Hard analysis meets critical knots (Stationary points of the Moebius energy are smooth)

TL;DR

This paper proves that stationary points of the Moebius energy for curves are smooth if the energy is finite, using purely analytical methods that are applicable in higher dimensions.

Contribution

It establishes the smoothness of stationary points of the Moebius energy without relying on Moebius invariance, broadening the scope to higher dimensions.

Findings

Stationary points of finite Moebius energy are smooth.

Analytical methods suffice without Moebius invariance.

Techniques generalize beyond one-dimensional curves.

Abstract

We prove that if a curve parametrized by arc length is a stationary point of the Moebius energy introduced by Jun O'Hara, then it is smooth whenever the Moebius energy is finite. Our methods, interestingly, only rely on purely analytical arguments, entirely without using Moebius invariance. Furthermore, the techniques involved are not fundamentally restricted to one-dimensional domains, but are generalizable to arbitrary dimensions.