Beyond ECH capacities

Hutchings, Michael

doi:10.2140/gt.2016.20.1085

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Beyond ECH capacities

Michael Hutchings

Geometry & Topology 20 (2016) 1085–1126

DOI: 10.2140/gt.2016.20.1085

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 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.

Keywords

embedded contact homology, symplectic embeddings

Mathematical Subject Classification 2010

Primary: 53D42

References

Publication

Received: 27 October 2014

Revised: 4 April 2015

Accepted: 10 May 2015

Published: 28 April 2016

Proposed: Leonid Polterovich

Seconded: Ciprian Manolescu, Ronald Stern

Authors

Michael Hutchings
Department of Mathematics University of California, Berkeley 970 Evans Hall Berkeley, CA 94720 USA
http://www.math.berkeley.edu/~hutching

Volume 20, issue 2 (2016)