On the Role of Riemannian Metrics in Conformal and Quasiconformal Geometry

Conformal geometry and the theory of quasiconformal mappings are branches of mathematics that have a broad spectrum of applications ranging from theories in modern physics to recent applications in the inverse conductivity problem. Notably, the inverse conductivity problem is an especially active area of research with significant contributions from Finnish researchers. The fundamental quantities in these branches of mathematics are Riemannian metrics that many times define, or emerge as solutions to, nonlinear partial differential equations. Partly due to the nonlinearity of the equations in question, these branches of mathematics are still far from being complete theories. This thesis focuses on the role of Riemannian metrics in conformal geometry and on the theory of quasiconformal mappings. In general the results of this thesis imply that there is no simple classification of conformal mappings on Riemannian manifolds. A new coordinate invariant definition of quasiconformal and quasiregular mappings is presented and the basic properties of the new class of mappings are established. It is shown that any countable quasiconformal group on a Riemannian manifold (in the introduced sense) can be regarded as a group of conformal mappings with respect to another, optimal, Riemannian metric. In a converse manner, another result of this thesis shows that any smooth manifold of dimension 3 or higher admits infinitely many Riemannian metrics such that there is no conformal diffeomorphisms on the manifold. The principle of how to find an optimal Riemannian metric for a group of mappings is developed further. It is shown that if the action of a volume form preserving diffeomorphism has a bounded orbit in the space of Riemannian metrics, then a new Riemannian metric can be found such that the diffeomorphism can be regarded as an isometry. The proof of this result relies on generalizations of Neumann's mean ergodic theorem and fixed point theorem to certain nonpositive curvature metric spaces. The generalizations are formulated and proven in this thesis. Finally, implications of the regularity of Riemannian metrics in conformal geometry are studied. A new proof of a regularity theorem of conformal mappings between two Riemannian manifolds is achieved. The proof is based on a new coordinate system that generalizes both the harmonic coordinates and the isothermal coordinates. The existence of such coordinates on any Riemannian manifold is established. Additionally, a convergence theorem for conformal mappings is given.