Uniqueness of quasi-Einstein metrics on 3-dimensional homogeneous manifolds

2012/05/28 by A. Barros, Barros, A., E. Ribeiro +3 · 1 citation
Mathematics · Physics and Astronomy · #53C20 #53C21 (Primary) 53C65 (Secondary) #53C25 #Advanced Differential Geometry Research #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds

paper · doi:10.48550/arxiv.1205.6168

openalex publication_date 2012/05/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The purpose of this article is to study the existence and uniqueness of quasi-Einstein structures on 3-dimensional homogeneous Riemannian manifolds. To this end, we use the eight model geometries for 3-dimensional manifolds identified by Thurston. First, we present here a complete description of quasi-Einstein metrics on 3-dimensional homogeneous manifolds with isometry group of dimension 4. In addition, we shall show the absence of such gradient structure on Sol³, which has 3-dimensional isometry group. Moreover, we prove that Berger's spheres carry a non-trivial quasi-Einstein structure with non gradient associated vector field, this shows that a theorem due to Perelman can not be extend to quasi-Einstein metrics. Finally, we prove that a 3-dimensional homogeneous manifold carrying a gradient quasi-Einstein structure is either Einstein or ℍ²_κ × ℝ.

Citations

• Ricci solitons: the equation point of view
• The Geometries of 3-Manifolds

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• Remarks on potential functions of noncompact quasi-Einstein manifolds

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