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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.

##### MSC:
 53D35 Global theory of symplectic and contact manifolds 57R12 Smooth approximations in differential topology 57R40 Embeddings in differential topology
##### Keywords:
packing; symplectic manifold; embedding
##### Citations:
Zbl 0592.53025; Zbl 0833.53028; Zbl 0861.52008
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