Symplectic embeddings of 4-dimensional ellipsoids into polydiscs

TL;DR

This paper investigates symplectic embeddings of 4-dimensional ellipsoids into polydiscs, establishing conditions under which volume is the only obstruction and proposing conjectures for specific cases, with partial verification.

Contribution

It introduces new criteria for symplectic embeddings into polydiscs and formulates conjectures on when volume is the sole obstruction, extending previous work on simpler shapes.

Findings

Existence of a constant C depending on d/c for embedding conditions

Volume obstruction is the only barrier when b/a exceeds C

Verification of conjecture for b = 13/2

Abstract

McDuff and Schlenk have recently determined exactly when a four-dimensional symplectic ellipsoid symplectically embeds into a symplectic ball. Similarly, Frenkel and M\"uller have recently determined exactly when a symplectic ellipsoid symplectically embeds into a symplectic cube. Symplectic embeddings of more complicated structures, however, remain mostly unexplored. We study when a symplectic ellipsoid $E(a,b)$ symplectically embeds into a polydisc $P(c,d)$. We prove that there exists a constant $C$ depending only on $d/c$ (here, $d$ is assumed greater than $c$) such that if $b/a$ is greater than $C$, then the only obstruction to symplectically embedding $E(a,b)$ into $P(c,d)$ is the volume obstruction. We also conjecture exactly when an ellipsoid embeds into a scaling of $P(1,b)$ for $b$ greater than or equal to $6$, and conjecture about the set of $(a,b)$ such that the only obstruction to embedding $E(1,a)$ into a scaling of $P(1,b)$ is the classical volume. Finally, we verify our conjecture for $b = \frac{13}{2}$.