## Involve: A Journal of Mathematics

- Involve
- Volume 10, Number 2 (2017), 219-242.

### Symplectic embeddings of four-dimensional ellipsoids into polydiscs

Madeleine Burkhart, Priera Panescu, and Max Timmons

#### 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 $d\u2215c$ (here, $d$ is assumed greater than $c$) such that if $b\u2215a$ 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 $b=\frac{13}{2}$.

#### Article information

**Source**

Involve, Volume 10, Number 2 (2017), 219-242.

**Dates**

Received: 8 September 2014

Revised: 21 June 2015

Accepted: 1 July 2015

First available in Project Euclid: 13 December 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.involve/1513135628

**Digital Object Identifier**

doi:10.2140/involve.2017.10.219

**Mathematical Reviews number (MathSciNet)**

MR3574298

**Zentralblatt MATH identifier**

1353.53084

**Subjects**

Primary: 53DXX

**Keywords**

symplectic geometry

#### Citation

Burkhart, Madeleine; Panescu, Priera; Timmons, Max. Symplectic embeddings of four-dimensional ellipsoids into polydiscs. Involve 10 (2017), no. 2, 219--242. doi:10.2140/involve.2017.10.219. https://projecteuclid.org/euclid.involve/1513135628