Paires de structures de contact sur les vari\'et\'es de dimension trois

TL;DR

This paper introduces a new concept of positive pairs of contact structures on 3-manifolds, generalizing previous definitions, and explores their properties, including integrability, tightness, and implications for Reebless foliations.

Contribution

It defines positive pairs of contact structures on 3-manifolds, proves their integrability and stability properties, and analyzes their impact on Reeb components and foliations.

Findings

If the plane field is uniquely integrable and both structures are tight, the foliation has no Reeb component with null-homologous core.

The ambient manifold admits a Reebless foliation under these conditions.

A Reeb-type stability theorem is established for positive pairs of tight contact structures.

Abstract

We introduce a notion of positive pair of contact structures on a 3-manifold which generalizes a previous definition of Eliashberg-Thurston and Mitsumatsu. Such a pair gives rise to a locally integrable plane field $\lambda$. We prove that if $\lambda$ is uniquely integrable and if both structures of the pair are tight, then the integral foliation of $\lambda$ doesn't contain any Reeb component whose core curve is homologous to zero. Moreover, the ambient manifold carries a Reebless foliation. We also show a stability theorem "\`a la Reeb" for positive pairs of tight contact structures.