Domains of injective holomorphy (Q2909636)

The authors call a domain \(\Omega\subset \mathbb C\) a domain of injective holomorphy if there exists an injective holomorphic function \(f:\Omega\to \mathbb C\) that is non-extendable beyond \(\Omega\). Various examples of domains that are and are not domains of injective holomorphy are given. For instance, a domain \(\Omega\) with finitely many complementary components in \(\overline{\mathbb C}\) is a domain of injective holomorphy if and only if at most one of these components is a singleton. Regular domains (i.e. \(\overline\Omega^0=\Omega\)) as well as simply-connected and doubly-connected domains are domains of injective holomorphy, while if \(\Omega'\) is a domain and \(E\subset\Omega'\) is a non-empty closed subset of zero analytic capacity that contains at least two points, then \(\Omega'\setminus E\) is not a domain of injective holomorphy.