Speed-controlling gradient flows for knot energies, and analyticity of critical knots

Steenebrügge, Daniel; von der Mosel, Heiko (Thesis advisor); Wagner, Alfred (Thesis advisor); Blatt, Simon (Thesis advisor)

Aachen : RWTH Aachen University (2023)
Dissertation / PhD Thesis

Dissertation, RWTH Aachen University, 2023

Abstract

In this thesis we deal with gradient flows and curves of maximal slope for various knot energies: We prove long time existence of such flows for the subfamilies of generalized integral Menger curvature and tangent-point energies associated to Hilbert spaces. Here, a projection ensures conservation of the curves' speed. For the Hilbert case of generalized integral Menger curvature, we furthermore show long time existence of a gradient flow where we reparametrize the curves proportionally to arc length at fixed points in time. In this case, we add the length of the curve to the energy. Furthermore, we prove existence for curves of maximal slope for the complete families of generalized integral Menger curvature and tangent-point energies, as well as O'Hara's energies, all in the cases where the energy is not scale-invariant and for the whole positive axis. To do so, we add a term controlling the speed of the curve to the energy and work in smaller spaces compactly embedded in the respective Banach spaces associated to the energies. Finally, we regard critical points with respect to fixed length of generalized integral Menger curvature associated to a Hilbert space. They are analytic which we prove via Cauchy's method of majorants. We briefly explain a few consequences of this result in the context of geometric knot theory. In preparation for the above results, we establish both $C^{1,1}_{\mathrm{loc}}$-regularity of the generalized tangent-point energies and continuous differentiability of O'Hara's knot energies. Furthermore this thesis contains an expansive collection of statements relevant in the context of knot energies, regarding periodic Sobolev-Slobodeckiǐ spaces.