Patterson-Sullivan measure and conformally flat metrics (Q1358242)

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.