Stationary Points of O'Hara's Knot Energies

TL;DR

This paper investigates the smoothness of stationary points of O'Hara's knot energies for curves, proving they are infinitely differentiable under certain conditions, which advances understanding of knot energy regularity.

Contribution

It establishes the $C^1$ regularity of O'Hara's knot energies and proves stationary points are smooth, providing new insights into the structure of these energies and their minimizers.

Findings

Stationary points of finite energy are $C^ $-smooth.

O'Hara's energies are $C^1$ on certain regular curves.

Local minimizers of the energy are smooth.

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.