IAS/Princeton/Montreal/Paris/Tel-Aviv Symplectic Geometry Zoominar

Consider a symplectic surface in a three-dimensional contact manifold with boundary on Reeb orbits. We assume that the rotation numbers of the boundary Reeb orbits satisfy a certain inequality, and we also make a technical assumption that the Reeb vector field has a particular "nice" form near the boundary of the surface. We then show that there exist Reeb orbits which intersect the interior of the surface, with a lower bound on the frequency of these intersections in terms of the symplectic area of the surface and the contact volume of the three-manifold. No genericity of the contact form is assumed. The proof uses "elementary" spectral invariants of contact three-manifolds. An application of this result gives a very general relation between mean action and the Calabi invariant for area-preserving surface diffeomorphisms.