Uniformization of a punctured compact Riemann surface
= Uniformization of a punctured compact Riemann surface
{title2=$2g-2+n$}
Let $R$ be a compact <Riemann surface> of <genus> $g$, and remove $n$ distinct points. The punctured surface is uniformized by the <unit disc> exactly when $2g-2+n>0$. It is uniformized by the <complex plane> exactly when $2g-2+n$ is $-1$ or $0$, corresponding respectively to $\mathbb C$, and to $\mathbb C^*$ or a <complex torus>. The sole spherical case is $g=n=0$.