Quasisymmetric spheres over Jordan domains

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 $n\ge 3$.

26 pages, accepted in the Transactions of the American Mathematical Society

References in corpus (1)

• Sets of constant distance from a Jordan curve

Cited by in corpus (3)

• Bi-Lipschitz embedding of the generalized Grushin plane in Euclidean spaces
• Quasispheres and metric doubling measures
• Weak chord-arc curves and double-dome quasisymmetric spheres