Bounded Geometry and Families of Meromorphic Functions with Two Asymptotic Values

TL;DR

This paper characterizes a class of universal covering maps with two asymptotic values, showing that those with finite post-singular sets are equivalent to meromorphic maps with constant Schwarzian derivative under a bounded geometry condition.

Contribution

It establishes a precise criterion linking bounded geometry to the combinatorial equivalence of certain transcendental meromorphic functions.

Findings

Maps with finite post-singular sets are equivalent to meromorphic maps with constant Schwarzian derivative under bounded geometry.

The paper provides a topological classification of AV2 class maps based on geometric conditions.

Bounded geometry is necessary and sufficient for the combinatorial equivalence in this context.

Abstract

In this paper we consider the topological class, denoted by AV2, of universal covering maps from the plane to the sphere with two removed points. We prove that an element f in AV2 with finite post-singular set is combinatorially equivalent to a meromorphic transcendental map g with constant Schwarzian derivative if any only if f satisfies a Bounded Geometry condition.