# Hyperbolic Metric on a Riemann Surface

From uniformization theorem, it is known that every conformal class of metrics on a genus-$g$ Riemann surface with $n$ punctures such that $2g+n\ge 3$ contains a unique hyperbolic metric. The punctures correspond to the fixed points of the parabolic elements of the associated Fuchsian group. The question is that: what is the explicit local expression of this unique hyperbolic metric for such a surface around a puncture, associated with the fixed point $x$ of a parabolic element? A good reference is highly appreciated.

• Thanks. I will remove my comments now, and you can do the same. – GH from MO Sep 25 '17 at 11:34

Choose an horocycle around the puncture. Then the end delimited by the horocycle is isometric to a cusp, which is obtained by quotienting the following domain of the Poincaré half-plane $$C_R = \{ z \in {\bf C} \mid Im(z) > R\}$$ by the translation $z\mapsto z+1$. In that model, the bounding horocycle is just the horizontal line $Im(z) = R$. The model $C_R$ is endowed with the usual hyperbolic metric ${|dz|^2 \over Im(z)^2} = {dx^2 + dy^2 \over y^2}$ and the geodesics are half circles orthogonal to $Im(z) = 0$ as always.
You can map $C_R$ to a punctured disk of radius $e^{-2\pi R}$ using the transform $w = e^{2\pi i z}$ if you want. The resulting metric is (the square of) ${|dw|\over |w|\ln(|w|)}$. But I think that the Poincaré half-plane model is nicer to work with.