×

zbMATH — the first resource for mathematics

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
PDF BibTeX XML Cite
Full Text: DOI EuDML
References:
[1] [E 1] Eliashberg, Ya.: On Symplectic Manifolds which are bounded by standard Contact Spheres and Exotic Contact Structures on Spheres of Dimension >3. J. Differ. Geom. (to appear)
[2] [E 2] Eliashberg, Ya.: Topological Characterization of Stein Manifolds of Dimension >2. Int. J. Math.1, 19-46 (1990) · Zbl 0699.58002 · doi:10.1142/S0129167X90000034
[3] [E 3] Eliashberg, Ya.: Filling by Holomorphic Discs and Its Applications. (preprint 1989)
[4] [EG] Eliashberg, Ya., Gromov, M. Convex Symplectic Manifolds, preprint (1990)
[5] [F] Floer, A.: Symplectic Fixed Points and Holomorphic Spheres. Commun. Math. Phys.120, 575-611 (1989) · Zbl 0755.58022 · doi:10.1007/BF01260388
[6] [G 1] Gromov, M.: Pseudo-holomorphic curves in symplectic manifolds. Invent. Math.82, 307-347 (1985) · Zbl 0592.53025 · doi:10.1007/BF01388806
[7] [G 2] Gromov, M.: Partial Differential Relations. Ergeb. Math. Vol. 3. Folgeband 9. Berlin Heidelberg New York, Springer 1986
[8] [H] Hofer, H. (private communication)
[9] [McD 1] McDuff, D.: Symplectic Diffeomorphisms and the Flux Homomorphism. Invent. Math. 77, 353-366 (1984) · Zbl 0546.58014 · doi:10.1007/BF01388450
[10] [McD 2] McDuff, D.: Examples of simply-connected non-K√§hlerian manifolds. J. Differ. Geom.20, 267-277 (1984) · Zbl 0567.53031
[11] [McD 3] McDuff, D.: Examples of symplectic structures. Invent. Math.89, 13-36 (1987) · Zbl 0625.53040 · doi:10.1007/BF01404672
[12] [McD 4] McDuff, D.: The Structure of Rational and Ruled symplectic 4-manifolds. J. Am. Math. Soc.3, 679-712 (1990) · Zbl 0723.53019
[13] [McD 5] McDuff, D.: Elliptic Methods in Symplectic Geometry. Bull. Am. Math. Soc.23, 311-358 (1990) · Zbl 0723.53018 · doi:10.1090/S0273-0979-1990-15928-2
[14] [McD 6] McDuff, D.: The Local Behaviour of Holomorphic Curves in Almost Complex 4-manifolds. J. Differ. Geom. (to appear) · Zbl 0736.53038
[15] [V] Viterbo, C. (private communication)
[16] [W] Wolfson, J.: Gromov’s compactness of pseudo-holomorphic curves and symplectic geometry. J. Differ. Geom.28, 383-405 (1988) · Zbl 0661.53024
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.