2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/120374We consider the problem of extending a conformal metric of negative curvature, given outside a neighbourhood of 0 in the unit disk $\DD$, to a conformal metric of negative curvature in $\DD$. 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 apply these considerations to compactification of Riemann surfaces.In Memory of Robert Brooks. An analysis of Gromov's example added. 20 pages, 5 figures. to appear in Journal d'Analyse MathematiqueDifferential Geometry53c21; 30f45text