Rigidity and Nonexistence Results for Perelman Singular Spaces

On a Riemannian manifold with a smooth function [Formula: see text], we consider the linearization of the Perelman scalar curvature [Formula: see text] and its [Formula: see text]-formal adjoint operator [Formula: see text]. A manifold endowed with a metric [Formula: see text] whose operator [Formula: see text] has a nontrivial kernel is called a Perelman singular space. In this paper, we present examples and apply general maximum principles to obtain rigidity or nonexistence results in this context. Specifically, we obtain rigidity results for the sphere and Euclidean space under suitable conditions.