Equivariant Stability Thresholds

• Equivariant Stability Threshold is a numerical invariant that delineates stable and unstable dynamics in systems with group symmetry using maximal weight functions and inequality criteria.
• It is critical across disciplines including Kähler geometry, nonlinear PDEs, algebraic geometry, and neural network design by providing precise, computable stability criteria.
• Its computation leverages spectral analysis, variational methods, and combinatorial techniques to robustly classify solution behaviors and guide system designs.

An equivariant stability threshold is a parameter or numerical invariant that separates stable from unstable behavior in systems with symmetry—typically manifesting as group equivariance. This threshold appears across several mathematical and physical contexts, providing a sharp criterion for stability of orbits under group actions, global behavior of solutions to PDEs with equivariant ansatz, geometric invariant theory, and representation learning in group-equivariant networks. The definition and computation of such thresholds rely on the structure imposed by equivariance, allowing fine-grained and often explicit stability criteria in settings ranging from Kähler reduction to equivariant neural network architectures.

1. Equivariant Stability Threshold in Hamiltonian and Kähler Geometry

Let $(Z,\omega)$ be a Kähler manifold with holomorphic Hamiltonian action of a complex reductive group $U^{\mathbb{C}}$, and $G \subset U^{\mathbb{C}}$ a real reductive subgroup with a Cartan decomposition $\mathfrak{g} = \mathfrak{k} \oplus \mathfrak{p}$. For a $G$-invariant real submanifold $X \subset Z$, a $G$-gradient map $\mu_\mathfrak{p}: X \to \mathfrak{p}$ arises from projecting the moment map.

A key tool is the maximal weight function

$$\lambda_x(\xi) = \lim_{t \to +\infty} \frac{\langle \mu_\mathfrak{p}(\exp(t\xi) \cdot x), \xi \rangle}{\|\xi\|},$$

defined for $x \in X$ and $\xi \in \mathfrak{p} \setminus \{0\}$. This function characterizes the asymptotic growth along 1-parameter subgroups in the noncompact direction. When the $G$-action is energy complete—i.e., flows with finite "energy"

$$E(c_{\xi,x}) = \int_0^\infty \|\xi_X(\exp(t\xi)x)\|^2 dt$$

have convergent limits—the stability of $x$ is completely determined by evaluating $\lambda_x(\xi)$:

• $x$ is stable if $\lambda_x(\xi) > 0$ for all $\xi \neq 0$,
• semistable if $\lambda_x(\xi) \geq 0$ for all $\xi$,
• polystable if $\lambda_x(\xi) \geq 0$ for all $\xi$ and any vanishing $\lambda_x(\xi)$ corresponds to orbit closure.

The precise stability threshold is thus the set of inequalities $\lambda_x(\xi) > 0$ (or $\geq 0$) over all admissible $\xi$. This generalizes the Hilbert–Mumford numerical criterion from GIT to real group actions and is foundational in modern Kähler geometry (Biliotti et al., 2022).

2. Threshold Phenomena in Equivariant Nonlinear PDEs

In invariant nonlinear dispersive and wave equations, the equivariant stability threshold typically refers to a sharp boundary in initial data (e.g., mass or energy) determining global behavior.

In the equivariant Chern–Simons–Schrödinger equation in 2D, the threshold is given by the $L^2$-mass of the ground state solution $Q$: $M[Q] = \|Q\|_{L^2}^2 = 8\pi(m+1).$

• For initial mass $M[\varphi_0] < M[Q]$, all solutions scatter.
• For $M[\varphi_0] = M[Q]$, non-scattering solutions are precisely those arising from symmetries of $Q$ (scaling, phase, pseudoconformal).
• The mass $M[Q]$ is the sharp equivariant stability threshold (Li et al., 2020).

In supercritical equivariant wave maps from $\mathbb{R}^{d+1} \to \mathbb{S}^d$ ($3 \leq d \leq 6$), the threshold between scattering and blowup is not a simple scalar but a codimension-one stable manifold $W^s(f_1)$ in phase space attached to the first excited self-similar solution $f_1(y)$; it is characterized via linearized spectral analysis as the set of initial data orthogonal to the unique unstable eigenmode (Biernat et al., 2016).

