Conformal extension of metrics of negative curvature

We consider the problem of extending a conformal metric of negative curvature, given outside of a neighbourhood of 0 in the unit disk double struck D sign, to a conformal metric of negative curvature in double struck D sigh. We give conditions under which such an extension is possible and also give obstructions to such an extension. The methods we use are based on a maximum principle and the Ahlfors-Schwarz Lemma. We also give an example in which no extension is possible, even when the conformality condition is dropped. We apply these considerations to the compactification of Riemann surfaces.