arXiv subjects

Eric Cochran

Publications and source records attributed to Eric Cochran.

3 recordsLinked to original sources

arXiv2026

Homogeneous spaces with two equivalent isotropy summands develop positive Ricci curvature under Ricci flow

We study normalized Ricci flow on simply connected homogeneous spaces G/H for which the isotropy representation splits into exactly two equivalent irreducible subrepresentations. We prove that every G-invariant metric evolves to one with positive Ricci curvature, and that the family of G-invariant metrics with positive Ricci curvature is forward-invariant under the flow. The proof relies on the fact that the phase portrait of the family of fixed-volume G-invariant metrics can be explicitly visualized.

math.DG↗

arXiv2025

On Compact Quasi-Einstein Metrics of Constant Scalar Curvature

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.

math.DG↗

arXiv2024

Killing Fields on Compact m-Quasi-Einstein Manifolds

We show that given a compact, connected $m$-quasi Einstein manifold $(M,g,X)$ without boundary, the potential vector field $X$ is Killing if and only if $(M, g)$ has constant scalar curvature. This extends a result of Bahuaud-Gunasekaran-Kunduri-Woolgar, where it is shown that $X$ is Killing if $X$ is incompressible. We also provide a sufficient condition for a compact, non-gradient $m$-quasi Einstein metric to admit a Killing field. We do this by following a technique of Dunajski and Lucietti, who prove that a Killing field always exists in this case when $m=2$. This condition provides an alternate proof of the aforementioned result of Bahuaud-Gunasekaran-Kunduri-Woolgar. This alternate proof works in the $m = -2$ case as well, which was not covered in the original proof.

math.DG↗