Conformal Rigidity of Polar Set Complements

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 Möbius 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 Möbius group, provided $|E|\ge 2$.

5 pages