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

Full-text: Access denied (no subscription detected)

However, an active subscription may be available with MSP at msp.org/involve.

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

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 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 .

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


Export citation

References

  • M. Beck and S. Robins, Computing the continuous discretely: Integer-point enumeration in polyhedra, Springer, New York, 2007.
  • O. Buse and R. Hind, “Ellipsoid embeddings and symplectic packing stability”, Compos. Math. 149:5 (2013), 889–902.
  • D. Cristofaro-Gardiner and A. Kleinman, “Ehrhart polynomials and symplectic embeddings of ellipsoids”, preprint, 2013.
  • D. Frenkel and D. Müller, “Symplectic embeddings of 4-dimensional ellipsoids into cubes”, preprint, 2012.
  • M. Hutchings, “Lecture notes on embedded contact homology”, pp. 389–484 in Contact and symplectic topology, edited by F. Bourgeois et al., Bolyai Soc. Math. Stud. 26, János Bolyai Math. Soc., Budapest, 2014.
  • B.-H. Li and T.-J. Li, “Symplectic genus, minimal genus and diffeomorphisms”, Asian J. Math. 6:1 (2002), 123–144.
  • D. McDuff, “The Hofer conjecture on embedding symplectic ellipsoids”, J. Differential Geom. 88:3 (2011), 519–532.
  • D. McDuff and F. Schlenk, “The embedding capacity of 4-dimensional symplectic ellipsoids”, Ann. of Math. $(2)$ 175:3 (2012), 1191–1282.