In this paper we give a characterization of the Gromov hyperbolicity of trains (a large class of Denjoy domains which contains the flute surfaces) in terms of the behavior of a real function. This function describes somehow the distances between some remarkable geodesics in the train. This theorem has several consequences; in particular, it allows to deduce a result about stability of hyperbolicity, even though the original surface and the modified one are not quasi-isometric. In order to obtain these results we also prove some trigonometric lemmas that are interesting by themselves, since they provide very simple estimates on some hyperbolic distances.
"A real variable characterization of Gromov hyperbolicity of flute surfaces." Osaka J. Math. 48 (1) 179 - 207, March 2011.