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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)

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