Entire curves avoiding given sets in \(\mathbb{C}^n\) (Q1880457)

Let \(F\subset\mathbb C^n\) be a proper closed subset of \(\mathbb C^n\) and \(A\subset\mathbb C^n\setminus F\) be at most countable, \(n\geq2\). The aim of this note is to discuss conditions for \(F\) and \(A\), under which there exists a holomorphic immersion (or a proper holomorphic embedding) \(\varphi: \mathbb C\to\mathbb C^n\) with \(A\subset\varphi(\mathbb C)\subset\mathbb C^n\setminus F\). The main tool for constructing such mappings is the Arakelian's approximation theorem. The authors prove: If \(F\) is a union of at most \(n-1\) \(\mathbb C\)-linearly independent complex hyperplanes in \(\mathbb C^n\), then for any discrete set of points in \(\mathbb C^n\setminus F\) there exists a proper holomorphic embedding of \(\mathbb C\) into \(\mathbb C^n\) avoiding \(F\) and passing through any of this points. If \(K\) is a polynomially convex compact set in \(\mathbb C^n\), then for any discrete set \(C\) of points in \(\mathbb C^n\setminus K\) there exists a proper holomorphic embedding \(H\) of \(\mathbb C\) into \(\mathbb C^n\) avoiding \(K\) and passing through any of these points. In addition, for a given point \(c\in\mathbb C\) and \(X\in\mathbb C^n\setminus\{0\}\) we can choose \(H\) such that \(H'(H^{-1}(c))=X\). In particular, the Lempert function and the Kobayashi pseudometric of \(\mathbb C^n\setminus K\) vanish. If \(F\) and \(G\) are two sets in \(\mathbb C^k\) and \(\mathbb C^l\), respectively, then for any countable set \(C\) of points in \(\mathbb C^{k+l}\) with dist\((C,F\times G)>0\), there exists a holomorphic immersion of \(\mathbb C\) into \(\mathbb C^{k+l}\) avoiding \(F\times G\) and passing through any point of \(C\). If \(F\) and \(G\) are two proper closed sets in \(\mathbb C^k\) and \(\mathbb C^l\), respectively, then for any point \(c\in\mathbb C^{k+l}\setminus(F\times G)\) and any vector \(X\in\mathbb C^{k+l}\) there exists a holomorphic mapping of \(\mathbb C\) into \(\mathbb C^{k+l}\setminus(F\times G)\) with \(f(0)=c\) and \(f'(0)=X\).