On symplectic capacities and their blind spots (Q6608224)

\textit{I. Ekeland} and \textit{H. Hofer} [Math. Z. 200, No. 3, 355--378 (1989; Zbl 0641.53035); Math. Z. 203, No. 4, 553--567 (1990; Zbl 0729.53039)] introduced the notion of a symplectic capacity and constructed a sequence of capacities, \(\{c^{\mathrm{EH}}_k\}_{k\in\mathbb{N}}\), for subsets of \((\mathbb{R}^{2n},\omega_0)\)) in terms of the closed orbits of autonomous Hamiltonian flows.\N\N\textit{J. Gutt} and \textit{M. Hutchings} [Algebr. Geom. Topol. 18, No. 6, 3537--3600 (2018; Zbl 1411.53062)] used \(S^1\)-equivariant Floer theory to construct another sequence of symplectic capacities, \(\{c_k^{\mathrm{GH}}\}_{k\in\mathbb{N}}\), for star-shaped domains in \((\mathbb{R}^{2n},\omega_0)\)) in terms of closed orbits of Hamiltonian flows.\N\NThe paper under review settles three basic questions concerning the above Gutt-Hutchings capacities.\N\NThe first result says that there is a smooth family \(V_\delta\) of toric star-shaped domains in \((\mathbb{R}^4,\omega_0)\) with smooth boundary such that \(\delta\mapsto c^{\mathrm{GH}}_k(V_\delta)\) is constant for all \(k\) and \(\operatorname{volume}(V_\delta) =V_0+\delta\). Therefore the capacities \(c_k^{\mathrm{CH}}\) cannot be used to detect the volume of domains with smooth boundaries.\N\NThe second result asserts that for every \(j\in\mathbb{N}\), there is a smooth family \(V^j_\delta\) of toric star-shaped domains in \(\mathbb{R}^4\) with smooth boundary such that \(\delta\mapsto c_k^{\mathrm{GH}}(V^j_\delta)\) is constant for all \(k\ne j\), and \(c_j^{\mathrm{GH}}(V_\delta^j)=c_j^{\mathrm{GH}}(V_0^j)+\delta\). This implies that the capacities \(c_k^{\mathrm{CH}}\) are not mutually independent.\N\NThe third result says that there is a smooth family \(V_\delta\) of toric star-shaped domains in \(\mathbb{R}^4\) with smooth boundary all of which have the same Gutt-Hutchings capacities and volume, but no two of which are symplectomorphic. This shows that the Gutt-Hutchings capacities together with the volume do not constitute a complete set of symplectic invariants for star-shaped domains with smooth boundary, and therefore negatively answers a version of the recognition question (Question 2 in Section 3.6 of [\textit{K. Cieliebak} et al., Math. Sci. Res. Inst. Publ. 54, 1--44 (2007; Zbl 1143.53341)]).\N\NThe key to prove these results is that in the presence of additional symmetry the authors give a significant simplification of the combinatorial formulas from [\textit{J. Gutt} and \textit{M. Hutchings}, Algebr. Geom. Topol. 18, No. 6, 3537--3600 (2018; Zbl 1411.53062)].\N\NSince the last paper appeared it was conjectured that the Ekeland-Hofer capacities and the Gutt-Hutchings capacties coincide.\N\NRecently, the exciting thing is made. \textit{S. Matijević} announced a proof that these two sequences of symplectic capacities agree for convex domains in [``Positive ($S^1$-equivariant) symplectic homology of convex domains, higher capacities, and Clarke's duality'', Preprint, \url{arXiv:2410.13673}], and \textit{J. Gutt} and \textit{V. G. B. Ramos} announced that these two sequences of symplectic capacities agree for on all star-shaped domains in [``The equivalence of Ekeland-Hofer and equivariant symplectic homology capacities'', Preprint, \url{arXiv:2412.09555}].\N\NTherefore three results in the paper under review also hold true for the Ekeland-Hofer capacities.