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

Gabriel P. Paternain·Leonid Polterovich·Karl Friedrich Siburg
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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.

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Gabriel P. Paternain, Leonid Polterovich, Karl Friedrich Siburg. 2003-02-14. Boundary rigidity for Lagrangian submanifolds, non-removable intersections, and Aubry-Mather theory. https://arxiv.org/abs/math/0207140

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