Weak chord-arc curves and double-dome quasisymmetric spheres

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.

References in corpus (2)

• Sets of constant distance from a Jordan curve
• Quasisymmetric spheres over Jordan domains