  We generalize the mixed tori which appear in the second author's JSJ-type
decomposition theorem for symplectic fillings of contact manifolds. Mixed tori
are convex surfaces in contact manifolds which may be used to decompose
symplectic fillings. We call our more general surfaces splitting surfaces, and
show that the decomposition of symplectic fillings continues to hold.
Specifically, given a strong or exact symplectic filling of a contact manifold
which admits a splitting surface, we produce a new symplectic manifold which
strongly or exactly fills its boundary, and which is related to the original
filling by Liouville surgery.
