  We prove a boundary version of the strong form of the Ahlfors-Schwarz lemma
with optimal error term. This result provides nonlinear extensions of the
boundary Schwarz lemma of Burns and Krantz to the class of negatively curved
conformal pseudometrics defined on arbitary hyperbolic domains in the complex
plane. Based on a new boundary Harnack inequality for solutions of the Gauss
curvature equation, we also establish a sharp rigidity result for conformal
metrics with isolated singularities. In the particular case of constant
negative curvature this strengthens classical results of Nitsche and Heins
about Liouville's equation $\Delta u=e^u$.
