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.
As explained in @cite_20 after Corollary 1.13, it follows that there exists no exact Lagrangian embedding into @math of a closed manifold @math of nonpositive curvature. (Corollary is a stabilized version of this without the nonpositive curvature assumption.)
@cite_20
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Given a closed orientable Lagrangian surface L in a closed symplectic four-manifold X together with a relative homology class d in H_2 (X , L ; Z) with vanishing boundary in H_1 (L ; Z), we prove that the algebraic number of J-holomorphic discs with boundary on L, homologous to d and passing through the adequate number of points neither depends on the choice of the points nor on the generic choice of the almost-complex structure J. We furthermore get analogous open Gromov-Witten invariants by counting, for every non-negative integer k, unions of k discs instead of single discs.