Quantitative embedded contact homology. (English) Zbl 1238.53061

The embedded contact homology is used to assign to each four-dimensional Liouville domain a sequence of real numbers which are called “ECH capacities”. Two kinds of ECH capacities are introduced: (i) distinguished; and (ii) full ECH capacities. Both the full and distinguished ECH capacities of a four-dimensional Liouville domain \((X, \omega)\) with boundary \(Y\) are defined in terms of the embedded contact homology of \((Y, \lambda)\), where \(\lambda\) is a contact form on \(Y\) with \(d\lambda = \omega\mid_Y\). In other words, they 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. It is shown that the ECH capacities are monotone with respect to embeddings. The ECH capacities are calculated for elipsoids, polydisks, certain subsets of the cotangent bundle of \(T^2\), and disjoint unions of structures for which the ECH capacities are known. A conjecture under which the asymptotics of ECH capacities of a Liouville domain recover its symplectic volume is also formulated.


53D35 Global theory of symplectic and contact manifolds
57R17 Symplectic and contact topology in high or arbitrary dimension
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