Weak Chord-Arc Curves and Double-Dome Quasisymmetric Spheres

Abstract

Let Ω be a planar Jordan domain and α > 0. We consider double-dome-like surfaces Σ(Ω, t α ) over Ω where the height of the surface over any point x ∈ Ω equals dist(x, ∂Ω) α . We identify the necessary and su cient conditions in terms of Ω and α so that these surfaces are quasisymmetric to S 2 and we show that Σ(Ω, t α ) is quasisymmetric to the unit sphere S 2 if and only if it is linearly locally connected and Ahlfors 2-regular.