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

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 $C^{\mathrm{\infty }}$ 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.