Injectivity Criteria for Holomorphic Curves in ${\mathbb{C}}^n$

Abstract

Combining the definition of Schwarzian derivative for conformal mappings between Riemannian manifolds given by Osgood and Stowe with that for parametrized curves in Euclidean space given by Ahlfors, we establish injectivity criteria for holomorphic curves $\varphi :\mathbb{D}\to {\mathbb{C}}^n$. The result can be considered a generalization of a classical condition for univalence of Nehari.