Sub-leading asymptotics of ECH capacities

Abstract

In previous work (Cristofaro-Gardiner et al. in Invent Math 199:187–214, 2015), the first author and collaborators showed that the leading asymptotics of the embedded contact homology spectrum recovers the contact volume. Our main theorem here is a new bound on the sub-leading asymptotics.

Remarks about the C∞-closing lemma for 3-dimensional Reeb flows

“…By setting e k (λ) := c σ k (λ) − 2k vol(Y, λ), the volume theorem is equivalent to e k = o(k 1/2 ). This estimate has been improved in recent papers [22], [7], and [13] (in chronological order). In particular, [13] proves e k = O(k 1/4 ) for contacttype hypersurfaces in C 2 ; actually, this is a special case of Theorem 1.1 in [13], which applies to any compact domain with C ∞ -boundary in C 2 .…”

Newton–Okounkov bodies and symplectic embeddings into nontoric rational surfaces

“…Most pertinent to this paper is the following volume property. There has been much study of the asymptotics of these error terms via symplectic methods [18,26,44] and algebraic methods [46,47].…”

“…We can capture the subleading asymptotic behavior of the ECH capacities by defining $$$$\begin{equation*} e_k(X,\omega ):=c^{\textnormal {ech}}_k(X,\omega )-\sqrt {4\operatorname{vol}(X,\omega )k}. \end{equation*}$$$$There has been much study of the asymptotics of these error terms via symplectic methods [18, 26, 44] and algebraic methods [46, 47].…”

Hyperbolicity, irrationality exponents and the eta invariant

“…It was originally motivated by the proof of the three dimensional Weinstein conjecture using Seiberg-Witten theory by Taubes [28]. The semiclassical limit formula for the eta invariant of [25] was recently used by the author in [9] to improve the remainder term in the asymptotics of embedded contact homology (ECH) capacities [8]. Our main theorem here as well as its corollaries could have further applications in this direction.…”