## Duke Mathematical Journal

### Strongly fillable contact manifolds and $J$-holomorphic foliations

Chris Wendl

#### 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 $T^3$ 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 $T^3$ 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 $T^*T^2$ 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

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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