Weak chord-arc curves and double-dome quasisymmetric spheres

TL;DR

This paper characterizes when double-dome surfaces over planar Jordan domains, with height depending on boundary distance raised to a power, are quasisymmetric to the sphere, based on geometric conditions of the domain.

Contribution

It provides necessary and sufficient conditions on the domain and exponent for the double-dome surfaces to be quasisymmetric to the sphere, linking geometric properties to quasisymmetry.

Findings

Double-dome surfaces are quasisymmetric to the sphere if and only if they are linearly locally connected and Ahlfors 2-regular.

The paper identifies precise geometric conditions on the domain and the exponent for quasisymmetry.

The results connect geometric domain properties with the quasisymmetric uniformization of the associated surfaces.

Abstract

Let $\Omega$ be a planar Jordan domain and $\alpha>0$. We consider double-dome-like surfaces $\Sigma(\Omega,t^{\alpha})$ over $\overline{\Omega}$ where the height of the surface over any point $x\in\overline{\Omega}$ equals $\text{dist}(x,\partial\Omega)^{\alpha}$. We identify the necessary and sufficient conditions in terms of $\Omega$ and $\alpha$ so that these surfaces are quasisymmetric to $\mathbb{S}^2$ and we show that $\Sigma(\Omega,t^{\alpha})$ is quasisymmetric to the unit sphere $\mathbb{S}^2$ if and only if it is linearly locally connected and Ahlfors $2$-regular.