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

TL;DR

This paper constructs a specific small set in symplectic space that cannot be embedded into a standard symplectic cylinder, and establishes bounds on the regular coisotropic capacity related to displacement energy.

Contribution

It introduces a new example of a small set with non-squeezability properties and provides sharp bounds on the regular coisotropic capacity in symplectic geometry.

Findings

Existence of a compact set with Hausdorff dimension n that does not embed into the symplectic cylinder.

A sharp lower bound on the d-th regular coisotropic capacity up to a factor of 3.

Capacity bounds serve as lower bounds on displacement energy in certain symplectic manifolds.

Abstract

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.