Conformal Rigidity of Polar Set Complements

TL;DR

This paper proves that any conformal map on the complement of a closed polar set in the complex plane must be a Möbius transformation, establishing rigidity and discreteness of automorphism groups for such complements.

Contribution

It demonstrates that conformal maps on complements of closed polar sets are Möbius maps, revealing a rigidity property using elementary potential theory.

Findings

Conformal maps on complements of closed polar sets are Möbius transformations.

The automorphism group of such complements is a discrete subgroup of the Möbius group.

The result applies when the polar set has at least two points.

Abstract

Let $E$ be a closed polar subset of $\mathbb{C}$. In this short note, we use elementary potential theoretic tools to show that any conformal map on $\mathbb{C}\setminus{E}$ is necessarily a M\"{o}bius map. As a consequence we obtain that the group of conformal automorphisms of the complement of a closed polar set $E$ is a discrete subgroup of the M\"{o}bius group, provided $\lvert E \rvert \ge 2$.