Every proper simply connected plane domain is conformally the unit disc; the whole plane is the only other case.Suggest a slogan
Every nonempty simply connected open set \(\Omega\subseteq\CC\) is conformally equivalent to exactly one of \(\CC\) and the unit disc \(\DD\): to \(\CC\) if \(\Omega=\CC\), and to \(\DD\) otherwise.
By the uniformization theorem, every simply connected Riemann surface is conformally equivalent to exactly one of \(\CP^1\), \(\CC\), and \(\DD\). For open subsets of \(\CC\) the case \(\CP^1\) does not occur because \(\CP^1\) is compact, and \(\CC\) is not conformally equivalent to \(\DD\) by Liouville’s theorem.