Positive isotopies of Legendrian submanifolds and applications

We show that there is no positive loop inside the component of a fiber in the space of Legendrian embeddings in the contact manifold $ST^*M$, provided that the universal cover of $M$ is $RM^n$. We consider some related results in the space of one-jets of functions on a compact manifold. We give an application to the positive isotopies in homogeneous neighborhoods of surfaces in a tight contact 3-manifold.

Paires de structures de contact sur les varietes de dimension trois

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 $lambda$ doesnt 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.

On the flux of pseudo-Anosov homeomorphisms

We exhibit a pseudo-Anosov homeomorphism of a surface S which acts trivially on the first homology group of S and whose flux is non zero

Finitude homotopique et isotopique des structures de contact tendues

Let V be a closed 3-manifold. In this paper we prove that the homotopy classes of plane fields on V that contain tight contact structures are in finite number and that, if V is atoroidal, the isotopy classes of tight contact structures are also in finite number.