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I was reading the book "Conformal Invariants" of L. Ahlfors, and seen (P.74) he says that "It is in fact obvious that the part of the teichmuller annulus (for R=1) in the upper plane is conformally ...

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 $f:\mathcal{P}\rightarrow \Sigma_3$ ...