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_20: We use a neck stretching argument for holomorphic curves to produce symplectic disks of small area and Maslov class with boundary on Lagrangian submanifolds of nonpositive curvature. Applications include the proof of Audin's conjecture on the Maslov class of Lagrangian tori in linear symplectic space, the construction of a new symplectic capacity, obstructions to Lagrangian embeddings into uniruled symplectic manifolds, a quantitative version of Arnold's chord conjecture, and estimates on the size of Weinstein neighbourhoods. The main technical ingredient is transversality for the relevant moduli spaces of punctured holomorphic curves with tangency conditions.
summary: Let @math be a bounded star-shaped domain with smooth boundary and @math a closed Legendrian submanifold of nonpositive curvature. The last condition means that @math that admits a Riemannian metric of nonpositive sectional curvature. In the recent preprint @cite_20 K. Cieliebak and K. Mohnke proved that @math possesses a Reeb chord of length bounded above by the (toroidal) Lagrangian capacity of @math , see [Corollary 1.12] CM . Using [Corollary 1.3] CM , they deduced that @math admits a Reeb chord of length bounded above by @math , if @math , and @math , i.e., @math is the unit sphere.