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Is there any work on manifolds (perhaps contact) with symplectic boundary (not asking about the boundary of a symplectic manifold)?

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Perhaps it would be good to make your question more precise : are you asking about manifolds whose boudary can be endowed with a symplectic structure ? – Samuel Tinguely May 8 '12 at 14:54
One way to start thinking about this would be to understand what type of structure $M\times\mathbb{R}$ can have if $M$ is equipped with a symplectic form. If $\omega=-d\theta$ is exact then $dt+\theta$ is a contact form on $M\times\mathbb{R}$, however this exclude compact $M$'s. On the other side, the good notion of boundary of contact manifolds seems to be that of convex hypersurfaces, which outside a dividing set are exact symplectic. – Noz May 8 '12 at 18:14
For closed symplectic manifold, here some symplectic manifold arises as boundary of quasi-symplectic manifold which are manifold whith a closed $2$-form with one dimensionnal kernel (in the case of $M\times\mathbb{R}$ it is simply $\omega$. I don't know whether or not those type of manifold have been extensively studied. – Noz May 8 '12 at 18:15
Maybe one could make a definition like this. A filling of symplectic manifold $(M,\omega)$ is a $2n+1$ manifold with boundary $M$ and with stable Hamiltonian structure $(\Omega,\lambda)$ such that $\Omega\vert_M=\omega$. – Noz May 8 '12 at 18:27

There is a book by Guillemin, Gizburg and Karshon: Moment Maps, Cobordisms, and Hamiltonian Group Actions.

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For contact manifolds there is the notion of a convex hypersurface due to Giroux, ie, a hypersurface that is transverse to a contact vector field. A convex hypersurface is composed by two exact symplectic domains (not necessarily connected) glued together along the dividing set (the dividing set is a hypersurface in the hypersurface) that carries a natural contact structure (see also this question).

I don't know if this helps, since the boundary is not a closed symplectic manifold, but it is a very natural object to consider in contact topology.

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