The asymptotics of ECH capacities
@article{CristofaroGardiner2012TheAO, title={The asymptotics of ECH capacities}, author={Daniel Cristofaro-Gardiner and Michael Hutchings and Vinicius G. B. Ramos}, journal={Inventiones mathematicae}, year={2012}, volume={199}, pages={187-214} }
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…
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References
SHOWING 1-10 OF 17 REFERENCES
Embedded contact homology and its applications
- Mathematics
- 2010
Embedded contact homology (ECH) is a kind of Floer homology for contact three-manifolds. Taubes has shown that ECH is isomorphic to a version of Seiberg-Witten Floer homology (and both are…
From one Reeb orbit to two
- Mathematics
- 2012
We show that every (possibly degenerate) contact form on a closed three-manifold has at least two embedded Reeb orbits. We also show that if there are only finitely many embedded Reeb orbits, then…
The embedded contact homology index revisited
- Mathematics
- 2008
Let Y be a closed oriented 3-manifold with a contact form such that all Reeb orbits are nondegenerate. The embedded contact homology (ECH) index associates an integer to each relative 2-dimensional…
Quantitative embedded contact homology
- Mathematics
- 2010
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…
Proof of the Arnold chord conjecture in three dimensions, II
- Mathematics
- 2013
In "Proof of the Arnold chord conjecture in three dimensions I", we deduced the Arnold chord conjecture in three dimensions from another result, which asserts that an exact symplectic cobordism…
Proof of the Arnold chord conjecture in three dimensions I
- Mathematics
- 2011
In “Proof of the Arnold chord conjecture in three dimensions, I” [12], we deduced the Arnold chord conjecture in three dimensions from another result, which asserts that an exact symplectic cobordism…
The Hofer conjecture on embedding symplectic ellipsoids
- Mathematics
- 2010
In this note we show that one open four dimensional ellipsoid embeds symplectically into another if and only the ECH capacities of the first are no larger than those of the second. This proves a…
Embedded contact homology and Seiberg-Witten Floer cohomology I
- Mathematics
- 2008
This is the third of five papers whose purpose is to prove that the embedded contact homology of a compact, oriented 3–dimensional manifold with contact 1–form is isomorphic to the manifold’s…
Recent progress on symplectic embedding problems in four dimensions
- MathematicsProceedings of the National Academy of Sciences
- 2011
Numerical invariants defined using embedded contact homology give general obstructions to symplectic embeddings in four dimensions which turn out to be sharp in the above cases.
The Weinstein conjecture for stable Hamiltonian structures
- Mathematics
- 2009
We use the equivalence between embedded contact homology and Seiberg-Witten Floer homology to obtain the following improvements on the Weinstein conjecture. Let Y be a closed oriented connected…