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Let $\mathcal{P}$ be a hyperbolic pair of pants with geodesic boundary and $\Sigma_3$ be the hyperbolic thrice punctures sphere. I want to construct a conformal map $f:\mathcal{P}\rightarrow \Sigma_3$ and apply Schwarz–Ahlfors–Pick theorem. I have the following questions.

1) Is there a version of Schwarz–Ahlfors–Pick theorem for Riemann surfaces with boundary as domain?

2) I have tried to construct the map $f$ in the following way. First attach a punctured euclidean disc at each boundary of $\mathcal{P}$ to get a smooth surface $\widetilde{\Sigma_3}$. Then construct a function $g$ on $\widetilde{\Sigma_3}$ to get a conformal Riemannian metric of negative curvature on the whole surface. Then apply Uniformization theorem. But I could not succeed to construct the function $g$. Is there an obstruction to construct such a function (like Gauss-Bonnet theorem)?

Any comment, suggestion or reference will be extremely helpful. Thanks in advance.

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  • $\begingroup$ 2) One way to construct f would be to notice that your hyperbolic pair of pants is a planar Riemann surface ,then apply Koebe uniformisation theorem to show that it is biholomorphic to a domain on the Riemann sphere . $\endgroup$ Jul 6, 2016 at 14:57

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