On the Role of Riemannian Metrics in Conformal and Quasiconformal Geometry

Tony Liimatainen

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arXiv · 1110.0639

On the Role of Riemannian Metrics in Conformal and Quasiconformal Geometry

Tony Liimatainen

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Abstract

This article is the introductory part of authors PhD thesis. The article presents a new coordinate invariant definition of quasiregular and quasiconformal mappings on Riemannian manifolds that generalizes the definition of quasiregular mappings on $\R^n$. The new definition arises naturally from the inner product structures of Riemannian manifolds. The basic properties of the mappings satisfying the new definition and a natural convergence theorem for these mappings are given. These results are applied in a subsequent paper, arXiv:1209.1285. In the current article, an application, likewise demonstrating the usability of the new definition, is given. It is proven that any countable quasiconformal group on a general Riemannian manifolds admits an invariant conformal structure. This result generalizes a classical result by Pekka Tukia in the countable case.

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Tony Liimatainen. 2013-10-14. On the Role of Riemannian Metrics in Conformal and Quasiconformal Geometry. https://arxiv.org/abs/1110.0639

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