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.

42 pages

References in corpus (1)

• A speed preserving Hilbert gradient flow for generalized integral Menger curvature

Cited by in corpus (1)

• Optimal planar immersions of prescribed winding number and Arnold invariants