Capping off open books and the Ozsváth-Szabó contact invariant

Abstract

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.