Optimal geodesics for boundary points of the Gardiner-Masur compactification
The Gardiner-Masur compactification of Teichmüller space is homeomorphic to the horofunction compactification of the Teichmüller metric. 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 which 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 geodesic converges to $ξ$ in forward direction and to $η$ in backward direction. As an application, we show that if $\mathbf{G}_n$ is a sequence of Teichmüller geodesics passing through $X_n$ and $Y_n$ such that $X_n \to ξ$ and $Y_n \to η$, then $\mathbf{G}_n$ converges to a unique Teichmüller geodesic.