Integral of scalar curvature on manifolds with a pole (Q6621296)

The paper deals with complete \(3\)-dimensional Riemannian manifolds with a pole and non-negative Ricci curvature. After proving estimates for the radial derivative of a suitable geodesic sphere area, and for the Ricci curvature along the radial direction, the author proves that the asymptotic scaling invariant integral of the scalar curvature is equal to a term determined by the asymptotic volume ratio of the Riemannian manifold. In particular, he infers that any 3-dimensional complete, non-compact Ricci pinched Riemannian manifold with a pole, is flat.