On the existence of symplectic barriers (Q6574268)

In the famous paper [Invent. Math. 82, 307--347 (1985; Zbl 0592.53025)] \textit{M. Gromov} gave his celebrated non-squeezing theorem which states that the \(2n\)-dimensional Euclidean (open) ball \(B^{2n}(r)\) with radius \(r\) can be symplectically embedded into the (open) cylinder \(B^2(1)\times\mathbb{R}^{2n-2}\) if and only if \(r\le 1\).\N\NMotivated by this rigidity result, \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 symplectic capacities to characterize the rigidity problem of symplectic embeddings, and by constructing a sequence of symplectic capacities using Hamiltonian dynamics they reproved the aforementioned theorem by M. Gromov [loc. cit.].\N\NOn the other hand, for a given compact subset \(Z\) in \(\mathbb {R}^{2 n}\) and for every \(\epsilon > 0\), \textit{A. B. Katok} [Math. USSR, Izv. 7, 535--571 (1974; Zbl 0316.58010); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 37, 539--576 (1973)] showed that there exists a Hamiltonian diffeomorphism \(\phi\) on \(\mathbb {R}^{2 n}\) with respect to the standard symplectic structure such that \(\mathrm{Vol} (\phi (X) \backslash (B^2(1)\times\mathbb{R}^{2n-2})) \leq \epsilon\).\N\NIn view of the above results, a natural question is: How much does one need to remove from the ball \(B^{2 n} (r)\) so that it symplectically embeds into \(B^2(1)\times\mathbb{R}^{2n-2}\)?\N\NIn a recent paper, \textit{K. Sackel} et al. [Geom. Topol. 28, No. 3, 1113--1152 (2024; Zbl 07851049)] proved that:\N\N(1) For a Lagrangian plane \(L\subset\mathbb{R}^4\) there exists a symplectic embedding from \(B^4(\sqrt{2})\setminus L\) into \(B^2(1)\times\mathbb{R}^2\); \N\N(2) For \(r>1\) the lower Minkowski dimension of a closed subset \(E\subset\mathbb{R}^4\) is at least \(2\) if \(B^4(r)\setminus E\) can be embeds symplectically into \(B^2(1)\times\mathbb{R}^2\).\N\NThe authors establish the existence of a new type of rigidity of symplectic embeddings related to the aforementioned question. Namely, for every \(\delta > 0\) and dimension \(2n > 2\) they prove that there exists a finite union of codimension two pairwise disjoint symplectic hyperplanes \(\Sigma\) such that for every symplectic capacity one has \(c\left(B^{2 n} (1) \backslash \Sigma \right) < \pi \delta^{2}\). This result shows the existence of so-called symplectic barriers. The latter is a notion analogous to that of a Lagrangian barrier due to \textit{P. Biran} [Geom. Funct. Anal. 11, No. 3, 407--464 (2001; Zbl 1025.57032)].