Symplectic embeddings of polydisks

Richard Hind; Samuel Lisi

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arXiv · 1304.3065

Symplectic embeddings of polydisks

Richard Hind·Samuel Lisi

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Abstract

In this note, we obtain new obstructions to symplectic embeddings of a product of disks (a polydisk) into a 4-dimensional ball. The polydisk P(r,s) is the product of the disk of area r with the disk of area s. The ball of capacity a, denoted B(a), is the ball with \pi r^2 \le a. We show P(1,2) embeds in B^4(a) if and only if a is at least 3. This shows the inclusion of P(1,2) in B^4(3) is optimal. The necessity of a \ge 3 implies that for this particular embedding problem neither the Ekeland-Hofer nor ECH capacities give a sharp obstruction. We contrast this with the case of ellipsoid embeddings into a ball when the ECH capacities give a complete list of obstructions [McDuff 2011]. Our obstruction does not come from a symplectic capacity, but instead from pseudoholomorphic foliations, thus the techniques seem to be special to dimension 4.

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Richard Hind, Samuel Lisi. 2013-04-10. Symplectic embeddings of polydisks. https://arxiv.org/abs/1304.3065

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