Stationary points of O’Hara’s knot energies

Manuscripta Mathematica volume 140, pages 29–50 (2013)Cite this article

Abstract

In this article we study the regularity of stationary points of the knot energies E (α) introduced by O’Hara (Topology 30(2):241–247, 1991; Topol Appl 48(2):147–161, 1992; Topol Appl 56(1):45–61, 1994) in the range \({\alpha\in(2,3)}\) . In a first step we prove that E (α) is C 1 on the set of all regular embedded curves belonging to \({{H^{(\alpha+1)/2,2}(\mathbb {R}{/}\mathbb {Z}, \mathbb {R}^n)}}\) and calculate its derivative. After that we use the structure of the Euler-Lagrange equation to study the regularity of stationary points of E (α) plus a positive multiple of the length. We show that stationary points of finite energy are of class C ∞—so especially all local minimizers of E (α) among curves with fixed length are smooth.

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Affiliations

  1. Mathematics Institute, University of Warwick, Zeeman Building, Coventry, CV4 7AL, UK

    Simon Blatt

  2. Abteilung für Angewandte Mathematik, Universität Freiburg, Hermann-Herder-Straße 10, 79104, Freiburg, Germany

    Philipp Reiter

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  1. Simon Blatt
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Correspondence to Philipp Reiter.

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Blatt, S., Reiter, P. Stationary points of O’Hara’s knot energies. manuscripta math. 140, 29–50 (2013). https://doi.org/10.1007/s00229-011-0528-8

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Mathematics Subject Classification (2000)

  • 42A45
  • 53A04
  • 57M25