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arXiv · 1310.6647

Symplectic embeddings into four-dimensional concave toric domains

Keon Choi·Daniel Cristofaro-Gardiner·David Frenkel·Michael Hutchings·Vinicius G. B. Ramos
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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.

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Keon Choi, Daniel Cristofaro-Gardiner, David Frenkel, Michael Hutchings, Vinicius G. B. Ramos. 2013-10-24. Symplectic embeddings into four-dimensional concave toric domains. https://doi.org/10.1112/jtopol%2Fjtu008

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