Results on abundance of global surfaces of section for Reeb flows

Abstract: One might ask if global surfaces of section (GSS) for Reeb flows in dimension 3 are abundant in two different senses. One might ask if GSS are abundant for a given Reeb flow, or if Reeb flows carrying some GSS are abundant in the set of all Reeb flows. In this talk, answers to these two questions in specific contexts will be presented. First, I would like to discuss a result, obtained in collaboration with Florio, stating that there are explicit sets of Reeb flows on which are right-handed in the sense of Ghys; in particular, for such a flow all finite (non-empty) collections of periodic orbits span a GSS. Then, I would like to discuss genericity results, obtained in collaboration with Colin, Dehornoy and Rechtman, for Reeb flows carrying a GSS; as a particular case of such results, we prove that a $C^\infty$-generic Reeb flow on an arbitrary closed 3-manifold carries a (rational) GSS.

differential geometrygeneral mathematicsgeneral topologygroup theorysymplectic geometry

Audience: researchers in the topic

Workshop on Singularity Theory, Geometry and Related Topics

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