Beyond ECH capacities

By Michael Hutchings. 2014

51 citations recorded by the source corpus.

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.

Subfields: Computational Theory and Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/2453f8da7df706401b95242f504307ba3fc40595

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Citations in the corpus

Symplectic embeddings from concave toric domains into convex ones (2014, methodology): “We can define such a path by using an analogue of the “corner rounding” operation from [15]: specifically, given Λ we define Λ1 by choosing any vertex other than the origin, removing the lattice point corresponding to that edge, and taking the convex hull of the remaining lattice points.”
Symplectic capacities from positive S1–equivariant symplectic homology (2017, methodology): “e e.g. [MS12, CGFS16]. The question of when a four-dimensional polydisk can be symplectically embedded into a polydisk or an ellipsoid is only partially understood; for some of the latest results see [Hut16]. The analogous questions in higher dimensions are more complicated, and much less is underst”