Symplectic embeddings of four-dimensional polydisks into half integer ellipsoids

Abstract

We obtain new sharp obstructions to symplectic embeddings of four-dimensional polydisks P (a, 1) into four-dimensional ellisoids E(bc, c) when 1 ≤ a < 2 and b is a halfinteger. We demonstrate that P (a, 1) symplectically embeds into E(bc, c) if and only if a + b ≤ bc, showing that inclusion is optimal and extending the result by Hutchings [Hu16] when b is an integer. Our proof is based on a combinatorial criterion developed by Hutchings [Hu16] to obstruct symplectic embeddings. We additionally show that the range of a and b cannot be extended further using the Hutchings criterion.

An elementary alternative to ECH capacities

“…1.19] that if X Ω and X Ω ′ are four-dimensional convex toric domains, and if there exists a symplectic embedding X Ω → X Ω ′ , then a certain combinatorial criterion holds. This leads to stronger symplectic obstructions in some cases where ECH capacities do not give sharp obstructions, for example to symplectically embedding a polydisk into a ball or ellipsoid; see [23,4,9]. The proof of [23,Thm.…”