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Periodic orbits and pseudo-holomorphic curves, application to the Weinstein conjecture in dimension 3 (after H. Hofer et al.). (Orbites périodiques et courbes pseudo-holomorphes, application à la conjecture de Weinstein en dimension 3 (d’après H. Hofer et al.).) (French) Zbl 0853.57013
Séminaire Bourbaki. Volume 1993/94. Exposés 775-789. Paris: Société Mathématique de France, Astérisque. 227, 309-333 (Exp. No. 786) (1995).
Let $$(W, \omega)$$ be a symplectic manifold, $$M$$ a hypersurface of $$W$$ and $$M_\omega$$ its characteristic foliation. If there is a Hamiltonian $$H: W\to \mathbb{R}$$ constant on $$M$$ and without critical points on a neighborhood of $$M$$ then the symplectic gradient of $$M$$ characterizes $$M_\omega$$ in every point of $$M$$. Thus $$M_\omega$$ is the typical object of the study of Hamiltonian dynamics. An important question is the existence of compact foliations (or periodic orbits when a Hamiltonian is given). Concerning this problem, A. Weinstein [J. Differ. Equations 33, 353-358 (1979; Zbl 0388.58020)] conjectured that if $$M$$ is compact and of contact type then $$M_\omega$$ has a compact foliation. An answer to this conjecture in dimension 3 is given by the theorem of Hofer which is the main subject of the present report. Many interesting comments about this theorem and relations with some special contact structures and pseudo-holomorphic curves are also presented.
For the entire collection see [Zbl 0811.00012].

MSC:
 57M50 General geometric structures on low-dimensional manifolds
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