The uniformization theorem states that every simply connected Riemann surface is conformally equivalent to exactly one of the Riemann sphere, the complex plane, or the unit disk. Consequently, the universal cover of every connected Riemann surface is one of these three surfaces.
The alternatives are called elliptic, parabolic, and hyperbolic uniformization, respectively. In particular, every Riemann surface admits a
complete conformal metric of constant Gaussian
curvature 1, 0, or 
after passage to its universal
cover.
