Quasiconformal Jordan Domains

IF 0.9 3区数学 Q2 MATHEMATICS Analysis and Geometry in Metric Spaces Pub Date : 2020-11-14 DOI:10.1515/agms-2020-0127

Toni Ikonen

引用次数: 2

Abstract

Abstract We extend the classical Carathéodory extension theorem to quasiconformal Jordan domains (Y, dY). We say that a metric space (Y, dY) is a quasiconformal Jordan domain if the completion ̄Y of (Y, dY) has finite Hausdorff 2-measure, the boundary ∂Y = ̄Y \ Y is homeomorphic to 𝕊1, and there exists a homeomorphism ϕ: 𝔻 →(Y, dY) that is quasiconformal in the geometric sense. We show that ϕ has a continuous, monotone, and surjective extension Φ: 𝔻 ̄ → Y ̄. This result is best possible in this generality. In addition, we find a necessary and sufficient condition for Φ to be a quasiconformal homeomorphism. We provide sufficient conditions for the restriction of Φ to 𝕊1 being a quasisymmetry and to ∂Y being bi-Lipschitz equivalent to a quasicircle in the plane.

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拟共形Jordan域

将经典的carath扩展定理推广到拟共形Jordan域(Y, dY)。如果(Y, dY)的补全Y具有有限的Hausdorff 2测度，边界∂Y =∈Y \ Y与𝕊1同胚，并且存在一个几何意义上拟共形的同胚φ: →(Y, dY)，则称度量空间(Y, dY)是拟共形的Jordan定义域。我们证明了Φ具有连续、单调和满射的扩展Φ: _→Y _。这个结果在这种一般性中是最好的。此外，我们还得到了Φ是拟共形同胚的一个充分必要条件。我们提供了限制Φ为准对称且∂Y为平面上的准圆的双lipschitz等价的充分条件。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Analysis and Geometry in Metric Spaces Mathematics-Geometry and Topology

CiteScore

1.80

自引率

0.00%

发文量

审稿时长

16 weeks

期刊介绍： Analysis and Geometry in Metric Spaces is an open access electronic journal that publishes cutting-edge research on analytical and geometrical problems in metric spaces and applications. We strive to present a forum where all aspects of these problems can be discussed. AGMS is devoted to the publication of results on these and related topics: Geometric inequalities in metric spaces, Geometric measure theory and variational problems in metric spaces, Analytic and geometric problems in metric measure spaces, probability spaces, and manifolds with density, Analytic and geometric problems in sub-riemannian manifolds, Carnot groups, and pseudo-hermitian manifolds. Geometric control theory, Curvature in metric and length spaces, Geometric group theory, Harmonic Analysis. Potential theory, Mass transportation problems, Quasiconformal and quasiregular mappings. Quasiconformal geometry, PDEs associated to analytic and geometric problems in metric spaces.

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