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

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 TP(p,q) in the sub-critical range p ∈ (q+2,2q+1) and see that those are all injective and regular curves in the Sobolev–Slobodeckiĭ space W(p-1)/q,q(ℝ/ℤ,ℝn). 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) + λ length, p ∈ (4,5), λ > 0, are smooth – so especially all local minimizers are smooth.