Non-squeezing and other global rigidity results in locally conformal symplectic geometry

Abstract

Using generating functions quadratic at infinity for Lagrangian submanifolds of twisted cotangent bundles, we define spectral selectors for compactly supported lcs Hamiltonian diffeomorphisms of the locally conformal symplectizations $S^1 \times \mathbb{R}^{2n+1}$ and $S^1 \times \mathbb{R}^{2n} \times S^1$ of $\mathbb{R}^{2n+1}$ and $\mathbb{R}^{2n} \times S^1$, and obtain several applications: the construction of a bi-invariant partial order on the group of compactly supported lcs Hamiltonian diffeomorphisms of $S^1 \times \mathbb{R}^{2n+1}$ and $S^1 \times \mathbb{R}^{2n} \times S^1$, of an integer-valued bi-invariant metric on the group of compactly supported lcs Hamiltonian diffeomorphisms of $S^1 \times \mathbb{R}^{2n} \times S^1$, and of an integer-valued lcs capacity for domains of $S^1 \times \mathbb{R}^{2n} \times S^1$. The lcs capacity is used to prove a lcs non-squeezing theorem in $S^1 \times \mathbb{R}^{2n} \times S^1$ analogous to the contact non-squeezing theorem in $\mathbb{R}^{2n} \times S^1$ discovered in 2006 by Eliashberg, Kim and Polterovich. Along the way we introduce for Liouville lcs manifolds the notions of essential Lee chords between exact Lagrangian submanifolds and of essential translated points of exact lcs diffeomorphisms. We prove that essential translated points always exist for time-$1$ maps of sufficiently $C^0$-small lcs Hamiltonian isotopies of compact Liouville lcs manifolds and for all compactly supported lcs Hamiltonian diffeomorphisms of $S^1 \times \mathbb{R}^{2n+1}$ and $S^1 \times \mathbb{R}^{2n} \times S^1$. We also obtain an existence result for essential Lee chords between the zero section of a twisted cotangent bundle with compact base and its image by any lcs Hamiltonian isotopy, which can be thought of as a lcs analogue of the Lagrangian and Legendrian Arnold conjectures on usual cotangent and $1$-jet bundles.