Instantaneous Hamiltonian displaceability and arbitrary symplectic squeezability for critically negligible sets

TL;DR

This paper demonstrates that certain negligible sets in symplectic geometry can be displaced or squeezed using Hamiltonian diffeomorphisms, with results sharp in the measure parameter, and introduces a folding method with potential broader applications.

Contribution

It establishes sharp conditions under which countably rectifiable negligible sets can be displaced or squeezed in symplectic space, introducing a folding technique for these results.

Findings

Countably $m$-rectifiable sets can be displaced from $(2n-m)$-negligible sets.

Countably $n$-rectifiable and $n$-negligible sets are arbitrarily symplectically squeezable.

The folding method may determine Gromov width for certain sets, indicating they are not barriers.

Abstract

We call a metric space $s$-negligible iff its $s$-dimensional Hausdorff measure vanishes. We show that every countably $m$-rectifiable subset of $\mathbb{R}^{2n}$ can be displaced from every $(2n-m)$-negligible subset by a Hamiltonian diffeomorphism that is arbitrarily $C^\infty$-close to the identity. As a consequence, every countably $n$-rectifiable and $n$-negligible subset of $\mathbb{R}^{2n}$ is arbitrarily symplectically squeezable. Both results are sharp w.r.t. the parameter $s$ in the $s$-negligibility assumption. The proof of our squeezing result uses folding. Potentially, our folding method can be modified to show that the Gromov width of $B^{2n}_1\setminus A$ equals $\pi$ for every countably $(n-1)$-rectifiable closed subset $A$ of the open unit ball $B^{2n}_1$. This means that $A$ is not a barrier.