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

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 or...

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Published inJournal of the European Mathematical Society : JEMS Vol. 25; no. 9; pp. 3365 - 3451
Main Authors Hryniewicz, Umberto L., Salomão, Pedro A. S., Wysocki, Krzysztof
Format Journal Article
LanguageEnglish
Published 2023
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Summary: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 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 2 -spheres.
ISSN:1435-9855
1435-9863
DOI:10.4171/jems/1220