Isometric disks are holomorphic

TL;DR

This paper proves that all totally-geodesic isometries from the unit disk to certain Teichmüller spaces are holomorphic or anti-holomorphic, extending to a broad class of disk-rigid domains.

Contribution

It establishes a rigidity result for isometries into Teichmüller spaces, showing they must be holomorphic or anti-holomorphic, including for various disk-rigid domains.

Findings

All such isometries are Teichmüller disks.

The result applies to strictly convex bounded domains.

It extends to a large class of disk-rigid domains.

Abstract

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