Perelman singular manifolds

On a Riemannian manifold with a smooth function f: M→ℝ, we consider the linearization of the Perelman scalar curvature ℛ and its L^2-formal adjoint operator δℛ^*. A manifold endowed with a metric g whose operator δℛ^* has a nontrivial kernel is called a Perelman singular manifold. In this paper, we present examples and apply general maximum principles to obtain rigidity or nonexistence results in the underlying setting.