An overtwisted convex hypersurface in higher dimensions

Chiang, River; Niederkrüger-Eid, Klaus

doi:10.2140/agt.2025.25.3813

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An overtwisted convex hypersurface in higher dimensions

River Chiang and Klaus Niederkrüger-Eid

Algebraic & Geometric Topology 25 (2025) 3813–3831

DOI: 10.2140/agt.2025.25.3813

Abstract

We show that the germ of the contact structure surrounding a certain kind of convex hypersurface is overtwisted. We then find such hypersurfaces close to any plastikstufe with toric core, thereby proving that the existence of a plastikstufe with toric core implies 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 $Σ$ .

Keywords

contact topology, overtwisted, convex hypersurface, dimension larger than 3

Mathematical Subject Classification

Primary: 53D10

References

Publication

Received: 7 October 2021

Revised: 8 March 2023

Accepted: 14 October 2024

Published: 29 October 2025

Authors

River Chiang
Department of Mathematics National Cheng Kung University Tainan City Taiwan

Klaus Niederkrüger-Eid
Institut Camille Jordan UMR5208 Université Claude Bernard Lyon 1 Villeurbanne France

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Volume 25, issue 7 (2025)