Convergence of gradient flows on knotted curves

TL;DR

This work proves full convergence of gradient flows for the arc-length restricted tangent-point energy $\mathrm{TP}^{(p,2)}$ on knotted curves by establishing a Łojasiewicz-Simon gradient inequality in the arc-length setting. Central to the approach is proving that $\mathrm{TP}^{(p,2)}$ is analytic on the manifold of immersed $H^s$-embeddings and that its Hessian on the arc-length submanifold $\mathcal{A}^s$ is Fredholm with index zero, enabling a local gradient inequality. As a consequence, gradient flows constrained to arc-length converge strongly to a critical point, with convergence rates governed by the Łojasiewicz exponent $\theta$, and the paper also shows the analytic nature of the arc-length metric. These results provide a rigorous framework for the long-time behavior of gradient flows for knot energies under topological constraints and reinforce the viability of analytic methods in the geometric analysis of curves.

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.

Theorems & Definitions (30)

• Theorem 1.1
• Corollary 1.2
• Theorem 1.3
• Theorem 1.4: Łojasiewicz-Simon gradient inequality
• Theorem 1.5
• Theorem 1.6: FeehanMaridakisLojasiewiczSimonGradientInequalities2020
• Definition 2.1: AnalyticFunctionsOnBanachSpacesWhittlesey
• Theorem 2.2: p. 1079, AnalyticFunctionsOnBanachSpacesWhittlesey
• Example 2.3
• Lemma 2.4