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We generalize the mixed tori which appear in the second authors 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.
We classify symplectically foliated fillings of certain contact foliated manifolds. We show that up to symplectic deformation, the unique minimal symplectically foliated filling of the foliated sphere cotangent bundle of the Reeb foliation in the 3-s
We use Menkes JSJ-type decomposition theorem for symplectic fillings to reduce the classification of strong and exact symplectic fillings of virtually overtwisted torus bundles to the same problem for tight lens spaces. For virtually overtwisted stru
An exact Lagrangian submanifold $L$ in the symplectization of standard contact $(2n-1)$-space with Legendrian boundary $Sigma$ can be glued to itself along $Sigma$. This gives a Legendrian embedding $Lambda(L,L)$ of the double of $L$ into contact $(2
We consider a large family F of torus bundles over the circle, and we use recent work of Li--Mak to construct, on each Y in F, a Stein fillable contact structure C. We prove that (i) each Stein filling of (Y,C) has vanishing first Chern class and fir
Let $H(q,p)$ be a Hamiltonian on $T^*T^n$. We show that the sequence $H_{k}(q,p)=H(kq,p)$ converges for the $gamma$ topology defined by the author, to $bar{H}(p)$. This is extended to the case where only some of the variables are homogenized, that is