Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Quasiconformal and bi-Lipschitz homeomorphisms, uniform domains and the quasihyperbolic metric


Author: Gaven J. Martin
Journal: Trans. Amer. Math. Soc. 292 (1985), 169-191
MSC: Primary 30C60
DOI: https://doi.org/10.1090/S0002-9947-1985-0805959-2
MathSciNet review: 805959
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ D$ be a proper subdomain of $ {R^n}$ and $ {k_D}$ the quasihyperbolic metric defined by the conformal metric tensor $ d{\overline s ^2} = \operatorname{dist} {(x,\partial D)^{ - 2}}d{s^2}$. The geodesics for this and related metrics are shown, by purely geometric methods, to exist and have Lipschitz continuous first derivatives. This is sharp for $ {k_D}$; we also obtain sharp estimates for the euclidean curvature of such geodesics. We then use these results to prove a general decomposition theorem for uniform domains in $ {R^n}$, in terms of embeddings of bi-Lipschitz balls. We also construct a counterexample to the higher dimensional analogue of the decomposition theorem of Gehring and Osgood.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 30C60

Retrieve articles in all journals with MSC: 30C60


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1985-0805959-2
Article copyright: © Copyright 1985 American Mathematical Society