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This question is related to another question.

If we have a Riemann surface with punctures of negative Euler characterstisc, how can one define a complete hyperbolic metric?

I know that in this case the universal cover is the hyperbolic plane and it has a complete metric. Do we project this metric to the puntured surface? If so, why is it complete?

I will deeply appreciate if somebody gives an example or a good reference.

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