Optimal geodesics for boundary points of the Gardiner–Masur compactification

The Gardiner–Masur compactification of Teichmüller space is naturally homeomorphic to the horofunction compactification of the Teichmüller metric, in the sense that the identity map on Teichmüller space extends to a homeomorphism. Let ξ and η be a pair of boundary points in the Gardiner–Masur compactification that fill up the surface. We show that there is a unique Teichmüller geodesic that is optimal for the horofunctions corresponding to ξ and η . In particular, when ξ and η are Busemann points that fill up the surface, the Teichmüller geodesic converges to ξ in the forward direction and to η in the backward direction. As an application, we show that if G n is a sequence of Teichmüller geodesics passing through X n and Y n such that X n → ξ and Y n → η in the Gardiner–Masur compactification, then G n converges to a unique Teichmüller geodesic.