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Symplectic manifolds with contact type boundaries. (English) Zbl 0719.53015
A symplectic manifold (Z,$$\omega$$) is said to have a contact type boundary $$\partial Z$$ if the restriction $$\sigma$$ of $$\omega$$ to $$\partial Z$$ equals $$d\alpha$$ for a certain contact 1-form $$\alpha$$. Then, if J is an $$\omega$$-compatible almost complex structure such that no J-holomorphic curve can touch the boundary from inside, $$\partial Z$$ is said to be J-convex, and this is analogous to the case of a complex manifold with pseudoconvex boundary. The author proves a number of results which include: a) there exists a compact 4-dimensional symplectic manifold with a disconnected boundary of contact type; b) conditions that ensure the connectedness of a contact type boundary; c) if $$\omega$$ vanishes on all spherical elements in $$H_ 2(Z;{\mathbb{R}})$$ and Z is contactomorphic to $$S^{2n-1}$$ with its standard contact structure, Z is diffeomorphic to the ball $$B^{2n}$$; d) under a certain positivity condition for the first Chern class $$c_ 1(Z)$$, if a neighbourhood of $$\partial Z$$ is symplectomorphic to a neighbourhood of the boundary of the ball of radius k in $${\mathbb{R}}^{2n}$$, then the symplectic radius of Z is $$\leq k$$.
Reviewer: I.Vaisman (Haifa)

MSC:
 53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.) 37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
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References:
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