An overtwisted convex hypersurface in higher dimensions

We show that the germ of the contact structure surrounding a certain kind of convex hypersurfaces is overtwisted. We then find such hypersurfaces close to any plastikstufe with toric core so that these imply overtwistedness. All proofs in this article are explicit, and we hope that the methods used here might hint at a deeper understanding of the size of neighborhoods in contact manifolds. In the appendix we reprove in a concise way that the Legendrian unknot is loose if the ambient manifold contains a large enough neighborhood of a 2-dimensional overtwisted disk. Additionally we prove the folklore result that the singular distribution induced on a hypersurface $\Sigma$ of a contact manifold $(M, \xi)$ determines the germ of the contact structure around $\Sigma$.