On the capacity of Lagrangians in the cotangent disc bundle of the torus
Résumé
This paper was withdrawn due to a critical error. For recent results in this direction, see Shelukhin's papers on the subject https://arxiv.org/abs/1904.06798 , https://arxiv.org/abs/1811.05552 and BIran-Cornea https://arxiv.org/abs/2008.04756
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Former abstract: We prove that a Lagrangian torus in $T^*T^n$ Hamiltonianly isotopic to the zero section and contained in the unit disc bundle has bounded $\gamma$-capacity, where $\gamma(L)$ is the norm on Lagrangian submanifold. On one hand this gives new obstructions to Lagrangian embeddings of a quantitative kind. On the other hand, it gives a certain control on the $\gamma$ topology in terms f the Hausdorff topology. Finally this result is a crucial ingredient in establishing symplectic homogenization theory.
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