Let @math be a contact form on a manifold @math , and @math a closed Legendrian submanifold. I prove that @math intersects some characteristic for @math at least twice if all characteristics are closed and of the same period, and @math embeds nicely into the product of @math and an exact symplectic manifold. As an application of the method of proof, the minimal action of a regular closed coisotropic submanifold of complex projective space is at most @math . This yields an obstruction to presymplectic embeddings, and in particular to Lagrangian embeddings.
In @cite_14 K. Cieliebak proved that Legendrian spheres in the boundaries of certain subcritical Weinstein domains intersect some characteristic for @math at least twice.
@cite_14
2027438934
Arnold conjectured that every Legendrian knot in the standard contact structure on the 3-sphere possesses a characteristic chord with respect to any contact form. I confirm this conjecture if the know has Thurston-Bennequin invariant −1. More generally, existence of chords is proved for a standard Legendrian unknot on the boundary of a subcritical Stein manifold of any dimension. There is also a multiplicity result which implies in some situations existence of infinitely many chords.¶The proof relies on the behaviour of symplectic homology under handle attaching. The main observation is that symplectic homology only changes in the presence of chords. Let (S, fλ) be the three sphere endowed with a contact form fλ, where λ is the standard tight contact form on S and f : S → R 0 is a smooth function. Moreover let L ⊂ S be a Legendrian knot. Then V.I. Arnold raised the conjecture in 1986 that there is a characteristic chord, i.e. an orbit x of the Reeb vectorfield that intersects the knot L at two different times. Characteristic chords are important in the theory of contact homology, see [6], [5]. Very little is known about their existence or non-existence . Using a deep result by H. Hofer, K. Wysocki and E. Zehnder about the existence of a global surface of section in strictly convex hypersurfaces in R, we observe that characteristic chords exist for certain Legendrian knots in strictly convex hypersurfaces in R. We study holomorphic spheres in certain symplectic cobordisms and derive information about periodic Reeb orbits in the concave end of these cobordisms from the non-compactness of the relevant moduli spaces. We use this to confirm the strong Weinstein conjecture (predicting the existence of null-homologous Reeb links) for various higher-dimensional contact manifolds, including contact type hypersurfaces in subcritical Stein manifolds and in some cotangent bundles. The quantitative character of this result leads to the definition of a symplectic capacity. We show that positive @math -equivariant symplectic homology is a contact invariant for a subclass of contact manifolds which are boundaries of Liouville domains. In nice cases, when the set of Conley-Zehnder indices of all good periodic Reeb orbits on the boundary of the Liouville domain is lacunary, the positive @math -equivariant symplectic homology can be computed; it is generated by those orbits. We prove a "Viterbo functoriality" property: when one Liouville domain is embedded into an other one, there is a morphism (reversing arrows) between their positive @math -equivariant symplectic homologies and morphisms compose nicely. These properties allow us to give a proof of Ustilovsky's result on the number of non isomorphic contact structures on the spheres @math . They also give a new proof of a Theorem by Ekeland and Lasry on the minimal number of periodic Reeb orbits on some hypersurfaces in @math . We extend this result to some hypersurfaces in some negative line bundles.