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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.)
##### Keywords:
blow-up; symplectic packing; pseudo-holomorphic curve
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