Global holomorphic extension of a local map and a Riemann mapping theorem for algebraic domains (Q1128279)

The main result of the paper is the following theorem. Let \(D, D'\subset\mathbb C^{n+1}\) be bounded domains with algebraic smooth boundaries (\(\partial D\) is algebraic if it is locally defined by real polynomials). Assume that \(\partial D\) is connected and \(D'\) is strongly pseudoconvex. Let \(a\in\partial D\), let \(U\) be an open neighborhood of \(a\), and let \(f\:U\rightarrow f(U)\) be a biholomorphic mapping such that \(f(U\cap\partial D)\subset\partial D'\). Then for any path \(\gamma\:[0,1]\rightarrow\overline D\) with \(\gamma(0)=a\), the mapping \(f\) extends holomorphically along \(\gamma\). As an application, the authors obtain the following classification theorem. Let \(\mathbb B\) denote the unit Euclidean ball in \(\mathbb C^{n}\). Let \(D\subset\mathbb C^{n+1}\) be a bounded algebraic spherical domain (\(D\) is called spherical if for any point \(p\in\partial D\) there exist an open neighborhood \(U\) and a biholomorphic mapping \(\Phi\:U\rightarrow\Phi(U)\) such that \(\Phi(U\cap\partial D)\subset\partial\mathbb B\) and \(\Phi(U\cap D)\subset\mathbb B\)). Then \(D\) is biholomorphic to \(\mathbb B\).