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arXiv · 1102.1474

Global properties of tight Reeb flows with applications to Finsler geodesic flows on $S^2$

Umberto Hryniewicz·Pedro A. S. Salomão
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Abstract

We show that if a Finsler metric on $S^2$ with reversibility $r$ has flag curvatures $K$ satisfying $(\frac{r}{r+1})^2 <K \leq 1$, then closed geodesics with specific contact-topological properties cannot exist, in particular there are no closed geodesics with precisely one transverse self-intersection point. This is a special case of a more general phenomenon, and other closed geodesics with many self-intersections are also excluded. We provide examples of Randers type, obtained by suitably modifying the metrics constructed by Katok \cite{katok}, proving that this pinching condition is sharp. Our methods are borrowed from the theory of pseudo-holomorphic curves in symplectizations. Finally, we study global dynamical aspects of 3-dimensional energy levels $C^2$-close to $S^3$.

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Umberto Hryniewicz, Pedro A. S. Salomão. 2011-10-16. Global properties of tight Reeb flows with applications to Finsler geodesic flows on $S^2$. https://doi.org/10.1017/s0305004112000333

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