2025/11/26 by Cochran, Eric
Mathematics · #53C25 #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Nonlinear Partial Differential Equations
paper · doi:10.48550/arxiv.2511.21892
openalex publication_date 2025/11/26 · openalex created_date 2025/12/03 · openalex updated_date 2026/07/28
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.