Univalent holomorphic mappings on a complex manifold with a C1 exhaustion function

Let M be a complex manifold with a exhaustion function ρ. Assume that ρ attains its minimum at a single point o and dρ≠ 0 on M\{o}. In this case, we will give a necessary condition and a sufficient condition for locally biholomorphic mappings on M to be biholomorphic and to have Φ-like images. Received: 15 May 1998 / Revised version: 8 March 1999