Constructions controlees de champs de Reeb et applications

Colin, Vincent¹ ; Honda, Ko²

¹ Université de Nantes, UMR 6629 du CNRS, 44322 Nantes, France
² University of Southern California, Los Angeles, California 90089, USA

Geometry & topology, Tome 9 (2005) no. 4, pp. 2193-2226.

Résumé

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 $\partial V$ 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 $\partial V$ is irreducible and different from the 3–sphere.

Résumé

On construit, sur toute variété $V$ de dimension trois orientable, compacte, irréductible, bordée par des tores ou toroïdale, une forme de contact dont le champ de Reeb $R$ est sans orbite périodique contractible et tangent au bord. De plus, si $\partial V$ est non vide, le champ $R$ est transversal à un feuilletage tendu. En utilisant des résultats de Hofer, Wysocki et Zehnder, on obtient sous certaines conditions que la variété obtenue par obturation de Dehn le long du bord de $V$ est irréductible et différente de la sphère $S^3$.

Détail

DOI : 10.2140/gt.2005.9.2193

Keywords: Reeb vector field, contact structure, taut foliation

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