Open books and the Weinstein conjecture (Q2922455)

Let \((M,\xi)\) be a contact closed oriented manifold. It is said to satisfy the Weinstein conjecture if for every choice of contact form \(\alpha\) the corresponding Reeb vector field \(R_\alpha\) has a closed orbit. NEWLINENEWLINENEWLINEAn open book decomposition of \(M\) is a pair \((B,\theta)\) consisting of an oriented codimension-2 submanifold \(B\subset M\) with trivial normal bundle and a locally trivial fibration \(\theta:M\setminus B\to S^1\) that in a neighbourhood \(B\times D^2\setminus B\times\{0\}\) is the projection to \(\partial D=S^1\). A contact structure \(\xi\) is supported by \((B,\theta)\) if a contact form \(\alpha\), \(\xi=\mathrm{Ker}(\alpha)\), can be chosen in such a way that the 2-form \(d\alpha\) induces a positive symplectic form on each fibre of \(\theta\) and the 1-form \(\alpha\) induces a positive contact form on \(B\). NEWLINENEWLINENEWLINEThe main theorem of this paper states the existence of a contractible periodic Reeb orbit for any contact structure \((M,\xi)\) supported by an open book whose binding \(B\) can be realized as a hypersurface of restricted contact type in a subcritical Stein manifold. A key ingredient in the proof is a higher-dimensional version of Eliashberg's theorem about symplectic cobordisms from a contact manifold to a symplectic fibration.