Interplay between Quantitative Aspects of Locally Conformally Symplectic Geometry and Contact Dynamics
Pacôme Van Overschelde
TL;DR
This paper develops a quantitative framework for exact locally conformally symplectic (LCS) manifolds by introducing elasticity $E(\lambda,\eta)$, the set of Lee-form scalings that preserve nondegeneracy, and shows how elasticity controls whether an exact LCS structure is of the first kind. It proves that exact LCS manifolds of rank $1$ with certain nondegeneracy and completeness hypotheses are isomorphic to LCS mapping tori, and in the compact case this is equivalent to a bounded elasticity complement. It further links the admissible values for LCS mapping tori to the Birkhoff averages of the conformal factor associated with a contactomorphism, providing explicit limits for $\mathcal{A}_{(\alpha,\psi)}$ in terms of $A_n(h)$. Consequently, the work yields lower bounds on maxima and upper bounds on minima for the conformal factors of contactomorphisms on closed manifolds, bridging LCS geometry with contact dynamics and mapping-torus constructions as a classification tool.
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.