Morse-Smale characteristic foliations and convexity in contact manifolds

TL;DR

This paper extends Giroux's convexity result for surfaces with Morse-Smale characteristic foliations to contact manifolds of any dimension, and demonstrates the approximation of certain hypersurfaces by convex ones.

Contribution

It generalizes a key convexity theorem from 3-dimensional contact topology to higher dimensions and applies it to approximate specific hypersurfaces by convex ones.

Findings

The convexity result holds in arbitrary-dimensional contact manifolds.

Certain hypersurfaces can be approximated arbitrarily closely by convex hypersurfaces.

The generalization broadens the applicability of convexity techniques in contact topology.

Abstract

We generalize a result of Giroux which says that a closed surface in a contact $3$-manifold with Morse-Smale characteristic foliation is convex. Specifically, we show that the result holds in contact manifolds of arbitrary dimension. As an application, we show that a particular closed hypersurface introduced by A. Mori is $C^{\infty}$-close to a convex hypersurface.