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Patterson-Sullivan measure and conformally flat metrics. (English) Zbl 0868.53024
We construct a canonical compatible Riemannian metric on a given conformally flat manifold of the form \(\Omega/\Gamma\), where \(\Omega\) is a domain of \(S^n\) and \(\Gamma\) is a Kleinian group. A major ingredient in our construction is the measure supported on the limit set of \(\Gamma\) which was introduced by Patterson and Sullivan. We study the curvature and symmetry of the constructed metric. We also use the metric to obtain a new cohomology vanishing theorem for compact conformally flat manifolds.

MSC:
53C20 Global Riemannian geometry, including pinching
30F40 Kleinian groups (aspects of compact Riemann surfaces and uniformization)
53A30 Conformal differential geometry (MSC2010)
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