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

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 in \cite{HWZ,HWZ1}. We show that a Finsler metric on $\mathbb S^2$ with curvature $K\geq 1$ and with all geodesic loops of length $>\pi$ 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 \cite{HW}.