Symplectic embeddings of polydisks

2013/04/10 by Richard Hind, Hind, Richard, Samuel Lisi +1 · 1 citation
Mathematics · #53D35 #57R17 #Advanced Algebra and Geometry #FOS: Mathematics #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #Symplectic Geometry (math.SG)

paper · pdf · doi:10.48550/arxiv.1304.3065

openalex publication_date 2013/04/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 πr² ≤ a. We show P(1,2) embeds in B⁴(a) if and only if a is at least 3. This shows the inclusion of P(1,2) in B⁴(3) is optimal. The necessity of a ≥ 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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