The aim of this paper is to develop the theory of a compactification of Teichmüller space given by F. Gardiner and H. Masur, which we call the Gardiner–Masur compactification of the Teichmüller space. We first develop the general theory of the Gardiner–Masur compactification. Secondly, we will investigate the asymptotic behaviors of Teichmüller geodesic rays under the Gardiner–Masur embedding. In particular, we will observe that the projective class of a rational measured foliation G can not be an accumulation point of every Teichmüller geodesic ray under the Gardiner–Masur embedding, when the support of G consists of at least two simple closed curves.
Extremal length Teichmüller space Geodesic Gardiner–Masur boundary Thurston boundary