Symplectic embeddings into four-dimensional concave toric domains

Abstract

ECH capacities give obstructions to symplectically embedding one symplectic four-manifold with boundary into another. We compute the ECH capacities of a large family of symplectic four-manifolds with boundary, called "concave toric domains". Examples include the (nondisjoint) union of two ellipsoids in ${\mathbb{R}}^4$. We use these calculations to find sharp obstructions to certain symplectic embeddings involving concave toric domains. For example: (1) we calculate the Gromov width of every concave toric domain; (2) we show that many inclusions of an ellipsoid into the union of an ellipsoid and a cylinder are "optimal"; and (3) we find a sharp obstruction to ball packings into certain unions of an ellipsoid and a cylinder.

31 pages, 2 figures; fixed one typo, updated references, to appear in Journal of Topology

References in corpus (1)

• Symplectic embeddings of 4-dimensional ellipsoids into cubes

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• Algebraic capacities
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• An elementary alternative to ECH capacities
• Combinatorial embedded contact homology for toric contact manifolds
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• Symplectic embeddings of the ${\ell }_p$-sum of two discs
• ECH embedding obstructions for rational surfaces
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• Special eccentricities of rational four-dimensional ellipsoids
• Towers of Looijenga pairs and asymptotics of ECH capacities
• On symplectic capacities and their blind spots
• Lattice Formulas For Rational SFT Capacities
• Newton-Okoukov bodies and symplectic embeddings into non-toric rational surfaces
• Coarse distance from dynamically convex to convex
• The special concave toric domain for the rotating Kepler problem