Fillability

Fillability is a concept in contact and symplectic geometry describing when a contact manifold can be realized as the boundary of a symplectic manifold in a compatible way. In the standard setting, one starts with a closed oriented 3-manifold Y equipped with a positive contact structure ξ. A symplectic manifold (W, ω) is called a filling of (Y, ξ) if ∂W = Y and the symplectic form ω interacts with ξ in a way that makes the boundary geometry compatible with the contact structure.

Fillability is categorized by varying degrees of compatibility:

- Weak fillability: ω restricted to the contact planes ξ is positive, but no extra structure near the boundary is required.

- Strong fillability: there exists a Liouville vector field near ∂W pointing outward, which endows a collar where ω = dλ and λ|_Y defines ξ.

- Exact fillability: the symplectic form ω is globally exact on W (ω = dλ), with λ|_Y giving the contact form for ξ.

- Stein fillability: the filling W is a Stein domain, i.e., a complex manifold with a proper plurisubharmonic function whose gradient-like behavior makes ∂W carry the induced contact structure.

Key relationships hold among these notions: Stein fillable implies strongly fillable, which implies weakly fillable. In all cases, fillable contact structures are tight, meaning they do not contain overtwisted disks; however, tightness does not imply fillability in general. The standard contact structure on the 3-sphere S^3 is Stein fillable (by the complex ball in C^2). Obstructions from tools like gauge theory and Heegaard Floer homology identify many contact structures that are not fillable.

Fillability connects boundary contact geometry to interior symplectic topology and serves as a central tool for distinguishing and classifying contact structures.