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

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 Sol3Sol3, 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 cannot 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 Hκ2×R.