Metric on a Riemannian manifold $(M,g)$ is *Einstein*, if for some function $\lambda\colon M\to \mathbb R$
$$
Ric(g)=\lambda g.
$$
It is well know, that such $\lambda$ is, in fact, a constant.

The notion of Einstein metric fits perfectly into the world of K\"ahler manifolds, in this case such a metric is called *K\"ahler-Einstein*. Existence of K\"ahler-Einstein metrics is related to many deep results in differential and algebraic geometry.

I wonder whether there is something interesting going on for an arbitrary complex Hermitian manifold $(M, g, J)$. More specifically, let $\nabla$ be the Chern connection on a Hermitian manifold $(M, g, J)$ and let $\Omega\in \Lambda^{1,1}T^*M\otimes\Lambda^{1,1}T^*M$ be the curvature form of this connection
$$
\Omega(\xi,\bar\eta,\zeta,\bar\nu):=g\Bigl(
(
\nabla_\xi\nabla_{\bar\eta}-\nabla_{\bar\eta}\nabla_\xi-
\nabla_{[\xi,\bar\eta]}
)\zeta, \bar\nu
\Bigr).
$$
Note that unlike the K\"ahler situation, $\nabla$ has torsion and $\Omega$ has less symmetries, then usually. In particular, we can define **two** different *Chern-Ricci forms*:
$$
\Theta^{(1)}_{i\bar j}=g^{k\bar l}\Omega_{i\bar jk\bar l},\quad
\Theta^{(2)}_{k\bar l}=g^{i\bar j}\Omega_{i\bar jk\bar l}.
$$

One might play the same game as in Riemannian case, and introduce two versions of Chern-Einstein metrics (is there a common name for these phenomena?) $$ g_{i\bar j}=\lambda_1 \Theta^{(1)}_{i\bar j}, \mbox{ and } g_{i\bar j}=\lambda_2 \Theta^{(2)}_{i\bar j}. $$

*Remark.* In the second case the metric $g$ on $T^{1,0}M$ is Hermitian-Einstein. Hermitian-Einstein is **not** necessarily Chern-Einstein for $\Theta^{(2)}$.

Very vaguely, I would like to know

**Q1** What is know about the existence of Chern-Einstein metrics on Hermitian manifolds (I am mostly interested in dimensions >2)?

**Q2** What are topological/geometrical obstructions to the existence of such metrics?

A more direct question is

**Q3** Is it true that $\lambda_1$ and $\lambda_2$ (whic are apriori functions on $M$) are necessarily constant?