I know next to nothing about Riemannian geometry, and have some trouble interpreting some things I read in relation to the uniformization theorem for Riemann surfaces.
To me the uniformization theorem says that the universal cover of any Riemann surfaces is biholomorphic to either $\mathbb{C}P^1$, $\mathbb{C}$ or $D$, the open unit disk with its complex structure given by that induced from $\mathbb{C}$(?).
In particular, for $S$ of genus $g>1$ we have a universal cover $D$, so $$\pi:D\to S$$ Now I do not see what this has to do with a metric of constant negative curvature? $D$ is just an open subset of $\mathbb{C}$, and so it seems to me that it carries a flat metric? I just don't see where in this story a metric of negative curvature shows up.