2021/02/11 by Yasha Savelyev, Savelyev, Yasha
Mathematics · #Differential Geometry (math.DG) #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Geometry and complex manifolds #Symplectic Geometry (math.SG)
paper · pdf · doi:10.48550/arxiv.2102.05820
openalex publication_date 2021/02/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 (ℚ \sqcup \± ∞\ 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.