3. Algebraic and Geometric Equivariant Stability Thresholds

In algebraic geometry, the equivariant stability threshold governs the (semi)stability of objects (sheaves, bundles, or varieties) under group action:

• For a $G$-equivariant sheaf $\mathcal{E}$ on a toric variety, the stability threshold is the set of inequalities on polarized slopes:

$\mu_H(\mathcal{F}) < \mu_H(\mathcal{E})$

for all proper $T$-invariant subsheaves $\mathcal{F} \subset \mathcal{E}$, expressed as a system of inequalities in the parameters of the polarization (Dasgupta et al., 2019).

• For $\mathbb{Q}$-Fano spherical varieties, the $G$-equivariant stability threshold is the infimum:

$$\delta_G(X) := \inf_{v \in \operatorname{DivVal}_X^G} \frac{A_X(v)}{S(-K_X; v)}$$

where $A_X(v)$ is the log discrepancy and $S(-K_X; v)$ the expected vanishing order of a canonical $B$-invariant divisor $D_X^B$ that computes this threshold. The variety is $G$-K-semistable if $\delta_G(X) \geq 1$ (Zheng, 31 Dec 2025). In toric cases, this reduces to combinatorial inequalities involving the barycenter of the moment polytope.

4. Explicit Computation and Criteria in Examples

• In vector bundles: On toric Fano varieties with Picard number 2 or 3, the threshold is a single slope inequality for the tangent bundle over all $T$-equivariant reflexive subsheaves. The region in the ample cone where all such inequalities are satisfied defines the (semi)stability locus (Dasgupta et al., 2019).
• In cohomogeneity-one harmonic maps: The stability threshold is explicitly computable as the crossing of the lowest eigenvalue $\lambda_1$ of the Jacobi operator through zero as external parameters (e.g., isotropy indices or group action order) are varied. For isoparametric spheres, these thresholds select the stable range for the identity map and related linear self-maps (Branding et al., 2021).

5. Equivariant Stability Thresholds in Group-Equivariant Neural Networks

In equivariant convolutional architectures, the stability threshold against deformations is explicit in network parameters. For an $N$-layer equivariant CKN or CNN, the threshold $\delta^{*}$ quantifying permissible deformation is

$\delta^{*} = \frac{\varepsilon}{C_1(1+N) + C_2}$

where $C_1$ and $C_2$ depend on patch diameter and pooling bandwidth. This guarantee controls the norm deviation of equivariant representations under geometric perturbations, linking network design (depth, pooling, patch size) directly to robustness margins. The threshold enables precise architecture tuning for desired stability (Chowdhury, 2024).

6. Spectral, Analytical, and Numerical Characterization

The computation or verification of the equivariant stability threshold is context-dependent:

• In finite or infinite-dimensional dynamical systems, it reduces to computation of spectral gaps or discrete eigenvalues separating neutral/stable spectrum from instability.
• In geometric invariant theory, it emerges from the maximization (or infimum) of suitable $\mathrm{G}$-equivariant numerical functions (maximal weights, slopes, or expected vanishing orders).
• In symmetric PDEs, threshold manifolds are verified using spectral linearization, orthogonality to unstable modes, and numerical bifurcation techniques.

The existence and sharpness of equivariant stability thresholds fundamentally structure phase transitions in solution space, phase diagrams in parameter space, and stability boundaries in moduli of invariant objects.

7. Significance and Universality

Equivariant stability thresholds unify stability theory across geometric invariant theory, nonlinear PDE, algebraic geometry, and representation learning. Their explicit, computable nature in equivariant contexts allows precise classification of solutions, informs numerical and architectural choices, and sharpens classical stability criteria. The presence of such thresholds also demarcates the domains in which nonlinear, dispersive, or geometric behavior can be rigorously controlled by perturbative or variational methods, with loss of control (eigenvalue crossing, blowup, non-dispersive behavior) exactly at threshold values.

References:

• Stability for real reductive Lie group actions: maximal weights and Hilbert–Mumford criterion (Biliotti et al., 2022).
• Threshold mass blowup/scattering dichotomy in equivariant CSS equations (Li et al., 2020).
• Codimension-one stable manifolds in equivariant wave maps (Biernat et al., 2016).
• Combinatorial slope criteria for equivariant bundle stability on toric Fanos (Dasgupta et al., 2019).
• Canonical divisors computing $G$-equivariant thresholds in spherical K-stability (Zheng, 31 Dec 2025).
• Explicit eigenvalue thresholds for harmonic map stability in isoparametric and compact group settings (Branding et al., 2021).
• Explicit stability bounds for group-equivariant CKNs and CNNs (Chowdhury, 2024).