Contact non-squeezing at large scale via generating functions

TL;DR

This paper proves a contact non-squeezing theorem at large scales using generating functions, providing a classical Morse theory approach that extends previous results obtained via SFT and microlocal sheaves.

Contribution

It develops an equivariant generating function homology framework for contact domains, generalizing contact non-squeezing results with a finite-dimensional Morse theory approach.

Findings

Established contact non-squeezing using generating functions.

Extended the theory with an equivariant homology framework.

Connected translated chains of contactomorphisms to the construction.

Abstract

Using SFT techniques, Eliashberg, Kim and Polterovich (2006) proved that if $\pi R_2^2 \leq K \leq \pi R_1^2$ for some integer $K$ then there is no contact squeezing in $\mathbb{R}^{2n} \times S^1$ of the prequantization of the ball of radius $R_1$ into the prequantization of the ball of radius $R_2$. This result was extended to the case of balls of radius $R_1$ and $R_2$ with $1 \leq \pi R_2^2 \leq \pi R_1^2$ by Chiu (2017) and the first author (2016), using respectively microlocal sheaves and SFT. In the present article we recover this general contact non-squeezing theorem using generating functions, a classical method based on finite dimensional Morse theory. More precisely, we develop an equivariant version, with respect to a certain action of a finite cyclic group, of the generating function homology for domains of $\mathbb{R}^{2n} \times S^1$ defined by the second author (2011). A key role in the construction is played by translated chains of contactomorphisms, a generalization of translated points.