Local exotic tori

Brendel, Joé

doi:10.1090/tran/9385

Local exotic tori
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by Joé Brendel;

Trans. Amer. Math. Soc.

DOI: https://doi.org/10.1090/tran/9385

Published electronically: January 30, 2025

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Abstract:

For a broad class of symplectic manifolds of dimension at least six, we find the following new phenomenon: there exist local exotic Lagrangian tori.

More specifically, let $X$ be a geometrically bounded symplectic manifold of dimension at least six. We show that every open subset of $X$ contains infinitely many Lagrangian tori which are distinct up to symplectomorphisms of $X$ while being Lagrangian isotopic and having the same classical invariants. The proof relies on a locality property of the displacement energy germ, which allows us to compute it in a Darboux chart.

Since these tori are not monotone, bubbling may occur and the count of Maslov index two $J$-holomorphic disks does not yield an invariant.

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