Regularity theory for tangent-point energies: The non-degenerate sub-critical case

TL;DR

This paper introduces a new family of knot energies, characterizes finite energy curves in the sub-critical range, and proves smoothness of stationary points for certain parameters, advancing the understanding of tangent-point energies.

Contribution

It defines a two-parameter family of knot energies, analyzes their properties, and establishes regularity results for stationary points in the sub-critical case.

Findings

Finite energy curves are injective and regular in Sobolev-Slobodecki
21 space.

The first variation formula is a non-degenerate elliptic operator for q=2.

Stationary points of the energy plus length are smooth for p in (4,5).

Abstract

In this article we introduce and investigate a new two-parameter family of knot energies $TP^{(p,q)}$ that contains the tangent-point energies. These energies are obtained by decoupling the exponents in the numerator and denominator of the integrand in the original definition of the tangent-point energies. We will first characterize the curves of finite energy in the sub-critical range $p\in(q+2,2q+1)$ and see that those are all injective and regular curves in the Sobolev-Slobodecki\u{i} space $W^{(p-1)/q,q}$. We derive a formula for the first variation that turns out to be a non-degenerate elliptic operator for the special case $q=2$ --- a fact that seems not to be the case for the original tangent-point energies. This observation allows us to prove that stationary points of $TP^{(p,2)}$ + \lambda length, $p\in(4,5)$, \lambda > 0, are smooth --- so especially all local minimizers are smooth.