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

TL;DR

This paper develops new injectivity criteria for holomorphic curves in complex n-space by extending classical univalence conditions, combining Schwarzian derivatives for conformal maps and curves.

Contribution

It generalizes classical univalence criteria like Nehari's condition to higher-dimensional holomorphic curves using a combined Schwarzian derivative approach.

Findings

Established new injectivity criteria for holomorphic curves in $\

Generalized Nehari's univalence condition to $\

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 $\phi:\mathbb{D}\to\mathbb{C}^n$. The result can be considered a generalization of a classical condition for univalence of Nehari.