Conformal mappings for domains whose complement is totally disconnected

Question

Consider a conformal homeomorphism $f \colon \Omega \to \Omega'$ between domains $\Omega, \Omega'$ in the plane. Let $E = \mathbb{R}^2 \setminus \Omega$ and $E' = \mathbb{R}^2 \setminus \Omega'$. My understanding is that you can have a situation where $E$ is totally disconnected and yet $E'$ is not. If we think of the map $f$ as extending to a map between the connected components of $E$ and the connected components of $E'$, then $f$ would take a point $x \in E$ to a non-trivial continuum in $E'$.

Is there a standard, more-or-less explicit example of such a set $E$ and map $f$?

Answer 1

The only example I know of a conformal map that "stretches" a point boundary component to a nondegenerate continuum was constructed by Gehring and Martio in Quasiextremal distance domains and extension of quasiconformal mappings, J. Analyse Math. 45 (1985), 181–206. See Theorem 4.1.

The authors use earlier results of Ahlfors and Beurling to construct a Cantor set $E$ of positive area whose complement is mapped conformally to a domain whose boundary components are single points and the unit circle. The construction is based on the following classical result relating the existence of such a map with extremal length considerations:

Theorem Let $\Omega \subset \mathbb{C}$ be a domain and let $z_0 \in \partial \Omega$ be a point boundary component. The following are equivalent :

• There is a conformal map on $\Omega$ that sends $z_0$ to non-degenerate boundary component of $f(\Omega)$;
• There exists $r>0$ such that $\operatorname{mod}(\Gamma) < \infty$, where $\Gamma$ is the family of all closed curves in $\Omega \cap B(z_0,r)$ having nonzero winding number around $z_0$.

The idea is to start with a single point, say $0$, and surround it by sufficiently many point boundary components in order to block the curves winding around $0$ so that the modulus is finite. This implies that $0$ is mapped by a nondegenerate continuum under some conformal map $f$.

Lastly, let me mention that your question, more specifically the existence of such a set $E$ which is small (at least zero area), is related to a very difficult conjecture of He and Schramm on the rigidity of circle domains in Koebe's uniformization conjecture. See Section 9 of my paper with Ntalampekos

D. Ntalampekos and M. Younsi, Rigidity theorems for circle domains, Invent. Math. 2019