Quasiconformal Mappings And Their Applications To Harmonic Mappings

The theory of quasiconformal mappings is one of the most important branches in the domain of complex analysis. It affects the developments of other mthematical subjects such as differential geometry, partial differential equation and topology. It also has extensive applications to many applied subjects such as elasticity theory, hydrodynamics, robotics, dynamical systems and biology. So it is meaningful to study the theory of quasiconformal mappings and its applicatioons. In this dissertation we aim to study the extrmal theory of quasiconformal mappings and the applications of quasiconformal mappings to harmonic mappings.First, for a uniquely extremal Beltrami coefficient, we give a necessary and sufficient condition to determine the Teichmuller equivalent calss induced by a truncation is a Strebel point, and use it to provide a method to construct a Hamilton sequence. By the Beltrami coefficents of the form of truncations, we get a sufficient condition to judge the extremality of the quasiconformal mappings and quasiconformal deformations determined by the Lowner equation is equivalent. Next, after studying the harmonicity of C~2-Teichmiiller mappings, we prove that there are no solutions to the Schoen conjecture among the class of C~2-Teichmuller mappings. Last, we build corresponding differential equations for different problems about harmonic quasiconformal mappings, and prove that the Beurling-Ahlfors extension of a quasisymmetric homeomorphism, which is odd and in C~2, is not the solution to the Schoen conjecture, and the inverse of aπ-harmonic quasiconformal mapping of the upper half plane onto itself is a solution to the Schoen conjecture if and only if it is conformal. This dissertation is organized as follows.In Chapter I we first give an introduction of the background of quasiconformal mappings, and then narrate the origin and developments of the problems what we study in this dissertation. We concisely enumerate the main results of the dissertation, too.In ChapterⅡwe study some extremal problems of quasiconformal mappings. We give a necessary and sufficient condition to decide whether the Teichmiiller equivalent class [α] of a truncationαinduced by a uniquely extremal Beltrami coefficient is a Strebel point in T. We also obtain a necessary and sufficient condition of the unique extremality ofα. Using the properties of truncations we provide a method to construct Hamilton sequences. We also get a sufficient condition for the extremality of F(w, t) to be equivalent to that of f(z, t) which is a solution determined by the Lowner equation. Some corresponding results in the infinitesimal case are obtained, too.In ChapterⅢwe study the relation between Teichmiiller mappings and harmonic mappings. We present a necessary and sufficient condition for a C~2-Teichmuller mapping to beρ-harmonic. By this result we show that there is no solution to the Schoen conjecture in the class of C~2-Teichmuller mappings. We also obtain two characterizations forπ-harmonic Teichmiiller mappings.In ChapterⅣwe study the connection between Beurling-Ahlfors extensions and harmonic mappings. We obtain a necessary condition for a Beurling-Ahlfors extension to be hyperbolic harmonic. Particularly, if a boundary correspondence h is in C~2 and odd, then the Beurling-Ahlfors extension of h is not hyperbolic harmonic. We also show that if h is in C~2 piecewisely, then the Beurling-Ahlfors extension of h is notπ-harmonic unless h(x)=ax+b,x∈R.In ChapterⅤwe study inversible harmonic quasiconformal mappings. Using the (?) and (?) energy densities, we build a partial differential equation for a reversible harmonic diffeomorphism with respect to (ρ,σ). As an application of this result we show that if f is aπ-harmonic mapping of the upper half plane onto itself then its inverse is harmonic with respect to the Poincare metric if and only if it is a conformal mapping. As another application of this result we obtain a new necessary and sufficient condition for the minimal surface lifted by a univalent harmonic function to be a plane.