## Involve: A Journal of Mathematics

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

### 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 $E(a,b)$ 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 $b ≥ 6$, 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 volume. Finally, we verify our conjecture for $b = 13 2$.

#### Article information

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

Dates
Revised: 21 June 2015
Accepted: 1 July 2015
First available in Project Euclid: 13 December 2017

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

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