A conformally removable set is a compact subset 
of the Riemann sphere such that every orientation-preserving
homeomorphism of the sphere which is conformal on its complement is a Möbius
transformation. The homeomorphism in this definition is already defined on the
whole sphere. For example, the empty set is removable,
and a singleton is removable by the removable-singularity theorem. Removability of
a circle-domain boundary instead has to control conformal equivalences initially
defined only on the domain.
A conformally rigid circle domain need not have a conformally removable boundary. Rajala (2025) constructed such a counterexample to one direction of the He-Schramm conjecture.
