Banach gradient flows for various families of knot energies

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.