Isometric disks are holomorphic

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