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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.

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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