On Compact Quasi-Einstein Metrics of Constant Scalar Curvature
Eric Cochran
TL;DR
The paper classifies compact quasi-Einstein metrics with constant scalar curvature, proving that in dimension three such metrics are locally homogeneous by leveraging a Sasakian reduction and the Tanno classification. It connects the quasi-Einstein condition to Killing fields, Sasakian geometry, and Riemannian submersion theory to derive a complete 3D picture, while also exploring higher-dimensional circle bundle constructions over Kähler–Einstein bases and the role of the Goldberg conjecture. A canonical variation analysis reveals when multiple quasi-Einstein metrics can exist and when Einstein metrics arise within the variation. Finally, the work provides an alternate proof that in 3D the integral curves of the quasi-Einstein vector field must be closed, establishing a rigidity that underlies the local homogeneity result.
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 Kähler-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 Kähler-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.