Annali di Matematica Pura ed Applicata (1923 -)

, Volume 198, Issue 5, pp 1639–1650 | Cite as

The biharmonic homotopy problem for unit vector fields on 2-tori

  • E. LoubeauEmail author
  • M. Markellos


The bienergy of smooth maps between Riemannian manifolds, when restricted to unit vector fields, yields two different variational problems depending on whether one takes the full functional or just the vertical contribution. Their critical points, called biharmonic unit vector fields and biharmonic unit sections, form different sets. Working with surfaces, we first obtain general characterizations of biharmonic unit vector fields and biharmonic unit sections under conformal change of the metric. In the case of a two-dimensional torus, this leads to a proof that biharmonic unit sections are always harmonic and a general existence theorem, in each homotopy class, for biharmonic unit vector fields.


Bienergy functional Biharmonic unit vector fields Biharmonic unit sections 2-Torus 

Mathematics Subject Classification

Primary 58E20 Secondary 53C20 



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Copyright information

© Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Département de Mathématiques, LMBA, UMR 6205Université de Bretagne OccidentaleBrest Cedex 3France
  2. 2.Department of Mathematics and StatisticsUniversity of CyprusNicosiaCyprus

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