Rigid circle domains with non-removable boundaries

TL;DR

This paper constructs a specific rigid circle domain with a boundary that cannot be conformally removed, providing a counterexample to a conjecture about boundary rigidity and non-removability.

Contribution

It presents a novel construction of a rigid circle domain with a non-removable boundary, disproving the rigidity conjecture of He and Schramm.

Findings

Constructed a rigid circle domain with non-removable boundary.

Used Cantor sets and a theorem of Wu for the construction.

Proved rigidity via a metric characterization of conformal maps.

Abstract

We give a negative answer to the rigidity conjecture of He and Schramm by constructing a rigid circle domain $\Omega$ on the Riemann sphere with conformally non-removable boundary. Here rigidity means that every conformal map from $\Omega$ onto another circle domain is a M\"obius transformation, and non-removability means that there is a homeomorphism of the Riemann sphere which is conformal off $\partial \Omega$ but not everywhere. Our construction is based on a theorem of Wu, which states that the product of any Cantor set $E$ with a sufficiently thick Cantor set $F$ is non-removable. We show that one can choose $E$ and $F$ so that the complement of the union of $E \times F$ 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, i.e., whether removability of the boundary implies rigidity, remains open.