Isometric disks are holomorphic (Q517434)

Let \(S_{g, n}\) be a hyperbolic Riemann surface of genus \(g\) with \(n\) punctures and \(T_{g, n}\) denote the Teichmüller space of \(S_{g, n}\) with Kobayashi-Teichmüller metric. The main result of this paper is: Let \(f\) be an isometric map from the hyperbolic unit disk \(\Delta\) into the Teichmüller space \(T_{g, n}\). Then \(f\) is either holomorphic or anti-holomorphic; in particular, the image domain \(f(\Delta)\) is a Teichmüller disk in \(T_{g, n}\).