A JSJ-type decomposition theorem for symplectic fillings

We establish a JSJ-type decomposition theorem for splitting exact symplectic fillings of contact 3-manifolds along \emph{mixed tori} – these are convex tori satisfying a particular geometric condition. As an application, we show that if $(M,\xi )$ is obtained from $(S^3,{\xi }_{\mathrm{s}\mathrm{t}\mathrm{d}})$ via Legendrian surgery along a knot which has been stabilized both positively and negatively, then $(M,\xi )$ has a unique exact filling.