Constructions controlees de champs de Reeb et applications

Abstract

On every compact, orientable, irreducible 3-manifold V which is toroidal or has torus boundary components we construct a contact 1-form whose Reeb vector field R does not have any contractible periodic orbits and is tangent to the boundary. Moreover, if bdry V is nonempty, then the Reeb vector field R is transverse to a taut foliation. By appealing to results of Hofer, Wysocki, and Zehnder, we show that, under certain conditions, the 3-manifold obtained by Dehn filling along bdry V is irreducible and different from the 3-sphere. Resumé On construit, sur toute variete V de dimension trois orientable, compacte, irreductible, bordee par des tores ou toroidale, une forme de contact dont le champ de Reeb R est sans orbite periodique contractible et tangent au bord. De plus, si le bord de V est non vide, le champ R est transversal a un feuilletage tendu. En utilisant des resultats de Hofer, Wysocki et Zehnder, on obtient sous certaines conditions que la variete obtenue par obturation de Dehn le long du bord de V est irreductible et differente de la sphere S^3.