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We prove a boundary version of the strong form of the Ahlfors-Schwarz lemmawith optimal error term. This result provides nonlinear extensions of theboundary Schwarz lemma of Burns and Krantz to the class of negatively curvedconformal pseudometrics defined on arbitary hyperbolic domains in the complexplane. Based on a new boundary Harnack inequality for solutions of the Gausscurvature equation, we also establish a sharp rigidity result for conformalmetrics with isolated singularities. In the particular case of constantnegative curvature this strengthens classical results of Nitsche and Heinsabout Liouville's equation $\Delta u=e^u$.

The strong form of the Ahlfors-Schwarz lemma at the boundary and a rigidity result for Liouville's equation