Abstract
We explore the relationship between contact forms on \({\mathbb{S}}^3\) defined by Finsler metrics on \({\mathbb{S}}^2\) and the theory developed by H. Hofer, K. Wysocki and E. Zehnder (Hofer etal. Ann. Math. 148, 197–289, 1998; Ann. Math. 157, 125–255, 2003). We show that a Finsler metric on \({\mathbb{S}}^2\) with curvature K ≥ 1 and with all geodesic loops of length > π is dynamically convex and hence it has either two or infinitely many closed geodesics. We also explain how to explicitly construct J-holomorphic embeddings of cylinders asymptotic to Reeb orbits of contact structures arising from Finsler metrics on \(\mathbb S^2\) with K = 1, thus complementing the results obtained in Harris and Wysocki (Trans. Am. Math. Soc., to appear).
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Communicated by: H.-B. Rademacher (Leipzig).
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Harris, A., Paternain, G.P. Dynamically convex Finsler metrics and J-holomorphic embedding of asymptotic cylinders. Ann Glob Anal Geom 34, 115–134 (2008). https://doi.org/10.1007/s10455-008-9111-2
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DOI: https://doi.org/10.1007/s10455-008-9111-2