Robustly non-convex hypersurfaces in contact manifolds

TL;DR

This paper constructs the first examples of non-convex hypersurfaces in high-dimensional contact manifolds that cannot be approximated by convex ones, introducing a new dynamical blender in contact geometry.

Contribution

It provides the first examples of non-convex hypersurfaces in contact manifolds of dimension 5 or higher that defy convex approximation, using novel dynamical techniques.

Findings

Existence of non-convex hypersurfaces in high-dimensional contact manifolds.

Introduction of a dynamical blender in the contact setting.

Contrasts with previous results in dimension 3 and the $C^0$ case.

Abstract

We construct the first examples of hypersurfaces in any contact manifold of dimension 5 and larger that cannot be $C^2$-approximated by convex hypersurfaces, contrasting sharply with the foundational results of Giroux in dimension $3$ and Honda-Huang in the $C^0$ case. The main technical step is the first construction of a dynamical blender in the contact setting.