Proper holomorphic mappings between real analytic domains in \(\mathbf \mathbb{C}^ n\)
Let \(D_1 \Subset \mathbb{C}^n\) and \(D_2 \Subset \mathbb{C}^n\) be two bounded domains with real analytic boundaries. Let \(f\) be a proper holomorphic mapping from \(D_1\) to \(D_2\) that can be extended smoothly up to \(\overline{D}_1\). The authors study the unique continuation property for the normal component of \(f\) at \(p\) in case \(f(p)\) is minimal but not minimally convex. Theorem 1. Let \(D_1, D_2 \subset \mathbb{C}^n\) be bounded domains with \(M_1\) and \(M_2\) as part of their boundaries, respectively. Assume that \(M_1\) and \(M_2\) are real analytic minimal hypersurfaces and \(f\) is a proper holomorphic mapping from \(D_1\) to \(D_1\), that is, \(C^\infty\) smooth up to \(M_1\) and maps \(M_1\) into \(M_2\). Then the normal component of \(f\) is not flat at any point of \(M_1\). Theorem 2. Let \(D_1\) and \(D_2\) be two smoothly bounded domains with real analytic boundaries in \(\mathbb{C}^n\) \((n > 1)\), and let \(f\) be a proper holomorphic mapping from \(D_1\) to \(D_2\). Suppose that \(f\) is \(C^\infty\) smooth up to \(\overline{D_1}\). Then, there exists a constant \(C\) depending only on \(D_1\) and \(D_2\) such that the branching degree of \(f\) is bounded by \(C\). As an immediate application of Theorem 2, one has the following corollary. Corollary. Let \(\Omega\) be smoothly bounded domains with real analytic boundary. Then any proper holomorphic self-map of \(\Omega\) that is smooth up to the boundary must be biholomorphic.
scientific article; zbMATH DE number 908697
scientific article