From one Reeb orbit to two. (English) Zbl 1338.53108

Cristofaro-Gardiner, Daniel; Hutchings, Michael

J. Differ. Geom. 102, No. 1, 25-36 (2016).

The main results of the paper, that have an implication for Hamiltonian dynamics, are the following two theorems:
Theorem 1. Every (possibly degenerate) contact form on a closed three-manifold has at least two embedded Reeb orbits.
Theorem 2. Let \((Y,\lambda)\) be a closed contact three-manifold having only finitely many embedded Reeb orbits \(\gamma_1,\gamma_2,\dots,\gamma_m\). Then their symplectic actions \[ A(j_1),A(j_2),\dots,A(j_m) \] are not all integer multiples of a single real number.

Reviewer: Mihail Banaru (Smolensk)

MSC:

53D10

Contact manifolds (general theory)

Keywords:

Reeb orbits; symplectic action; contact three-manifold; contact form