Asymptotic Geometry and Conformal Types of Carnot-Carathéodory Spaces |
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Authors: | VA Zorich |
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Institution: | (1) Department of Mathematics and Mechanics, Moscow State University, 119899 Moscow, Russia, e-mail: vzor@glas.apc.org, RU |
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Abstract: | An intrinsic definition in terms of conformal capacity is proposed for the conformal type of a Carnot—Carathéodory space
(parabolic or hyperbolic). Geometric criteria of conformal type are presented. They are closely related to the asymptotic
geometry of the space at infinity and expressed in terms of the isoperimetric function and the growth of the area of geodesic
spheres. In particular, it is proved that a sub-Riemannian manifold admits a conformal change of metric that makes it into
a complete manifold of finite volume if and only if the manifold is of conformally parabolic type. Further applications are
discussed, such as the relation between local and global invertibility properties of quasiconformal immersions (the global
homeomorphism theorem).
Submitted: November 1997, revised: November 1998. |
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