From one Reeb orbit to two

2012/02/22 by Daniel Cristofaro-Gardiner, Cristofaro-Gardiner, Daniel, Michael Hutchings +1 · 4 citations
Mathematics · #FOS: Mathematics #Symplectic Geometry (math.SG) #math.SG

paper · pdf · doi:10.48550/arxiv.1202.4839

13 pages; minor corrections, updated references

arxiv created 2014/01/05 · arxiv updated 2014/01/07

Abstract

We show that every (possibly degenerate) contact form on a closed three-manifold has at least two embedded Reeb orbits. We also show that if there are only finitely many embedded Reeb orbits, then their symplectic actions are not all integer multiples of a single real number; and if there are exactly two embedded Reeb orbits, then the product of their symplectic actions is less than or equal to the contact volume of the manifold. The proofs use a relation between the contact volume and the asymptotics of the amount of symplectic action needed to represent certain classes in embedded contact homology, recently proved by the authors and V. Ramos.

Cited by

• Strong closing lemmas in Hamiltonian dynamics
• Symplectic Weyl Laws
• Two irrationally elliptic closed orbits of Reeb flows on the boundary of star-shaped domain in ℝ²ⁿ
• On the mean indices of closed characteristics on dynamically convex star-shaped hypersurfaces in ℝ²ⁿ

Related

• The asymptotics of ECH capacities
• Perfect Reeb flows and action-index relations
• Multiplicity of closed Reeb orbits on contact manifolds with periodic equivariant symplectic homology
• Reeb orbits frequently intersecting a symplectic surface
• A note on Reeb dynamics on the tight 3-sphere