Journal of Symplectic Geometry

Volume 11 (2013)

Number 4

A symplectically non-squeezable small set and the regular coisotropic capacity

Pages: 509 – 523



Jan Swoboda (Max–Planck–Institut für Mathematik, Bonn, Germany)

Fabian Ziltener (Korea Institute for Advanced Study, Seoul, Korea)


We prove that for $n\geq2$ there exists a compact subset $X$ of the closed ball in $\mathbb{R}^{2n}$ 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 Audin and Polterovich.

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