Geometric invariants associated with projective structures and univalence criteria

TL;DR

This paper explores geometric invariants linked to projective structures on Riemann surfaces, establishing relations between Schwarzian derivatives and deriving univalence criteria for meromorphic functions.

Contribution

It introduces invariant Schwarzian derivatives of general virtual order and relates them to projective Schwarzian derivatives, providing new univalence criteria.

Findings

Relation between invariant and projective Schwarzian derivatives for virtual orders 2 and 3

Univalence criteria for meromorphic functions based on projective Schwarzian derivatives

New geometric invariants for projective structures on Riemann surfaces

Abstract

For a nonconstant holomorphic map between projective Riemann surfaces with conformal metrics, we consider invariant Schwarzian derivatives and projective Schwarzian derivatives of general virtual order. We show that these two quantities are related by the "Schwarzian derivative" of the metrics of the surfaces (at least for the case of virtual orders 2 and 3). As an application, we give univalence criteria for a meromorphic function on the unit disk in terms of the projective Schwarzian derivative of virtual order 3.