Rigid circle domains with non‐removable boundaries

We give a negative answer to the rigidity conjecture of He and Schramm by constructing a rigid circle domain on the Riemann sphere with conformally non‐removable boundary. Here, rigidity means that every conformal map from onto another circle domain is a Möbius transformation, and non‐removability means that there is a homeomorphism of which is conformal on but not everywhere. Our construction is based on a theorem of Wu, which states that the product of any Cantor set with a sufficiently thick Cantor set is non‐removable. We show that one can choose and so that the complement of the union of and suitably placed disks is rigid. The proof of rigidity involves a metric characterization of conformal maps, which was recently proved by Ntalampekos. The other direction of the rigidity conjecture, that is, whether removability of the boundary implies rigidity, remains open.