Quasisymmetric spheres over Jordan domains

Abstract

Let Ω \Omega be a planar Jordan domain. We consider double-dome-like surfaces Σ \Sigma defined by graphs of functions of dist ⁡ ( ⋅ , ∂ Ω ) \operatorname {dist}(\cdot ,\partial \Omega ) over Ω \Omega . The goal is to find the right conditions on the geometry of the base Ω \Omega and the growth of the height so that Σ \Sigma is a quasisphere or is quasisymmetric to S 2 \mathbb {S}^2 . An internal uniform chord-arc condition on the constant distance sets to ∂ Ω \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 R n \mathbb {R}^n , for any n ≥ 3 n\ge 3 .

Weak Chord-Arc Curves and Double-Dome Quasisymmetric Spheres

“…We claim that it is enough to verify (4.1) only for those points a ∈ Σ(Ω, t α ) + whose projection satisfies dist(π(a), ∂Ω) ≤ 0 and for r < 0 /3. Indeed, following the notation in [12], let ∆ 0 /3 be the set of all points in Ω whose distance from ∂Ω is greater than 0 /3 and ∆ + 0 /3 be the subset of Σ(Ω, t α ) + whose projection on R 2 × {0} is ∆ 0 /3 . Since ∂∆ 0 /3 = γ 0 /3 is a quasicircle, the domain ∆ 0 /3 is 2-regular.…”

“…The LQC property and Väisälä's method. The connection between the LQC property of Ω and the LLC property of Σ(Ω, t α ) is established in the following proposition from [12]. (0, 1) and Ω is a Jordan domain whose boundary ∂Ω is a quasicircle.…”

“…When α > 1, Theorem 1.1 is trivial since, for any Jordan domain Ω, the surface Σ(Ω, t α ) is not linearly locally connected and hence not quasisymmetric to S 2 . If α = 1, Σ(Ω, t) is quasisymmetric to S 2 if and only if ∂Ω is a quasicircle [12,Theorem 1.1]. This result, combined with the fact that the projection of Σ(Ω, t) on Ω is a bi-Lipschitz mapping, and the fact that Ω is 2-regular if ∂Ω is a quasicircle, gives Theorem 1.1 when α = 1.…”

“…In this paper we consider a special case of the double dome-like surfaces constructed over planar Jordan domains Ω in [12]. For a number α > 0 and a Jordan domain Ω ⊂ R 2 consider the 2-dimensional surface in R 3 Σ(Ω, t α ) = {(x, z) : x ∈ Ω , z = ±(dist(x, ∂Ω)) α }.…”

“…We say that Ω has the level quasicircle property (or LQC property) if there exist 0 > 0 and K > 1 such that for all ∈ [0, 0 ], the -level set of Ω γ = {x ∈ Ω : dist(x, ∂Ω) = } is a K-quasicircle. A consequence of Theorem 1.2 in [12] is that if a planar Jordan domain Ω has the LQC property and ∂Ω is a chord-arc curve then Σ(Ω, t α ) is quasisymmetric to S 2 for all α ∈ (0, 1).…”