Quasisymmetric spheres over Jordan domains

TL;DR

This paper characterizes when certain double-dome-like surfaces over Jordan domains are quasispheres, providing geometric conditions on the base domain and height growth, and introduces new examples of quasispheres in higher dimensions.

Contribution

It establishes precise geometric and growth conditions under which these surfaces are quasispheres, extending the theory to higher dimensions with new examples.

Findings

Conditions on the base domain ensure quasisphere property.

A mild growth condition on the height function is sufficient.

New examples of quasispheres in dimensions n ≥ 3 are constructed.

Abstract

Let $\Omega$ be a planar Jordan domain. We consider double-dome-like surfaces $\Sigma$ defined by graphs of functions of $dist( \cdot ,\partial \Omega)$ over $\Omega$. The goal is to find the right conditions on the geometry of the base $\Omega$ and the growth of the height so that $\Sigma$ is a quasisphere, or quasisymmetric to $\mathbb{S}^2$. An internal uniform chord-arc condition on the constant distance sets to $\partial \Omega$, coupled with a mild growth condition on the height, gives a close-to-sharp answer. Our method also produces new examples of quasispheres in $\mathbb{R}^n$, for any $n\ge 3$.