Repulsive knot energies and pseudodifferential calculus for O’Hara’s knot energy family E(α), α ∈ [2, 3)

Abstract

AbstractWe develop a precise analysis of J. O’Hara’s knot functionals E(α), α ∈ [2, 3), that serve as self‐repulsive potentials on (knotted) closed curves. First we derive continuity of E(α) on injective and regular H2 curves and then we establish Fréchet differentiability of E(α) and state several first variation formulae. Motivated by ideas of Z.‐X. He in his work on the specific functional E(2), the so‐called Möbius Energy, we prove C∞‐smoothness of critical points of the appropriately rescaled functionals \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\tilde{E}^{(\alpha )}= {\rm length}^{\alpha -2}E^{(\alpha )}$\end{document} by means of fractional Sobolev spaces on a periodic interval and bilinear Fourier multipliers.