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.



Hyperbolic Metric on a Riemann Surface
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.


A reference is the book of Hubbard, *Teichmüller Theory and Applications to Geometry, Topology, and Dynamics, Volume 1*.


