Beyond ECH capacities

Abstract

ECH (embedded contact homology) capacities give obstructions to symplectically embedding one four-dimensional symplectic manifold with boundary into another. These obstructions are known to be sharp when the domain and target are ellipsoids (proved by McDuff), and more generally when the domain is a "concave toric domain" and the target is a "convex toric domain" (proved by Cristofaro-Gardiner). However ECH capacities often do not give sharp obstructions, for example in many cases when the domain is a polydisk. This paper uses more refined information from ECH to give stronger symplectic embedding obstructions when the domain is a polydisk, or more generally a convex toric domain. We use these new obstructions to reprove a result of Hind-Lisi on symplectic embeddings of a polydisk into a ball, and generalize this to obstruct some symplectic embeddings of a polydisk into an ellipsoid. We also obtain a new obstruction to symplectically embedding one polydisk into another, in particular proving the four-dimensional case of a conjecture of Schlenk.

41 pages; v4: a couple of minor corrections and clarifications

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• Combinatorial embedded contact homology for toric contact manifolds
• An elementary alternative to ECH capacities
• Taming the pseudoholomorphic beasts in $\mathbb{R}\times (S^1\times S^2)$
• Symplectic embeddings of four-dimensional polydisks into balls
• On the embedding complexity of Liouville manifolds
• Equidistributed periodic orbits of $C^{\infty }$-generic three-dimensional Reeb flows
• The stabilized symplectic embedding problem for polydiscs
• Closed-open morphisms on periodic Floer homology
• Cobordism maps on periodic Floer homology induced by elementary Lefschetz fibrations