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

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 $So{l}^3,$ 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 ${\mathbb{H}}_{\kappa }^2\times \mathbb{R}.$

To appear in Differential Geometry and its Applications

References in corpus (4)

• Recent Progress on Ricci Solitons
• On the nonexistence of quasi-Einstein metrics
• On the Bartnik extension problem for the static vacuum Einstein equations
• Locally conformally flat Lorentzian quasi-Einstein manifolds

Cited by in corpus (1)

• Characterizations and integral formulae for generalized quasi-Einstein metrics