Polyfolds, cobordisms, and the strong Weinstein conjecture (Q340444)

Given a closed (co-orientable) contact manifold \((M,\xi)\) and a defining contact form \(\alpha\) (i.e., \(\xi=\text{ker\,}\alpha\)), Weinstein conjectured that the Reeb vector field \(R\), which is uniquely defined by \(i_Rd\alpha=0\) and \(\alpha(R)=1\), admits a periodic solution. Here, a good reference is the article [\textit{F. Pasquotto}, Jahresber. Dtsch. Math.-Ver. 114, No. 3, 119--130 (2012; Zbl 1263.37003)]. A stronger conjecture was given by \textit{C. Abbas} et al. [Comment. Math. Helv. 80, No. 4, 771--793 (2005; Zbl 1098.53063)], that asks for a finite collection of periodic solutions of \(R\), a so-called null-homologous Reeb link, that are oriented by \(R\) and eventually counted with a positive multiple of a period represents the trivial class in the homology of \(M\).NEWLINENEWLINE The main result of the paper under review is a confirmation of the strong Weinstein conjecture for closed contact manifolds \((M, \xi)\) that appear as the concave end of symplectic cobordisms admitting an essential local foliation by holomorphic spheres. This generalizes the results obtained in an article by \textit{H. Geiges} and the second author [Math. Proc. Camb. Philos. Soc. 153, No. 2, 261--279 (2012; Zbl 1262.53080)].