Non-squeezing property of contact balls

TL;DR

This paper proves a contact non-squeezing theorem in prequantization spaces, confirming a conjecture by Eliashberg, Kim, and Polterovich using microlocal category methods.

Contribution

It establishes the non-squeezing property for contact balls in prequantization spaces, solving a long-standing conjecture with novel microlocal techniques.

Findings

Contact non-squeezing holds for certain radii in prequantization spaces.

Microlocal category methods effectively prove the non-squeezing conjecture.

The result confirms the rigidity of contact embeddings in this setting.

Abstract

In this paper we solve a contact non-squeezing conjecture proposed by Eliashberg, Kim and Polterovich. Let $B_R$ be the open ball of radius $R$ in $\mathbf{R}^{2n}$ and let $\mathbf{R}^{2n}\times\mathbf{S}^1$ be the prequantization space equipped with the standard contact structure. Following Tamarkin's idea, we apply microlocal category methods to prove that if $R$ and $r$ satisfy $1\leq\pi r^2<\pi R^2$, then it is impossible to squeeze the contact ball $B_R\times\mathbf{S}^1$ into $B_r\times\mathbf{S}^1$ via compactly supported contact isotopies.