Generalized quasidisks and conformality

Abstract

In this notes, we survey the recent developments on theory of generalized quasidisks. Based on the more or less standard techniques used earlier, we also provide some minor improvements on the recorded results. A few nature questions were posed.2010 Mathematics Subject Classification. 30C62,30C65. Key words and phrases. homeomorphism of finite distortion, generalized quasidisk, local connectivity, three point property, cusps.C.-Y.

Uniform continuity of quasiconformal mappings onto generalized John domains

“…By Proposition 3.8, quasihyperbolic geodesics starting from the center in Ω are ϕ-dist John curves and note that the quasihyperbolic metric is comparable to the number of Whitney cubes that intersect the quasihyperbolic geodesic. The proof of [11,Lemma 3.7] applies with obvious modifications.…”

“…We need a weaker version of this condition, which was introduced in [11]. We say that Ω is (ϕ, ψ)-locally connected ((ϕ, ψ)-LC) if…”

“…The class of John domains includes all smooth domains, Lipschitz domains and certain fractal domains (for example the snowflake domain). Motivated by the recent studies on generalized quasidisks [8,11], Guo and Koskela [9] have introduced the class of ϕ-John domains, which form a natural generalization of John domains. Let ϕ be a continuous, increasing function with ϕ(0) = 0 and ϕ(t) ≥ t for all t > 0.…”

Generalized John disks

“…The only substantial change is that, in the estimate for Ω ( (0)), we apply the proof of [7,Lemma 3.7] to conclude that…”

“…The recent studies on mappings of finite distortion motivate one to relax the above definition of a John domain, see [1,6,7] for this theory. For example, the image of the unit disk D under a homeomorphism of the entire plane with locally exponentially integrable distortion is not necessarily a John domain; even a polynomial exterior cusp is possible [18].…”