On the Palais-Smale condition in geometric knot theory

TL;DR

This paper demonstrates that various energies in geometric knot theory satisfy the Palais-Smale condition, enabling the existence and analysis of critical knots and their gradient flows within prescribed isotopy classes.

Contribution

It establishes the Palais-Smale condition for a broad class of knot energies, including non-local and integral curvature functionals, and proves existence and smoothness of critical knots.

Findings

Palais-Smale condition holds for multiple knot energies.

Existence of minimizing and critical knots in any isotopy class.

Long-time existence and convergence of gradient flows to critical knots.

Abstract

We prove that various families of energies relevant in geometric knot theory satisfy the Palais-Smale condition (PS) on submanifolds of arclength para\-metrized knots. These energies include linear combinations of the Euler-Bernoulli bending energy with a wide variety of non-local knot energies, such as O'Hara's self-repulsive potentials $E^{\alpha,p}$, generalized tangent-point energies $\TP^{(p,q)}$, and generalized integral Menger curvature functionals $\intM^{(p,q)}$. Even the tangent-point energies $\TP^{(p,2)}$ for $p\in (4,5)$ alone are shown to fulfill the (PS)-condition. For all energies mentioned we can therefore prove existence of minimizing knots in any prescribed ambient isotopy class, and we provide long-time existence of their Hilbert-gradient flows, and subconvergence to critical knots as time goes to infinity. In addition, we prove $C^\infty$-smoothness of all arclength constrained critical knots, which shows in particular that these critical knots are also critical for the energies on the larger open set of regular knots under a fixed-length constraint.