Banach gradient flows for various families of knot energies

TL;DR

This paper proves the long-time existence of Banach gradient flows for various knot energies, using the theory of curves of maximal slope and establishing differentiability of O'Hara's energies.

Contribution

It introduces a framework for analyzing gradient flows of knot energies in Banach spaces and proves differentiability of O'Hara's energies, enabling long-term flow analysis.

Findings

Established long-time existence of gradient flows for knot energies.

Proved continuous differentiability of O'Hara's energies.

Developed a new analytical approach using curves of maximal slope.

Abstract

We establish long-time existence of Banach gradient flows for generalised integral Menger curvatures and tangent-point energies, and for O'Hara's self-repulsive potentials $E^{\alpha,p}$. In order to do so, we employ the theory of curves of maximal slope in slightly smaller spaces compactly embedding into the respective energy spaces associated to these functionals, and add a term involving the logarithmic strain, which controls the parametrisations of the flowing (knotted) loops. As a prerequisite, we prove in addition that O'Hara's knot energies $E^{\alpha,p}$ are continuously differentiable.