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

Abstract

We 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 H 2 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 functionalsẼ (α ) = length α −2 E (α ) by means of fractional Sobolev spaces on a periodic interval and bilinear Fourier multipliers.

Integro-Differential Harmonic Maps into Spheres

“…For several of these curvature energies, regularity even of minimizers is not understood, and arguments as in [19] seem not to work, since also the invariance class is not known. See [7,9,31,32]. Thus, the present work is also intended to deliver a framework which hopefully will lead to substantial progress in this area.…”

The gradient flow of O’Hara’s knot energies

“…To estimate the terms containing Q α , we will use the fact that Q α = c α D α+1 +R whereR is a bounded operator from W s+2, p to W s, p [24,Proposition 2.3]. For k 1 , k 2 ∈ N 0 with k 1 + k 2 = k we estimate using Hölder's inequality and the GagliardoNirenberg-Sobolev inequality…”

“…Whether or not the same is true for composite knot classes is an open problem, though there are some numerical experiments that indicate that this might not be the case in every such knot class [19]. Furthermore, they derived a formula for the L 2 -gradient of the Möbius energy [14,Equation 6.12] which was extended by Reiter [24,Theorem 1.45] to the energies E α for α ∈ [2,3). They showed that the first variation of these functionals can be given by Using the Möbius invariance of E 2 , Freedman, He, and Wang showed that local minimizers of the Möbius energy are of class C 1,1 [14]-and thus gave a first answer to the question about the niceness of the optimal shapes.…”

How nice are critical knots? Regularity theory for knot energies

“…The subsequent article by He on the Euler-Lagrange equation and heat flow for the Möbius energy [2000] was the starting point both for investigating the corresponding gradient flow [Blatt, 2010[Blatt, , 2012b and the regularity of stationary points [Reiter, 2010[Reiter, , 2012, both for the oneparameter sub-family E α,1 , α ∈ [2, 3). After a first attempt [Blatt & Reiter, 2008], the identification of the energy spaces [Blatt, 2012a] led to a significant improvement of the latter result [Blatt & Reiter, 2013a;.…”