Quasiconformal Homogeneity of Genus Zero Surfaces

TL;DR

This paper proves the existence of a universal quasiconformal constant for hyperbolic genus zero surfaces, establishing a lower bound for their quasiconformal homogeneity, and answering a longstanding question.

Contribution

It demonstrates a universal lower bound for the quasiconformal constant of hyperbolic genus zero surfaces, excluding the disk, advancing understanding of their geometric structure.

Findings

Existence of a universal constant K_0 > 1 for hyperbolic genus zero surfaces

If K-quasiconformally homogeneous, then K ≥ K_0

Answers a question posed by Gehring and Palka

Abstract

A Riemann surface $M$ is said to be $K$-quasiconformally homogeneous if for every two points $p,q \in M$, there exists a $K$-quasiconformal homeomorphism $f \colon M \rightarrow M$ such that $f(p) = q$. In this paper, we show there exists a universal constant $K_0 > 1$ such that if $M$ is a $K$-quasiconformally homogeneous hyperbolic genus zero surface other than the disk $\mathbb{D}$, then $K \geq K_0$. This answers a question by Gehring and Palka.