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Symplectic packing constructions. (English) Zbl 0830.52011
Let \(B(r)\) be the closed four-dimensional ball of radius \(r\) with the standard symplectic structure \(\omega_0\). A symplectic packing of \(B(1)\) with \(k\) balls of packing radius \(r\) is a set of \(k\) symplectic embeddings of \(B(r)\) into the interior of \(B(1)\), \(\{\psi_i\}\), \(i = 1, \ldots, k\), such that the images of \(\psi_i\) and \(\psi_j\) are disjoint, \(i \neq j\). For a fixed \(k\), there is an upper bound to the set of possible packing radii since symplectic maps preserve volume. Via the theory of pseudo-holomorphic curves, Gromov proved that for certain values of \(k\), there are obstructions to obtaining this upper bound of the packing radii. By combining pseudo-holomorphic curve theory with the theory of symplectic blow-ups, McDuff and Polterovich precisely determined the least upper bound of the set of packing radii for \(k = 1, \ldots, 9\) embeddings. In the following, explicit and elementary constructions are given for these maximal packings when \(k = 1, \ldots, 6,9\). Constructions are also given for full packings of the \(2n\)- dimensional ball by \(k^n\) balls. In addition, the techniques are applied to study alternate packing problems.

MSC:
52C17 Packing and covering in \(n\) dimensions (aspects of discrete geometry)
53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
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