Asymptotic Conformality of the Barycentric Extension of Quasiconformal Maps

Katsuhiko Matsuzaki, Masahiro Yanagishita

We first remark that the complex dilatation of a quasiconformal homeomorphism of a hyperbolic Riemann surface $R$ obtained by the barycentric extension due to Douady--Earle vanishes at any cusp of $R$. Then we give a new proof, without using the Bers embedding, of a fact that the quasiconformal homeomorphism obtained by the barycentric extension from an integrable Beltrami coecient on $R$ is asymptotically conformal if $R$ satisfies a certain geometric condition.