Symplectic Ball Packing Stability

• Symplectic ball packing stability is the phenomenon by which small balls are embedded in symplectic manifolds, with the total volume being the sole limiting factor.
• It leverages relative blow-up techniques and anti-symplectic involutions to preserve geometric and homological symmetry, ensuring consistent packing thresholds.
• Cohomological criteria and inflation methods guide the construction of symplectic forms that stabilize ball embeddings even in complex topological settings.

Symplectic ball packing stability refers to the phenomenon that, in many symplectic manifolds, sufficiently small balls can be embedded in such a way that the only barrier to full packing is the volume of the target manifold. Ball packing stability is intimately tied to deep geometric, topological, and analytic structures in symplectic geometry, including blow-up techniques, Lagrangian submanifolds, anti-symplectic involutions, cohomological constraints, singular curve theory, the structure of symplectic cones, and the behavior of symplectic capacities. This article synthesizes the theoretical framework, tools, quantitative criteria, and recent developments relevant to symplectic ball packing stability, with particular focus on constructions defined relative to Lagrangians and anti-symplectic symmetry.

1. Relative Blow-Up and Blow-Down Procedures

Ball packing stability in symplectic manifolds is elegantly encoded via blow-up and blow-down operations along Lagrangian submanifolds or relative to anti-symplectic involutions. Given a symplectic manifold $(M, \omega)$ and a Lagrangian submanifold $L \subset M$, a relative symplectic embedding is defined so that under a ball embedding

$$\psi : (B(r), \omega_0, B_\mathbb{R}(r)) \rightarrow (M, \omega, L)$$

the pre-image of $L$ is exactly the "real part" $B_\mathbb{R}(r)$ of the ball. Blowing up along the image of such an embedding replaces the ball neighborhood with an exceptional divisor; locally this is modeled by the tautological complex line bundle $\mathcal{L} \rightarrow \mathbb{CP}^{n-1}$ and its disk bundle $\mathcal{L}(r)$ (with real part $\mathcal{R}(r)$ over $\mathbb{RP}^{n-1}$).

A family of symplectic forms $\widetilde{\tau}(\epsilon, \lambda)$ can be constructed on $\mathcal{L}$ so that:

• Away from $\mathcal{L}(1+\epsilon)$: $$\widetilde{\tau}(\epsilon,\lambda) = \pi^*(\lambda^2 \omega_0)$$
• Near the zero section: $$\widetilde{\tau}(\epsilon,\lambda) = \rho(1, \lambda)$$ with $$\rho(\kappa, \lambda) = \kappa^2 \pi^*\omega_0 + \lambda^2 \theta^*\sigma$$

The cohomology class of the new symplectic form $\widetilde{\omega}$ on the blow-up $\tilde{M}$ is given by

$$[\widetilde{\omega}] = [\pi^*\omega] - \sum_{j=1}^k \lambda_j^2 e_j$$

where $e_j$ is the Poincaré dual of the exceptional divisor. The blow-down operation contracts exceptional spheres and controls changes to the homology and topology of both $M$ and the Lagrangian $L$. If $M$ carries an anti-symplectic involution $\phi$ and $L = \mathrm{Fix}(\phi)$, then the constructions can be made equivariant under $\phi$ and standard complex conjugation, ensuring the inherited symmetry: $$\phi^*\widetilde{\omega} = -\widetilde{\omega}, \quad \phi_*E_j = -E_j$$

2. Packing Numbers, Cohomology, and Stability Thresholds

Symplectic ball packing questions are reduced to cohomological criteria via blow-up techniques. The $k$-th packing number for $(M,\omega)$ is

$$p_k = \sup_\psi \left( \frac{\mathrm{Vol}(\sqcup_{i=1}^k B(\lambda))}{\mathrm{Vol}(M)} \right)$$

In the real/relative setting,

$$p_{L,k} = \sup_\psi \left( \frac{\mathrm{Vol}(\sqcup_{i=1}^k B(\lambda))}{\mathrm{Vol}(M)} \right), \quad \psi^{-1}(L) = \sqcup_{i=1}^k B_\mathbb{R}(\lambda)$$

After blowing up, the sizes $\lambda_j$ of the balls appear explicitly in the cohomology class of the blown-up symplectic form. The existence of a symplectic form with cohomology class $$[\widetilde{\omega}] = [\pi^*\omega] - \sum_{j=1}^k \lambda_j^2 e_j$$ is both a necessary and in many cases sufficient condition for the symplectic embedding. Thus, the packing problem reduces to the representability of certain cohomology classes by symplectic forms on the blow-up.

