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 Möbius energy case ${\mathcal{E}}^{2,1}$ in \cite{BV19}.