Geometric characterizations of inner uniformity through Gromov hyperbolicity

2017/06/17 by Manzi Huang, Antti Rasila, Huang, Manzi +5
Mathematics · #30F45 #30L10 #Analytic and geometric function theory #Complex Variables (math.CV) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Primary: 30C65 #Secondary: 30C20

paper · pdf · doi:10.48550/arxiv.1706.05494

openalex publication_date 2017/06/17 · openalex created_date 2017/06/30 · openalex updated_date 2026/07/28

Abstract

In this paper, we study the characterization of inner uniformity of bounded domains G in \IRⁿ, and prove that the following three conditions are equivalent: (1) G is inner uniform; (2) G is Gromov hyperbolic and its inner metric boundary is naturally quasisymmetrically equivalent to the Gromov boundary; (3) G is Gromov hyperbolic and linearly locally connected with respect to the inner metric. The equivalence between the conditions (1) and (2), and the implication from (2) to (3) affirmatively answer three questions raised by Bonk, Heinonen, and Koskela in 2001.

Citations

• Quasisymmetric parametrizations of two-dimensional metric spheres
• The boundary correspondence under quasiconformal mappings
• Quasiconformal maps in metric spaces with controlled geometry

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