Local Persistence of Lagrangian Intersections

Given a Lagrangian L, I will discuss the existence of a neighborhood W of L with the following property: for any Hamiltonian diffeomorphism f, if f(L) is contained inside W, then f(L) intersects L. On the one hand, for any symplectic manifold of dimension at least 6, I will construct Lagrangians which do not admit any such neighborhoods. On the other hand, I will give conditions which ensure the existence of such neighborhoods for a large class of Lagrangians. These conditions actually ensure the exactness of certain nearby Lagrangians, and I will discuss further applications of this phenomenon. This is based on joint work with Marcelo Atallah, Jean-Philippe Chassé, and Egor Shelukhin.