Perelman singular manifolds

On a Riemannian manifold with a smooth function $f:M\to \mathbb{R}$, we consider the linearization of the Perelman scalar curvature $\mathcal{R}$ and its $L^2$-formal adjoint operator $\delta {\mathcal{R}}^{\ast }$. A manifold endowed with a metric $g$ whose operator $\delta {\mathcal{R}}^{\ast }$ 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.