Universal locally univalent functions and universal conformal metrics with constant curvature

TL;DR

This paper establishes universality theorems for locally univalent functions and conformal metrics with constant curvature, demonstrating the existence of universal functions and metrics in complex analysis.

Contribution

It introduces new universality results for locally univalent functions and conformal metrics, extending previous work and providing explicit constructions in hyperbolic domains.

Findings

Existence of universal bounded locally univalent functions on the unit disk

Universal conformal metrics with prescribed constant curvature in hyperbolic domains

Refinement of a result by M. Heins on locally univalent functions

Abstract

We prove Runge-type theorems and universality results for locally univalent holomorphic and meromorphic functions. Refining a result of M. Heins, we also show that there is a universal bounded locally univalent function on the unit disk. These results are used to prove that on any hyperbolic simply connected plane domain there exist universal conformal metrics with prescribed constant curvature.