Volume 20, issue 2 (2016)

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Beyond ECH capacities

Michael Hutchings

Geometry & Topology 20 (2016) 1085–1126

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 and 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.

embedded contact homology, symplectic embeddings
Mathematical Subject Classification 2010
Primary: 53D42
Received: 27 October 2014
Revised: 4 April 2015
Accepted: 10 May 2015
Published: 28 April 2016
Proposed: Leonid Polterovich
Seconded: Ciprian Manolescu, Ronald Stern
Michael Hutchings
Department of Mathematics
University of California, Berkeley
970 Evans Hall
Berkeley, CA 94720