## Packing stability for symplectic 4-manifolds.(English)Zbl 1354.53081

Let $$P \subset \mathbb R^2$$ be a subset of the first quadrant, and denote by $$U(P) \subset \mathbb C^2$$ the interior of $$\{(z,w) : (\pi |z|^2, \pi |w|^2) \in P\}$$. When $$P$$ is the convex hull of $$\{(0,0), (a, 0), (0, b) \}$$, for some positive real numbers $$a,b$$, then $$U(P)$$ is a symplectic ellipsoid. In particular, for $$a = b = \lambda$$, $$U(P)$$ is a symplectic ball of capacity $$\lambda$$ and it is denoted by $$B(\lambda)$$. The authors also define pseudoballs as those $$U(P)$$, where $$P$$ is the convex hull of $$\{(0,0), (0,a), (b,0), (\alpha, \beta) \}$$ for some $$a >\alpha$$, $$b > \beta$$, and $$a,b< \alpha + \beta$$. A symplectic $$4$$-manifold $$X$$ has strong packing stability if there exists a positive real number $$\lambda$$ such that, for all collections of real numbers $$(\lambda_i)$$, with $$\lambda_i< \lambda$$, a symplectic embedding $$\sqcup_i B(\lambda_i) \hookrightarrow X$$ exists iff $$\tfrac{1}{2} \sum_i \lambda^2_i \leq \mathrm{Vol} (X)$$. The authors first prove that certain open symplectic $$4$$-manifolds have strong packing stability: namely, all symplectic ellipsoids and pseudo-balls. Then they use these results to prove their main Theorem: All closed symplectic $$4$$-manifolds have strong packing stability. This generalizes older results by P. Biran [Geom. Funct. Anal. 7, No. 3, 420–437 (1997; Zbl 0892.53022)].

### MSC:

 53D05 Symplectic manifolds (general theory) 57R17 Symplectic and contact topology in high or arbitrary dimension

### Keywords:

packing stability; symplectic 4-manifolds

Zbl 0892.53022
Full Text:

### References:

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