Symplectic embeddings of four-dimensional ellipsoids into polydiscs

Abstract

McDuff and Schlenk recently determined exactly when a four-dimensional symplectic ellipsoid symplectically embeds into a symplectic ball. Similarly, Frenkel and Müller recently determined exactly when a symplectic ellipsoid symplectically embeds into a symplectic cube. Symplectic embeddings of more complicated sets, however, remain mostly unexplored. We study when a symplectic ellipsoid $E\left(a,b\right)$ symplectically embeds into a polydisc $P\left(c,d\right)$. 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 $E\left(a,b\right)$ into $P\left(c,d\right)$ is the volume obstruction. We also conjecture exactly when an ellipsoid embeds into a scaling of $P\left(1,b\right)$ for $b\ge 6$, and conjecture about the set of $\left(a,b\right)$ such that the only obstruction to embedding $E\left(1,a\right)$ into a scaling of $P\left(1,b\right)$ is the volume. Finally, we verify our conjecture for $b=\frac{13}{2}$.