Dynamically convex Finsler metrics and J-holomorphic embedding of asymptotic cylinders

Abstract

Abstract. We explore the relationship between contact forms on S 3 defined by Finsler metrics on S 2 and the theory developed by H. Hofer, K. Wysocki and E. Zehnder in [9,10]. We show that a Finsler metric on 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 S 2 with K = 1 thus complementing the results obtained in [8].

The existence of two closed geodesics on every Finsler 2-sphere

“…This alternative also holds for Finsler metrics with K ≥ 1 for which every geodesic loop is longer than π . This was proved in 2006 by Harris and Paternain [16]. Their proof is based on [19].…”

Magnetic Katok examples on the two-sphere

“…We recall the precise statement here for the convenience of the reader. Before doing that, we remark that this result can be thought of as a generalisation of a theorem by Harris and Paternain [11] asserting that suitably pinched Finsler metrics have dynamically convex geodesic flows. Theorem 1.1 ([3]).…”

“…By Lemma 3.3 the condition c > s(1 + α) ensures that it is enough to prove the corollary assuming s = 0. In this case the statement holds by [11,Section 5].…”

“…They were introduced by Katok [15] and studied further by Ziller [20]. The double cover of the sphere bundle of the Katok examples was proved to be strictly contactomorphic to an ellipsoid of the type above by Harris and Paternain [11,Section 4 and 5]. Their result yields the case s = 0 of the theorem above, since it was also known that such a sphere bundle is contactomorphic to an exact magnetic system on S 2 (see [18]).…”

The length of a shortest geodesic loop

“…Proposition 2 (Harris-Paternain [HP,Theorem B,Section 6]) Let F be a non-reversible Finsler metric on the 2-sphere with reversibility λ and flag curvature…”