Manuscripta Mathematica

, Volume 140, Issue 1–2, pp 29–50 | Cite as

Stationary points of O’Hara’s knot energies



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.

Mathematics Subject Classification (2000)

42A45 53A04 57M25 


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Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  1. 1.Mathematics InstituteUniversity of WarwickCoventryUK
  2. 2.Abteilung für Angewandte MathematikUniversität FreiburgFreiburgGermany

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