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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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