Packing stability for symplectic 4-manifolds (Q2821686)

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 \textit{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 \textit{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 \textit{symplectic ellipsoids} and \textit{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 \textit{P. Biran} [Geom. Funct. Anal. 7, No. 3, 420--437 (1997; Zbl 0892.53022)].