An overtwisted convex hypersurface in higher dimensions
River Chiang, Klaus Niederkrüger-Eid
TL;DR
The paper addresses the problem of detecting overtwistedness in high-dimensional contact manifolds by constructing explicit overtwisted convex disks near plastikstufes with toric cores. It defines a model hypersurface $\Sigma_C$ with singular distribution $\mathcal{D}_C$ via $\beta = r\sin r\, d\vartheta - \sum t_j ds_j$ and proves a universal threshold $C_{OT}$ such that, when the ambient germ induces $\mathcal{D}_C$, the germ is overtwisted. The authors show that any contact manifold containing a plastikstufe $\mathbb{D}_{OT} \times \mathbb{T}^n$ admits an embedding of an overtwisted convex disk, with a neighborhood modeled on $B(\varepsilon) \times \mathbb{D}_{<\delta}(T^*\mathbb{T}^n)$, and provide a dimension-free construction by unwinding the toric core. A central tool is a product-structure argument that squeezes height while controlling fiber directions, yielding a uniform bound $G(r,t) \le \ln(7/6)$ that allows the 3D model to extend to higher dimensions; the appendix also proves the looseness of the Legendrian unknot in large overtwisted charts and establishes the germ-determined-by-hypersurface principle. Overall, the work offers explicit, constructive methods to certify overtwistedness in high dimensions and clarifies how neighborhood size governs contact-germ flexibility.
Abstract
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 $Σ$ of a contact manifold $(M, ξ)$ determines the germ of the contact structure around $Σ$.