Nonsqueezing property of contact balls

Abstract

In this paper we solve a contact nonsqueezing conjecture proposed by Eliashberg, Kim, and Polterovich. Let $B_R$ be the open ball of radius $R$ in ${\mathbb{R}}^{2n}$, and let ${\mathbb{R}}^{2n}\times {\mathbb{S}}^1$ 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 $1\le \pi r^2<\pi R^2$, then it is impossible to squeeze the contact ball $B_R\times {\mathbb{S}}^1$ into $B_r\times {\mathbb{S}}^1$ via compactly supported contact isotopies.