## 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

#### Abstract

A toric domain is a subset of $\left({\u2102}^{n},{\omega}_{std}\right)$ which is invariant under the standard rotation action of ${\mathbb{T}}^{n}$ on ${\u2102}^{n}$. 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 $c\left(U\right)=c\left(V\right)$ 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 ${\mathbb{T}}^{2}$-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

**Source**

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

**Dates**

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**

https://projecteuclid.org/euclid.involve/1511370917

**Digital Object Identifier**

doi:10.2140/involve.2015.8.665

**Mathematical Reviews number (MathSciNet)**

MR3366017

**Zentralblatt MATH identifier**

1322.53081

**Subjects**

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

**Keywords**

symplectic capacities toric domain moment space axes

#### Citation

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. https://projecteuclid.org/euclid.involve/1511370917