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From uniformization theorem, it is known that every conformal class of metrics on a Riemann surface contains a unique hyperbolic metric. For a genus-$g$ Riemann surface with $n$ punctures, the punctures correspond to 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.

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  • $\begingroup$ Did you mean to start from $X \cong \Gamma/\mathbb{H}$ with $\Gamma \subset \text{SL}(2,\mathbb{R})$, set $E = \Gamma \cap \left(\begin{smallmatrix}1 & * \\0 & 1\end{smallmatrix}\right)$ and take the points of $X$ fixed by $E$ ? Then the hyperbolic metric on $\mathbb{H}$ is $\frac{dx^2+dy^2}{y^2}$ so it should be the same as finding a covering $\mathbb{H} \to X$. $\endgroup$ – reuns Sep 23 '17 at 10:10
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In the hyperbolic plane model $\mathbb{H}$, set the fixed point of the parabolic element at $\infty$, and choose the parabolic element to be $\gamma(z)=z+1$. We know that the infinitesimal hyperbolic metric for $\mathbb{H}$ is given by $|ds|=\frac{|d\xi|}{\operatorname{Im}(\xi)}$, where $\operatorname{Im}(\xi)$ denotes the imaginary part of $\xi \in \mathbb{H}$ and $|d\xi|$ is the Euclidean infinitesimal element of length. There is a holomorphic map $\phi: \mathbb{H} \to \mathbb{C} \backslash\{0\}$ given by $\xi \mapsto e^{2\pi i \xi}$. If we restrict to the fundamental domain of $\langle \gamma \rangle$ given by $F:=\{ \xi \in \mathbb{H} : 0<\operatorname{Re}(\xi) <1 \}$, then $\phi(F)=\mathbb{D}\backslash\{ 0\}$ biholomorphically. Let $\psi:=(\phi|_{F})^{-1} : \mathbb{D}\backslash\{ 0\} \to F \subset \mathbb{H}$. This map is $z \mapsto \xi=\frac{\log(z)}{2\pi i}$. We can compute the hyperbolic metric in $\mathbb{D}\backslash\{ 0\}$ by pulling back the metric in $F \subset \mathbb{H}$ via $\psi$.

First, note that

$$\frac{\log(z)}{2\pi i}=\frac{\log|z|}{2\pi i} + \frac{1}{2\pi}\arg(z).$$

So that $\operatorname{Im}(\psi(z))=-\frac{\log|z|}{2\pi}$.

Moreover, $\psi'(z)=\frac{1}{2\pi i z}$.

If we compute the pullback mentioned above, we get the local expression of the hyperbolic metric around a cusp, as we desired.

$$|dr|=\psi^*|ds|=\psi^*\frac{|d\xi|}{|\operatorname{Im}(\xi)|}=\frac{|\psi'(z)||dz|}{|\operatorname{Im}(\psi(z))|}=\frac{1}{2\pi|z|}\frac{2\pi}{\log|z|}=\frac{1}{|z|\log|z|}.$$

Depending on the kind of computations you want to do, there might be other convenient coordinates to describe this metric. Another example are horocyclic coordinates.

enter image description here

Consider again the hyperbolic plane model, and pick a horocycle centered at some point $p_0$ of the ideal boundary $\mathbb{R} \cup \infty$. In figure A, we are choosing $p_0=0$ and the horocycle is drawn in red, and parametrized arc length via $h(t)$, with the convention that $p_0$ lies on the left-hand-side of $h$. For an arbitrary point $p$ (blue in figure A), consider the unique geodesic intersecting $h$ orthogonally and going through $p$ (the geodesic is green in figure A). We define two coordinates: 1. Denote $\rho$ the distance from $h$ to $p$ along this geodesic, with the convention of $\rho$ being positive if it lies on the right-hand-side of $h$. 2. Let $h(t)$ be the intersection point of $h$ with the green geodesic, and let the second coordinate be $t$.

The coordinates $(\rho,t)$ for a fixed horocycle with center in the cusp are called the horocyclic coordinates. In figure B we represent these coordinates choosing the cusp to be at $\infty$ and thus $p_0=\infty$, as we did in the beginning of this post. Now horocycles look like horizontal lines. We choose the specific horocycle at height $y=1$ (in red), with the parametrization $h(t)=t+i$, and we define $\rho$ and $t$ as before. With this specific choice of horocycle, the metric tensor is

$$dr^2=d\rho^2 + e^{2\rho} dt^2$$

and this gives another expression for the local metric around a cusp.

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  • $\begingroup$ Thank you very much for the detailed answer. Can you please refer to a reference on this topic where I can find more details about hyperbolic structure on a Riemann surface? $\endgroup$ – QGravity Apr 22 '18 at 4:29
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    $\begingroup$ Chapters 1, 2 and 3 in John Hubbard's book "Teichmüller Theory and Applications to Geometry, Topology, and Dynamics, volume 1". I would also recommend the first chapters in "Geometry and Spectra of Compact Riemann Surfaces" by Peter Buser. He gives a detailed description of horocyclic coordinates. $\endgroup$ – DMG Apr 22 '18 at 13:47

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