Old and New on the Quasihyperbolic Metric

Pekka Koskela
Additional contact information
Pekka Koskela: University of Jyväskylä, Department of Mathematics

A chapter in Quasiconformal Mappings and Analysis, 1998, pp 205-219 from Springer

Abstract: Abstract Let D be a proper subdomain of $$ {\mathbb{R}^d}$$ . Following Gehring and Palka [GP] we define the quasihyperbolic distance between a pair x 1, x 2 of points in D as the infimum of $$ {\smallint _\gamma }\frac{{ds}}{{D\left( {x,\partial D} \right)}}$$ over all rectifiable curves γ joining x 1, x 2 in D. We denote the quasihyperbolic distance between x 1, x 2 by k D (x 1, x 2). As pointed out by Gehring and Osgood [GO], x 1 and x 2 can be joined by a quasihyperbolic geodesic; also see [Mr]. The quasihyperbolic metric is comparable to the usual hyperbolic metric in a simply connected plane domain by the Koebe distortion theorem. For a multiply connected plane domain D these two metrics are comparable if and only if the boundary of D is uniformly perfect as shown by Beardon and Pommerenke [BP]. Gehring and Palka introduced the quasihyperbolic metric as a tool for the study of quasiconformal homogeneity and since then this metric has found a number of applications. Let us simply mention here that this metric plays an important role in the famous extension theorem of Jones [J1] for functions in BMO and that this metric has by now become a standard tool in the study of quasiconformal mappings.

Keywords: Hausdorff Dimension; Quasiconformal Mapping; Quasiregular Mapping; Uniform Domain; John Domain (search for similar items in EconPapers)
Date: 1998
References: Add references at CitEc
Citations:

There are no downloads for this item, see the EconPapers FAQ for hints about obtaining it.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-1-4612-0605-7_13

Ordering information: This item can be ordered from
http://www.springer.com/9781461206057

DOI: 10.1007/978-1-4612-0605-7_13

Access Statistics for this chapter

More chapters in Springer Books from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().