Critical metrics of the volume functional on complete manifolds

TL;DR

The article classifies complete V-static metrics (critical points of the volume functional) on manifolds without boundary under two main regimes. First, with κ ≠ 0 and ∇Ric = 0, it proves rigidity to standard space forms or to warped products with an Einstein fiber, using direct differential identities rather than boundary integrals. Second, under Bach-flat (or radially flat) conditions with proper potential f, it achieves a similar global classification and provides explicit ODEs for the warping function, with dimension-specific corollaries in 3D and 4D. These results advance the understanding of volume comparison rigidity and static-type spaces in geometric analysis and general relativity. The work also supplies boundary-case analogues, yielding streamlined proofs of known rigidity theorems in a boundary-free setting.

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 $\mathbb{R} \times_{\varphi} Σ_c$, where $Σ_c$ 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.

Theorems & Definitions (32)

• Definition 1
• Theorem 1
• Remark 1
• Remark 2
• Theorem 2
• Remark 3
• Corollary 1
• Theorem 3
• Remark 4
• Example 1: miao2011einstein