In concrete examples (e.g., for $(\mathbb{CP}^2, \sigma, \mathbb{RP}^2)$), the relative packing numbers $p_{\mathbb{RP}^2, k}$ coincide with the absolute ones $p_k$ for $k \geq 9$. For real, rank-1 symplectic four-manifolds with non-Seiberg-Witten simple type, one can pack $k$ balls provided

$\sum_{q=1}^k \lambda_q^4 < \mathrm{Vol}(M,\omega)$

and the radii satisfy $\lambda_q < \sqrt{d_\omega}$, with the critical constant $$d_\omega = \inf_{B \in D_\omega} \frac{\omega(B)}{c_1(B)}$$.

3. Anti-Symplectic Involution and Homological Symmetry

When the manifold $M$ admits an anti-symplectic involution $\phi: M \to M$ ($\phi^2=1$, $\phi^*\omega=-\omega$), whose fixed point set is $L$, the blow-up and packing constructions must respect this symmetry. The relative embedding must satisfy

$\psi \circ c = \phi \circ \psi$

where $c$ is complex conjugation. After blowing up, the exceptional classes $E_j$ satisfy $\phi_* E_j = -E_j$. Correspondingly, the modified symplectic form and almost complex structure on the blow-up satisfy: $$\phi^*\widetilde{\omega} = -\widetilde{\omega}, \quad \phi_* \widetilde{J} = -\widetilde{J} \phi_*$$ This refinement allows one to prove that for sufficiently many balls (e.g., $k\geq 9$ in $\mathbb{CP}^2$), the maximal relative and absolute packing numbers are equal.

The construction uses equivariant local symplectic forms, symplectic inflation adapted to the real setting, and homological constraints—such as the adjunction formula and intersection numbers—that are essential in controlling the topology of the blown-up manifold and its Lagrangian submanifold.

4. Impact of Homological and Topological Constraints

Cohomological tracking via the formula

$$[\widetilde{\omega}] = [\pi^*\omega] - \sum_{j=1}^k \lambda_j^2 e_j$$

links the symplectic geometry of embedding balls with the topology of the ambient manifold. The exceptional spheres, their self-intersections, and the relationship between their homology classes and symmetry (as mandated by anti-symplectic involution) provide algebraic obstructions or enablements for packing stability.

In particular, the interplay between symmetrized forms (e.g., $\bar{\omega} = \frac{1}{2}(\omega - \phi^*\omega)$, so that $\phi^* \bar{\omega} = -\bar{\omega}$), the behavior of exceptional classes under involution, and the parametrization of the symplectic cone (the set of cohomology classes representable by symplectic forms) ensures that for large $k$, ball packing efficiency stabilizes and becomes volume-dominated.

5. Maximal Packing, Inflation Techniques, and Thresholds

Maximal packing in both absolute and relative cases is proven using a combination of blow-up/blow-down procedures, inflation along curves, and precise control of symplectic cohomology. The inflation step, made possible via the explicit understanding of symplectic forms on the blow-up, tames the almost complex structure and allows for "inflation" of pseudo-holomorphic spheres of negative self-intersection (exceptional curves).

The threshold phenomenon—packings reach the full-volume bound, and relative obstructions vanish at or above a critical number of balls—has been thoroughly verified for various rational and real symplectic four-manifolds. The symmetry acquired from involutive automorphisms, together with the use of symplectic inflation and the examination of cohomological obstructions, ensures that relative packing numbers converge to their absolute counterparts in the regime where $k$ is sufficiently large.

6. Applications and Future Prospects

The relative blow-up and packing stability method provides a robust framework for understanding symplectic ball embeddings with Lagrangian constraints. It generalizes the packing problem to settings with additional symmetries (anti-symplectic involution). These techniques yield precise results for the value of packing numbers, inform the construction of symplectic forms with desired representability properties, and facilitate the paper of real symplectic manifolds and their Lagrangian submanifolds.

Prospective developments include generalization to higher dimensions, extensions to more complex Lagrangian topologies, and further exploration of the relationship between packing stability thresholds and the algebraic structure of symplectic cones and homological invariants.

Table: Key Formulas in Relative Blow-Up Packing Stability

Construction	Formula / Condition	Purpose
Cohomology class of blow-up	$$[\widetilde{\omega}] = [\pi^*\omega] - \sum_j \lambda_j^2 e_j$$	Encodes loss through ball packing embedding
Relative packing number equivalence	$p_{L,k} = p_k$ for $k \geq k_0$	Ball embedding efficiency for large $k$
Symmetry under anti-symplectic involution	$\phi^*\widetilde{\omega} = -\widetilde{\omega}$, $\phi_* E_j = -E_j$	Ensures compatibility across operations
Threshold condition (e.g., $\mathbb{CP}^2$)	$k \geq 9$	Packing stability holds above threshold

These formulas encapsulate the geometric, cohomological, and symmetry principles underlying symplectic ball packing stability when using relative blow-up and blow-down constructions in real or symmetric settings.