Isometric disks are holomorphic

Abstract

This paper shows that every totally-geodesic isometry from the unit disk to a finite-dimensional Teichmüller space for the intrinsic Kobayashi metric is either holomorphic or anti-holomorphic; in particular, it is a Teichmüller disk. Additionally, a similar result is proved for a large class of $\textit{disk-rigid}$ domains, which includes strictly convex domains, as well as finite-dimensional Teichmüller spaces.