Volume comparison of conformally compact manifolds with scalar curvature \(R\geq -n(n-1)\) (Q270238)

The present paper concerns a version of the well-known Schoen's conjecture in terms of renormalized volume, for conformally compact manifolds of arbitrary dimension. So, an affirmative answer is given for the case when the manifold is a slight perturbation of the hyperbolic space or other strictly stable conformally compact Einstein manifolds. The main tool is the normalized Ricci-DeTurck flow. As an application, a local volume comparison of conformally compact manifolds of scalar curvature \(R\geq -n(n-1)\) is obtained.