The Geometric Properties Of Quasiconformal Mappings And The Optimization Problem On Riemannian Manifolds

In this paper, we study the geometric properties of quasiconformal mappings andthe optimization problem on Riemannian manifolds. At the same time, we also study anapplications of the quasiconformal mappings on Teichmüller space. The paper consistsof five chapters:In Chapter 1, based on the history and application of quasiconformal mappingtheory, Teichmüller space theory and the optimization theory, we state the backgroundand significance of our research topic.In Chapter 2, we study the Schwarz type theorems of the homeomorphisms be-tween unit circles. We generalize the Schwarz lemma of holomorphic mappings, andobtain the Schwarz type theorems of quasiconformal mappings under the area distor-tion conditions, and the Schwarz type theorems of the homeomorphisms normalized onorigin under some conformal modulus conditions.In Chapter 3, we study the conformally natural extension to unit disk of the home-omorphism between unit circles. Firstly, we contruct a family of conformally natural,parametrized extension Dηfrom two known extensions F1,F2, and study its proper-ties. At the same time, we give a sufficient condition to insure that it is a family ofglobal homeomorphisms. Secondly, we define an inverse extension from the Douady-Earle extension, and obtain some good properties which is similar to the Douady-Earle extension. But the inverse extension is not always the same to the Douady-Earle exten-sion.In Chapter 4, we give another proof of that the asymtotic Bers mapping is an em-bedding from the asymtotic Teichmüller space to the asymtotic holomorphic quadraticdifferentials space by the inverse extension defined in Chapter 3.In Chapter 5, we study the optimization problem on Riemannian manifolds. Wedefine a variational inequality problem on finite-dimensional complete Riemannianmanifold, and obtain the equivalence relation with the optimization problem. We givethe conditions which grantee the existence and uniqueness of solution to this problem.Finally, we study the properties of solutions and solutions set.