Relative embeddings of discs into convex domains (Q1823349)

Let \(\Omega\) be a domain in \({\mathbb{C}}^ n\). \(S\subset \Omega\) is called a discrete subset of \(\Omega\) if S has no limit points in \(\Omega\). The author proves a number of interesting theorems concerning the existence of holomorphically embedded discs containing such sets. Denote by \({\mathbb{D}}\) the open unit disc in the complex plane \({\mathbb{C}}.\) Theorem 1. Let \(\Omega\) be a convex domain in \({\mathbb{C}}^ N\), \(N\geq 3\). Every discrete subset of \(\Omega\) is contained in a holomorphically embedded disc, i.e. in a submanifold of \(\Omega\) of the form F(\({\mathbb{D}})\), where F: \({\mathbb{D}}\to \Omega\) is a holomorphic embedding. For \(N=2\) the result is weaker: in this case only the existence of a variety \(V=F({\mathbb{D}})\) is guaranteed, V containing the discrete subset and F: \({\mathbb{D}}\to \Omega\) being a proper holomorphic immersion. Theorem 3. Let \(\Omega\) be a bounded strictly convex domain. Then the map F in the theorems above can be chosen to extend continuously to the closure \({\bar {\mathbb{D}}}.\) By choosing a discrete subset of \(\Omega\) the closure of which contains the boundary \(\partial \Omega\) the following interesting corollary is obtained. Corollary. Let \(\Omega \subset {\mathbb{C}}^ N\) be a bounded strictly convex domain. There is a continuous map F: \({\bar {\mathbb{D}}}\to {\bar \Omega}\), holomorphic in \({\mathbb{D}}\) and such that \(F(\partial {\mathbb{D}})=\partial \Omega\).