Duke Mathematical Journal

Strongly fillable contact manifolds and J-holomorphic foliations

Chris Wendl

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Abstract

We prove that every strong symplectic filling of a planar contact manifold admits a symplectic Lefschetz fibration over the disk, and every strong filling of T3 similarly admits a Lefschetz fibration over the annulus. It follows that strongly fillable planar contact structures are also Stein fillable, and all strong fillings of T3 are equivalent up to symplectic deformation and blowup. These constructions result from a compactness theorem for punctured J-holomorphic curves that foliate a convex symplectic manifold. We use it also to show that the compactly supported symplectomorphism group on T*T2 is contractible, and to define an obstruction to strong fillability that yields a non-gauge-theoretic proof of Gay's recent nonfillability result [G] for contact manifolds with positive Giroux torsion

Article information

Source
Duke Math. J., Volume 151, Number 3 (2010), 337-384.

Dates
First available in Project Euclid: 8 February 2010

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1265637657

Digital Object Identifier
doi:10.1215/00127094-2010-001

Mathematical Reviews number (MathSciNet)
MR2605865

Zentralblatt MATH identifier
1207.32022

Subjects
Primary: 32Q65: Pseudoholomorphic curves
Secondary: 57R17: Symplectic and contact topology

Citation

Wendl, Chris. Strongly fillable contact manifolds and $J$ -holomorphic foliations. Duke Math. J. 151 (2010), no. 3, 337--384. doi:10.1215/00127094-2010-001. https://projecteuclid.org/euclid.dmj/1265637657


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