Quasiconformal and Bi-lipschitz Homeomorphisms, Uniform Domains and the Quasihyperbolic Metric

Let D be a proper subdomain of R" and kD the quasihyperbolic metric defined by the conformal metric tensor ds2 = dist(x, dD)~2ds2. The geodesies for this and related metrics are shown, by purely geometric methods, to exist and have Lipschitz continuous first derivatives. This is sharp for kD; we also obtain sharp estimates for the euclidean curvature of such geodesies. We then use these results to prove a general decomposition theorem for uniform domains in R", 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.