Bi-Lipschitz extension from boundaries of certain hyperbolic spaces |
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Authors: | Anton Lukyanenko |
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Institution: | 1. University of Illinois Urbana-Champaign, 1407 W. Green Street, Urbana, IL, 61802, USA
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Abstract: | Tukia and Väisälä 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. |
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