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arXiv · 1912.01078

Genus zero global surfaces of section for Reeb flows and a result of Birkhoff

Umberto L. Hryniewicz·Pedro A. S. Salomão·Krzysztof Wysocki
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Abstract

We exhibit sufficient conditions for a finite collection of periodic orbits of a Reeb flow on a closed $3$-manifold to bound a positive global surface of section with genus zero. These conditions turn out to be $C^\infty$-generically necessary. Moreover, they involve linking assumptions on periodic orbits with Conley-Zehnder index ranging in a finite set determined by the ambient contact geometry. As an application, we reprove and generalize a classical result of Birkhoff on the existence of annulus-like global surfaces of section for geodesic flows on positively curved two-spheres.

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Umberto L. Hryniewicz, Pedro A. S. Salomão, Krzysztof Wysocki. 2019-12-02. Genus zero global surfaces of section for Reeb flows and a result of Birkhoff. https://arxiv.org/abs/1912.01078

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