ρ-Einstein solitons and critical metrics of the volume functional on complete manifolds

Abstract

The purpose of this work is to investigate analytic and geometric properties of ρ-Einstein solitons and V -static metrics on complete Riemanninan manifolds. In the first part, we study geometric and analytical features of complete non-compact ρ-Einstein solitons, which are self-similar solutions of the Ricci–Bourguignon flow. We study the spectrum of the drifted Laplacian operator for complete gradient shrinking ρ-Einstein solitons. Moreover, similar to classical results due to Calabi–Yau and Bishop for complete Riemannian manifolds with nonnegative Ricci curvature, we prove new volume growth estimates for geodesic balls of complete noncompact ρ-Einstein solitons. In particular, the rigidity case is discussed. In addition, we establish weighted volume growth estimates for geodesic balls of such manifolds. For the second part, we investigate critical metrics of the volume functional (V -static metrics) on complete manifolds without boundary. We prove that every critical metric of the volume functional on a connected complete manifold with parallel Ricci tensor is isometric to one of the standard models. Moreover, we show that a Bach-flat critical metric of the volume functional on a complete, simply connected manifold with proper potential function is isometric to one of the following: the standard sphere Sn, Euclidean space Rn, hyperbolic space Hn, or a warped product R×ϕ Σc, where Σc is a regular level set of the potential function. In particular, we establish classification results in dimensions three and four under weaker assumptions on the Bach tensor.