Higher symplectic capacities and the stabilized embedding problem for integral ellipsoids (Q6601738)

Let \(X_{1}\)\ and \(X_{2}\)\ be four-dimensional symplectic manifolds. There has recently been considerable interest in understanding the stabilized symplectic embedding problem, namely the question of whether or not there exists a symplectic embedding\N\[\NX_{1}\times\mathbb{C}^{N}\overset{s}{\hookrightarrow}X_{2}\times\mathbb{C}^{N}\N\]\Nfor some \(N\). The embedding problem is already quite subtle even when \(X_{1}\)\ and \(X_{2}\)\ are simple shapes like elliipsoids \(E\left( a,b\right) \), balls \(B\left( c\right) :=E\left( c,c\right) \), polydiscs \(P\left( a,b\right) \), and cubes \(C\left( c\right) :=P\left( c,c\right) \). In contrast, the stabilized polydisc-into-ball problem is completely solved [\textit{K. Siegel}, Int. Math. Res. Not. 2022, No. 16, 12402--12461 (2022; Zbl 07573380), Theorem 1.3.5] in terms of a piecewise linear function with two pieces.\N\NSymplectic embedding problems are profitably investigated by symplectic capacities [\textit{K. Cieliebak} et al., Math. Sci. Res. Inst. Publ. 54, 1--44 (2007; Zbl 1143.53341)]. The third named author [Int. Math. Res. Not. 2022, No. 16, 12402--12461 (2022; Zbl 07573380)] has been developing a theory of higher symplectic capacities, which are invariant under taking products, and so are well suited for studying the stabilized embedding problem. \N\NThis paper aims to apply this theory, assuming the expected properties, to settle the stablized embedding problem for integral ellipsoids, when the eccentricity of the domain has the opposite parity of the eccentricity of the target and the target is not a ball. For the other parity, the embedding constructed here is definitely not always optimal, while in the ball case the authors' methods recover previous results in [\textit{D. McDuff}, J. Differ. Geom. 88, No. 3, 519--532 (2011; Zbl 1239.53109); \textit{R. Hind} and \textit{E. Kerman}, Invent. Math. 196, No. 2, 383--452 (2014; Zbl 1296.53160); Invent. Math. 214, No. 2, 1023--1029 (2018; Zbl 1407.53094)]. There is a similar story, without any condition on the eccentricity of the target, when the target is a polydisc, a special case of which implies a conjecture concerning the rescaled polydisc limit function in [\textit{D. Cristofaro-Gardiner} et al., Algebr. Geom. Topol. 17, No. 2, 1189--1260 (2017; Zbl 1362.53085)].\N\NFor the entire collection see [Zbl 1515.53004].