Niederkrüger, Klaus; Rechtman, Ana The Weinstein conjecture in the presence of submanifolds having a Legendrian foliation. (English) Zbl 1241.53064 J. Topol. Anal. 3, No. 4, 405-421 (2011). Let \((M,\xi)\) be a contact manifold with contact form \(\alpha\). The associated Reeb field \(R_\alpha\) is the unique vector field satisfying \(\alpha(R_\alpha)=1\) and \(\imath_{_{R_\alpha}}d\alpha=0\). The A. Weinstein conjecture [J. Differ. Equations 33, 353–358 (1979; Zbl 0388.58020)] states that if \((M,\xi)\) is a closed contact manifold with \(\xi=\ker\alpha\), then its Reeb field \(R_\alpha\) has a closed orbit. In [Invent. Math. 114, No.3, 515–563 (1993; Zbl 0797.58023)], H. Hofer proved this conjecture for a closed contact \(3\)-manifold \((M,\xi)\) in the following three cases: if \(M\) is diffeomorphic to \(\mathbb S^3\), if \(\xi\) is over-twisted, or if the second homotopy group \(\pi_2(M)\) is not trivial. The generalization of the second case to higher dimensions was proven by P. Albers and H. Hofer in [Comment. Math. Helv. 84, No. 2, 429–436, (2009; Zbl 1166.53052)].In this paper, the authors try to generalize the third case. In their generalization of Hofer’s theorem for contact \((2n+1)\)-manifolds the \(2\)-sphere is replaced by an embedded \((n+1)\)-submanifold such that the contact structure restricts to an open book decomposition. It is proven that if such a submanifold represents a non-trivial homology class then there exists a periodic contractible Reeb orbit. The authors present some examples of such submanifolds. Among these examples are the connected sum of any contact manifold \(M\) with the projective space with its standard contact structure or with many subcritically fillable manifolds as \(\mathbb S^n\times\mathbb S^{n+l}\) or \(\mathbb T^n\times\mathbb S^{n+l}\). Reviewer: Andrew Bucki (Edmond) Cited in 1 ReviewCited in 5 Documents MSC: 53D10 Contact manifolds (general theory) 22E46 Semisimple Lie groups and their representations 53C35 Differential geometry of symmetric spaces 57S20 Noncompact Lie groups of transformations Keywords:Weinstein conjecture; periodic orbit; Reeb vector field; Legendrian open book Citations:Zbl 0388.58020; Zbl 0797.58023; Zbl 1166.53052 PDFBibTeX XMLCite \textit{K. Niederkrüger} and \textit{A. Rechtman}, J. Topol. Anal. 3, No. 4, 405--421 (2011; Zbl 1241.53064) Full Text: DOI arXiv References: [1] Albers P., Comment. Math. Helv. 84 pp 429– [2] Epstein D., Proc. London Math. Soc. (3) 16 pp 369– [3] DOI: 10.1007/BF01232679 · Zbl 0797.58023 [4] DOI: 10.2140/agt.2006.6.2473 · Zbl 1129.53056 [5] DOI: 10.2140/gt.2010.14.719 · Zbl 1186.57020 [6] DOI: 10.1002/cpa.3160310203 [7] DOI: 10.1007/s000390050106 · Zbl 0954.57015 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.