ECH capacities and the Ruelle invariant

TL;DR

This paper investigates the asymptotic behavior of ECH capacities in symplectic geometry, proposing a conjecture involving the Ruelle invariant and proving it for toric domains, with implications for symplectic embedding obstructions.

Contribution

It introduces a conjecture linking ECH capacity asymptotics to the Ruelle invariant and proves it for a broad class of toric domains, advancing understanding of symplectic embedding obstructions.

Findings

Proved the conjecture for a large class of toric domains.

Established a bound on the error term with an improved exponent from 2/5 to 1/4.

Derived a general obstruction to symplectic embeddings of open toric domains with equal volume.

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 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 R^4, we bound the error term with an improvement on the previously known exponent from 2/5 to 1/4.