2020/08/15 by Clark Butler, Butler, Clark
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analytic and geometric function theory #FOS: Mathematics #Geometric Analysis and Curvature Flows #Metric Geometry (math.MG)
paper · pdf · doi:10.48550/arxiv.2008.06806
openalex publication_date 2020/08/15 · openalex created_date 2020/08/21 · openalex updated_date 2026/07/28
We define ∂-biLipschitz homeomorphisms between uniform metric spaces and show that these maps are always quasimöbius. We also show that a homeomorphism being ∂-biLipschitz is equivalent to the map biLipschitz in the quasihyperbolic metrics on these spaces. The proofs of these claims require us to uniformize the quasihyperbolic metric. We further show that all admissible uniformizations of a Gromov hyperbolic space are quasimöbius to one another by the identity map, including those uniformizations that are based at a point of the Gromov boundary. Using the main results we then show that the sphericalization and inversion operations are compatible with uniformization of hyperbolic spaces in a natural sense.