On the regularity of critical points for O'Hara's knot energies: From smoothness to analyticity

Abstract

We prove the analyticity of smooth critical points for O'Hara's knot energies $\mathcal{E}^{\alpha,p}$, with $p=1$ and $2<\alpha< 3$, subject to a fixed length constraint. This implies, together with the main result in \cite{BR13}, that bounded energy critical points of $\mathcal{E}^{\alpha,1}$ subject to a fixed length constraint are not only $C^\infty$ but also analytic. Our approach is based on Cauchy's method of majorants and a decomposition of the gradient that was adapted from the M\"obius energy case $\mathcal{E}^{2,1}$ in \cite{BV19}.