For a given ring (domain) in \(\overline{\mathbb {R}}^n\), we discuss whether its boundary components can be separated by an annular ring with modulus nearly equal to that of the given ring. In particular, we show that, for all \(n\ge 3\), the standard definition of uniformly perfect sets in terms of the Euclidean metric is equivalent to the boundedness of the moduli of the separating rings. We also establish separation theorems for a “half” of a ring. As applications of those results, we will prove boundary Hölder continuity of quasiconformal mappings of the ball or the half space in \(\mathbb {R}^n\).

中文翻译：

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高维Teichmüller定理及其应用

对于\（\ overline {\ mathbb {R}} ^ n \）中的给定环（域），我们讨论其边界分量是否可以由模数几乎等于给定环的环形环分隔。特别地，我们表明，对于所有\（n \ ge 3 \），关于欧几里德度量的均匀完美集的标准定义等同于分隔环的模的有界性。我们还为环的“一半”建立了分离定理。作为这些结果的应用，我们将证明\（\ mathbb {R} ^ n \）中球或半空间的拟保形映射的边界Hölder连续性。