Ricci iterations of well-behaved K\"ahler metrics

Abstract

We introduce a large class of canonical K\"ahler metrics, called in this paper well-behaved, extending metrics induced by complex space forms. We study K\"ahler--Ricci iterations of well-behaved metrics on compact and non-compact K\"ahler manifolds. That is, we are interested in well-behaved metrics for which the iteration of the Ricci operator is a multiple of a K\"ahler 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 K\"ahler-Ricci soliton $\Omega$ arises as K\"ahler--Ricci iteration of a metric $\omega$ induced by a complex space form, then the K\"ahler--Ricci soliton is forced to be trivial, that is, K\"ahler--Einstein. These three theorems extend well known results on K\"ahler--Einstein metrics to higher iterations of the Ricci operator and a larger class of metrics.