Convergence of gradient flows on knotted curves

TL;DR

This paper proves the full convergence of gradient flows for arc-length restricted tangent point energies towards critical points, using a ojasiewicz-Simon inequality, and establishes analyticity properties of related energies and metrics.

Contribution

It demonstrates the convergence of gradient flows for tangent point energies and proves analyticity of the energy and metric on the manifold of immersed curves.

Findings

Gradient flows converge to critical points.

Tangent point energies are analytic on the manifold.

The Hessian of the energy is Fredholm with index zero.

Abstract

We prove full convergence of gradient-flows of the arc-length restricted tangent point energies in the Hilbert-case towards critical points. This is done through a {\L}ojasiewicz-Simon gradient inequality for these energies. In order to do so, we prove, that the tangent-point energies are anlytic on the manifold of immersed embeddings and that their Hessian is Fredholm with index zero on the manifold of arc-length parametrized curves. As a by-product, we also show that the metric on the manifold of embedded immersed curves, defined by the first author, is analytic.