Genus zero global surfaces of section for Reeb flows and a result of Birkhoff
Umberto L. Hryniewicz
RWTH Aachen, GermanyPedro A. S. Salomão
University of São Paulo, Brazil; NYU Shanghai, ChinaKrzysztof Wysocki
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Abstract
We exhibit sufficient conditions for a finite collection of periodic orbits of a Reeb flow on a closed -manifold to bound a positive global surface of section with genus zero. These conditions turn out to be -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 re-prove and generalize a classical result of Birkhoff on the existence of annulus-like global surfaces of section for geodesic flows on positively curved -spheres.
Cite this article
Umberto L. Hryniewicz, Pedro A. S. Salomão, Krzysztof Wysocki, Genus zero global surfaces of section for Reeb flows and a result of Birkhoff. J. Eur. Math. Soc. 25 (2023), no. 9, pp. 3365–3451
DOI 10.4171/JEMS/1220