Weak Capacity and Critical Exponents

Abstract

We investigate critical exponents relating to weak capacity in Ahlfors regular metric measure spaces. This allows a proof of a weak capacity version of a result by Bonk and Kleiner about the uniformization of metric $2$-spheres. Using our result, we promote local quasisymmetric equivalence with $\mathbb{S}^2$ to global quasisymmetric equivalence. We also use our weak capacity version to derive conditions for quasisymmetric equivalence to $\mathbb{S}^2$ in the presence of a group action. We investigate the relation between our defined critical exponents and Ahlfors regular conformal dimension, particularly in the cases where the Combinatorial Loewner Property is present or the space attains its Ahlfors regular conformal dimension.