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

TL;DR

This paper generalizes Nitsche's classical results on the behavior of solutions to the Liouville equation near isolated singularities to the Gaussian curvature equation with a negative H"older continuous function, and applies this to a higher-order Yau–Ahlfors–Schwarz lemma.

Contribution

It extends the understanding of solution behavior near singularities for the Gaussian curvature equation and derives a higher-order Yau–Ahlfors–Schwarz lemma for complete conformal metrics.

Findings

Generalization of Nitsche's result to Gaussian curvature equation with variable curvature

Derivation of a higher-order Yau–Ahlfors–Schwarz lemma

Insights into the structure of solutions near isolated boundary points

Abstract

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