Rotationally Symmetric Fillings

• Rotationally symmetric fillings are structures defined by invariance under rotations, simplifying complex PDEs to tractable ODEs and allowing explicit classification.
• They are pivotal in diverse fields including differential geometry, tiling theory, and quantum error correction, where symmetry reduction streamlines analysis and construction.
• Leveraging ansatz frameworks and group-theoretic methods, these fillings offer robust models for geometric flows, stability assessments, and combinatorial optimization.

Rotationally symmetric fillings are geometric and analytic structures—metrics, mappings, tilings, sets, or codes—whose defining data or solutions possess invariance under a discrete or continuous group of rotations about a central axis or point. Such fillings arise naturally in differential geometry, geometric flows, symplectic and contact topology, enumerative combinatorics, quantum error correction, and physical models. The concept subsumes many distinct types: Kähler–Ricci solitons, biharmonic maps, self-shrinkers and self-expanders in curvature flow, convex Cheeger sets, symmetric Lagrangian fillings, rotationally invariant tilings, and codes in multimode quantum systems. Technical construction methods leverage symmetry to reduce complex equations or combinatorial problems to tractable ODEs, explicit tiling rules, or group-theoretic principles, leading to canonical models or explicit classification results.

1. Geometric Construction and Ansatz Frameworks

Rotational symmetry streamlines both analytic and combinatorial structures. In geometric analysis, this is exemplified by Calabi’s method for building metrics of the form

$$\omega = \pi^* \omega_M + \partial \bar{\partial} P(s),$$

on the total space of a vector bundle over a Kähler–Einstein base (e.g., $L^{\oplus n} \to M$), where $s$ is a radial coordinate in the fibre direction and $P$ is a smooth potential function (Li, 2010). This ansatz ensures the metric is invariant under $\mathsf{U}(n)$ action (rotation in fibres), drastically reducing the soliton or Ricci flow equations to ODEs in the radial variable. The reduction yields, for instance, the first-order nonlinear ODE for the potential

$$F’(\varphi) + d \frac{\varepsilon}{1+\varepsilon\varphi} F(\varphi) + (n-1)\frac{F(\varphi)}{\varphi} - \mu F(\varphi) = n - (\tau-n\varepsilon)\varphi,$$

which has explicit integrating factor solutions. The remaining construction details (boundary, smoothness, compactification) follow from symmetry-induced boundary conditions (e.g., $F(\varphi) = \varphi + O(\varphi^2)$ as $\varphi \to 0$ and matched vanishing conditions at infinity for projective compactification).

2. Classification and Stability of Rotationally Symmetric Maps

Symmetry reduction also facilitates the classification of higher-order mappings. In the context of biharmonic maps between model manifolds, rotationally symmetric proper biharmonic conformal maps admit full classification in dimension $m=4$, yielding explicit formulas:

• For maps $\mathbb{R}^4 \to S_4(d^2)$,

$$\alpha(r) = \frac{2}{d}\arctan(c^2 r),\qquad \alpha'(r) = \frac{1}{d}\frac{\sin(d\alpha)}{r}$$

• For maps $\mathbb{R}^4 \to H_4(-d^2)$,

$$\alpha(r) = \frac{2}{d} \tanh^{-1}(c^2 r),\qquad \alpha'(r) = \frac{1}{d} \frac{\sinh (d\alpha)}{r}$$

(Montaldo et al., 2015). The classification is derived from reduction of the bienergy functional to an ODE subject to symmetry constraints. The stability analysis under equivariant variations (i.e., symmetric perturbations of the profile) shows strict positivity of the second variation, confirming robustness of such fillings with respect to geometric flow or variational deformations.

3. Rotational Symmetry and Rigidity in Geometric Flows

In curvature flows (Ricci, mean curvature, inverse mean curvature), rotational symmetry plays a fundamental role in both existence and rigidity theory. Complete self-expanders to the inverse mean curvature flow with cylindrical ends must be rotationally symmetric (Drugan et al., 2016), as shown via kernel analysis of the linearized operator and application of the strong maximum principle:

• The rotation function $\langle R, \nu \rangle$ (induced by rotation vector fields) and support function $\langle F, \nu \rangle$ both solve the linearized soliton equation.
• Asymptotic analysis at infinity shows the quotient $$h = \langle R, \nu \rangle / \langle F, \nu \rangle$$ vanishes, forcing $R$ to be tangent everywhere (full rotational invariance). Similar symmetry phenomena appear in Ricci flows generated from warped product metrics on $\mathbb{R}^{n+1}$, where the initial data may have a cone-like singularity yet the flow instantaneously regularizes the metric outside the tip (Hsiao, 29 May 2025). Short-time existence and scaling-invariant curvature bounds are contingent on the absence of minimal hyperspheres and on noncollapsed geometry at infinity.

