Symplectic embeddings into four‐dimensional concave toric domains

By Keon Choi, Michael Hutchings, Daniel Cristofaro-Gardiner, D. Frenkel, V. G. B. Ramos. 2013

65 citations recorded by the source corpus.

ECH (embedded contact homology) 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 R4 . 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.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/46eea3e3baf2b75b2f3bc1f48eba047c5489890e

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In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Beyond ECH capacities (2014, background): “ ε > 0. Let λ and λ′ be contact forms on Y and Y′ respectively provided by Lemma 5.4. If ε is suﬃciently small, then there is also a weakly exact symplectic cobordism (X,ω), with X diﬀeomorphic to [0,1]×S3, from (Y′,λ′) to (Y,λ), because multiplying the contact form λ std|Y by a function f on Y i”
Symplectic embeddings from concave toric domains into convex ones (2014, methodology): “1 The concave case Weight sequences in the concave case were defined by Choi, the author, Frenkel, Hutchings, and Ramos in [3].”
Symplectic capacities from positive S1–equivariant symplectic homology (2017, background): “ECH capacities can be computed in many examples, such as four-dimensional “concave toric domains” [CCGF`14] and “convex toric domains” [CG14, Hut16], defined in §1.”