A rigidity result for holomorphic quadratic differentials of finite norm in the unit disk

Let phi dz2$\varphi dz<^>2$ be a holomorphic quadratic differential of finite norm in the unit disk D$\mathbb {D}$. Let D$\mathbb {D}$ be equipped with the metric |phi||dz2|$| \varphi || dz<^>2 |$. In this paper, we prove that a geodesic-preserving map of the unit disk into itself is affine. A map is called geodesic-preserving if the image of each geodesic is a geodesic. We make no assumptions on the continuity of such a map.