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**Symplectic packing in dimension 4.**
*(English)*
Zbl 0892.53022

We say that the closed symplectic 4-manifold \((M,\Omega)\) admits a symplectic packing by \(N\) balls of radii \(\lambda_1, \dots, \lambda_N\) if there exists a symplectic embedding of the disjoint union \(\cup^N_{q=1} (B(\lambda_q), \omega_0)\) of standard closed 4-balls of radius \(\lambda_q\) endowed with the canonical Darboux form \(\omega_0\) on \(\mathbb{R}^4\). Following M. Gromov [Invent. Math. 82, 307-347 (1985; Zbl 0592.53025)] and D. McDuff and L. Polterovich [ibid. 115, 405-429 (1994; Zbl 0833.53028)], the author investigates the manifolds which admit full packing by \(N\) equal balls (i.e., the supremum of volumes, which can be filled by symplectic embeddings of \(N\) disjoint equal balls, equals the volume of \(M)\) for large enough \(N\). Using the inflation procedure and Gromov’s invariant theory, he gives a homological condition for recognizing such manifolds and presents a direct method for computing values of \(N_0\) such that for all \(N\geq N_0\) there exist full packings by \(N\) equal balls. Finally, he works out several examples which admit full packing for every \(N\geq 9\).

Reviewer: St.Janeczko (Warszawa)

### MSC:

53C23 | Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces |

53C15 | General geometric structures on manifolds (almost complex, almost product structures, etc.) |

### Online Encyclopedia of Integer Sequences:

Unreduced numerator of volume of best symplectic packing of n balls in 4-dimensional ball.Unreduced denominator of volume of best symplectic packing of n balls in 4-dimensional ball.

Numerator of volume of best symplectic packing of n balls in 4-dimensional ball.