An elementary alternative to ECH capacities

Hutchings, Michael

doi:10.1073/pnas.2203090119

Cited by 2 publications

(1 citation statement)

References 37 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Remark 1.2.7 (generalizations) The approach taken in this paper to define fz g k g naturally generalizes to define various other families of capacities, eg by replacing the local tangency constraint <-.k/ p> with k generic point constraints, and/or by allowing curves of higher genus. In this spirit, the very recent preprint of Hutchings [23] adapts our approach to define (without relying on Seiberg-Witten theory) a sequence of four-dimensional capacities, which agree in many cases with the ECH capacities.…”

Section: Introduction 1overviewmentioning

confidence: 99%

Symplectic capacities, unperturbed curves and convex toric domains

McDuff,

Siegel

2024

Geom. Topol.

View full text Add to dashboard Cite

Higher-dimensional symplectic embeddings remain rather poorly understood, but there has been considerable recent interest in so-called "stabilized symplectic embedding problems", in which one studies symplectic embeddings of the form X C N s ,! X 0 C N for four-dimensional Liouville domains X and X 0 , and for N 2 Z 1 ; see for instance Hind and Kerman [17], Cristofaro-Gardiner and Hind [9], Cristofaro-Gardiner, Hind and McDuff [10], McDuff [29], Siegel [37; 38] and Irvine [26]. In order to systematize and generalize these results, the second author introduced in [37] a sequence of symplectic capacities g 1 ; g 2 ; g 3 ; : : : which are "stable" in the sense that g k .X C N / D g k .X / for any Liouville domain X and k; N 2 Z 1 . These capacities are defined using symplectic field theory (SFT), more specifically the (chain level) filtered ᏸ 1 structure on linearized contact homology, and their definition also involves curves satisfying local tangency constraints. As a proof of concept, [37] shows that these capacities perform quite well in toy problems, for instance they recover the sharp obstructions from [29] and they often outperform the Ekeland-Hofer capacities. In fact, the capacities g 1 ; g 2 ; g 3 ; : : : are a specialization of a more general family of capacities fg b g which are expected to give sharp obstructions to the stabilized ellipsoid embedding problem.However, two broad questions naturally become apparent:

show abstract

Section: Introduction 1overviewmentioning

confidence: 99%