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**Packing symplectic manifolds by hand.**
*(English)*
Zbl 1101.53056

A symplectic \(k\) packing of a symplectic manifold \((M^{2n}, \omega)\) by equal balls consists of \(k\) symplectic embeddings of the standard symplectic open \(2n\)-ball \(B^{2n}(a)\) of rdius \(\sqrt{a/\pi}\). The problem has its origin in the theory of pseudo-holomorphic curves invented by M. L. Gromov [Invent. Math. 82, 307–347 (1985; Zbl 0592.53025)] and is developed by D. McDuff and L. Polterovich [Invent. Math. 115, 405–429 (1994; Zbl 0833.53028)] in the context of algebraic geometry. In the present paper along earlier constructions by L. Traynor [J. Differ. Geom. 42, 411–429 (1995; Zbl 0861.52008)] and others, the author gives explicit realizations of maximal packings of symplectic manifolds, such as ruled surfaces \(\pi: \Sigma_g\times S^2\to \Sigma_g\) (trivial bundle) and \(\Sigma_g\ltimes S^2\to\Sigma_g\) (nontrivial bundle) by making use of his flexible symplectomorphism from the open disk \(D(a)\) of area \(a\) in \(\mathbb R^2\) to \(\mathbb R(a)=\{(x, y)\in\mathbb R^2\mid 0<x<a,\;0<y<1\}\), which is obtanied by a “hand made” family of concentric loops in \(\mathbb R(a)\). The author gets more maximal packings and relative maximal packings which are different from the earlier constructions.

Reviewer: Haruo S. Suzuki (Sapporo)