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

TL;DR

This paper proves that smooth critical points of a certain class of generalized integral Menger curvature energies are analytic, extending regularity results to show these curves are infinitely differentiable and analytic.

Contribution

It establishes the analyticity of critical points for generalized integral Menger curvature energies in the Hilbert case, using a novel recursive estimate approach.

Findings

Critical points are shown to be analytic.

Finite-energy critical curves are infinitely differentiable.

The method extends regularity results to analyticity.

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.