Boundary rigidity for Lagrangian submanifolds, non-removable   intersections, and Aubry-Mather theory
  We consider Lagrangian submanifolds lying on a fiberwise strictly convex hypersurface in some cotangent bundle or, respectively, in the domain bounded by such a hypersurface.   We establish a new boundary rigidity phenomenon, saying that certain Lagrangians on the hypersurface cannot be deformed (via Lagrangians having the same Liouville class) into the interior domain.   Moreover, we study the "non-removable intersection set" between the Lagrangian and the hypersurface, and show that it contains a set with specific dynamical behavior, known as Aubry set in Aubry-Mather theory. 
Gabriel P. Paternain, Leonid Polterovich, Karl Friedrich Siburg