Optimal geodesics for boundary points of the Gardiner-Masur compactification

The Gardiner-Masur compactification of Teichm\"uller space is homeomorphic to the horofunction compactification of the Teichm\"uller metric. Let $\xi$ and $\eta$ be a pair of boundary points in the Gardiner-Masur compactification that fill up the surface. We show that there is a unique Teichm\"uller geodesic which is optimal for the horofunctions corresponding to $\xi$ and $\eta$. In particular, when $\xi$ and $\eta$ are Busemann points that fill up the surface, the geodesic converges to $\xi$ in forward direction and to $\eta$ in backward direction. As an application, we show that if $\mathbf{G}_n$ is a sequence of Teichm\"uller geodesics passing through $X_n$ and $Y_n$ such that $X_n \to \xi$ and $Y_n \to \eta$, then $\mathbf{G}_n$ converges to a unique Teichm\"uller geodesic.