The Weinstein conjecture for stable Hamiltonian structures
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In this article, the equivalence between embedded contact homology and Seiberg-Witten Floer homology was used to obtain the following improvements on the Weinstein conjecture: if Y is a closed oriented connected 3-manifold with a stable Hamiltonian structure, then R denotes the associated Reeb vector field on Y.Abstract:
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 3–manifold with a stable Hamiltonian structure, and let R denote the associated Reeb vector field on Y . We prove that if Y is not a T 2 –bundle over S , then R has a closed orbit. Along the way we prove that if Y is a closed oriented connected 3–manifold with a contact form such that all Reeb orbits are nondegenerate and elliptic, then Y is a lens space. Related arguments show that if Y is a closed oriented 3–manifold with a contact form such that all Reeb orbits are nondegenerate, and if Y is not a lens space, then there exist at least three distinct embedded Reeb orbits.read more
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Journal ArticleDOI
Quantitative Embedded Contact Homology
TL;DR: In this paper, the authors define the ECH capacities of Liouville domains as the spectrum of the amount of action needed to represent certain classes in embedded contact homology, and show that these capacities are monotone with respect to symplectic embeddings.
Journal ArticleDOI
Embedded contact homology and Seiberg–Witten Floer cohomology II
TL;DR: In this article, the authors constructed an isomorphism between embedded contact homology and Seiberg-Witten Floer cohomology of a compact 3-manifold with a given contact 1-form.
Book ChapterDOI
Lecture Notes on Embedded Contact Homology
TL;DR: In this paper, the authors give an introduction to embedded contact homology (ECH) of contact three-manifolds, gathering together many basic notions which are scattered across a number of papers.
Posted Content
The embedded contact homology index revisited
TL;DR: The ECH index as mentioned in this paper assigns an integer to each relative 2-dimensional homology class of surfaces whose boundary is the difference between two unions of Reeb orbits, which determines the relative grading on ECH.
Journal ArticleDOI
From one Reeb orbit to two
TL;DR: In this article, it was shown that every degenerate contact form on a closed three-manifold has at least two embedded Reeb orbits and that if there are only finitely many Reeb orbit, then their actions are not all integer multiples of a single real number.
References
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Book
Symplectic Invariants and Hamiltonian Dynamics
Helmut Hofer,Eduard Zehnder +1 more
TL;DR: In this article, the authors consider the class of all symplectic manifolds (M, ω) possibly with boundary, but of fixed dimension 2n, where ω is a symplectic structure, i.e. a two-form on M which is closed and nondegenerate.
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An introduction to contact topology
TL;DR: A comprehensive introduction to contact topology is given in this article, including recent striking applications in geometric and differential topology: Eliashberg's proof of Cerf's theorem via the classification of tight contact structures on the 3-sphere, and the Kronheimer-Mrowka proof of property P for knots via symplectic fillings of contact 3-manifolds.
Journal ArticleDOI
Compactness results in Symplectic Field Theory
TL;DR: In this article, a series of compactness results for moduli spaces of holomorphic curves arising in Symplectic field theory is presented. But these results generalize Gromov's compactness theorem in (8) as well as compactness theorems in Floer homology theory, and in contact geometry, (9, 19).
Journal ArticleDOI
Pseudoholomorphic curves in symplectizations with applications to the Weinstein conjecture in dimension three.
TL;DR: In this paper, the main results of the Weinstein conjecture are given. But they do not consider the local properties of the manifold and local fillings, which is the case in this paper.
Book
Monopoles and Three-Manifolds
TL;DR: The Seiberg-Witten equations and compactness of Hilbert manifolds have been studied in this paper, with a focus on compactness and non-exact perturbations.