Spinning Conical Defect Geometry

• Spinning conical defect geometry is a framework describing manifolds with conical singularities combined with spinning or torsional deformations, governed by isometric bending energy.
• The stability analysis employs a fourth-order self-adjoint operator whose Fourier spectrum reveals zero modes and paired unstable modes under global isometry constraints.
• This model elucidates practical instability and reconfiguration pathways in systems such as soft condensed matter, cosmic strings, and flexible electronic materials.

A spinning conical defect geometry describes regions in a manifold (often of physical, geometric, or material interest) where conical curvature singularities are augmented or intertwined with additional geometric features such as intrinsic angular momentum, global spinning symmetry, or defect-induced torsion. The archetypal example is a two-dimensional or thin three-dimensional sheet (such as a piece of unstretchable material, a brane configuration, or a physical model of a cosmic string) exhibiting a conical singularity together with nontrivial bending, spinning, or torsional excitations. These geometries play a crucial role in analyzing instabilities, dynamical reconfiguration, and symmetry-breaking effects in both soft condensed matter systems and gravitating backgrounds. Their stability properties, mode structure, and sensitivity to constraint conditions (such as unstretchability) determine the observable features of spinning sheets, defects in topological insulators, high-spin holographic states, and even quantum information metrics in holographic duals.

1. Bending Energy, Isometry Constraints, and Defect Construction

The fundamental physical principle for spinning conical defect geometries in unstretchable sheets is that the only allowed deformations are isometric—preserving the reference metric. The elastic energy for a thin sheet, neglecting stretching modes entirely, is controlled by the bending energy: $H_B = \frac{1}{2} \int dA\, K^2$ where $K = C_1 + C_2$ is the mean curvature. For defects (departure from flatness), the surplus or deficit angle quantifies the curvature concentrated at the defect apex.

In the presence of a fixed reference metric $g_{ab}^{(0)}$, the full energy functional includes Lagrange multipliers enforcing the isometry: $$H[T^{ab}] = H_B - \frac{1}{2} \int dA\, T^{ab}(g_{ab} - g_{ab}^{(0)})$$ Isometric deformations ($\delta g_{ab} = 0$) restrict admissible variations, leading to a global, nonlocal constraint on possible deformations.

Upon expansion to quadratic order in normal deformations, the second variation of the bending energy focuses on the normal component: $$\delta^2 H = \int dA\, \delta \mathcal{E}_\perp (\cdot \delta)$$ The constraint manifests as a requirement on the allowed normal modes, which, in the case of a conical defect, reduces to a global integral constraint involving the equilibrium curvature.

2. Fourth-Order Self-Adjoint Operator and Mode Spectrum

The nature of the allowed deformation modes is encoded in the eigenstates of a self-adjoint fourth-order linear operator, which arises from the expansion above (after projecting out the tangential component using the isometric constraint): $$\delta^2 H = \langle \Phi | L | \Phi \rangle, \quad L = \frac{d^4}{ds^4} + \frac{d}{ds}[V_1(s)]\frac{d}{ds} + V_2(s) + \frac{1}{2} V_1''(s)$$ where $s$ parametrizes the arc-length along a closed curve on the unit sphere (encoding the conical geometry), and $V_1$, $V_2$ are functions of the equilibrium curvature $\kappa(s)$ and tangential stress. The variable $\Phi(s)$ denotes the normal component of the mode.

Allowed modes must also satisfy the global isometric constraint: $\oint ds\, \kappa(s) \Phi(s) = 0$ Spectral stability is determined by the eigenvalues $\lambda$ of $L$ subject to this constraint.

3. Linear Approximation, Fourier Modes, and Symmetry

For small surplus angles (nearly flat conical defects), the curvature is well-approximated by $\kappa(s) \approx \kappa_0 \sin(ns)$, where $n$ is an integer (fold symmetry). The operator $L$ and its potentials simplify: $$L = \frac{d^4}{ds^4} + (n^2+1)\frac{d^2}{ds^2} + n^2$$ The periodic boundary condition ($\Phi(s)$ is defined on a closed curve) renders the natural basis for analysis as Fourier modes. The eigenvalues for the quadratic stability problem are: $\lambda_m = (m^2 - n^2)(m^2 - 1)$ for integer $m$. Notably:

• The constant mode ($m=0$) is always stable.
• Two zero modes at $m=1$ and $m=n$ correspond to infinitesimal rotations—the signature of broken continuous symmetry in the defect.
• For $n=2$, the spectrum is gapped and stable.
• For $n \geq 3$, there are $2(n-2)$ negative eigenvalues in the interval $m = 2, ..., n-1$, indicating $2(n-2)$ unstable modes (arranged in degenerate even-odd pairs).

