On Compact Quasi-Einstein Metrics of Constant Scalar Curvature

TL;DR

This paper proves that all compact three-dimensional quasi-Einstein metrics with constant scalar curvature are locally homogeneous, linking them to Sasakian geometry, and explores their construction as circle bundles over Einstein spaces.

Contribution

It establishes the local homogeneity of three-dimensional compact quasi-Einstein metrics with constant scalar curvature and characterizes their structure as circle bundles over K"ahler-Einstein bases.

Findings

All 3D compact quasi-Einstein metrics with constant scalar curvature are locally homogeneous.

Such metrics can be constructed as circle bundles over K"ahler-Einstein manifolds.

Existence of non-homogeneous examples in higher dimensions.

Abstract

We show that all compact quasi-Einstein metrics of constant scalar curvature in dimension three are locally homogeneous. We accomplish this by using the equivalence of constant scalar curvature quasi-Einstein metrics $(M,g,X)$ and quasi-Einstein metrics with $X$ Killing in the compact case to make a connection to Sasakian geometry in dimension three. In higher dimensions, there are examples which are non-locally homogeneous with constant scalar curvature. Such examples were constructed by Kunduri-Lucietti as circle bundles over a compact K\"ahler-Einstein base. We then ask when compact quasi-Einstein metrics of constant scalar curvature can be constructed as circle bundles over Einstein metrics, and prove that the base must in fact be K\"ahler-Einstein, assuming a conjecture due to Goldberg. These spaces, in fact, admit one parameter families of quasi-Einstein metrics by considering the canonical variation, which we study further.