Weak Capacity and Critical Exponents

2017/10/25 by Jeff Lindquist, Lindquist, Jeff
Mathematics · #Analytic and geometric function theory #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Metric Geometry (math.MG)

paper · pdf · doi:10.48550/arxiv.1710.09181

openalex publication_date 2017/10/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 \mathbbS² to global quasisymmetric equivalence. We also use our weak capacity version to derive conditions for quasisymmetric equivalence to \mathbbS² 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.

Citations

• Quasisymmetric parametrizations of two-dimensional metric spheres
• Quasiconformal maps in metric spaces with controlled geometry

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