Symplectic Packing Stability

• Symplectic packing stability is defined as the property that, beyond a threshold, the only obstruction to embedding collections of small symplectic domains is the volume constraint.
• It has been rigorously established for closed rational 4-manifolds and manifolds with smooth boundaries using algebraic decompositions of Hamiltonian diffeomorphisms.
• The concept underpins sharp Weyl law estimates and spectral invariant computations, linking boundary regularity to quantitative packing and capacity asymptotics.

Symplectic packing stability refers to phenomena in symplectic geometry where the only obstruction to embedding collections of small symplectic domains (such as balls or ellipsoids) into a target manifold is the classical volume constraint. Historically, this concept has had deep connections to rigidity and flexibility in symplectic topology, quantitative embeddings, spectral invariants, and the algebraic structure of symplectic and Hamiltonian groups. Packing stability has played a central role in the paper of four-dimensional symplectic manifolds, their ball and ellipsoid packing numbers, and the asymptotics of symplectic spectral sequences. Recent advances extend these results to manifolds with boundary, provide sharp Weyl law estimates for symplectic capacities, and clarify the interplay between boundary regularity and stability thresholds.

1. Formal Definitions and Stability Criteria

A symplectic manifold $(M^{2n}, \omega)$ is said to have packing stability if, for some integer $k_0$, every collection of $k \geq k_0$ disjoint symplectic balls $B(\lambda)$ of equal capacity can be symplectically embedded into $M$ provided the volume constraint

$$\frac{1}{2}\sum_{i=1}^k \lambda^2 \leq \operatorname{Vol}(M)$$

is satisfied (Buse et al., 2014, Edtmair, 18 Sep 2025). An analogous notion exists for ellipsoid packings: there is a threshold $a_0$ so that $E(\lambda, \lambda a)$ embeds fully for all $a \geq a_0$.

Packing stability means the "packing number"

$$p_k(M) = \sup\left\{ \lambda \in [0,1] : \bigsqcup_{i=1}^k B(\lambda) \overset{s}{\hookrightarrow} M \right\}$$

equals 1 for all $k\geq k_0$. For ellipsoid embeddings,

$$p^E_a(M) = \sup\left\{ \lambda \in [0,1] : E(\lambda, \lambda a) \overset{s}{\hookrightarrow} M \right\}$$

equals 1 for all $a \geq a_0$ (Edtmair, 18 Sep 2025).

2. Positive Results: Manifolds with Smooth Boundary and Closed Manifolds

Packing stability was first rigorously established for closed rational symplectic 4-manifolds and their generalizations (Buse et al., 2011, Buse et al., 2014). The key principle is that, for sufficiently small balls, volume is the only obstruction. This extends to irrational symplectic forms and more general domains ("pseudo-balls"). The major advance in (Edtmair, 18 Sep 2025) is proving that every compact, connected symplectic 4-manifold with smooth boundary enjoys packing stability: for large enough $k$, the packing number $p_k(M) = 1$ and full volume-filling is possible. Furthermore, "ellipsoid packing stability" holds—one can fully fill any such $M$ by a single ellipsoid of sufficiently high aspect ratio.

This result provides not only qualitative assurance of flexibility, but also quantitative control used to prove sharp spectral asymptotics (see Section 4 below). The mechanism behind the proof is a detailed analysis of the algebraic structure of the Hamiltonian diffeomorphism group $\operatorname{Ham}(M,\omega)$, exploiting Banyaga's simplicity and quantifying perfectness: any Hamiltonian diffeomorphism can be decomposed into finitely many elementary pieces (rotations, translations, etc.), giving precise geometric control needed for packing constructions.

3. Failure of Packing Stability: Influence of Boundary Regularity

While packing stability holds for smooth boundaries, (Cristofaro-Gardiner et al., 2023) and (Edtmair, 18 Sep 2025) show it can fail dramatically for domains whose boundary regularity drops below $C^2$. The construction in (Edtmair, 18 Sep 2025) provides a star-shaped domain $X \subset \mathbb{R}^4$, arbitrarily $C^{1}$-close to the unit ball, but with boundary in $C^{1,\alpha}$ for every $\alpha \in (0,1)$ (and smooth away from a single point), for which packing stability fails: for all $k$, $p_k(X) < 1$.

