Convergence of gradient flows on knotted curves

2025/11/10 by Döhrer, Elias, Freches, Nicolas
Mathematics · #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Geometric and Algebraic Topology

paper · doi:10.48550/arxiv.2511.07214

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 Łojasiewicz-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.

Citations

• On a Complete Riemannian Metric on the Space of Embedded Curves
• Scale-invariant tangent-point energies for knots
• The Gradient Flow of O'Hara's Knot Energies
• Global existence and convergence of solutions to gradient systems and applications to Yang-Mills gradient flow
• Regularity theory for tangent-point energies: The non-degenerate sub-critical case
• The gradient flow of the Möbius energy near local minimizers

Related

• Thom's gradient conjecture for parabolic systems and the Yang-Mills and Ricci flows
• Scale-invariant tangent-point energies for knots
• The curve shrinking flow, compactness and its relation to scale manifolds
• The Gradient Flow of O'Hara's Knot Energies
• On a Complete Riemannian Metric on the Space of Embedded Curves