Paires de structures de contact sur les variétés de dimension trois

Abstract

On introduit une notion de paire positive de structures de contact sur les variétés de dimension trois qui généralise celle de Eliashberg and Thurston [5] and Mitsumatsu [13;14]. Une telle paire "normale" donne naissance à un champ de plans continu et localement intégrable . On montre que si est uniquement intégrable et si les structures de contact sont tendues, alors le feuilletage intégral de est sans composante de Reeb d'âme homologue à zéro. De plus, dans ce cas, la variété ambiante porte un feuilletage sans composante de Reeb. On démontre également un théorème de stabilité "à la Reeb" pour les paires positives de structures tendues.We introduce the notion of a positive pair of contact structures of a three dimensional manifold which generalises that of [5; 13; 14]. A normal such pair gives rise to a continuous, locally integrable plane field . We show that if is uniquely integrable and if the contact structures are tight then the integral foliation of has no Reeb component whose core is homologous to zero. Moreover, in this case, the ambient manifold carries a foliation without a Reeb component. We also show a Reeb stability theorem for positive pairs of tight structures.

Engel structures and weakly hyperbolic flows on four-manifolds

“…This correspondence is analogous to the correspondence between bi-contact structures and projectively or conformally Anosov flows on three-manifolds found by Eliashberg-Thurston and by Mitsumatsu.1We found the explanations in those references to be somewhat elliptical. Related arguments also appear in [6].…”

“…This kernel is 1 We found the explanations in those references to be somewhat elliptical. Related arguments also appear in [6]. a line field W ⊂ E giving rise to the characteristic foliation of E, and the quotient bundle E/W carries a conformal symplectic structure.…”

“…To achieve this we fix a mollifier, i.e. a smooth function h : R−→R + 0 with support in [−1, 1] and R h(s)ds = 1, and consider the usual convolution (6) h * X ± (p) = R h(s)Dϕ s (X ± (ϕ −s (p))) ds .…”

On Anosovity, divergence and bi-contact surgery

“…It is worth mentioning that although projectively Anosov flows have been previously studied in various contexts, such as foliation theory [15,6,33,13], Riemannian geometry [9,10,34,27], hyperbolic dynamics [25,3,35,36] and Reeb dynamics [26], their primary significance for us is that they serve as bridge between Anosov dynamics and contact and symplectic geometry [32] (see Section 2.2), eventually yielding a complete characterization of Anosov flows in terms of such geometries [28]. We also remark that such flows are also called by different names in the literature, including conformally Anosov flows or flows with dominated splitting.…”

Taut foliations

“…This study was inspired by an observation of Vincent Colin, who pointed out to us that if T F is only continuous, then T F is not uniquely integrable and this impacts the existence of transversals. We thank Vincent for helpful conversations, and in particular for explaining his work, [5], where these issues arise in the search for a foliation approximated by a pair of contact structures. We also thank the Banff International Research Station for their hospitality, which made these conversations possible.…”