Interplay between Quantitative Aspects of Locally Conformally Symplectic Geometry and Contact Dynamics

TL;DR

This paper explores the quantitative properties of locally conformally symplectic (LCS) manifolds, characterizing their elasticity and establishing a link with contact dynamics, including bounds on conformal factors and a classification of LCS mapping tori.

Contribution

It introduces the notion of elasticity for exact LCS pairs, characterizes LCS manifolds of the first kind, and links LCS mapping tori with contact dynamics through elasticity bounds.

Findings

Characterization of LCS manifolds of the first kind.

LCS mapping tori over closed contact manifolds are characterized by bounded elasticity.

Bounds on the conformal factors of contactomorphisms are established.

Abstract

We investigate quantitative properties of exact locally conformally symplectic (LCS) manifolds, namely the homotheties of the Lee form that still produce an exact LCS form. This gives the notion of elasticity of an exact LCS pair. Using this, we characterize LCS manifolds of the first kind. We then generalize a result of Bazzoni and Marrero on the latter, by showing that an exact LCS manifold of rank one admitting an exact LCS pair, whose complementary of elasticity is bounded, is isomorphic to an LCS mapping torus. Conversely, we show that any LCS mapping tori over a closed contact manifold satisfies this condition, thereby providing a characterization of LCS mapping tori over closed contact manifolds. In doing so, we establish a link between the limit values of the elasticity of an LCS mapping torus over a closed contact manifold and the Birkhoff average of the conformal factor of its contactomorphism. As a consequence, we obtain a lower bound on the maxima and an upper bound on the minima of the various conformal factors associated with a contactomorphism on a closed contact manifold.