On Compact Quasi-Einstein Metrics of Constant Scalar Curvature
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866917165286817792 |
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| author | Cochran, Eric |
| author_facet | Cochran, Eric |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_21892 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On Compact Quasi-Einstein Metrics of Constant Scalar Curvature Cochran, Eric Differential Geometry 53C25 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. |
| title | On Compact Quasi-Einstein Metrics of Constant Scalar Curvature |
| topic | Differential Geometry 53C25 |
| url | https://arxiv.org/abs/2511.21892 |