The asymptotics of ECH capacities

By Daniel Cristofaro-Gardiner, Michael Hutchings, V. G. B. Ramos. 2012

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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.

Subfields: Computational Theory and Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/4250bfa2f3c5b7b577be895b95a76b4fe605127f

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Citations in the corpus

Lecture Notes on Embedded Contact Homology (2013, methodology): “g-Witten Floer cohomology (with Z/2 coeﬃcients). 16 r<0. We will prove the Polydisk property at the end of §4.3. The proof of the Volume property is beyond the scope of these notes; it is given in [10], using ingredients from Taubes’s proof of the Weinstein conjecture [58]. 2 Origins of ECH One o”
Dense existence of periodic Reeb orbits and ECH spectral invariants (2015, background): “Next we recall the following remarkable result from [2], namely ECH spectral invariants recover the volume.”
Sharp systolic inequalities for Reeb flows on the three-sphere (2015, background): “nsion, and its proof is based on a reduction to normal forms. A natural question is whether the inequality (2) holds also for contact forms which are far from Zoll ones (see also [Hut11, Remark 8.5], [CGHR15] and [APB14, Section 2.3] for related questions and results about the relationship between t”
Torsion contact forms in three dimensions have two or infinitely many Reeb orbits (2017, unknown): “The key new insight of our paper is that one may use the “volume property” of ECH from Cristofaro-Gardiner, Hutchings and Ramos [5] to control both the genus and the potential breakings of these curves.”
Density of minimal hypersurfaces for generic metrics. (2017, background): “The argument of [7] is based on a different kind of asymptotic law, involving spectral invariants in Embedded Contact Homology ([3]).”