Recognition through and representability of symplectic capacities and squeezing of small sets

Abstract

This thesis is about recognition of objects through symplectic capacities, representability of symplectic capacities as target-embedding capacities, and arbitrary symplectic squeezing of small subsets of $mathbb{R}^{2n}$. More precisely, the following are the main results of this thesis. • We prove that the generalized capacities recognize objects in symplectic categories whose objects are symplectic manifolds $(M,omega)$ such that $M$ is compact and 1-connected, $omega$ is exact, and there exists a boundary component of $M$ with negative helicity. The set of generalized capacities is thus a complete invariant for such categories. This answers a question by K. Cieliebak, H. Hofer, J. Latschev, and F. Schlenk. • We give the first concrete examples of symplectic capacities that are not target-representable. On the other hand, we show that the restriction of any capacity on the category of compact connected exact symplectic manifolds to the objects at which it is continuous is (disconnectedly) target-representable. In particular, continuous capacities on this category are target-representable. We also show that the volume capacity is connectedly target-representable on the category of compact exact symplectic manifolds. These results provide some answers to another question of K. Cieliebak, H. Hofer, J. Latschev, and F. Schlenk. • We show that every countably $m$-rectifiable subset of $mathbb{R}^{2n}$ can be displaced from every $(2n − m)$-Hausdorff negligible subset of $mathbb{R}^{2n}$ by a Hamiltonian diffeomorphism that is arbitrarily $C^{infty}$-close to the identity. Using this result, we prove that every countably $n$-rectifiable and $n$-Hausdorff negligible subset of $mathbb{R}^{2n}$ is arbitrarily symplectically squeezable. Both results are sharp with respect to the parameter $s$ in the $s$-Hausdorff negligibility assumption.