Quasisymmetric spheres over Jordan domains

Let Ω be a planar Jordan domain. We consider double-dome-like surfaces Σ defined by graphs of functions of dist(·, ∂Ω) over Ω. The goal is to find the right conditions on the geometry of the base Ω and the growth of the height so that Σ is a quasisphere or is quasisymmetric to S². An internal uniform chord-arc condition on the constant distance sets to ∂Ω, coupled with a mild growth condition on the height, gives a close-to-sharp answer. Our method also produces new examples of quasispheres in Rⁿ, for any n ≥ 3.