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

For a given ring (domain) in ${\bar{\mathbb{R}}}^n$ 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 ${\mathbb{R}}^n\mathrm{.}$