Teichmullers theorem in higher dimensions and its applications

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 $nge 3,,$ 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 Holder continuity of quasiconformal mappings of the ball or the half space in $mathbb{R}^n.$