Quasiconformal maps, analytic capacity, and non linear potentials

In this paper we prove that if φ:ℂ→ℂ is a K-quasiconformal map, with K >1, and E ⊂ ℂ is a compact set contained in a ball B. As a consequence, if E is not K-removable (i.e., removable for bounded K-quasiregular maps), it has positive capacity C C 2K/2K+1 , 2K+1/K+1. This improves previous results that assert that E must have non-σ-finite Hausdorff measure of dimension 2/K+1 . We also show that the indices 2K/2K+1 , 2K+1/K+1 are sharp, and that Hausdorff gauge functions do not appropriately discriminate which sets are K-removable. So essentially we solve the problem of finding sharp "metric" conditions for K-removability. © 2013.