Reeb dynamics detects odd balls (Q2807115)

The authors of the paper under review consider compact, connected \((2n + 1)\)-dimensional manifold \((M, \alpha)\) with the property that the boundary of \(M\) is diffeomorphic to the \(2n\)-dimensional sphere and they provide a criterion for \(M\) to be diffeomorphic to a ball. More precisely, they prove that if the infimum of all positive periods of contractible closed orbits of the Reeb vector field associated to \(\alpha\) is greater or equal than \(\pi\), and \(\partial M\) diffeomorphic to \(S^{2n}\) admits a contact embedding into the contact cylinder NEWLINE\[NEWLINE\left(\mathbb R\times D\times \mathbb C^{n-1}, db+\frac{1}{2}(x_0dy_0-y_0dx_0)-\sum_{j=1}^{n-1}y_jdx_j\right),NEWLINE\]NEWLINE where \(D\subset \mathbb R^2\) is the closed unit disc, then \(M\) is diffeomorphic to a ball.NEWLINENEWLINENote that in dimension three this result has already been proven in [\textit{Y. Eliashberg} and \textit{H. Hofer}, Differ. Integral Equ. 7, No. 5--6, 1303--1324 (1994; Zbl 0803.58045)]. In the same paper, Eliashberg and Hofer have also announced a higher-dimensional version of it under the assumption that \(H_2(M,\mathbb R)\) vanishes, but a proof has never been published.NEWLINENEWLINEIn addition, observe that the bound \(\pi\) in the result of the paper under review is optimal, and that in dimension three the main result can be strengthened, i.e., given that the infimum of all positive periods of contractible closed orbits of the Reeb vector field associated to \(\alpha\) is greater of equal than \(\pi\), one gets that there are no closed Reeb orbits at all (this formulation appears in [loc. cit.]).