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

Abstract

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.

Apollonian metric, uniformity and Gromov hyperbolicity

“…Since g(x) → ∞ as x → ∞, g is θ-QS, see [22,Theorem 3.10]. Moreover, by [17,Corollary 3,12] we have that g is θ 1 -QS with θ 1 (t) = C max{t λ , t 1/λ }, where C ≥ 1 and λ ≥ 1 depend only on θ. Hence, we get that f is θ 1 -QM with θ 1 (t) = C max{t λ , t 1/λ } as desired.…”

“…Moreover, one can obtain the following invariance of ϕ-uniformity of domains in R n under quasimöbius mappings. Recently, Hästö, Klén, Sahoo and Vuorinen [12] studied the geometric properties of ϕ-uniform domains in R n . They proved that ϕ-uniform domains are preserved under quasiconformal mappings of R n .…”

On the Removability Property of φ‐Natural Domains in Real Banach Spaces

“…To investigate the relationships between conformal invariants and quasi hyperbolic metrics, in [14], Vuorinen introduced the so-called ψ-uniform domains, which are a generalization of uniform domains. See [1,[15][16][17][18][19] for the development of ψ-uniform domains in recent years.…”

Mapping Properties of a Scale Invariant Cassinian Metric and a Gromov Hyperbolic Metric

“…These metrics are also known as hyperbolic-type metrics in the literature. The quasi-invariance or distortion properties of the metrics that are not Möbius invariant and the quasi-invariance properties under quasiconformal maps are of recent interest (see [13,23]). Note that the quasihyperbolic and the distance ratio metrics do satisfy the bilipschitz property with bilipschitz constant 2 under Möbius maps (see [32, page 36], [8,Corollary 2.5] and [7,Proof of Theorem 4]).…”

“…Proof. For all x, y ∈ D, τ D ( f (x), f (y)) ≤j D ( f (x), f (y)) ≤ C max{j D (x, y),j D (x, y) α } ≤ C max{2τ D (x, y), 2 ατ D (x, y) α } ≤ C 1 max{τ D (x, y),τ D (x, y) α },where the first and third inequalities follow from[18, Theorems 4.2, 4.3], the second from[13, Lemma 2.3], and the constant C 1 depends only on n and K.…”