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Let $X$ be a hyperbolic compact connected Riemann surface. Let $U\subset X$ be an open subset. Assume that $U\neq X$.

We can uniformize $X$ by $\mathbf{H}$ directly to obtain it as a quotient of $\mathbf{H}$ by some cofinite Fuchsian group $\Gamma$ without cusps nor elliptic elements.

But we can also uniformize $U$ in the same way and then obtain $X$ by adding the set $X-U$ of cusps.

Can these uniformizations be related in some sense? Even abstractly speaking? Have such "different" uniformizations been studied in some sense?

It's a bit of a vague question, I admit. I'm just wondering what exactly can be done in this context.

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The Schwarz lemma implies that lengths of curves in the uniformization of $U$ are strictly larger than the length of the curve in the uniformization of $X$.

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