-9947-00-02366-7
PII: S 0002-9947(00)02366-7
Keywords: Willmore-Chen submanifold, Kaluza-Klein metric, elastic curves
Received by editor(s): November 11, 1997
Received by editor(s) in revised form: June 25, 1998
Published electronically: February 24, 2000
Additional Notes: This research has been partially supported by DGICYT grant PB97-0784 and Fundación Séneca (C.A.R.M.) grant PB/5/FS/97.
Dedicated: Dedicated to the memory of Alfred Gray
Article copyright: © Copyright 2000 American Mathematical Society