Geometrically Distinct Homothetic Expander Curves

• The paper demonstrates that 'jellyfish' curves, a family of homothetic expanders with dihedral symmetry, provide infinitely many self-similar solitons for elastic flow.
• The construction employs perturbations of Euler’s elastica and the Implicit Function Theorem to enforce precise boundary conditions and achieve dihedral gluing.
• The results reveal that variations in rotation index and symmetry order yield non-equivalent curves, enriching the classification of curvature-driven flows.

Geometrically distinct homothetic expanders, referred to as "jellyfish" curves, constitute an infinite family of closed, smoothly immersed planar curves exhibiting nontrivial self-similar expansion under the free elastic flow. Each member is distinguished by dihedral symmetry and rotation index, with the set containing no two curves equivalent up to similarities for distinct symmetry orders or rotation indices. These curves embody self-similar solutions to the fourth-order geometric evolution equation associated with the L²(ds)-gradient flow of elastic energy, structurally varying in curvature and global geometry. Their existence demonstrates a richer landscape of solitons for curvature-driven flows than previously recognized, going beyond simple circles and lemniscates (Andrews et al., 29 Jan 2026).

1. Elastic Flow and Homothetic Expander Definition

The elastic flow evolves a family of closed curves $\gamma : S^1 \times [0, \infty) \to \mathbb{R}^2$ by the equation

$$(EF)\quad\partial_t \gamma = -\nabla^{L^2(ds)} E[\gamma] \cdot \nu = -\left(k_{ss} + \frac{1}{2} k^3\right)\nu,$$

with elastic energy $E[\gamma] = \frac{1}{2}\int_\gamma k^2\, ds$ and $k$ the curvature.

A curve $\gamma$ is a homothetic expander if

$$\gamma(s,t) = \rho(t)\hat\gamma(s),\quad \rho(0) = 1, \quad \rho'(t) > 0,$$

where the profile $\hat\gamma$ satisfies the homothetic balance:

$$k_{ss} + \frac{1}{2}k^3 = -\sigma\langle\hat\gamma, \nu\rangle,\quad \sigma > 0$$

and the expansion rate is encoded by $\rho(t)^4 = 1 + 4\sigma t$.

This formulation leads to a profile ODE of fourth order, derived from the Frenet frame in arc-length gauge. The scalar Euler–Lagrange–homothetic equation is:

$$k'' + \frac{1}{2}k^3 = -\sigma\langle\hat\gamma(s),\nu(s)\rangle,$$

accompanied by the system

$\gamma'(s) = T(s),\quad T'(s) = k(s)\nu(s),$

where $T$ is the unit tangent, $\nu$ is the normal, and $\sigma$ is the homothetic parameter.

2. Existence Theorem and Construction Mechanism

The existence theorem (Theorem 1.1) states that there exists an $m_0$ such that for every integer $m > m_0$, a closed curve $\gamma^j_m$ with dihedral symmetry $D_m$ exists satisfying:

• (i) $\gamma^j_m$ solves the homothetic expander equation for (EF),
• (ii) $\sigma(\gamma^j_m)>0$, achieving self-similar expansion,
• (iii) $\gamma^j_m, \gamma^j_{m'}$ are inequivalent for $m\neq m'$.

The construction proceeds as follows:

• Base Arc: Start with a half-period of Euler’s elastica, solution to $k'' + \frac{1}{2}k^3 = 0$ on $[0, L_0]$, yielding $k$ oscillating between $1$ and $-1$ and fixed tangent change.
• Perturbation: The arc-length gauge ODE system is introduced for $(x, y, \theta, k, v)$ and parameters $(\alpha, \epsilon)$:

$$\begin{aligned} &x' = -\sin\theta,\quad y' = \cos\theta, \ &\theta' = k,\quad k' = v,\quad v' = -\tfrac{1}{2}k^3 - \alpha\cos\theta - \alpha\epsilon\langle(x, y), T\rangle. \end{aligned}$$

• Boundary Map and Implicit Function Theorem (IFT): Define $B(s; \alpha, \epsilon) = [b_1, b_2]^T$ with $b_1$ orthogonality to a center, $b_2 = v(s)$. Unique zeros in $B$ via IFT produce a family of fundamental arcs.
• Dihedral Gluing: The fundamental arc, constructed to meet two radial lines orthogonally with $v = k' = 0$, is reflected and concatenated $m$ times, rotated by $2\pi/m$, yielding a $D_m$ symmetric closed curve. The seam-angle $\Theta(\epsilon)$ at terminal point varies through rational intervals, with $\Theta = \pi/m$ specifying the solution for each $m$.
• Non-equivalence Verification: The symmetry group is exactly $D_m$, without additional hidden symmetries; the rotation index $p$ distinguishes further.

3. Indexing, Geometric Distinction, and Classification

Curves are indexed by their dihedral order $m$ or, equivalently, by the rational seam-angle $\Theta = \pi/m$. Simple "one-wavelike" series acquire turning index $p = 1$, generalized as $p$ for $\Theta = p\pi/m$.

Geometric distinction criteria:

• Two curves differ under similarity iff dihedral orders or rotation indices differ.
• The symmetry group is $D_m$ for each curve, with the full group realized geometrically (no hidden symmetries).
• Rotation index and symmetry order are similarity invariants for classification.

4. Profile ODEs, Energy, and Scaling Properties

The governing elastic energy functional $E[\gamma] = \frac{1}{2}\int_\gamma k^2\,ds$ is minimized by the evolution. The L²(ds)-gradient flow produces the geometric PDE,

$$\partial_t\gamma = -\delta E / \delta\gamma\,\nu = -(k_{ss}+\frac{1}{2}k^3)\nu,$$

where curvature terms dominate at high oscillation.

Self-similar homothetic motion requires $$k_{ss} + \frac{1}{2}k^3 = -\sigma\langle\hat\gamma,\nu\rangle$$, with expansion rate determined by $\rho'(t)/\rho(t)^2 = \sigma > 0$.

Introduction of small parameters $(\alpha, \epsilon)$ per the construction permits perturbation from the elastica case and explicit application of the Implicit Function Theorem to boundary maps, ensuring solvability under prescribed seam-conditions $b_1 = b_2 = 0$.

5. Qualitative Geometry and Asymptotic Behavior

"Jellyfish" expanders exhibit $m$-fold symmetric geometry reminiscent of a central "body" with $m$ oscillatory "tentacles." Each fundamental arc has curvature $k(s)$ monotonic from $1$ to $-1$, extended evenly across the glued structure.

For large $m$, numerical evidence indicates the curves are embedded and approach a circular geometry perturbed by sinusoidal components, remaining congruent under isotropic scaling throughout expansion.

6. Mathematical and Dynamical Significance

The abundance and geometric diversity of jellyfish expanders reveal previously unexplored richness in the set of homothetic solutions to fourth-order curvature flows. Their existence introduces infinitely many nontrivial solitons, challenging the classification of self-similar solutions and pointing to additional complexity in global dynamics. The construction methodology generalizes to epicyclic shrinkers for the curve diffusion flow and epicyclic expanders for the ideal flow, suggesting a broader framework for symmetry-rich geometric evolution phenomena (Andrews et al., 29 Jan 2026).