Symplectic embeddings of polydisks

TL;DR

This paper establishes new obstructions to symplectic embeddings of polydisks into 4-dimensional balls, showing that certain embeddings are only possible when the ball's capacity reaches a specific threshold, which is not detected by traditional capacities.

Contribution

It introduces a novel obstruction method using pseudoholomorphic foliations for polydisk embeddings, highlighting limitations of existing capacity-based obstructions.

Findings

P(1,2) embeds into B^4(a) only if a ≥ 3

The embedding threshold a=3 is optimal for P(1,2)

Capacity-based obstructions like Ekeland-Hofer and ECH are not sharp here

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.