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 D, to a conformal metric of negative curvature in D. 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 compactification of Riemann surfaces.