Holomorphic embeddings of planar domains in \(\mathbb{C}^ 2\) (Q1906491)

This is a very interesting embedding result. The authors prove that every bounded finitely-connected domain in \(\mathbb{C}\) with no isolated boundary points can be properly holomorphically embedded in \(\mathbb{C}^2\). The authors motivate their construction by describing a procedure which yields an embedding of the unit disc \(\Delta\) into \(\mathbb{C}^2\) (it was previously known that \(\Delta\) and the annulus admit such embeddings). In this case they produce a sequence of maps which are compositions of polynomial shears (considered as defined on domains which are small perturbations of \(\Delta)\), and which converge to a map which gives a proper holomorphic embedding of a small perturbation of \(\Delta\) into \(\mathbb{C}^2\). Of course this perturbed domain is biholomorphic to \(\Delta\) by the Riemann mapping theorem. In the \(M\)-connected case one needs only consider the case of domains bounded by \(M\) circles. The proof in this case consists of a more subtle application of perturbation techniques.