Elliptic curves in lcs manifolds and metric invariants

We study invariants defined by count of charged, elliptic $J$-holomorphic curves in locally conformally symplectic manifolds. We use this to define $\mathbb{Q}$-valued deformation invariants of certain complete Riemann-Finlser manifolds and their isometries and this is used to find some new phenomena in Riemann-Finlser geometry. In contact geometry this Gromov-Witten theory is used to study fixed Reeb strings of strict contactomorphisms. Along the way, we state an analogue of the Weinstein conjecture in lcs geometry, directly extending the Weinstein conjecture, and discuss various partial verifications. A counterexample for a stronger, also natural form of this conjecture is given.