Uniqueness results for holomorphic mappings on the disc (Q1987346)

In this paper, holomorphic mappings from the disc into compact Riemann surfaces as well as into the complex projective space \(\mathbb{P}^n(\mathbb{C})\) are investigated. This paper derives several unicity results for such mappings. For instance, let \(\Delta(R)\) denote the disc of radius \(R\) with the convention that \(\Delta(\infty)=\mathbb{C}\). Let \(M\) be a compact Riemann surface and let \(\omega\) be a smooth positive \((1,1)\) (metric) form on \(M\) such that its Gauss curvature is constant and \(\int_M \omega=1\). Let \(f,g:\Delta(R)\rightarrow M\) be two holomorphic maps with the growth indexes with respect to \(\omega\) (according to Ru-Sibony) \(c_{f,\omega}<+\infty\) and \(c_{g,\omega}<+\infty\). If \(q>4+c_{f,\omega}+c_{g,\omega}\) and there are \(q\) distinct points \(a_1,\dots,a_q\) on \(M\) such that \(f(z)=a_j\) if and only if \(g(z)=a_j\) for \(j=1,\dots,q\), then \(f\equiv g\).