Sorry this blog has been quiet for a while. There are a lot of things I would like to write about, but I have been trying to finish some papers.
In particular, Dan Cristofaro-Gardiner, Dan Pomerleano, and I recently completed the paper “Torsion contact forms in three dimensions have two or infinitely many Reeb orbits“. I think the paper pretty much speaks for itself, so it is not necessary to say a lot more in a blog post. But if anyone wants to post questions or comments about the paper, they can do so here.
The main theorem of the paper asserts that given a nondegenerate contact form on a closed connected three-manifold, if the associated contact structure has torsion first Chern class, then there are either two or infinitely many simple Reeb orbits. (An earlier theorem of myself and Taubes says that a nondegenerate contact form on a closed the three-manifold which is not or a lens space has at least three simple Reeb orbits. So a corollary is that a nondegenerate contact form, whose contact structure has torsion first Chern class, on a closed connected three-manifold which is not or a lens space has infinitely many simple Reeb orbits.) The strategy of the proof is to assume that there are only finitely many simple Reeb orbits, and then show that one of the holomorphic curves counted by the U-map on ECH projects to an embedded curve in the three-manifold which is a genus zero global surface of section for the Reeb flow. One can then conclude the proof using a theorem of Franks, as Hofer-Wysocki-Zehnder did in their work on two or infinitely many Reeb orbits in tight .
Obvious questions for future research:
- Remove the assumption that the contact structure has torsion first Chern class. We used this assumption to control the “ index” on ECH, which in turn bounds the genus of holomorphic curves and allows us to find the genus zero curve we want. To remove this assumption, while continuing using our proof strategy, one would need to do more research on the index to see if it can still provide a genus zero curve in the non-torsion case.
- Remove the assumption that the contact form is nondegenerate. This might require some new technology.
- What can one say about the case where there are exactly two simple Reeb orbits? The only examples I know of contact forms on closed connected three-manifolds with only finitely many Reeb orbits are the contact form on induced by an irrational ellipsoid, and quotients thereof on lens spaces (which are nondegenerate and have exactly two simple Reeb orbits). Are there any other examples?
- Higher dimensions??? For example, a famous conjecture asserts that any star-shaped hypersurface in has at least simple Reeb orbits. (There are some partial results on this.) One could also ask if there are either or infinitely many simple Reeb orbits. Technology other than ECH is needed, since ECH is only defined for three-dimensional contact manifolds.
That’s enough for now, but I plan to write some new posts soon even before finishing other papers.