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The asymptotics of ECH capacities

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Abstract

In a previous paper, the second author used embedded contact homology (ECH) of contact three-manifolds to define “ECH capacities” of four-dimensional symplectic manifolds. In the present paper we prove that for a four-dimensional Liouville domain with all ECH capacities finite, the asymptotics of the ECH capacities recover the symplectic volume. This follows from a more general theorem relating the volume of a contact three-manifold to the asymptotics of the amount of symplectic action needed to represent certain classes in ECH. The latter theorem was used by the first and second authors to show that every contact form on a closed three-manifold has at least two embedded Reeb orbits.

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Notes

  1. This definition of “Liouville domain” is slightly weaker than the usual definition, which would require that \(\omega \) have a primitive \(\lambda \) on \(X\) which restricts to a contact form on \(Y\).

  2. One can define ECH with integer coefficients [9, Sect. 9], and the isomorphism (2) also exists over \({\mathbb Z}\), as shown in [17]. However \({\mathbb Z}/2\) coefficients will suffice for this paper.

  3. In the non-\(L\)-flat case, there may be several Seiberg–Witten solutions corresponding to the same ECH generator, and/or Seiberg–Witten solutions corresponding to sets of Reeb orbits with multiplicities which are not ECH generators because they include hyperbolic orbits with multiplicity greater than one. See [15, Sect. 5.c, Part 2].

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Correspondence to Daniel Cristofaro-Gardiner.

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D. Cristofaro-Gardiner was partially supported by NSF grant DMS-0838703. M. Hutchings and V. G. B. Ramos were partially supported by NSF grant DMS-1105820.

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Cristofaro-Gardiner, D., Hutchings, M. & Ramos, V.G.B. The asymptotics of ECH capacities. Invent. math. 199, 187–214 (2015). https://doi.org/10.1007/s00222-014-0510-7

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  • DOI: https://doi.org/10.1007/s00222-014-0510-7

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