A note on the asymptotic behavior of conformal metrics with negative curvatures near isolated singularities

TL;DR

This paper refines the understanding of the asymptotic behavior of conformal metrics with negative curvature near isolated singularities, extending previous estimates to higher derivatives using potential theory.

Contribution

It improves existing estimates for higher order derivatives near singularities and introduces new limits for SK-metrics near the origin.

Findings

Enhanced estimate for higher order derivatives near singularities.

New limits of Minda-type for SK-metrics near the origin.

Insights into SK-metrics using Ahlfors' lemma.

Abstract

The asymptotic behavior of conformal metrics with negative curvatures near an isolated singularity for at most second order derivatives was described by Kraus and Roth in one of their papers in 2008. Our work improves one estimate of theirs and shows the estimate for higher order derivatives near an isolated singularity by means of potential theory. We also give some limits of Minda-type for SK-metrics near the origin. Combining these limits with the Ahlfors' lemma, we provide some observations SK-metrics.