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 intM (p,2) , with p ∈ ( 7 3 , 8 3 ), subject to a fixed length constraint. This implies, together with already well-known regularity results, that finite-energy, critical C 1 -curves γ : R/Z → R n of generalized integral Menger curvature intM (p,2) subject to a fixed length constraint are not only C ∞ 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.