Suppose $(M,g, \omega)$ is a Kähler manifold with $\text{Ric}(g) = g$, i.e., $M$ is a Fano manifold. Is $M$ necessarily compact? If not, perhaps complete and Fano implies compact? I'd like to build a catalog of examples of Fano manifolds, so any input is appreciated.



Do non-compact Fano manifolds exist?
By the [Bonnet Myers theorem](https://en.wikipedia.org/wiki/Myers%27s_theorem), bounded positive Ricci curvature and complete Riemannian metric forces compact. David Wraith once explained to me that if the Ricci decays more slowly than quadratically in distance from a given point, on a complete Riemannian manifold, then the manifold is compact, a stronger result than Bonnet-Myers. Wraith's result does not have a published proof, as far as I know.


