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

TL;DR

This paper investigates the existence and uniqueness of quasi-Einstein metrics on 3-dimensional homogeneous manifolds, revealing specific structures and limitations across Thurston's model geometries.

Contribution

It provides a complete classification of quasi-Einstein metrics on certain 3D homogeneous manifolds and shows the non-existence or special cases on others.

Findings

Complete description of quasi-Einstein metrics on 3D homogeneous manifolds with 4D isometry group

Absence of gradient quasi-Einstein structures on Sol^3

Existence of non-trivial quasi-Einstein structures on Berger's spheres

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