Open Access
June 2011 Quantitative Embedded Contact Homology
Michael Hutchings
J. Differential Geom. 88(2): 231-266 (June 2011). DOI: 10.4310/jdg/1320067647


Define a “Liouville domain” to be a compact exact symplectic manifold with contact-type boundary. We use embedded contact homology to assign to each four-dimensional Liouville domain (or subset thereof) a sequence of real numbers, which we call “ECH capacities”. The ECH capacities of a Liouville domain are defined in terms of the “ECH spectrum” of its boundary, which measures the amount of symplectic action needed to represent certain classes in embedded contact homology. Using cobordism maps on embedded contact homology (defined in joint work with Taubes), we show that the ECH capacities are monotone with respect to symplectic embeddings. We compute the ECH capacities of ellipsoids, polydisks, certain subsets of the cotangent bundle of $T^ 2$, and disjoint unions of examples for which the ECH capacities are known. The resulting symplectic embedding obstructions are sharp in some interesting cases, for example for the problem of embedding an ellipsoid into a ball (as shown by McDuff-Schlenk) or embedding a disjoint union of balls into a ball. We also state and present evidence for a conjecture under which the asymptotics of the ECH capacities of a Liouville domain recover its symplectic volume.


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Michael Hutchings. "Quantitative Embedded Contact Homology." J. Differential Geom. 88 (2) 231 - 266, June 2011.


Published: June 2011
First available in Project Euclid: 31 October 2011

zbMATH: 1238.53061
MathSciNet: MR2838266
Digital Object Identifier: 10.4310/jdg/1320067647

Rights: Copyright © 2011 Lehigh University

Vol.88 • No. 2 • June 2011
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