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Asymptotic conformality of the barycentric extension of quasiconformal maps

Matsuzaki Katsuhiko (Waseda University, School of Education, Department of Mathematics, Tokyo, Japan)
Yanagishita Masahiro (Yamaguchi University, Graduate School of Sciences and Technology for Innovation, Department of Applied Science, Yamaguchi Prefecture, Japan)

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 coefficient on R is asymptotically conformal if R satisfies a certain geometric condition.

Keywords: integrable Teichmüller space, barycentric extension, complex dilatation, quasiconformal, asymptotically conformal, Teichmüller projection, Bers embedding