  We prove several results on weak symplectic fillings of contact 3-manifolds,
including: (1) Every weak filling of any planar contact manifold can be
deformed to a blow up of a Stein filling. (2) Contact manifolds that have fully
separating planar torsion are not weakly fillable - this gives many new
examples of contact manifolds without Giroux torsion that have no weak
fillings. (3) Weak fillability is preserved under splicing of contact manifolds
along symplectic pre-Lagrangian tori - this gives many new examples of contact
manifolds without Giroux torsion that are weakly but not strongly fillable.
  We establish the obstructions to weak fillings via two parallel approaches
using holomorphic curves. In the first approach, we generalize the original
Gromov-Eliashberg "Bishop disk" argument to study the special case of Giroux
torsion via a Bishop family of holomorphic annuli with boundary on an "anchored
overtwisted annulus". The second approach uses punctured holomorphic curves,
and is based on the observation that every weak filling can be deformed in a
collar neighborhood so as to induce a stable Hamiltonian structure on the
boundary. This also makes it possible to apply the techniques of Symplectic
Field Theory, which we demonstrate in a test case by showing that the
distinction between weakly and strongly fillable translates into contact
homology as the distinction between twisted and untwisted coefficients.
