Holomorphic open book decompositions

Abbas, Casim

doi:10.1215/00127094-1276301

Cited by 19 publications

(31 citation statements)

References 40 publications

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“…Theorem 1. 1. Let (M, ξ) be a contact 3-manifold with contact structure induced by a nondegenerate contact form λ, and let J ∈ J (M, λ) be a complex multiplication for which the triple (M, λ, J) admits a stable finite energy foliation F of energy E(F).…”

Section: Introduction and The Main Resultsmentioning

confidence: 99%

“…Since varying the points at which the connected sum occurs leads to contactomorphic contact manifolds, we are free to choose the points at each step to lie on an index-2 curve in the foliation. As is shown in Section 6, each time we apply our construction we add one even orbit to the foliation with two planes (modulo the R-action) asymptotic to it and two index-1 curves (modulo 1 This can be seen from the following argument which was explained to the second author by O. van Koert. Forming a contact connected sum at two points in a given connected manifold (M, ξ) is the same as forming a contact connected sum of (M, ξ) with (S 1 × S 2 , ξ 0 ) where ξ 0 is the contact structure arising as the kernel of the S 1 -invariant contact form λ 0 = cos θ dt + sin 2 θ dφ with t ∈ R/Z the coordinate along S 1 , and with φ ∈ R/2πZ and θ ∈ [0, π], respectively, the polar and azimuthal coordinates on S 2 .…”

Section: Introduction and The Main Resultsmentioning

confidence: 99%

“…with Ψ ∈ Σ(1) the path (2.6), and µ cz (Ψ) the Conley-Zehnder index of the path Ψ as characterized in Theorem 2. 1. We note that, as a result of the homotopy invariance axiom from Theorem 2.1, the Conley-Zehnder index of an orbit is invariant under homotopies of the trivialization.…”

Section: Background In Contact Geometry and Reeb Dynamicsmentioning

confidence: 93%

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Connected sums and finite energy foliations I: Contact connected sums

Fish

Siefring

2018

Journal of Symplectic Geometry

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We consider a 3-manifold M equipped with nondegenerate contact form λ and compatible almost complex structure J. We show that if the data (M, λ, J) admits a stable finite energy foliation, then for a generic choice of distinct points p, q ∈ M , the manifold M formed by taking the contact connected sum at p and q admits a nondegenerate contact form λ and compatible almost complex structure J so that the data (M , λ , J ) also admits a stable finite energy foliation. Along the way, we develop some general theory for the study of finite energy foliations.

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Section: Introduction and The Main Resultsmentioning

confidence: 99%

Section: Introduction and The Main Resultsmentioning

confidence: 99%

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Connected sums and finite energy foliations I: Contact connected sums

Fish

Siefring

2018

Journal of Symplectic Geometry

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“…The first step is thus to define the moduli space and show that it is nonempty. This rests on a construction known as the holomorphic open book ; the following result was first stated in [ACH05], and two proofs later appeared in independent work of Abbas [Abb11] and the author [Wen10c]. (1) Each connected component γ ⊂ B is a nondegenerate Reeb orbit with µ τ CZ (γ) = 1, where τ is any trivialization in which the pages approach γ with winding number 0.…”

Section: A Classification Theorem For Symplectic Fillingsmentioning

confidence: 99%

Lectures on Contact 3-Manifolds, Holomorphic Curves and Intersection Theory

Wendl

2020

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This is a slight revision of some expository lecture notes written originally for a 5-hour minicourse on the intersection theory of punctured holomorphic curves and its applications in 3-dimensional contact topology, given by the author as part of the the LMS Short Course "Topology in Low Dimensions" at Durham University, August 26-30, 2013. The revision includes a new appendix that has been added for the benefit of researchers in search of a "quick reference" on the basic facts of Siefring's intersection theory.Intersection theory has played a prominent role in the study of closed symplectic 4manifolds since Gromov's paper [Gro85] on pseudoholomorphic curves, leading to a myriad of beautiful rigidity results that are either not accessible or not true in higher dimensions. In recent years, the highly nontrivial extension of this theory to the case of punctured holomorphic curves, due to Siefring [Sie08, Sie11], has led to similarly beautiful results about contact 3-manifolds and their symplectic fillings. These notes begin with an overview of the closed case and an easy application (McDuff's characterization of symplectic ruled surfaces), and then explain the essentials of Siefring's intersection theory and how to use it in the real world. As a sample application, we discuss the classification of symplectic fillings of planar contact manifolds via Lefschetz fibrations [Wen10b]. Prior knowledge of holomorphic curves may be helpful but is not assumed-the required definitions and most of the hard analytical results are stated as black boxes, so as to focus on topological rather than analytical issues.

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“…Choose a sequence ǫ n such that ǫ n > 0, ǫ n → 0 and ǫ n R n → +∞. Now, apply Hofer's metric lemma [10] to the continuous sequence of functions du n (s, t) g eucl. ,g Pns defined on [−R n , R n ] × S 1 .…”

Section: Center Actionmentioning

confidence: 99%

Asymptotic behavior for $${\mathcal {H}}$$-holomorphic cylinders of small area

Doicu

Fuchs

2019

J. Fixed Point Theory Appl.

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H−holomorphic curves are solutions of a modified pseudoholomorphic curve equation involving a harmonic 1−form as perturbation term. Following [7] we establish an asymptotic behavior of a sequence of finite energy H−holomorphic cylinders with small dα−energies. Our results can be seen as a first step toward establishing the compactness of the moduli space of H−holomorphic curves, which in turns, due to the program initiated in [3], can be used for proving the generalized Weinstein conjecture.

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Holomorphic open book decompositions

Cited by 19 publications

References 40 publications

Connected sums and finite energy foliations I: Contact connected sums

Connected sums and finite energy foliations I: Contact connected sums

Lectures on Contact 3-Manifolds, Holomorphic Curves and Intersection Theory

Asymptotic behavior for $${\mathcal {H}}$$-holomorphic cylinders of small area

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