A criterion for reduction of variables in the Willmore-Chen variational problem and its applications
HTML articles powered by AMS MathViewer
- by Manuel Barros, Angel Ferrández, Pascual Lucas and Miguel A. Meroño PDF
- Trans. Amer. Math. Soc. 352 (2000), 3015-3027 Request permission
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.References
- Manuel Barros, Willmore tori in non-standard $3$-spheres, Math. Proc. Cambridge Philos. Soc. 121 (1997), no. 2, 321–324. MR 1426526, DOI 10.1017/S0305004196001466
- M. Barros, Free elasticae and Willmore tori in warped product spaces, Glasgow Math. J., 40, 265–272, 1998.
- Manuel Barros and Bang-Yen Chen, Stationary $2$-type surfaces in a hypersphere, J. Math. Soc. Japan 39 (1987), no. 4, 627–648. MR 905629, DOI 10.2969/jmsj/03940627
- M. Barros, A. Ferrández, P. Lucas, and M. A. Meroño, Willmore tori and Willmore-Chen submanifolds in pseudo-Riemannian spaces, J. Geom. Phys., 28, 45-66, 1998.
- Manuel Barros and Oscar J. Garay, Free elastic parallels in a surface of revolution, Amer. Math. Monthly 103 (1996), no. 2, 149–156. MR 1375059, DOI 10.2307/2975109
- Manuel Barros and Oscar J. Garay, Hopf submanifolds in $S^7$ which are Willmore-Chen submanfolds, Math. Z. 228 (1998), no. 1, 121–129. MR 1617967, DOI 10.1007/PL00004404
- M. Barros, O. Garay, and D. Singer, New examples of Willmore surfaces, Preprint, 1995.
- M. Barros and M. A. Meroño, Willmore tori in Kaluza-Klein conformal structures on the three sphere, Preprint, 1997.
- Bang-yen Chen, Some conformal invariants of submanifolds and their applications, Boll. Un. Mat. Ital. (4) 10 (1974), 380–385 (English, with Italian summary). MR 0370436
- Norio Ejiri, A counterexample for Weiner’s open question, Indiana Univ. Math. J. 31 (1982), no. 2, 209–211. MR 648171, DOI 10.1512/iumj.1982.31.31018
- Alfred Gray, Pseudo-Riemannian almost product manifolds and submersions, J. Math. Mech. 16 (1967), 715–737. MR 0205184
- Gary R. Jensen, Einstein metrics on principal fibre bundles, J. Differential Geometry 8 (1973), 599–614. MR 353209
- Shoshichi Kobayashi and Katsumi Nomizu, Foundations of differential geometry. Vol I, Interscience Publishers (a division of John Wiley & Sons, Inc.), New York-London, 1963. MR 0152974
- Joel Langer and David Singer, Curves in the hyperbolic plane and mean curvature of tori in $3$-space, Bull. London Math. Soc. 16 (1984), no. 5, 531–534. MR 751827, DOI 10.1112/blms/16.5.531
- Joel Langer and David A. Singer, The total squared curvature of closed curves, J. Differential Geom. 20 (1984), no. 1, 1–22. MR 772124
- Richard S. Palais, The principle of symmetric criticality, Comm. Math. Phys. 69 (1979), no. 1, 19–30. MR 547524
- U. Pinkall, Hopf tori in $S^3$, Invent. Math. 81 (1985), no. 2, 379–386. MR 799274, DOI 10.1007/BF01389060
- Alfonso Romero and Miguel Sánchez, New properties and examples of incomplete Lorentzian tori, J. Math. Phys. 35 (1994), no. 4, 1992–1997. MR 1267937, DOI 10.1063/1.530584
- Alan Weinstein, Fat bundles and symplectic manifolds, Adv. in Math. 37 (1980), no. 3, 239–250. MR 591728, DOI 10.1016/0001-8708(80)90035-3
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
- 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.
- © Copyright 2000 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 352 (2000), 3015-3027
- MSC (2000): Primary 53C40, 53A30, 58E30
- DOI: https://doi.org/10.1090/S0002-9947-00-02366-7
- MathSciNet review: 1621713
Dedicated: Dedicated to the memory of Alfred Gray