Positive Ricci curvature through Cheeger deformation

Abstract

Let $(M,g)$ be a Riemannian manifold with an isometric $G$-action. If a principal orbit has finite fundamental group and $\mathrm{Ric}_{M^{reg}/G}\geq 1$, Searle--Wilhelm proved that $M$ admits a new metric $\tilde g$ of positive Ricci curvature. $\tilde g$ is obtained after a conformal change followed by a Cheeger deformation. The question remained on whether it is sufficient to consider only the Cheeger deformation to attain positive Ricci curvature on the new metric $\tilde g$. Here we approach this question by giving necessary and sufficient conditions on the $G$-action. In particular, we construct an infinite family of manifolds satisfying the hypothesis of Searle--Wilhelm that do not develop positive Ricci curvature after Cheeger deformation. Further exploring the theory, we give a alternative proofs for Lawson--Yau result on positive scalar curvature under non-abelian symmetry, among others.Comment: 26 pages. Exposition reviewed. Applications include