Involve: A Journal of Mathematics

  • Involve
  • Volume 8, Number 4 (2015), 665-676.

On symplectic capacities of toric domains

Michael Landry, Matthew McMillan, and Emmanuel Tsukerman

Full-text: Access denied (no subscription detected)

However, an active subscription may be available with MSP at

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


A toric domain is a subset of (n,ωstd) which is invariant under the standard rotation action of Tn on n. For a toric domain U from a certain large class for which this action is not free, we find a corresponding toric domain V where the standard action is free and for which c(U) = c(V ) for any symplectic capacity c. Michael Hutchings gives a combinatorial formula for calculating his embedded contact homology symplectic capacities for certain toric four-manifolds on which the T2-action is free. Our theorem allows one to extend this formula to a class of toric domains where the action is not free. We apply our theorem to compute ECH capacities for certain intersections of ellipsoids and find that these capacities give sharp obstructions to symplectically embedding these ellipsoid intersections into balls.

Article information

Involve, Volume 8, Number 4 (2015), 665-676.

Received: 20 June 2014
Revised: 30 July 2014
Accepted: 2 August 2014
First available in Project Euclid: 22 November 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 53D05: Symplectic manifolds, general 53D20: Momentum maps; symplectic reduction 53D35: Global theory of symplectic and contact manifolds [See also 57Rxx]

symplectic capacities toric domain moment space axes


Landry, Michael; McMillan, Matthew; Tsukerman, Emmanuel. On symplectic capacities of toric domains. Involve 8 (2015), no. 4, 665--676. doi:10.2140/involve.2015.8.665.

Export citation


  • K. Choi, D. Cristofaro-Gardiner, D. Frenkel, M. Hutchings, and V. Ramos, “Symplectic embeddings into four-dimensional concave toric domains”, J. Topology (online publication May 2014).
  • K. Cieliebak, H. Hofer, J. Latschev, and F. Schlenk, “Quantitative symplectic geometry”, pp. 1–44 in Dynamics, ergodic theory, and geometry, edited by B. Hasselblatt, Math. Sci. Res. Inst. Publ. 54, Cambridge Univ. Press, 2007.
  • D. Frenkel and D. Müller, “Symplectic embeddings of 4-dimensional ellipsoids into cubes”, preprint, 2012.
  • M. Gromov, “Pseudoholomorphic curves in symplectic manifolds”, Invent. Math. 82:2 (1985), 307–347.
  • R. Hind and S. Lisi, “Symplectic embeddings of polydisks”, Selecta Math. (online publication January 2014).
  • M. Hutchings, “Quantitative embedded contact homology”, J. Differential Geom. 88:2 (2011), 231–266.
  • M. Hutchings, “Lecture notes on embedded contact homology”, pp. 389–484 in Contact and symplectic topology, edited by F. Bourgeois et al., Bolyai Society Mathematica Studies 26, Springer, New York, 2014.
  • D. McDuff, “The Hofer conjecture on embedding symplectic ellipsoids”, J. Differential Geom. 88:3 (2011), 519–532.
  • F. Schlenk, Embedding problems in symplectic geometry, de Gruyter Expositions in Mathematics 40, Walter de Gruyter, Berlin, 2005.
  • L. Traynor, “Symplectic packing constructions”, J. Differential Geom. 42:2 (1995), 411–429.