Quasiconformal Homogeneity of Genus Zero Surfaces

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 $f\text{\hspace{0.17em}⁣}:M\to M$ 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 $K\ge K_0$. This answers a question by Gehring and Palka.