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

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 Δu = eu.