From one Reeb orbit to two

Details From one Reeb orbit to two From one Reeb orbit to two

2012-02-22

arxiv.org/abs/1202.4839v2

Mathematics

Symplectic Geometry

13 pages

Scientific paper

We show that every (possibly degenerate) contact form on the three-sphere giving the tight contact structure has at least two embedded Reeb orbits. The same holds for any closed contact three-manifold satisfying a weak version of the "volume conjecture" in embedded contact homology. More generally, the weak volume conjecture implies that if there are only finitely many embedded Reeb orbits, then their symplectic actions are not all integer multiples of a single real number. The volume conjecture itself, which is expected to hold for every closed contact three-manifold, implies that either there are at least three embedded Reeb orbits, or there are two embedded Reeb orbits with an explicit upper bound on the product of their symplectic actions.

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