nLab Gromov's non-squeezing theorem

Gromov’s non-squeezing theorem is an analogue of Heisenberg uncertainty relations in symplectic geometry proved in seminal work of Gromov,

• Mikhail Gromov, Pseudo holomorphic curves in symplectic manifolds, Invent. Math. 82 (1985) 307–347

Some generalizations and consequences are studied in terms of symplectic capacities. A survey:

• Maurice de Gosson, Franz Luef, Symplectic capacities and the geometry of uncertainty: The irruption of symplectic topology in classical and quantum mechanics, Physics Reports 484:5, (2009) 131–179 doi

• A. Sukhov, A. Tumanov, Gromov’s non-squeezing theorem and Beltrami type equation, Commun. Part. Diff. Eq. 39:10 (2014) doi

Non-squeezing theorems in contact geometry

• Sheng-Fu Chiu, Non-squeezing property of contact balls, doi arXiv:1405.1178

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

• Peter Albers, Will J. Merry, Orderability, contact non-squeezing, and Rabinowitz Floer homology, arxiv:1302.6576

category: geometry