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Stationary curves under the Möbius-Plateau energy

Max Lipton, Gokul Nair

TL;DR

This work couples Möbius energy with a Plateau-area term to form the Möbius-Plateau energy and proves the existence of minimizers within fixed irreducible knot types for boundary curves of fixed length. It develops compactness and lower semicontinuity results, establishing convergence of minimizing sequences and the preservation of total length in the limit. The authors then analyze stationary configurations of helicoidal boundary strips, distinguishing screw-like and ribbon-like geometries; screws are plentiful while ribbons face stringent parametric constraints, including high frequency and near-axis confinement. Numerical and analytical exploration of the Euler–Lagrange equations yields explicit stationary screws and tight conditions for ribbons, illustrating rich interactions between boundary self-repulsion and spanning-surface tension. The results provide a rigorous variational framework for self-avoiding boundary problems and offer insight into the structure of minimizers in Möbius-Plateau-type energies.

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 Möbius Energy introduced by O'Hara due to its myriad regularity properties shown by Freedman et al. We first prove an existence theorem for this Möbius-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 Möbius-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.

Stationary curves under the Möbius-Plateau energy

TL;DR

This work couples Möbius energy with a Plateau-area term to form the Möbius-Plateau energy and proves the existence of minimizers within fixed irreducible knot types for boundary curves of fixed length. It develops compactness and lower semicontinuity results, establishing convergence of minimizing sequences and the preservation of total length in the limit. The authors then analyze stationary configurations of helicoidal boundary strips, distinguishing screw-like and ribbon-like geometries; screws are plentiful while ribbons face stringent parametric constraints, including high frequency and near-axis confinement. Numerical and analytical exploration of the Euler–Lagrange equations yields explicit stationary screws and tight conditions for ribbons, illustrating rich interactions between boundary self-repulsion and spanning-surface tension. The results provide a rigorous variational framework for self-avoiding boundary problems and offer insight into the structure of minimizers in Möbius-Plateau-type energies.

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 Möbius Energy introduced by O'Hara due to its myriad regularity properties shown by Freedman et al. We first prove an existence theorem for this Möbius-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 Möbius-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.
Paper Structure (9 sections, 11 theorems, 56 equations, 3 figures)

This paper contains 9 sections, 11 theorems, 56 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Energy Minimization
  4. Lower Semicontinuity of Plateau Energy
  5. Equicontinuity at the Boundary
  6. Convergence in the Interior
  7. Necessary Conditions of Möbius-Plateau Critical Helix Pairs
  8. Acknowledgements
  9. Estimating the negative contribution

Key Result

Lemma 2

Let $\gamma:X\rightarrow\mathbb R^3$ be a rectifiable curve in $\mathbb R^3$ parametrized by arc length. If $E_M[\gamma]$ is finite, then $\gamma$ is $C$ bi-Lipschitz with constant $C$ depending only on $E_M[\gamma]$. Furthermore, $C \to 1$ as $E_M[\gamma] \to 0$.

Figures (3)

  • Figure 1: Two stationary helicoidal screws with identical parameters $A = -1, \omega = 2$, except $B \approx 6.15375$ in the figure on the left, whilst $B = 1$ in the figure on the right. In these examples, we set $\alpha = 2, \beta = 1$
  • Figure 2: A graph of the integrand in \ref{['ribbonvar']} with $A = 0.1, B = 1,$ and $\omega = 10$, as a function of $v$.
  • Figure 3: Contours in the $AB$-plane for \ref{['ribbonvar']} and \ref{['ribbondiff']}, with $\alpha = \beta = 1, \omega = 10$, and approximate region of integration $[-10000,10000]$. The intersection of the contour curves is $(A,B) \approx (0.810515,0.818991)$ which is in the valid parameter space $\{(A,B): 0 < A < B, A + B < 2 \}$.

Theorems & Definitions (19)

  • Lemma 2: freedman1994mobius Lemma 1.2
  • Lemma 3: freedman1994mobius Corollary 1.3
  • Lemma 4: freedman1994mobius Lemma 4.2
  • Theorem 5: freedman1994mobius Theorem 4.3
  • Lemma 6
  • proof
  • Theorem 7
  • Lemma 8
  • proof
  • Lemma 9: Courant-Lebesgue
  • ...and 9 more