12. November 2024
Nicolas Freches (RWTH Aachen)
The Palais-Smale condition in geometric knot theory
In geometric knot theory, we investigate questions from classical (i.e. topological) knot theory using methods from analysis. We study self-repulsive potentials on spaces of closed, injective and regular curves. By understanding the energy landscape of such potentials, we aim to gain information about the topology of the underlying knot classes. We construct the manifold of closed, embedded curves parametrized by arc length and show, that the famous Palais-Smale condition holds on this manifold for two families of knot energies. An application is given by a longtime existence and strong subconvergence result for the respective Sobolev gradient flow.