Torsion contact forms in three dimensions have two or infinitely many Reeb orbits

Abstract

We prove that every nondegenerate contact form on a closed connected three-manifold, such that the associated contact structure has torsion first Chern class, has either two or infinitely many simple Reeb orbits. By previous results it follows that under the above assumptions, there are infinitely many simple Reeb orbits if the three-manifold is not the three-sphere or a lens space. We also show that for non-torsion contact structures, every nondegenerate contact form has at least four simple Reeb orbits.

arXiv comment42 pages, minor corrections and expository improvements

References in corpus (4)

• The Seiberg-Witten equations and the Weinstein conjecture
• The Weinstein conjecture for stable Hamiltonian structures
• Proof of the Arnold chord conjecture in three dimensions II
• Fast finite-energy planes in symplectizations and applications

Cited by in corpus (7)

• Contact three-manifolds with exactly two simple Reeb orbits
• Quantitative Heegaard Floer cohomology and the Calabi invariant
• Sub-leading asymptotics of ECH capacities
• Generalized common index jump theorem with applications to closed characteristics on star-shaped hypersurfaces and beyond
• Singular cotangent models and complexity in fluids with dissipation
• Proof of the $C^2$-stability conjecture for geodesic flows of closed surfaces
• Invariant Sets and Hyperbolic Closed Reeb Orbits