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Uniformizing Gromov Hyperbolic Spaces

出版	University of Jyväskylä. Department of Mathematics, 2001
主題	Mathematics / Geometry / Non-Euclidean
ISBN	28562909819782856290989
URL	http://books.google.com.hk/books?id=-mFZSQAACAAJ&hl=&source=gbs_api

註釋The unit disk in the complex plane has two conformally related lives: one as an incomplete space with the metric inherited from $\mathbb{R}^2$, the other as a complete Riemannian $2$-manifold of constant negative curvature. Consequently, problems in conformal analysis can often be formulated in two equivalent ways depending on which metric one chooses to use. The purpose of this volume is to show that a similar choice is available in much more generality. Here, the authors replace the incomplete disk by a uniform metric space (defined as a generalization of a uniform domain in $\mathbb{R}^n$) and the space of constant negative curvature by a general Gromov hyperbolic space. They then prove that there is a one-to-one correspondence between quasiisometry classes of (proper, geodesic, and roughly starlike) Gromov hyperbolic spaces and the quasisimilarity classes of bounded locally compact uniform spaces. They study Euclidean domains that are Gromov hyperbolic with respect to the quasihyperbolic metric and the Martin boundaries of such domains. A characterization of planar Gromov hyperbolic domains is given. They also study quasiconformal homeomorphisms of Gromov hyperbolic spaces of bounded geometry; under mild conditions on the spaces we prove that such maps are rough quasiisometries. They employ a version of the classical Gehring-Hayman theorem, and methods from analysis on metric spaces such as modulus estimates on Loewner spaces.