Planarity in Higher-Dimensional Contact Manifolds

We obtain several results for (iterated) planar contact manifolds in higher dimensions. (1) Iterated planar contact manifolds are not weakly symplectically co-fillable. This generalizes a 3D result of Etnyre [ ] to a higher-dimensional setting, where the notion of weak fillability is that due to Massot-Niederkruger-Wendl [ ]. (2) They do not arise as nonseparating weak contact-type hypersurfaces in closed symplectic manifolds. This generalizes a result by Albers-Bramham-Wendl [ ]. (3) They satisfy the Weinstein conjecture, that is, every contact form admits a closed Reeb orbit. This is proved by an alternative approach as that of [ ] and is a higher-dimensional generalization of a result of Abbas-Cieliebak-Hofer [ ]. The results follow as applications from a suitable symplectic handle attachment, which bears some independent interest.