Boundary Behaviors for Liouville's Equation in Planar Singular Domains

TL;DR

This paper investigates the asymptotic boundary behaviors of constant curvature metrics in planar domains with singularities, providing optimal estimates by comparing with tangent cone metrics, emphasizing the role of conformal structure.

Contribution

It introduces precise asymptotic estimates for boundary behaviors of Liouville's equation in singular domains, highlighting the importance of conformal structure.

Findings

Established optimal boundary estimates for metrics near singular points

Linked boundary behavior to tangent cone metrics

Highlighted the role of conformal structure in boundary asymptotics

Abstract

We study asymptotic behaviors near the boundary of complete metrics of constant curvature in planar singular domains and establish an optimal estimate of these metrics by the corresponding metrics in tangent cones near isolated singular points on boundary. The conformal structure plays an essential role.