Planarity in higher-dimensional contact manifolds

TL;DR

This paper explores higher-dimensional contact manifolds, establishing their non-fillability, non-embedding as certain hypersurfaces, and confirming the Weinstein conjecture for them, thus extending several 3D results to higher dimensions.

Contribution

It generalizes key 3D contact topology results to higher dimensions, including non-fillability, non-embedding, and the Weinstein conjecture for iterated planar contact manifolds.

Findings

Iterated planar contact manifolds are not weakly symplectically semi-fillable.

They do not appear as nonseparating weak contact-type hypersurfaces in closed symplectic manifolds.

They satisfy the Weinstein conjecture, ensuring the existence of closed Reeb orbits.

Abstract

We obtain several results for (iterated) planar contact manifolds in higher dimensions: (1) Iterated planar contact manifolds are not weakly symplectically semi-fillable. This generalizes a 3-dimensional result of Etnyre to a higher-dimensional setting. (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, i.e. every contact form admits a closed Reeb orbit. This is proved by an alternative approach as that of [Acu], 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.