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

TL;DR

This paper extends the Ahlfors-Schwarz lemma to boundary cases with optimal error bounds, introduces a boundary Harnack inequality for the Gauss curvature equation, and strengthens classical Liouville's equation results.

Contribution

It provides a boundary version of the strong Ahlfors-Schwarz lemma, a new boundary Harnack inequality, and a rigidity result for conformal metrics with isolated singularities.

Findings

Boundary Ahlfors-Schwarz lemma with optimal error term

New boundary Harnack inequality for Gauss curvature solutions

Strengthened classical results on Liouville's equation

Abstract

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 $\Delta u=e^u$.