A conformal symplectic Weinstein conjecture

TL;DR

This paper generalizes the Weinstein conjecture to locally conformally symplectic manifolds, introduces Reeb 2-curves, and provides partial verifications and applications in Reeb dynamics using Gromov-Witten theory.

Contribution

It extends the Weinstein conjecture to new geometric settings and develops Gromov-Witten methods for studying Reeb dynamics in these manifolds.

Findings

The conjecture holds for all closed exact lcs surfaces.

Partial verifications of the conjecture in higher dimensions.

New conditions for contactomorphisms to fix Reeb orbits.

Abstract

We introduce a direct generalization of the Weinstein conjecture to closed, Lichnerowicz exact, locally conformally symplectic manifolds, (for short $\lcs$ manifolds). This conjectures existence of certain 2-curves in the manifold, which we call Reeb 2-curves. The conjecture readily holds for all closed exact lcs surfaces. In higher dimensions, we give partial verifications of this conjecture, based on certain extended ($\mathbb{Q} ^{} \sqcup \{\pm \infty\}$ valued) Gromov-Witten, elliptic curve counts in $\lcs$ manifolds. As a basic application we get some novel results in classical Reeb dynamics. The most basic such result gives sufficient conditions for a strict contactomorphism to fix the image of some closed Reeb orbit on a closed contact manifold. Along the way we give a Gromov-Witten theoretic construction of the classical dynamical Fuller index (for Reeb vector field), which among other things explains its rationality.