# Quasispheres and metric doubling measures

## TL;DR

This paper characterizes quasispheres among metric two-spheres using the Bonk-Kleiner criterion, showing they are precisely those that are linearly locally connected and admit a weak metric doubling measure, linking geometric and measure-theoretic properties.

## Contribution

It provides a new characterization of quasispheres via weak metric doubling measures and linear local connectivity, extending the understanding of their geometric structure.

## Key findings

- Quasispheres are characterized by linear local connectivity and weak metric doubling measures.
- The Bonk-Kleiner criterion is effectively applied to identify quasispheres.
- The measure deformation condition is key to understanding quasisphere geometry.

## Abstract

Applying the Bonk-Kleiner characterization of Ahlfors 2-regular quasispheres, we show that a metric two-sphere $X$ is a quasisphere if and only if $X$ is linearly locally connected and carries a weak metric doubling measure, i.e., a measure that deforms the metric on $X$ without much shrinking.