Quasiconformal mappings of \(Y\)-pieces.
In order to deform a Riemann surface by shortening a closed geodesic, the author gives a quasiconformal mapping between \(Y\)-pieces. A Riemann surface with boundary is called a generalized \(Y\)-piece if it is homeomorphic to a 2-sphere minus three disks (or points) and the boundary consists of three closed geodesics (or punctures). When \(Y\)-pieces \(Y_1\), \(Y_2\) have three boundary lengths \((a_1,b_1,c_1)\), \((a_2,b_1,c_1)\), respectively, a quasiconformal mapping \(f:Y_1\to Y_2\) is considered such that it is isometric on two boundary components \(\gamma_b\), \(\gamma_c\) and multiplies length by \(a_2/a_1\) on \(\gamma_a\). Furthermore the Beltrami coefficient of \(f\) decays exponentially according to the distance from \(\gamma_a\). The quasiconformal mappings in this paper are used in another paper by the author [Rev. Mat. Iberoam. 18, No. 3, 653--684 (2002; Zbl 1064.30041)] to construct quasi-Fuchsian groups whose limit sets are non-rectifiable curves of dimension 1.
quasiconformal deformation, critical exponent
scientific article; zbMATH DE number 1931493
scientific article