text: 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.
  @cite_19: Let M be a closed three-dimensional manifold with contact form λ so that Ker λ is tight. In this paper, we will present a first application of the filling method by pseudoholomorphic curves recently developed by the author. We will show that Legendrian knots L ⊂ M satisfying suitable assumptions admit a Reeb Chord, that is, there is a trajectory x(t) of the Reeb vector field and T > 0 such that x(0),x(T) ∈ L and x(0) ≠ x(T).
  @cite_5: 
  @cite_17: 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.
summary: C. Abbas @cite_17 @cite_5 @cite_19 proved the strict chord property for certain Legendrian knots in tight closed contact 3-manifolds.