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

Barros, Manuel; Ferrández, Angel; Lucas, Pascual; Meroño, Miguel

doi:10.1090/S0002-9947-00-02366-7

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

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