Quantitative correspondence between quasi-symmetric mappings on complete metric spaces and rough quasi-isometric mappings on their hyperbolic fillings

2025/10/31 by Manzi Huang, Xiantao Wang, Huang, Manzi +5
Mathematics · #Analytic and geometric function theory #Fixed Point Theorems Analysis #Geometric and Algebraic Topology

paper · pdf · doi:10.48550/arxiv.2510.27523

Abstract

In this paper, we establish a quantitative correspondence between power quasi-symmetric mappings on complete metric spaces and rough quasi-isometric mappings on their hyperbolic fillings. In particular, we prove that the exponents in the power quasi-symmetric mappings coincide with the coefficients in the rough quasi-isometric mappings. This shows that the obtained correspondence is both sharp and consistent. In this way, we generalize the corresponding result by Björn, Björn, Gill, and Shanmugalingam (J. Reine Angew. Math., 2017) from the setting of rooted trees to that of hyperbolic fillings.

Citations

• Quasi-symmetries between metric spaces and rough quasi-isometries between their infinite hyperbolic cones
• Characterizations for the existence of traces of first-order Sobolev spaces on hyperbolic fillings
• Extension and trace theorems for noncompact doubling spaces
• Busemann functions and uniformization of Gromov hyperbolic spaces
• Uniformizing Gromov hyperbolic spaces with Busemann functions
• Traces of Besov, Triebel-Lizorkin and Sobolev spaces on metric spaces
• Besov spaces via hyperbolic fillings
• Weak capacity and modulus comparability in Ahlfors regular metric spaces
• Triebel-Lizorkin spaces on metric spaces via hyperbolic fillings
• Sobolev spaces and hyperbolic fillings
• On the conformal gauge of a compact metric space
• Quasi-isometries between visual hyperbolic spaces
• Boundaries of hyperbolic groups

Related

• A metric between quasi-isometric trees
• Real valued functions and metric spaces quasi-isometric to trees
• Quasi-symmetries between metric spaces and rough quasi-isometries between their infinite hyperbolic cones
• Bilipschitz Equivalence of Trees and Hyperbolic Fillings
• On the quantitative quasi-isometry problem: transport of Poincaré inequalities and different types of quasi-isometric distortion growth