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
- Strong fillability: there exists a Liouville vector field near ∂W pointing outward, which endows a collar
- Exact fillability: the symplectic form ω is globally exact on W (ω = dλ), with λ|_Y giving the contact
- Stein fillability: the filling W is a Stein domain, i.e., a complex manifold with a proper plurisubharmonic
Key relationships hold among these notions: Stein fillable implies strongly fillable, which implies weakly fillable. In
Fillability connects boundary contact geometry to interior symplectic topology and serves as a central tool for