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

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.