# Boundary behaviour of some conformal invariants on planar domains

## TL;DR

This paper investigates how certain conformal invariants behave near the boundary of planar domains using the scaling principle, focusing on the Aumann–Carathéodory rigidity constant, higher order curvatures, and recently defined conformal metrics.

## Contribution

It introduces a boundary analysis of conformal invariants on planar domains, applying the scaling principle to new and existing metrics and curvature measures.

## Key findings

- Boundary behaviour characterized for the Aumann–Carathéodory rigidity constant
- Higher order curvatures of the Carathéodory metric analyzed near boundaries
- Properties of two recently defined conformal metrics elucidated

## Abstract

The purpose of this note is to use the scaling principle to study the boundary behaviour of some conformal invariants on planar domains. The focus is on the Aumann--Carath\'{e}odory rigidity constant, the higher order curvatures of the Carath\'{e}odory metric and two conformal metrics that have been recently defined.