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Transactions of the American Mathematical Society

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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
MathSciNet review: 805959
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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.

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