Bi-Lipschitz extension from boundaries of certain hyperbolic spaces

Abstract

Tukia and Vaisala showed that every quasi-conformal map of \({\mathbb{R}^n}\) extends to a quasi-conformal self-map of \({\mathbb{R}^{n+1}}\). The restriction of the extended map to the upper half-space \({\mathbb{R}^n \times \mathbb{R}_+}\) is, in fact, bi-Lipschitz with respect to the hyperbolic metric. More generally, every simply connected homogeneous negatively curved manifold decomposes as \({M = N \rtimes \mathbb{R}_+}\) where N is a nilpotent group with a metric on which \({\mathbb{R}_+}\) acts by dilations. We show that under some assumptions on N, every quasi-symmetry of N extends to a bi-Lipschitz map of M. The result applies to a wide class of manifolds M including non-compact rank one symmetric spaces and certain manifolds that do not admit co-compact group actions. Although M must be Gromov hyperbolic, its curvature need not be strictly negative.