A Symplectically Non-Squeezable Small Set and the Regular Coisotropic Capacity

Details A Symplectically Non-Squeezable Small Set and the Regular Coisotropic Capacity A Symplectically Non-Squeezable Small Set and the Regular Coisotropic Capacity

2012-03-12

arxiv.org/abs/1203.2395v3

Mathematics

Symplectic Geometry

16 pages, v2: added references to articles by H. Geiges and K. Zehmisch, v3: added "for $n\geq2$" in the abstract

Scientific paper

We prove that for $n\geq2$ there exists a compact subset $X$ of the closed ball in $R^{2n}$ of radius $\sqrt{2}$, such that $X$ has Hausdorff dimension $n$ and does not symplectically embed into the standard open symplectic cylinder. The second main result is a lower bound on the $d$-th regular coisotropic capacity, which is sharp up to a factor of 3. For an open subset of a geometrically bounded, aspherical symplectic manifold, this capacity is a lower bound on its displacement energy. The proofs of the results involve a certain Lagrangian submanifold of linear space, which was considered by M. Audin and L. Polterovich.

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