Boundary localization of the normal family of holomorphic mappings, and remarks on existence of bounded holomorphic functions on complex manifolds (Q1262975)

Consider domains \(D_ 1\), \(D_ 2\) on taut manifolds \(X_ 1\) and \(X_ 2\) respectively. \(D_ 1\), \(D_ 2\) admit compact quotients. We prove that \(D_ 1\) is biholomorphic to \(D_ 2\) provided \(D_ 1\) is locally biholomorphic to \(D_ 2\) at two totally real boundary points. We also make some observations concerning an open problem to characterize those Stein manifolds with nontrivial bounded holomorphic functions.