Stationary curves under the M\"obius-Plateau energy

TL;DR

This paper introduces a new class of Plateau problems with self-repulsive energies, specifically using the Möbius energy, and proves existence of solutions while analyzing special cases like helicoidal strips.

Contribution

It establishes an existence theorem for the Möbius-Plateau problem and characterizes solutions for helicoidal strips, revealing differences between screw-like and ribbon-like configurations.

Findings

Existence of solutions for the Möbius-Plateau problem in certain knot classes.

Screw-like solutions are abundant and flexible.

Ribbon-like solutions require high frequency, low pitch, and proximity to the axis.

Abstract

Plateau problems with elastic boundary energies have been of recent theoretical and applied interest. However, strong assumptions have to be made to avoid self-intersections of the boundary curve during energy minimization. We introduce a class of Plateau problems for boundaries with self-repulsive energies that obviates self-contact in energy minimization problems. For the self-repulsive energy, we choose the M\"obius Energy introduced by O'Hara due to its myriad regularity properties shown by Freedman et al. We first prove an existence theorem for this M\"obius-Plateau problem in the class of closed Lipschitz curves of a given irreducible knot-type spanned by immersed discs. We then turn our attention to M\"obius-Plateau variations of helicoidal strips, which are classified as "screw-like" or "ribbon-like" based on the signs of the radii of the boundary helices. By analyzing the Euler-Lagrange equations, we show that screw-like solutions are plentiful, whilst ribbon-like solutions impose strong constraints on their parameters: they must have high frequency (equivalently, low pitch), thin width in comparison to the frequency, and remain close to the axis.