Critical metrics of the volume functional on complete manifolds

TL;DR

This paper classifies critical metrics of the volume functional on complete manifolds, showing they are isometric to standard models like spheres, Euclidean, hyperbolic spaces, or warped products, under various curvature conditions.

Contribution

It provides a classification of critical volume metrics on complete manifolds with specific curvature conditions, extending known results to broader settings.

Findings

Critical metrics with parallel Ricci tensor are isometric to standard models.

Bach-flat critical metrics are isometric to spheres, Euclidean, hyperbolic spaces, or warped products.

Classification results in dimensions three and four under weaker curvature assumptions.

Abstract

In this article, we investigate critical metrics of the volume functional on complete manifolds without boundary. We prove that any 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 $\mathbb{S}^n$, Euclidean space $\mathbb{R}^n$, hyperbolic space $\mathbb{H}^n$, or a warped product $\mathbb{R} \times_{\varphi} \Sigma_c$, where $\Sigma_c$ 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.