Teichm\"uller's theorem in higher dimensions and its applications

Anatoly Golberg; Toshiyuki Sugawa; Matti Vuorinen

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arXiv · 2006.01565

Teichm\"uller's theorem in higher dimensions and its applications

Anatoly Golberg·Toshiyuki Sugawa·Matti Vuorinen

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Abstract

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 Euclidean metric is equivalent to the boundedness of moduli of separating rings. We also establish separation theorems for a "half" of a ring. As applications of those results, we will prove boundary H\"older continuity of quasiconformal mappings of the ball or the half space in $\mathbb{R}^n.$

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Anatoly Golberg, Toshiyuki Sugawa, Matti Vuorinen. 2020-06-02. Teichm\"uller's theorem in higher dimensions and its applications. https://arxiv.org/abs/2006.01565

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