Vol. 10, No. 2, 2017

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Symplectic embeddings of four-dimensional ellipsoids into polydiscs

Madeleine Burkhart, Priera Panescu and Max Timmons

Vol. 10 (2017), No. 2, 219–242

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 dc (here, d is assumed greater than c) such that if ba 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 .

symplectic geometry
Mathematical Subject Classification 2010
Primary: 53DXX
Supplementary material

Code that checks through terms of $N$ and $M$ and Mathematica code

Received: 8 September 2014
Revised: 21 June 2015
Accepted: 1 July 2015
Published: 10 November 2016

Communicated by Bjorn Poonen
Madeleine Burkhart
Mathematics Department
University of Washington
Padelford Hall C-110
Seattle, WA 98195
United States
Priera Panescu
University of California
Santa Cruz, CA 95064
United States
Max Timmons
Massachusetts Institute of Technology
450 Memorial Drive
Cambridge, MA 02139
United States