Variants of Ahlfors' lemma and properties of the logarithmic potentials

TL;DR

This paper explores variants of Ahlfors' lemma within SK-metric spaces and derives a higher order derivative formula for logarithmic potentials, aiding in estimates near singularities of negatively curved conformal metrics.

Contribution

It introduces new variants of Ahlfors' lemma tailored for SK-metric spaces and provides a higher order derivative formula for logarithmic potentials.

Findings

Derived a higher order derivative formula for logarithmic potentials.

Extended Ahlfors' lemma variants to SK-metric spaces.

Enhanced estimates near singularities of negatively curved conformal metrics.

Abstract

As a special class of conformal metrics with negative curvatures, SK-metrics play a crucial role in metric spaces. This paper concerns the variants of Ahlfors' lemma in an SK-metric space and gives a higher order derivative formula for the logarithmic potential function, which can be applied for the estimates near the singularity of a conformal metric with negative curvatures.