Strongly Fillable Contact Manifolds and J-holomorphic Foliations

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 the 3-torus 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 the 3-torus 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 the cotangent bundle of the 2-torus is contractible, and to define an obstruction to strong fillability that yields a non-gauge-theoretic proof of Gay's recent nonfillability result for contact manifolds with positive Giroux torsion.

44 pages, 2 figures; v.3 has a few significant improvements to the main results: We now classify all strong fillings and exact fillings of T^3 (without assuming Stein), and also show that a planar contact manifold is strongly fillable if and only if all its planar open books have monodromy generated by right-handed Dehn twists. To appear in Duke Math. J

References in corpus (3)

• The Seiberg-Witten equations and the Weinstein conjecture
• Relative asymptotic behavior of pseudoholomorphic half-cylinders
• Four-dimensional symplectic cobordisms containing three-handles

Cited by in corpus (32)

• Weak and strong fillability of higher dimensional contact manifolds
• Planar open books, monodromy factorizations, and symplectic fillings
• A Hierarchy of Local Symplectic Filling Obstructions for Contact 3-Manifolds
• Cylindrical contact homology and topological entropy
• Algebraic Torsion in Contact Manifolds
• Maximal contact and symplectic structures
• Calabi-Yau Caps, Uniruled Caps and Symplectic Fillings
• Simplifying Weinstein Morse functions
• Planar open books with four binding components
• Fillings of unit cotangent bundles
• Positive factorizations of mapping classes
• Surgery along star shaped plumbings and exotic smooth structures on 4-manifolds
• Symplectic fillings of asymptotically dynamically convex manifolds I
• Holomorphic Open Book Decompositions
• Classification of tight contact structures on small Seifert fibered L-spaces
• Diffeomorphism type of symplectic fillings of unit cotangent bundles
• On filings of $\partial (V\times \mathbb{D})$
• Fillings of unit cotangent bundles of nonorientable surfaces
• Symplectic 4-manifolds admit Weinstein trisections
• On fillings of contact links of quotient singularities
• Combinatorial Reeb dynamics on punctured contact 3-manifolds
• On contact type hypersurfaces in 4-space
• Higher dimensional contact topology via holomorphic disks
• Fillability of small Seifert fibered spaces
• Planarity in higher-dimensional contact manifolds
• Stein fillable contact 3-manifolds and positive open books of genus one
• SFT computations and intersection theory in higher-dimensional contact manifolds
• Unexpected Stein fillings, rational surface singularities, and plane curve arrangements
• Contact surgery graphs
• A landscape of contact manifolds via rational SFT
• Symplectic fillings of unit cotangent bundles of hyperbolic surfaces
• A remark about weak fillings