Stationary points of O’Hara’s knot energies

Abstract

In this article we study the regularity of stationary points of the knot energies E (α) introduced by O'Hara in [14,15,16] in the range α ∈ (2, 3). In a first step we prove that E (α) is C 1 on the set of all regular embedded curves belonging to H (α+1)/2,2 (R/Z, R n ) and calculate its derivative. After that we use the structure of the Euler-Lagrange equation to study the regularity of stationary points of E (α) plus a positive multiple of the length. We show that stationary points of finite energy are of class C ∞ -so especially all local minimizers of E (α) among curves with fixed length are smooth.

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.…”

“…A more elementary approach, but less effective for limit estimates in Hardy-space and BMO was introduced in [37]. The following estimate can be deduced from both arguments, see also [7,Lemma A.5,16].…”

How nice are critical knots? Regularity theory for knot energies

“…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;.…”

“…Formally this is a straight-forward calculation, but one has to take into account that this involves interchanging differentiation and integration. Especially in the case of O'Hara's energies this is a quite subtle process which can be tackled by a careful analysis of approximating energy functionals, see [Blatt & Reiter, 2013a]. In this particular situation, the first variation turns out to be a principal value integral of type lim ε 0 |w|≥ε · · · .…”

“…Theorem (Stationary points are smooth [Blatt & Reiter, 2013a, 2013b). Any stationary point of E α,1 , intM (p,2) , and TP (p,2) in the non-degenerate sub-critical case with respect to fixed length, being injective and parametrized by arc-length, is C ∞ -smooth.…”