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Quasiconformality and hyperbolic skew

Part of: Geometric function theory Analysis on metric spaces

Published online by Cambridge University Press:  18 December 2020

COLLEEN ACKERMANN  and
ALASTAIR FLETCHER
COLLEEN ACKERMANN
Affiliation:
Montgomery College, Rockville Campus, 51 Mannakee Street, Rockville, MD20850, U.S.A. e-mail: colleen.ackermann@montgomerycollege.edu
ALASTAIR FLETCHER
Affiliation:
Department of Mathematical Sciences, Northern Illinois University, DeKalb, IL60115-2888, U.S.A. e-mail: fletcher@math.niu.edu
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Abstract

We prove that if $f:\mathbb{B}^n \to \mathbb{B}^n$ , for n ≥ 2, is a homeomorphism with bounded skew over all equilateral hyperbolic triangles, then f is in fact quasiconformal. Conversely, we show that if $f:\mathbb{B}^n \to \mathbb{B}^n$ is quasiconformal then f is η-quasisymmetric in the hyperbolic metric, where η depends only on n and K. We obtain the same result for hyperbolic n-manifolds. Analogous results in $\mathbb{R}^n$ , and metric spaces that behave like $\mathbb{R}^n$ , are known, but as far as we are aware, these are the first such results in the hyperbolic setting, which is the natural metric to use on $\mathbb{B}^n$ .

MSC classification

Primary: 30C62: Quasiconformal mappings in the plane
Secondary: 30L10: Quasiconformal mappings in metric spaces
Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 171 , Issue 3 , November 2021 , pp. 449 - 466
Copyright
© Cambridge Philosophical Society 2020

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Footnotes

†

Supported by a grant from the Simons Foundation, #352034.

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