Stationary Points of O'Hara's Knot Energies

Abstract

In this article we study the regularity of stationary points of the knot energies $E^\alpha$ introduced by O'Hara in the range $\alpha \in (2,3)$. In a first step we prove that $E^\alpha$ is $C^1$ on the set of all regular embedded closed curves belonging to $H^{(\alpha +1)/2,2}$ and calculate its derivative. After that we use the structure of the Euler-Lagrange equation to study the regularity of stationary points of $E^\alpha$ plus a positive multiple of the length. We show that stationary points of finite energy are of class $C^\infty$ - so especially all local minimizers of $E^\alpha$ among curves with fixed length are smooth.