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Jellyfish exist

Ben Andrews, Glen Wheeler

TL;DR

The paper proves that each of the three planar curvature flows admits rich families of self-similar solutions obtained by a unified two-stage method: first construct a fundamental arc via a boundary-value ODE problem, then close the arc into a closed curve using dihedral reflection to enforce symmetry. Jellyfish expanders arise from perturbations of a half-elastica and dihedral gluing; epicyclic shrinkers for the curve-diffusion flow and epicyclic expanders for the ideal flow are obtained by analogous constructions with base arcs and higher-order seam conditions. By controlling the seam-turning angles and leveraging the implicit-function theorem, the authors produce infinitely many geometrically distinct solutions with increasing symmetry orders, each non-equivalent under similarity transformations. These results reveal a much richer landscape of homothetic solutions than previously known, suggesting robust dynamical basins of attraction and motivating further classification and stability studies. They provide rigorous existence results for large families and establish a framework likely applicable to broader geometric-flow settings.

Abstract

We show the existence of infinitely many geometrically distinct homothetic expanders (jellyfish) for the elastic flow, epicyclic shrinkers for the curve diffusion flow, and epicyclic expanders for the ideal flow.

Jellyfish exist

TL;DR

The paper proves that each of the three planar curvature flows admits rich families of self-similar solutions obtained by a unified two-stage method: first construct a fundamental arc via a boundary-value ODE problem, then close the arc into a closed curve using dihedral reflection to enforce symmetry. Jellyfish expanders arise from perturbations of a half-elastica and dihedral gluing; epicyclic shrinkers for the curve-diffusion flow and epicyclic expanders for the ideal flow are obtained by analogous constructions with base arcs and higher-order seam conditions. By controlling the seam-turning angles and leveraging the implicit-function theorem, the authors produce infinitely many geometrically distinct solutions with increasing symmetry orders, each non-equivalent under similarity transformations. These results reveal a much richer landscape of homothetic solutions than previously known, suggesting robust dynamical basins of attraction and motivating further classification and stability studies. They provide rigorous existence results for large families and establish a framework likely applicable to broader geometric-flow settings.

Abstract

We show the existence of infinitely many geometrically distinct homothetic expanders (jellyfish) for the elastic flow, epicyclic shrinkers for the curve diffusion flow, and epicyclic expanders for the ideal flow.
Paper Structure (46 sections, 57 theorems, 291 equations, 5 figures)

This paper contains 46 sections, 57 theorems, 291 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Three curvature flow
  3. Main results
  4. Jellyfish expanders
  5. Existence of a fundamental arc
  6. ODE formulation and boundary map
  7. The base arc at $(\alpha,\varepsilon)=(0,0)$
  8. Solving $b_1=0$ for $L=L(\alpha,\varepsilon)$
  9. The fundamental arc
  10. Abundance of seam angles
  11. Dihedral gluing and proof of Theorem \ref{['thm:intro:cdf-epi']}
  12. Reflection principle and dihedral concatenation
  13. Maximality of the symmetry order and geometric distinctness
  14. An abundance of angles
  15. Dihedral gluing and proof of Theorem \ref{['thm:intro:jellyfish']}
  16. ...and 31 more sections

Key Result

Theorem 1.1

There is an $m_0<\infty$ such that jellyfish expanders $\gamma^j_m$ exist for all $m>m_0$.

Figures (5)

  • Figure 1: Representative closed self-similar solutions from each family constructed in this paper. Values of $\varepsilon$ are accurate to five significant figures. Colours are used to denote doubled fundamental arcs.
  • Figure 2: Closed jellyfish-like expanders for the free elastic flow that do not possess dihedral symmetry. The indicated point is the centre of expansion.
  • Figure 3: Free elastic flow jellyfish solution with $\omega=1$ and $\varepsilon=0.19378$ (to five significant figures). The closed curve is assembled from $5$ doubled arcs.
  • Figure 4: Curve diffusion flow (CDF) epicyclic shrinker for the dihedral gluing data $p/q=4/5$: $\omega=4$ and $\varepsilon=0.19672$ (to five significant figures). The closed curve is assembled from $5$ doubled arcs.
  • Figure 5: Ideal-flow epicyclic shrinker with $\omega=50$ and $\varepsilon=0.13507$ (to five significant figures). The closed curve is assembled from $51$ doubled arcs.

Theorems & Definitions (120)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Conjecture 1.4
  • Conjecture 1.5
  • Definition 2.1: Fundamental arc
  • Remark 2.2: Geometric meaning of $b_1$
  • Lemma 2.3: Smooth dependence
  • Lemma 2.4
  • proof
  • ...and 110 more