Conformal removability is hard

Abstract

A planar compact set $E$ is called conformally removable if every homeomorphism of the plane to itself that is conformal off $E$ is conformal everywhere, and hence linear. Characterizing such sets is notoriously difficult, and in this paper, we partially explain this by showing that the collection of conformally removable subsets of $S=[0,1{]}^2$ is not a Borel subset of the space of compact subsets of $S$ with its Hausdorff metric. We give some similar results for other classes of removable sets and pose a number of open problems related to removability and conformal welding, using the language of descriptive set theory.