Critical metrics of the volume functional on complete manifolds

2025/12/03 by Coimbra, Caio, Rafael Diógenes, Ernani Ribeiro +2
Mathematics · #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.2512.03731

openalex publication_date 2025/12/03 · openalex created_date 2025/12/05 · openalex updated_date 2026/07/31

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 \mathbbSⁿ, Euclidean space ℝⁿ, hyperbolic space ℍⁿ, or a warped product ℝ ×_φ 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.

Citations

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• On Bach-flat gradient shrinking Ricci solitons
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• Large-sphere and small-sphere limits of the Brown-York mass
• The entropy formula for the Ricci flow and its geometric applications
• On Some Static Solutions of Einstein's Gravitational Field Equations in a Spherically Symmetric Case

Related

• On the volume functional of compact manifolds with boundary with harmonic Weyl tensor
• Bach-flat critical metrics of the volume functional on 4-dimensional manifolds with boundary
• Critical Metrics of the Volume Functional on Manifolds with Boundary
• Critical metrics of the volume functional with pinched curvature
• Volume functional of compact 4-manifolds with a prescribed boundary metric