Dan Cristofaro-Gardiner, Vinicius Gripp, and myself just posted a new preprint, “The asymptotics of ECH capacities”. This paper proves three conjectures that were made in my paper “Quantitative embedded contact homology”.
The three conjectures are as follows. (Here I am referring to the published version of “Quantitative ECH”; the arXiv version contains an additional conjecture, Conjecture 8.5 there, which is false, as explained near the end of this post, and also in Remark 8.5 in the published version.) The first conjecture, Conjecture 1.12, asserts that for a Liouville domain with all ECH capacities finite, the asymptotics of the ECH capacities recover the symplectic volume. The second, stronger conjecture, Conjecture 8.1, asserts that for a contact three-manifold with finite ECH spectrum, the asymptotics of the ECH spectrum recover the contact volume. The third, strongest conjecture, Conjecture 8.7, asserts that for any contact three-manifold, the contact volume is recovered by the asymptotics of the amount of symplectic action needed to represent classes in ECH corresponding to a torsion spin-c structure. Also, Dan and I showed previously that Conjecture 8.7 implies that every (possibly degenerate) contact form on a closed three-manifold has at least two embedded Reeb orbits, see this post.
The proof of the volume conjecture has two parts: an upper bound on the asymptotics, and a lower bound on the asymptotics. The two proofs are independent and use completely different methods, but by some miracle, the upper and lower bounds agree! I can’t say that I really grasp what is going on here. The proof of the lower bound uses properties of ECH cobordism maps and seems fairly straightforward. The proof of the upper bound, on the other hand, uses Seiberg-Witten theory, in particular ingredients from Taubes’s proof of the Weinstein conjecture, and is very tricky. My coauthors understand this better than I do. (The upper bound is all that is needed to prove the existence of two Reeb orbits.)
It remains unknown, and one of my favorite questions, whether there is an upper bound on the length of the shortest Reeb orbit in terms of the contact volume and the contact structure; see the discussion in this post.