A stability property of symplectic packing. (English) Zbl 0930.53052

Let \(B(\lambda)\) denote the standard 4-dimensional closed ball of radius \(\lambda\) in \(\mathbb{R}^4\) endowed with the standard symplectic structure. Let \((M,\Omega)\) be a closed symplectic 4-manifold. The symplectic embedding of a disjoint union of \(N\) equal standard 4-dimensional balls into \((M,\Omega)\) is called a symplectic packing with \(N\) equal balls. If the volume that can be filled by such embeddings is arbitrarily close to the volume of \((M,\Omega)\), then one says that \((M,\Omega)\) admits full symplectic packing by \(N\) equal balls.
Following [S. K. Donaldson, J. Differ. Geom. 44, 666-705 (1996; Zbl 0883.53032)] and [C. H. Taubes, J. Am. Math. Soc. 9, 845-918 (1996; Zbl 0867.53025)], the author proves that for any symplectic 4-manifold with \([\Omega]\in H^2(M,\mathbb{Q})\) there exists \(N_0\) such that, for every \(N\geq N_0\), \((M,\Omega)\) admits full symplectic packing by \(N\) equal balls.
He calculates \(N_0\) showing that, if for some \(k_0\in\mathbb{Q}\) the Poincaré dual to \(k_0[\Omega]\) can be represented by a symplectic submanifold of genus at least 1, then one can assume that \(N_0=2k^2_0\text{Vol}(M,\Omega)\).


53D35 Global theory of symplectic and contact manifolds
52C17 Packing and covering in \(n\) dimensions (aspects of discrete geometry)
57R17 Symplectic and contact topology in high or arbitrary dimension
32Q65 Pseudoholomorphic curves
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