On the Role of Riemannian Metrics in Conformal and Quasiconformal Geometry

Source: arXiv

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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    ABSTRACT: We establish that the infinitesimal “ H -definition” for quasiconformal mappings on Carnot groups implies global quasisymmetry, and hence the absolute continuity on almost all lines. Our method is new even in R n where we obtain that the “limsup” condition in the H -definition can be replaced by a “liminf” condition. This leads to a new removability result for (quasi)conformal mappings in Euclidean spaces. An application to parametrizations of chord-arc surfaces is also given. Peer Reviewed
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