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arXiv · 1910.08260

ECH capacities and the Ruelle invariant

Michael Hutchings
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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.

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Michael Hutchings. 2022-01-29. ECH capacities and the Ruelle invariant. https://arxiv.org/abs/1910.08260

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