## From one Reeb orbit to two

The paper by Dan Cristofaro-Gariner and myself, “From one Reeb orbit to two”, is now available on the ArXiv.  This paper proves that any 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” relating asymptotics of ECH capacities to volume (and this conjecture is expected to hold for every closed contact three-manifold).  This result may seem rather modest, since it was previously shown (by myself and Taubes) that any nondegenerate contact form on a closed three-manifold has at least two embedded Reeb orbits.  Also there are lots of recent papers proving infinitely many Reeb orbits under various circumstances.  However our understanding is that the result on the three-sphere was not previously known.  (Now that the paper is out there, maybe someone can correct us if we are wrong about this.)

The idea of the proof is to show that if there is only one embedded Reeb orbit, then certain ECH capacities grow at least linearly.  However the Volume Conjecture in one of its forms asserts that the $k^{th}$ of the above ECH capacities is asymptotically $\sqrt{k}$ times a constant determined by the volume of the contact manifold, which is a contradiction.

By comparison, the proof in the nondegenerate case showed that if there is only one embedded Reeb orbit, then the grading of the ECH generators grows quadratically, with the result that the ECH is “too small” to agree with Seiberg-Witten Floer cohomology.

Our proof of the existence of least two embedded Reeb orbits for the tight contact structure on $S^3$ in the degenerate case also uses Seiberg-Witten theory, but only really in one place, namely where we invoke the existence of ECH cobordism maps (proved in my most recent paper with Taubes using Seiberg-Witten theory) on cobordisms of the form $[0,1]\times S^3$.

We also prove that the Volume Conjecture implies that either there are at least three embedded Reeb orbits, or there are two embedded Reeb orbits such that the product of their symplectic actions is at least the volume of the contact three-manifold.  Note that in previous posts I suggested two conjectures (I may not have spelled them out exactly like this): (1) Any contact form on a closed three-manifold which is not a lens space has at least three embedded Reeb orbits, and (2) Any contact form on a closed three-manifold has a Reeb orbit with an upper bound on its symplectic action in terms of the volume of the contact manifold.  So the Volume Conjecture implies that every contact form on a closed three-manifold satisfies Conjecture (1) or (2).  How many conjecture-proving credits is that worth?