On the Analyticity of Critical Points of the Generalized Integral Menger Curvature in the Hilbert Case

Abstract

We prove the analyticity of smooth critical points for generalized integral Menger curvature energies $\mathrm{intM}^{(p,2)}$, with $p \in (\tfrac 73, \tfrac 83)$, subject to a fixed length constraint. This implies, together with already well-known regularity results, that finite-energy, critical $C^1$-curves $\gamma: \mathbb{R}/\mathbb{Z} \to \mathbb{R}^n$ of generalized integral Menger curvature $\mathrm{intM}^{(p,2)}$ subject to a fixed length constraint are not only $C^\infty$ but also analytic. Our approach is inspired by analyticity results on critical points for O'Hara's knot energies based on Cauchy's method of majorants and a decomposition of the first variation. The main new idea is an additional iteration in the recursive estimate of the derivatives to obtain a sufficient difference in the order of regularity.