Duality, uniformity, and linear local connectivity

Abstract

Using Väisälä’s metric duality on joinability of sets, we show, among other things, that in $\mathbb{R}^3$ if the complementary domains of a surface are LLC, they are also uniform. As an application, we show that an Ahlfors regular topological sphere that admits a quasiconformal reflection is quasisymmetrically equivalent to the standard sphere.