[Untitled]

The theory of quasiconformal mappings originated at the beginning of the twentieth century. At that time, it arose out of geometric reasons in the works of Grotzsch on the one hand and as solutions to special types of elliptic systems of differential equations in the works of Lavrentiev on the other hand. Important applications in various fields of mathematics such as discrete-group theory, mathematical physics, and complex differential geometry led to the great development of the theory of quasiconformal mappings, which at present is a branch of complex analysis. Great contributions to this theory were made by M. A. Lavrentiev, H. Grotzsch, L. Ahlfors (who became one of the first Fields laureates (1936)), L. Bers, O. Teichmuller, P. Belinskii, and L. Volkovyskii in the past and C. Earle, I. Kra, V. Zorich, S. Krushkal, Yu. Reshetnyak, A. Sychev, F. Gehring, O. Lehto, C. Andreyan-Cazacu, R. Kuhnau, V. Aseev, V. Gutlyanskii, and V. Sheretov more recently. Many methods, such as the area method, the variational method, the method of parametric representations, the length–area principle, and the extremal-length method, were developed for solving problems on quasiconformal mappings. Many of them became more or less transformations of the ideas of classical methods of conformal analysis. However, not all of them give us a similar effect in solution of specific problems. Therefore, we can compare the Lowner–Kufarev method [2,54] for univalent functions in the conformal case and the parametric method of Shah Dao-Shing [67] and Gehring–Reich [28] in the quasiconformal case. Only a few problems were solved by the latter method, but the main extremal problems such as the Bieberbach conjecture in the conformal case involved the Lowner method. Thus, the problems in the quasiconformal case are much more difficult. At midcentury, it was established that the classical methods of geometric function theory could be extended to complex hyperbolic manifolds. Teichmuller spaces became the most important of them. In 1939, O. Teichmuller [81] proposed and partially realized an adventurous program of investigations in a moduli problem for Riemann surfaces. In the present paper, we often use the word “modulus” in a different way. Therefore, we have to clarify that the classical moduli problem starts from the works of B. Riemann [61] and is based on the fact that the conformal structure of a compact Riemann surface of genus g > 1 depends on 3g − 3 complex parameters (moduli). The problem is to investigate the nature of these parameters for general Riemann surfaces and to induce a real or a complex structure into the corresponding space of Riemann surfaces. O. Teichmuller connected the moduli problem with extremal quasiconformal mappings and with relevant quadratic differentials. This led him to the wellknown theory of Teichmuller spaces. Teichmuller’s ideas were strictly substantiated then in works by L. Ahlfors and L. Bers [6] and other authors. Investigations in Teichmuller spaces were carried out in different directions: the topological one connected with homotopy classes of diffeomorphisms of surfaces of finite conformal type (W. Abikoff, R. Fricke, J. Nielsen, and W. Thurston), mathematical physics connected with applications in conformal-gauge string theory (L. Takhtadzhyan and P. Zograf), the theory of discontinuous groups (H. Zieschang, E. Fogt, H.-D. Coldewey, I. Kra, S. Krushkal, B. Apanasov, and N. Gusevskii), the theory of dynamical systems (R. Devaney, A. Douady, J. Hubbard, L. Keen