Boundary rigidity for Lagrangian submanifolds, non-removable intersections, and Aubry-Mather theory
By Gabriel P. Paternain, L. Polterovich, K. Siburg. 2002
71 citations recorded by the source corpus.
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.
View Paper: https://www.semanticscholar.org/paper/24f86a9738d090033f4fa715c25426e65ee7b4a9
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Citations in the corpus
- Partial quasi-morphisms and quasi-states on cotangent bundles, and symplectic homogenization (2011, methodology): “onjugation invariance. The reader can find the proof of the symplectic invariance of the alpha function in [Ber] and the references therein, in the case of Tonelli Hamiltonians. It is also implicit in [PPS], in case His autonomous. The advantage of our approach is that this invariance follows formall”
- On the Homogenization of the Hamilton-Jacobi Equation (2019, background): “4) It was proved in [29] that H(c) represents the infimum of the energy values k’s such that the energy sublevel {H(x, p) ≤ k} contains in its interior a smooth Lagrangian graph of cohomology class c.”