Quasiconformal mappings and sets of finite perimeter

Abstract:

Let $D$ be a domain in ${R^n},n \geqslant 2,f$ a quasiconformal mapping on $D$. We give a definition of bounding surface of codimension one lying in $D$, and show that, given a system $\Sigma$ of such surfaces, the image of the restriction of $f$ to “almost every” surface is again a surface. Moreover, on these surfaces, $f$ takes ${H^{n - 1}}$ (Hausdorff $(n - 1)$-dimensional) null sets to ${H^{n - 1}}$ null sets. “Almost every” surface is given a precise meaning via the concept of the module of a system of measures, a generalization of the concept of extremal length.