On the existence of symplectic barriers
type: Item
instance of: scholarly article
title: On the existence of symplectic barriers
author: Pazit Haim-Kislev
author: R. Hind
author: Yaron Ostrover
published in: Selecta Mathematica. New Series
publication date: 2024-07-18T00:00:00Z
review text: 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…
reviewed by: Guangcun Lu
Mathematics Subject Classification ID: 53D05
Mathematics Subject Classification ID: 53D35
Mathematics Subject Classification ID: 53C24
zbMATH DE Number: 7882866
zbMATH Keywords: symplectic embeddings
zbMATH Keywords: symplectic barriers
zbMATH Keywords: symplectic capacities
MaRDI profile type: Publication
cites work: Lagrangian barriers and symplectic embeddings
cites work: Rigidity and uniruling for Lagrangian submanifolds
cites work: Pinwheels as Lagrangian barriers
cites work: Symplectic topology and Hamiltonian dynamics
cites work: A theorem on the structure of wave fronts and its applications in symplectic topology
cites work: Pseudo holomorphic curves in symplectic manifolds
cites work: Symplectic capacities of domains in C2
cites work: Q3480936
cites work: Q4850937
cites work: ERGODIC PERTURBATIONS OF DEGENERATE INTEGRABLE HAMILTONIAN SYSTEMS
cites work: Asymptotic behavior of exotic Lagrangian tori \(T_{a,b,c}\) in \(\mathbb{C}P^2\) as \(a+b+c\to\infty\)
cites work: Fibrations in symplectic topology
cites work: Symplectic packings and algebraic geometry
cites work: Embedding problems in symplectic geometry
cites work: Symplectic embedding problems, old and new
cites work: Isotropic isotopy and symplectic null sets
cites work: Symplectic packing constructions
DOI: 10.1007/S00029-024-00958-Y
description: scientific article; zbMATH DE number 7882866