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.

We solve the classical problem of Plateau in every metric space which is 1-complemented in an ultra-completion of itself. This includes all proper metric spaces as well as many locally non-compact metric spaces, in particular, all dual Banach spaces, some non-dual Banach spaces such as L 1 , all Hadamard spaces, and many more. Our results generalize corresponding results of Lytchak and the second author from the setting of proper metric spaces to that of locally non-compact ones. We furthermore solve the Dirichlet problem in the same class of spaces. The main new ingredient in our proofs is a suitable generalization of the Rellich-Kondrachov compactness theorem, from which we deduce a result about ultra-limits of sequences of Sobolev maps.

Abstract. We study uniform continuity of quasiconformal mappings onto δ-Gromov-hyperbolic ϕ-John domains. The general ϕ-John case is also investigated.

We establish the basic analytic properties of mappings of finite distortion between proper Ahlfors regular metric measure spaces that support a (1, 1)-Poincaré inequality. As applications, we prove that under certain integrability assumption for the distortion function, the branch set of a mapping of finite distortion between generalized n-manifolds of type A has zero Hausdorff n-measure.

Abstract. We introduce a weaker variant of the concept of three point property, which is equivalent to a non-linear local connectivity condition introduced in [12], sufficient to guarantee the extendability of a conformal map f : D → Ω to the entire plane as a homeomorphism of locally exponentially integrable distortion. Sufficient conditions for extendability to a homeomorphism of locally p-integrable distortion are also given.