Non-squeezing and other global rigidity results in locally conformal symplectic geometry
Mélanie Bertelson, Pranav Chakravarthy, Sheila Sandon
TL;DR
This work develops a comprehensive lcs rigidity framework by translating generating function methods from symplectic and contact topology into locally conformal symplectic geometry on $S^1\times\mathbb{R}^{2n+1}$ and $S^1\times\mathbb{R}^{2n}\times S^1$. Central tools are generating functions quadratic at infinity, which yield spectral selectors $c_{+}$ and $c_{-}$ for compactly supported lcs diffeomorphisms and enable the construction of a bi-invariant partial order and an integer-valued bi-invariant metric on Hamiltonian groups, as well as a robust lcs capacity. These lead to a locally conformal symplectic non-squeezing theorem for integer capacities and establish existence results for essential translated points and Lee chords, paralleling Arnold-type conjectures in the lcs setting. The untwisting and jet-embedding techniques connect lcs invariants to their symplectic/contact counterparts, allowing strong orderability results for key lcs manifolds and a Rokhlin-property obstruction for the Hamiltonian group of $S^1\times\mathbb{R}^{2n}\times S^1$. Overall, the paper extends rigidity phenomena from symplectic and contact topology to a broad lcs context with explicit quantitative invariants and constructions.
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.