Non-removability of Sierpinski spaces

Abstract

We prove that all Sierpiński spaces in ${\mathbb{S}}^n$, $n\ge 2$, are non-removable for (quasi)conformal maps, generalizing the result of the first named author arXiv:1809.05605. More precisely, we show that for any Sierpiński space $X\subset {\mathbb{S}}^n$ there exists a homeomorphism $f:{\mathbb{S}}^n\to {\mathbb{S}}^n$, conformal in ${\mathbb{S}}^n\setminus X$, that maps $X$ to a set of positive measure and is not globally (quasi)conformal. This is the first class of examples of non-removable sets in higher dimensions.

10 pages