Quasihyperbolic geodesics in John domains.

“…A domain G ⊂ R n is said to satisfy the twisted cone condition, if for every x, y ∈ G there exists a rectifiable path γ ⊂ G joining x and y such that Note that the path γ in the definition of the twisted cone condition may be replaced by a quasihyperbolic geodesic (see [GHM89]). We observe from Section 1 that a ϕ-uniform domain need not be uniform (or quasi-convex).…”

Section: The ϕ-Uniform Domains Which Are Uniformmentioning

confidence: 99%

Geometric properties of $$\varphi $$ φ -uniform domains

Hästö

Клен

Sahoo

et al. 2016

J Anal

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Abstract. We consider proper subdomains G of R n and their images G = f (G) under quasiconformal mappings f of R n . We compare the distance ratio metrics of G and G ; as an application we show that ϕ-uniform domains are preserved under quasiconformal mappings of R n . A sufficient condition for ϕ-uniformity is obtained in terms of the quasi-symmetry condition. We give a geometric condition for uniformity: If G ⊂ R n is ϕ-uniform and satisfies the twisted cone condition, then it is uniform. We also construct a planar ϕ-uniform domain whose complement is not ψ-uniform for any ψ.

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“…Let D & C be a bounded c-John disk and let z 0 be the euclidean midpoint of a hyperbolic geodesic with harmonic measure and hyperbolic distance in john disks 287 endpoints z 1 and z 2 in D. By [GHM,2.16 Lemma], there exists a crosscut of D containing z 0 which separates the components of n fz 0 g in D and…”

Section: We Say That a Domainmentioning

confidence: 99%