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Harmonic analysis meets critical knots. Critical points of the Möbius energy are smooth


Authors: Simon Blatt, Philipp Reiter and Armin Schikorra
Journal: Trans. Amer. Math. Soc. 368 (2016), 6391-6438
MSC (2010): Primary 46E35, 57M25
DOI: https://doi.org/10.1090/tran/6603
Published electronically: December 3, 2015
MathSciNet review: 3461038
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Abstract: Motivated by the Coulomb potential of an equidistributed charge on a curve, Jun O'Hara introduced and investigated the first geometric knot energy, the Möbius energy. We prove that every critical curve of this Möbius energy is of class $ C^\infty $ and thus extend the corresponding result due to Freedman, He, and Wang for minimizers of the Möbius energy.

In contrast to the techniques used by Freedman, He, and Wang, our methods do to not use the Möbius invariance of the energy, but rely on purely analytic methods motivated from a formal similarity of the Euler-Langrange equation to the half harmonic map equation for the unit tangent.


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Additional Information

Simon Blatt
Affiliation: Karlsruher Institut für Technologie, Institute for Analysis, Kaiserstraße 12, 76131 Karlsruhe, Germany
Address at time of publication: Fachbereich Mathematik, Universität Salzburg, Hellbrunner Strasse 34, 5020 Salzburg, Austria
Email: simon.blatt@kit.edu, simon.blatt@sbg.ac.at

Philipp Reiter
Affiliation: Fakultät für Mathematik, Universität Duisburg-Essen, Forsthausweg 2, 45117 Essen, Germany
Email: philipp.reiter@uni-due.de

Armin Schikorra
Affiliation: Max-Planck Institute for Mathematics in the Sciences, Inselstraße 22, 04103 Leipzig, Germany
Address at time of publication: Fachbereich Mathematik, Basel University, Spiegelgasse 1, 4051 Basel, Switzerland
Email: armin.schikorra@mis.mpg.de, armin.schikorra@unibas.ch

DOI: https://doi.org/10.1090/tran/6603
Received by editor(s): April 19, 2013
Received by editor(s) in revised form: August 20, 2014
Published electronically: December 3, 2015
Additional Notes: The first author was supported by Swiss National Science Foundation Grant Nr. 200020_125127 and the Leverhulm trust. The second author was supported by DFG Transregional Collaborative Research Centre SFB TR 71. The third author received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013) / ERC grant agreement no. 267087, DAAD PostDoc Program (D/10/50763) and the Forschungsinstitut für Mathematik, ETH Zürich. He would like to thank Tristan Rivière and the ETH for their hospitality.
Article copyright: © Copyright 2015 American Mathematical Society