Symplectic embeddings of four-dimensional polydisks into half integer ellipsoids

TL;DR

This paper establishes new sharp criteria for symplectic embeddings of four-dimensional polydisks into half-integer ellipsoids, extending previous results and demonstrating optimal inclusion conditions.

Contribution

It provides the first sharp obstructions for embedding polydisks into ellipsoids when the parameter b is a half-integer, extending Hutchings' work for integer b.

Findings

Embedding condition a + b ≤ bc is both necessary and sufficient under certain bounds.

The results extend Hutchings' criteria from integer to half-integer b.

The work confirms the optimality of the inclusion condition.

Abstract

We obtain new sharp obstructions to symplectic embeddings of four-dimensional polydisks $P(a,1)$ into four-dimensional ellipsoids $E(bc,c)$ when $1\le a< 2$ and $b$ is a half-integer. When $1 \leq a < 2-O(b^{-1})$ we demonstrate that $P(a,1)$ symplectically embeds into $E(bc,c)$ if and only if $a+b\le bc$. Our results show that inclusion is optimal and extend the result by Hutchings \cite{H} when $b$ is an integer. Our proof is based on a combinatorial criterion developed by Hutchings \cite{H} to obstruct symplectic embeddings.