Nicolas Freches | Searcharxiv

arXiv2025

Convergence of gradient flows on knotted curves

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.

math.CA↗

arXiv2025

On the Palais-Smale condition in geometric knot theory

We prove that various families of energies relevant in geometric knot theory satisfy the Palais-Smale condition (PS) on submanifolds of arclength para\-metrized knots. These energies include linear combinations of the Euler-Bernoulli bending energy with a wide variety of non-local knot energies, such as O'Hara's self-repulsive potentials $E^{\alpha,p}$, generalized tangent-point energies $\TP^{(p,q)}$, and generalized integral Menger curvature functionals $\intM^{(p,q)}$. Even the tangent-point energies $\TP^{(p,2)}$ for $p\in (4,5)$ alone are shown to fulfill the (PS)-condition. For all energies mentioned we can therefore prove existence of minimizing knots in any prescribed ambient isotopy class, and we provide long-time existence of their Hilbert-gradient flows, and subconvergence to critical knots as time goes to infinity. In addition, we prove $C^\infty$-smoothness of all arclength constrained critical knots, which shows in particular that these critical knots are also critical for the energies on the larger open set of regular knots under a fixed-length constraint.

math.CA↗