Discs in Stein manifolds containing given discrete sets
Let \(\Delta\) denote the unit disc in \({\mathbb C}.\) The following two results are known: i) Given a discrete subset \(S\) of a convex domain \(D \subset {\mathbb C}^N\) there exists a proper holomorphic map \(f\colon \Delta \rightarrow D\) such that \(S \subset f(D).\) ii) Given a Stein manifold \(M\) of dimension at least two, a point \(z \in M\) and a direction \(X \in T_zM - \{0\}\) there exists a proper holomorphic map \(f\colon \Delta \rightarrow M\) such that \(f(0)=z\) and \(f'(0)=\lambda X,\) with \(\lambda \neq 0.\) The author generalizes these two results by showing that, given a discrete subset \(\{z_n\}\) of a Stein manifold \(M\) with dim\(M \geq 2,\) there exists a proper holomorphic immersion \(f\colon \Delta \rightarrow M\) such that \(z_n \in f(\Delta)\) for all \(n,\) and with prescribed higher order contact with given one dimensional submanifolds of \(M\) at the points \(z_n.\) If \(\dim M \geq 3\) she shows that \(f\) can be chosen to be a proper embedding.
scientific article; zbMATH DE number 1761070