On removable sets for quasiconformal mappings
Let \(QCH(E)=\{\)homeomorphisms of \(\mathbb{R}^n\to\mathbb{R}^n\) which are quasiconformal off \(E\}\); call \(E\) removable if every \(f\) in \(QHC(E)\) is quasiconformal in \(\mathbb{R}^n\). The authors extend a result of \textit{L. Carleson} [Ark. Mat. 1, 311-318 (1951; Zbl 0042.30902)] and \textit{F. W. Gehring} [Ann. Acad. Sci. Fenn., Ser. AI 281, 1-28 (1960; Zbl 0090.05303)] on removable sets for \(QHC\) in \(\mathbb{R}^2\) to \(\mathbb{R}^n\). This theorem also generalizes a recent result of \textit{J. Heinonen} and \textit{P. Koskela} [Definitions of quasiconformality (to appear)] regarding the removability of certain Cantor type sets in \(\mathbb{R}^n\) to a broader class of sets. Given \(\{\alpha_k\}\), \(0<\alpha_k<1\), a set \(S\subset\mathbb{R}^n\) is called \(\{\alpha_k\}\)-porous if there exists a sequence of coverings \(C_k=\{B_{k,j}=B(x_{k,j},r_{k,j})\}\) of \(S\) by balls with mutually disjoint interiors, so that each \(B_{k,j}\backslash S\) contains a ring \(R_{k,j}=\{x:(1-\alpha_k)r_{k,j}<|x-x_{k,j}|<r_{k,j}\}\), \(\bigcup_{C_{k+1}}B_{k+1,j}\subset\bigcup_{C_k}(B_{k,j}\backslash R_{k,j})\) and \(\sup_j r_{k,j}\to 0\) as \(k\to\infty\). The authors prove: Theorem 1: If \(n\geq m\geq 2\) and \(S\) is an \(\{\alpha_k\}\)-porous set in \(\mathbb{R}^m\) with \(\sum\alpha_k=\infty\), then \(S\times\mathbb{R}^{n-m}\) is removable for \(QCH\) in \(\mathbb{R}^n\). The hypothesis \(\sum \alpha_k=\infty\) assures that \(S\) has zero \(n\)-dimensional Hausdorff measure. In the planar setting, they prove a related specialized result for extending a meromorphic function across a porous set. Note that the \(QCH\) definition assumes the maps are homeomorphisms as an a priori condition. Work of Martio and Näkki on the homeomorphic continuation of quasiconformal maps in conjunction with present results allow the authors to prove the following extension theorem as well: If \(n\geq 2\) and \(S\) is an \(\{\alpha_k\}\)-porous set in \(\mathbb{R}^n\) with \(\sum\alpha_k=\infty\) and if \(f\) is quasiconformal on \(\mathbb{R}^n\backslash S\), then \(f\) can be extended to be quasiconformal on \(\mathbb{R}^n\). Finally, the authors provide a partial answer to a question of \textit{C. Bishop} [Ann. Acad. Sci. Fenn., Ser. AI, Math. 19, 323-338 (1994; Zbl 0810.30007)] regarding \(CH(E)=\{\)homeomorphisms on \(\mathbb{R}^2\), conformal off \(E\}\). Given any set \(E\subset\mathbb{R}^2\) of positive area, they construct a subset \(F\) of \(E\) with positive area and a map \(G\in CH(E)\) such that \(G(F)\) has zero area.
scientific article; zbMATH DE number 896437