Asymptotic Conformality of the Barycentric Extension of Quasiconformal Maps

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 coe ffi cient on R is asymptotically conformal if R satisfies a certain geometric condition.