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.


53D35 Global theory of symplectic and contact manifolds
57R12 Smooth approximations in differential topology
57R40 Embeddings in differential topology
Full Text: DOI arXiv Euclid