Boundary rigidity for Lagrangian submanifolds, non-removable intersections, and Aubry-Mather theory

Abstract

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.

The main new point of this revised and substantially enlarged version, with G.P. Paternain as new co-author, is the relation between non-removable intersections and Aubry-Mather theory