Bounded analytic functions and metrics of constant curvature on Riemann surfaces (Q1119773)

Let f: \(\Omega\) \(\to \Delta\) be a locally schlicht holomorphic function where \(\Omega\) is a simply connected Riemann surface and \(\Delta\) is the unit disc. It is well known that the pull-back \[ f^*\lambda_{\Delta}(z)=| f'(z)| | dz| /(1-| f(z)|^ 2) \] is a metric on \(\Omega\) of constant curvature -4 where \(\lambda_{\Delta}\) denotes the Poincaré metric of \(\Delta\). Theorem 1 of this paper shows the converse. That is, any \(C^{\infty}\) of constant curvature -4 on \(\Omega\) has the above representation with some locally schlicht holomorphic function f: \(\Omega\) \(\to \Delta\). Some applications and related results are discussed in the rest of the paper. Among these, a counter example of a result of \textit{Ch. Pommerenke} [J. Lond. Math. Soc., II. Ser. 2, 689-695 (1970; Zbl 0199.398)] is given.