A problem on the convexity of the Teichmüller metric (Q1312963)

The paper shows that the Teichmüller metric on the Teichmüller space \(T(G)\), where \(G\) is a Fuchsian group of the second kind, is not strictly convex with respect to the geodesics. The precise result is the following. Consider any sphere in \(T(G)\) around the origin. The author shows by explicitly constructing extremal Beltrami differentials that there are pairs of points on the sphere, together with a geodesic joining them, so that the entire geodesic lies on the same sphere. The problem in the case when \(G\) is of the first kind remains open.