Symplectic manifolds with contact type boundaries.

*(English)*Zbl 0719.53015A 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 |

##### Keywords:

symplectic manifold; contact type boundary; pseudoconvex boundary; connectedness; first Chern class; symplectic radius##### References:

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