Ricci iterations on Kahler classes

Abstract

In this paper we consider the dynamical system involved by the Ricci operator on the space of K\"ahler metrics. A. Nadel has defined an iteration scheme given by the Ricci operator for Fano manifold and asked whether it has some nontrivial periodic points. First, we prove that no such periodic points can exist. We define the inverse of the Ricci operator and consider the dynamical behaviour of its iterates for a Fano K\"ahler-Einstein manifold. In particular we show that the iterates do converge to the existing K\"ahler-Ricci soliton on a toric manifold. Finally, we define a finite dimensional procedure to give an approximation of K\"ahler-Einstein metrics using this iterative procedure and apply it for $\mathbb{CP}^2$ blown up in 3 points.