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Global properties of tight Reeb flows with applications to Finsler geodesic flows on \(S^2\). (English) Zbl 1326.53053

This paper consider Finsler metrics on \(S^2\) satisfying a pinching condition for the flag curvatures, where the pinching constant depends on the reversibility of the metric. It is proved that under such hypothesis there are no closed geodesics with only one self intersection generalizing a theorem due to W. Ballmann [Invent. Math. 71, 593–597 (1983; Zbl 0505.53020)]. The novelty is the use of methods from the theory of pseudo-holomorphic curves in order to exclude closed geodesics from families of weak flat knot types, a weakened version of the notion discussed by S. B. Angenent [Ann. Math. (2) 162, No. 3, 1187–1241 (2005; Zbl 1137.53330)]. The pinching condition is the same that ensures in [A. Harris and G. P. Paternain, Ann. Global Anal. Geom. 34, No. 2, 115–134 (2008; Zbl 1149.53045)] that the contact form in the unit sphere bundle is dynamically convex. Some generalisations to more general Hamiltonian systems of the “short-long” dichotomy are also considered.

MSC:

53C22 Geodesics in global differential geometry
53D25 Geodesic flows in symplectic geometry and contact geometry
53C60 Global differential geometry of Finsler spaces and generalizations (areal metrics)
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