Ricci iterations on Kahler classes

TL;DR

This paper studies the Ricci operator's iterative behavior on Kähler classes, proving the absence of nontrivial periodic points, analyzing convergence to Kähler-Ricci solitons, and proposing an approximation method for Kähler-Einstein metrics.

Contribution

It introduces a new analysis of Ricci iterations on Kähler classes, including nonexistence of periodic points and convergence results, plus a practical approximation procedure.

Findings

No nontrivial periodic points for Ricci iteration on Fano manifolds.

Iterates of the inverse Ricci operator converge to Kähler-Ricci solitons.

A finite-dimensional approximation method for Kähler-Einstein metrics demonstrated on -blowup of .

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.