Ricci iterations of well-behaved Kähler metrics

We introduce a large class of canonical Kähler metrics, called in this paper well-behaved, extending metrics induced by complex space forms. We study Kähler–Ricci iterations of well-behaved metrics on compact and non-compact Kähler manifolds. That is, we are interested in well-behaved metrics for which the iteration of the Ricci operator is a multiple of a Kähler metric, i.e., ${\rho }_{\omega }^k=\lambda \Omega$. In particular, when $k=1$, under some condition on the maximal domain of definition of canonical coordinates, we show that $\lambda$ is forced to be positive. Moreover, for arbitrary $k$, we prove two additional results. Namely, if $\omega$ and $\Omega$ are induced by a flat metric, then $\omega$ is Ricci-flat. Finally, if a Kähler-Ricci soliton $\Omega$ arises as Kähler–Ricci iteration of a metric $\omega$ induced by a complex space form, then the Kähler–Ricci soliton is forced to be trivial, that is, Kähler–Einstein. These three theorems extend well known results on Kähler–Einstein metrics to higher iterations of the Ricci operator and a larger class of metrics.