Given a compact Riemann surface on which one is given a conformal metric $ds^2$. In terms of a local holomorphic coordinate $w$ on $S$, $ds^2 = h(w)|dw|^2$ and the Gaussian curvature is $K(w) = \frac{-4}{h(w)}\frac{\partial^2 logh(w)}{\partial w \partial \bar w}$.
Are there any references on the conformal metric of a Riemann surface? I am familiar with Rieamnnian geometry and one-variable complex analysis. We have been through some very basics of the Rieamann surface in class.