ECH capacities and the Ruelle invariant

Abstract

The ECH capacities are a sequence of real numbers associated to any symplectic four-manifold, which are monotone with respect to symplectic embeddings. It is known that for a compact star-shaped domain in \({{\mathbb {R}}}^4\), the ECH capacities asymptotically recover the volume of the domain. We conjecture, with a heuristic argument, that generically the error term in this asymptotic formula converges to a constant determined by a “Ruelle invariant” which measures the average rotation of the Reeb flow on the boundary. Our main result is a proof of this conjecture for a large class of toric domains. As a corollary, we obtain a general obstruction to symplectic embeddings of open toric domains with the same volume. For more general domains in \({{\mathbb {R}}}^4\), we bound the error term with an improvement on the previously known exponent from 2/5 to 1/4.