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On the conformal type of a Riemannian manifold. (English. Russian original) Zbl 0873.53025

Funct. Anal. Appl. 30, No. 2, 106-117 (1996); translation from Funkts. Anal. Prilozh. 30, No. 2, 40-55 (1996).
The authors define an open Riemannian manifold to be of conformally parabolic or conformally hyperbolic type if the conformal capacity of the (absolute) ideal boundary of the manifold vanishes or is positive, respectively. Using the properties of conformal capacity and modulus, they develop geometric tests and criteria for the conformal type of a Riemannian manifold. A Riemannian manifold is conformally parabolic if and only if there is a conformal change of the metric that transforms the given manifold into a complete Riemannian manifold of finite volume. Integral relations on a Riemannian manifold that are connected with conformal-type problems including the Ahlfors-Gromov lemma are discussed in §4.

MSC:

53C20 Global Riemannian geometry, including pinching
53A30 Conformal differential geometry (MSC2010)
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