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. 



 Complete metric on a Riemann surface with punctures
A good reference for this is, say, Kobayashi and Nomizu "Foundations of Differential Geometry". The result you are looking for is: If $M$ is a complete Riemannian manifold and 
$p: M\to M'$ is a (locally) isometric covering map to another Riemannian manifold, then $M'$ is also complete. To prove this note that every geodesic in $M'$ lifts to a geodesic in $M$. Since geodesics in $M$ extend to bi-infinite geodesics, you conclude that $M'$ is geodesically complete. Now, use Hopf-Rinow Theorem. Actually, several converse statements to this are also true, e.g., if both $M, M'$ are complete and $p: M\to M'$ is a locally-isometric map, then $p$ is a covering map. A proof of this is a bit more difficult. 