The geometric mechanism is subtle: volume decay near the boundary is slow, leading to high inner Minkowski dimension (see formula

$$\dim_M(\partial U) = 4 - \liminf_{d\to0} \frac{\ln V(d)}{\ln d}$$

from (Cristofaro-Gardiner et al., 2023)), and the subleading term in the Weyl law for symplectic capacities diverges to $-\infty$. No configuration of balls or ellipsoids can fully fill such domains, indicating a sharp threshold of boundary regularity for stability.

4. Applications to Symplectic Spectral Invariants and Weyl Laws

Packing stability has immediate implications for the sharpness of symplectic Weyl laws for spectral invariants such as embedded contact homology (ECH) capacities, periodic Floer homology (PFH) invariants, and link spectral invariants (Buse et al., 2011, Edtmair, 18 Sep 2025). For a smooth compact 4-manifold $M$, the ECH capacities satisfy

$$c_k(M) = 2\sqrt{\operatorname{vol}(M)\, k} + e_k(M)$$

with bounded error $e_k(M) = O(1)$ due to packing stability. For domains with irregular boundary, the error diverges, as in the star-shaped counterexamples where

$\lim_{k \to \infty} e_k^{\rm alt}(X) = -\infty.$

Additionally, in toric domains (Cristofaro-Gardiner et al., 2023), the "fractal Weyl law"

$$d_{ECH}(Z) := 2 + \frac{4\limsup_{k\to\infty} \ln(-e_k(Z))}{\ln k} \le \dim_M(\partial Z)$$

relates the growth rate of spectral remainders to the Minkowski dimension of the boundary, demonstrating a quantitative connection between domain geometry and packing properties.

5. Algebraic and Group-Theoretic Foundations

The algebraic structure of symplectic and Hamiltonian groups strongly governs packing stability phenomena. Quantitative factorization results for Hamiltonian diffeomorphisms (decomposition into elementary transformations with controlled error and number) underpin the local and global geometric constructions needed for stability proofs (Edtmair, 18 Sep 2025). Banyaga's simplicity and perfectness are essential in decomposing arbitrary diffeomorphisms into products of "tame" pieces (balls, frusta, cuboids, polydisks) that admit explicit embeddings and packings.

Homological stability results for symplectic groups further illuminate stabilization phenomena in packing problems: the invariance of group homology under "rank one stabilization" provides an algebraic analogue of geometric packing stability under addition of handles, balls, or boundary components (Sierra et al., 12 Nov 2024).

6. Thresholds, Future Directions, and Counterexamples

The existence of thresholds for capacity and boundary regularity is critical. Packing stability holds for smooth boundaries (the threshold between $C^{1,\alpha}$ and $C^2$ is sharp), and for balls or ellipsoids of sufficiently small size (thresholds in $k$ or aspect ratio $a$), but fails otherwise (Edtmair, 18 Sep 2025). Counterexamples such as wild toric domains (Cristofaro-Gardiner et al., 2023) and star-shaped constructions illustrate the necessity of these demands.

A plausible implication is that the interplay between regularity, spectral invariants, and group-theoretic structure will inform further studies on higher-dimensional packing stability, rigidity versus flexibility, and quantitative capacity asymptotics.

7. Summary Table: Packing Stability Scenarios

Domain Type	Packing Stability	Error Term Behavior
Closed or smooth-boundary 4-manifold	Yes	$O(1)$ in Weyl law
Star-shaped, $C^{1,\alpha}$ boundary	No	$\to -\infty$
Toric domain with wild boundary	No	Divergence (fractal law)

These findings clarify when symplectic geometry exhibits flexibility and full volume-filling, and when fine boundary properties or algebraic obstructions induce persistent rigidity. The connection with spectral asymptotics and capacity theory provides a quantitative bridge between packing theory and symplectic topology.