Every connected Riemann surface is a quotient of one simply connected model: $\CP^1$, $\CC$, or $\DD$.Suggest a slogan
Let \(S\) be a connected Riemann surface with universal cover \(\widetilde S\to S\). Then \(\widetilde S\) is a simply connected Riemann surface, and its group \(G\) of deck transformations acts on \(\widetilde S\) freely and properly discontinuously by biholomorphisms, with \(S\) biholomorphic to \(\widetilde S/G\). Moreover, \(\widetilde S\) is biholomorphic to exactly one of \(\CP^1\), \(\CC\), and \(\DD\).