4. Rotationally Symmetric Tilings and Combinatorial Fillings

A broad class of plane tilings exploits rotational symmetry for monohedral, edge-to-edge, or spiral arrangements. Convex pentagons and hexagons with explicit angle and side constraints (e.g., $|b| = |c| = |a| + |d|$, $D + E = 180^\circ$, $B = 360^\circ / n$) admit constructions yielding $C_n$ (pure rotation) or $D_n$ (rotation plus reflection) symmetry (Klaassen, 2015). The combination of tile reflection and pairing (e.g., via line symmetry, bisected hexagons) delivers rich families of tilings, including spiral arms and designs with regular polygonal holes at the center (Sugimoto, 2020, Sugimoto, 2020, Sugimoto, 2020). In disc tilings, any monohedral filling with $k \leq 3$ congruent pieces is necessarily rotationally generated—putting a lower bound on the number of tiles needed to “avoid” the center (Kurusa et al., 2019).

Table: Rotationally Symmetric Filling Constructions

Class	Geometric Principle	Construction Method
Kähler–Ricci solitons	Differential symmetry	Calabi ansatz, ODE reduction
Biharmonic/conformal maps	Variational symmetry	Bienergy ODE, Hamiltonian analysis
Self-shrinkers/expanders	Flow symmetry/rigidity	Kernel functions, max principle
Tiling patterns	Discrete symmetry	Angle/edge constraints, pairing
Cheeger sets	Isoperimetric symmetry	Minkowski sum, regularity proof
Quantum codes	Algebraic (group) symmetry	Group-theoretic design, mode mixing

5. Symmetry, Entropy, and Finiteness in Fillings

The addition of symmetry to the paper of fillings strengthens compactness and uniqueness results. Rotationally symmetric self-shrinkers of entropy $<$ 2 in $\mathbb{R}^{n+1}$ form a compact moduli space, and, with additional convexity or reflection symmetry, the space becomes finite up to rigid motion (Mramor, 2020). In orbifold geometry, rotational symmetry maximizes systolic ratios precisely at Besse metrics (where all geodesics are closed) (Lange et al., 2021). This variational extremality provides a pathway for classifying “optimal” fillings in both smooth and singular settings.

6. Symmetric Fillings in Topology and Quantum Codes

Combinatorics, topology, and algebra combine in constructing rotationally symmetric Lagrangian fillings of Legendrian torus links. Weakly separated collections of $k$-element subsets of $[n]$ invariant under cyclic shifts (addition by $\ell$ modulo $n$) provide a combinatorial backbone for plabic graphs and Legendrian weaves fixed by rotations (Chen et al., 23 Sep 2025). These fillings, via the T-shift procedure, produce twist-spun surfaces whose symmetry is indexed by group-theoretic congruence conditions on $k$, with implications in cluster algebras and symplectic fillability.

In quantum error correction, rotationally symmetric bosonic codes are designed using group-theoretic frameworks that enforce symmetry across multiple modes, with practical consequences for linear optics gate implementation and error protection. Multimode codes admit logical gate sets implemented via passive linear operations (beam-splitters, phase shifters), preserve rotational symmetry under mode mixing, and outperform single-mode codes against dephasing and photon loss (Ahmed et al., 28 Aug 2025).

7. Physical and Analytical Implications

Imposing rotational symmetry in physical and PDE models alters analytic properties and practical outcomes. For instance, scattering of electromagnetic waves by an anisotropic medium can yield rotationally symmetric far-zone momentum flow, provided the medium’s structural parameters and incident source satisfy balanced correlation and polarization conditions (Ding, 2023). In convexity-based analyses, Cheeger sets of planar convex bodies inherit $k$-rotational symmetry and guarantee symmetric edge-touching, optimizing isoperimetric ratios and simplifying numerical algorithms (Cañete, 1 Feb 2024).

Summary

Rotationally symmetric fillings encapsulate a wide spectrum of geometric, analytic, combinatorial, and quantum structures unified by their invariance under rotation. This symmetry provides powerful constraints that facilitate explicit constructions, reductions of PDEs to ODEs, classification of canonical models, and combinatorial optimization, with direct implications in geometric flows, topological fillings, tiling theory, material design, and fault-tolerant quantum systems. The cross-disciplinary presence of rotational symmetry highlights its fundamental role in both pure theory and applied domains.