The isometric constraint removes certain Fourier components; in particular, modes proportional to the equilibrium curvature do not satisfy the admissibility condition.

4. Nonlinear Stability, Mode Mixing, and Ground State Selection

Beyond the small surplus (linear) regime, the fourth-order operator retains its general structure but with potentials $V_1(s)$ and $V_2(s)$ that depend nonlinearly on $\kappa(s)$. The spectral problem is analyzed numerically using a Gram–Schmidt process to construct an orthonormal basis of trial deformations satisfying the constraint.

The nonlinear mode analysis confirms and refines the linear picture:

• The $n=2$ ("2-fold") ground state is nonlinearly stable—there are no negative eigenvalues once the constraint is enforced.
• For $n \geq 3$ ("excited states"), there are precisely $2(n-2)$ negative eigenvalues, and the corresponding unstable modes ("decay channels") remain paired as even/odd under $n$-fold symmetry.

As the surplus angle increases, the Fourier content of unstable modes broadens—higher harmonics mix in, and the node structure becomes more intricate. The dominant unstable mode for a given symmetry sector $n$ has a characteristic frequency $$m_0 = \mathrm{round}\left(\sqrt{(1 + n^2)/2}\right)$$, indicating which Fourier component grows most rapidly in a destabilized configuration.

5. Physical Consequences: Instabilities, Cascades, and Reconfiguration

The existence and multiplicity of unstable modes for $n \geq 3$ reflects a rich cascaded instability structure. In physically realizable systems (crumpled sheets, growing tissues, spinning conical shells in soft matter, or flexible electronics), the system will generically relax toward the uniquely stable $n = 2$ ground state, unless symmetries or boundary conditions constrain otherwise.

The interplay of geometry (surplus/deficit angle), intrinsic metric (unstretchability), and the spectrum of the fourth-order operator determines the allowed relaxation pathways. The even and odd parity of unstable modes reveals the possible morphologies of decay and symmetry breaking, influenced by intrinsic and extrinsic geometric constraints.

The identification of zero modes (rotational invariance) clarifies the moduli of rigid global motion, while the spectrum of negative modes elucidates the dominant buckling or wrinkling patterns accompanying defect evolution.

6. Quantitative Summary and Key Formulas

The stability and structure of spinning conical defect geometries are governed by the following:

• Bending energy including isometric constraint:

$$H[T^{ab}] = \frac{1}{2} \int dA\, K^2 - \frac{1}{2} \int dA\, T^{ab}(g_{ab} - g_{ab}^{(0)})$$

• Global constraint on admissible normal deformations:

$\oint ds\, \kappa(s) \Phi(s) = 0$

• Stability operator for normal modes:

$$L = \frac{d^4}{ds^4} + \frac{d}{ds}[V_1(s)]\frac{d}{ds} + V_2(s) + \frac{1}{2} V_1''(s)$$

• Linear regime Fourier eigenvalues:

$\lambda_m = (m^2 - n^2)(m^2 - 1)$

• Number of unstable modes: $2(n-2)$ (for $n \geq 3$).

A table summarizing the mode structure by $n$:

$n$	Stable Modes	Unstable Modes	Zero Modes
2	All	0	$m=1,2$ (rotations)
$\geq 3$	Partial	$2(n-2)$	$m=1,n$

7. Broader Implications and Experimental Relevance

The rigorous connection between geometry, elasticity, and stability in spinning conical defect geometries affords predictive control over the formation and evolution of defects in practical systems. This framework underscores essential aspects:

• Explains observed transitions and cascades among buckling/unstable modes in crumpled plates, biological morphogenesis, and advanced manufactured materials.
• Provides a model for defect-mediated dynamical response in flexible micro- and nano-structures subject to spinning or angular excitation.
• Informs the interpretation of decay channels, energy barriers, and the role of symmetry and global constraints in defect-rich environments.

The analysis presented clarifies that, despite the apparent continuum of possible conical configurations (infinite family of $n$-fold states), the strong constraint of unstretchability and the detailed spectral structure select uniquely stable and unstable morphologies, with implications for nonlinear pattern formation, induced reconfiguration, and ultimately the controlled manipulation of defect-driven functionality in physical and engineered systems (Guven et al., 2011).