Normal families of covering maps
Let \(\mathbb{C}\) denote the complex plane and let \(C^*=\mathbb{C}\cup\{\infty\}\). The kernel of a family of sets \(\{E_\alpha: \alpha\in I\}\) is defined as the set of points \(w\in C^*\) for which there exists a neighborhood \(W\) of \(w\) with \(W\subset E_\alpha\) for all \(\alpha\in I\setminus F_W\), where \(F_W\) is a finite subset of \(I\). If \(f\) is a meromorphic covering map, we denote the domain of \(f\) by \(D_f\) and the range of \(f\) by \(R_f\). The author proves a number of results relating to when a sequence of covering maps converges locally uniformly to a covering map. Some of the basic results are as follows. Theorem 1. If \(\{f_n\}\) is a sequence of meromorphic covering maps that converges uniformly in a neighborhood of a point \(z\in C^*\) to a function that is locally non-constant, and if \(G\) and \(X\) denote the connected components of the kernels of \(\{D_{f_n}\}\) containing \(z\) and of \(\{R_{f_n}\}\) containing \(w= \lim_{n\to\infty} f_n(z)\), respectively, then \(\{f_n\}\) converges locally uniformly to a meromorphic covering map of \(G\) onto \(X\). Theorem 2. If \({\mathcal F}\) is a family of holomorphic covering maps such that the point \(z= 0\) is in the kernel of \(\{D_f: f\in{\mathcal F}\}\), and if \(f(0)= 0\) and \(f'(0)= 1\) for each \(f\in{\mathcal F}\), then either (1) \({\mathcal F}\) is a normal family on the connected component of the kernel of \(\{D_f: f\in{\mathcal F}\}\) containing the point \(z= 0\), or (2) there exist sequences \(\{\lambda_n\}\subset\mathbb{C}\setminus\{0\}\) and \(\{f_n\}\subset{\mathcal F}\) such that \(\lambda_n\to 0\) and \[ {1\over\lambda_n} f_n(\lambda_n z)\to e^z-1,\quad n\to\infty \] locally uniformly on \(\mathbb{C}\). Theorem 2 is related to a result of \textit{L. Zalcman} [Am. Math. Mon. 82, No. 8, 813-817 (1975; Zbl 0315.30036)]. The author uses these results to prove a version of the Carathéodory kernel theorem, and also to give a short proof of the Riemann uniformization theorem. In addition, there is an application of Theorem 1 to Baker domains.
scientific article; zbMATH DE number 1765849
scientific article