Quasiconformal extension from curvilinear triangles

Abstract

We consider the problem of extension of a quasi-Mobius (quasisymmetric) mapping from a planar curvilinear triangle to a quasiconformal automorphism of the extended plane. Existence of an extension of the sort from the polygons whose sides are quasiarcs was proven by A. K. Varisov but without any upper estimates for the quasiconformality coefficient of the extended mapping. The application of another method makes it possible to obtain, for triangular domains, an upper estimate of dilation of the quasiconformal extension depending only on the distortion function of the initial quasi-Mobius mapping, on the upper estimate of the turning of the sides and the Mobius-invariant characteristic of nondegeneracy of a curvilinear triangle which is introduced in this article and is analogous to the relative radius of the inscribed circle.