UCLA

Let $(Z,d,\mu)$ be a compact, connected, Ahlfors $Q$-regular metric space with $Q>1$. Using a hyperbolic filling of $Z$, we define the notions of the $p$-capacity between certain subsets of $Z$ and of the weak covering $p$-capacity of path families $\Gamma$ in $Z$. We show comparability results and quasisymmetric invariance. We reprove a result due to Tyson on the geometric quasiconformality of quasisymmetric maps between compact, connected, Ahlfors $Q$-regular metric spaces. Under certain conditions, we identify the Ahlfors regular conformal dimension of $Z$ with critical exponents arising from weak capacity. Following an approach by Mario Bonk and Bruce Kleiner, we prove a necessary and sufficient condition involving weak capacity for an Ahlfors regular metric space that is topologically $\mathbb{S}^2$ to be quasisymmetrically equivalent to $\mathbb{S}^2$.