Conformal metrics and size of the boundary

We establish a lower bound for the Hausdorff dimension of the boundary associated with a conformal deformation of the Euclidean metric on the unit ball in [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i" /] The deformations we consider are motivated by quasiconformal maps. Our abstract approach leads to new results on the boundary behavior of these maps. We prove a conjecture by Hanson on the compression of sets and obtain an improved version of the so-called wall theorem. We also establish a Riesz-Privalov theorem in higher dimensions.