The behaviour of solutions of the Gaussian curvature equation near an isolated boundary point

Abstract

A classical result of Nitsche \cite{Nit57} about the behaviour of the solutions to the Liouville equation $\Delta u=4e^{2u}$ near isolated singularities is generalized to solutions of the Gaussian curvature equation $\Delta u=-\kappa (z)e^{2u}$ where $\kappa$ is a negative Hölder continuous function. As an application a higher--order version of the Yau--Ahlfors--Schwarz lemma for complete conformal Riemannian metrics is obtained.