Quantitative Embedded Contact Homology

Hutchings, Michael

doi:10.4310/jdg/1320067647

Cited by 106 publications

(181 citation statements)

References 29 publications

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“…ECH capacities are a sequence of capacities which are defined for four-dimensional symplectic manifolds [7]. For a symplectic manifold (X 4 , ω), there is a sequence of real numbers: 0 = c 0 (X, ω) < c 1 (X, ω) ≤ c 2 (X, ω) ≤ · · · ≤ ∞.…”

Section: Introductionmentioning

confidence: 99%

Symplectic embeddings and the Lagrangian bidisk

Ramos¹

2017

Duke Math. J.

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In this paper we obtain sharp obstructions to the symplectic embedding of the lagrangian bidisk into four-dimensional balls, ellipsoids and symplectic polydisks. We prove, in fact, that the interior of the lagrangian bidisk is symplectomorphic to a concave toric domain using ideas that come from billiards on a round disk. In particular, we answer a question of Ostrover [12]. We also obtain sharp obstructions to some embeddings of ellipsoids into the lagrangian bidisk.

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Section: Introductionmentioning

confidence: 99%

Symplectic embeddings and the Lagrangian bidisk

Ramos¹

2017

Duke Math. J.

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“…Proof: Copy the above proof of Theorem 1, using Proposition 10 and [11] in place of Proposition 7 and [10].…”

Section: More Examples Of Ech Capacitiesmentioning

confidence: 99%

“…26, as explained in ref. 11. The first step is to define a "completion" X of X by attaching cylindrical ends ½0;∞Þ × Y þ to the positive boundary and ð−∞;0 × Y − to the negative boundary.…”

Section: Embedded Contact Homologymentioning

confidence: 99%

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Recent progress on symplectic embedding problems in four dimensions

Hutchings

2011

Proc. Natl. Acad. Sci. U.S.A.

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We survey some recent progress on understanding when one fourdimensional symplectic manifold can be symplectically embedded into another. In 2010, McDuff established a number-theoretic criterion for the existence of a symplectic embedding of one fourdimensional ellipsoid into another. This result is related to previously known criteria for when a disjoint union of balls can be symplectically embedded into a ball. Numerical invariants defined using embedded contact homology give general obstructions to symplectic embeddings in four dimensions which turn out to be sharp in the above cases.

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“…The ECH capacities were introduced in [5]; see also the exposition in [7]; we will review the definition in the cases relevant to this paper in § 3.1.…”

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confidence: 99%

Symplectic embeddings into four-dimensional concave toric domains

Choi

Cristofaro-Gardiner

Frenkel

et al. 2014

Journal of Topology

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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 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.Contents c k (X, rω) = rc k (X, ω).

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Quantitative Embedded Contact Homology

Cited by 106 publications

References 29 publications

Symplectic embeddings and the Lagrangian bidisk

Symplectic embeddings and the Lagrangian bidisk

Recent progress on symplectic embedding problems in four dimensions

Symplectic embeddings into four-dimensional concave toric domains

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