On the analyticity of critical points of the M\"obius energy

TL;DR

This paper proves that smooth critical points of the M"obius energy parametrized by arc-length are analytic, extending regularity results and being the first to establish analyticity in non-local differential equations.

Contribution

It demonstrates the analyticity of critical points of the M"obius energy, combining Cauchy's method of majorants with a novel gradient decomposition.

Findings

Critical points are analytic if smooth and parametrized by arc-length.

Critical points with bounded energy are $C^ abla$ and analytic.

First analyticity result for non-local differential equations.

Abstract

We prove that smooth critical points of the M\"obius energy parametrized by arc-length are analytic. Together with the main result in \cite{BRS16} this implies that critical points of the M\"obius energy with merely bounded energy are not only $C^\infty$ but also analytic. Our proof is based on Cauchy's method of majorants and a decomposition of the gradient which already proved useful in the proof of the regularity results in \cite{BR13} and \cite{BRS16}. To best of the authors knowledge, this is the first analyticity result in the context of non-local differential equations.