Conformal Rigidity of Polar Set Complements

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 $\lvert E \rvert \ge 2$.