Ricci iteration on homogeneous spaces

TL;DR

This paper investigates the Ricci iteration on non-Kähler homogeneous spaces, revealing new phenomena such as ancient Ricci iterations and their relation to Einstein metrics, expanding understanding of discrete Ricci flow analogues.

Contribution

It is the first study of Ricci iteration on non-Kähler homogeneous spaces, introducing the concept of ancient Ricci iterations and analyzing their convergence and collapsing behavior.

Findings

Existence of ancient Ricci iterations on non-Kähler spaces.

Ancient Ricci iterations can emerge from collapsed Einstein metrics.

In compact homogeneous spaces, collapsing is excluded due to a relative compactness result.

Abstract

The Ricci iteration is a discrete analogue of the Ricci flow. We give the first study of the Ricci iteration on a class of Riemannian manifolds that are not K\"ahler. The Ricci iteration in the non-K\"ahler setting exhibits new phenomena. Among them is the existence of so-called ancient Ricci iterations. As we show, these are closely related to ancient Ricci flows and provide the first nontrivial examples of Riemannian metrics to which the Ricci operator can be applied infinitely many times. In some of the cases we study, these ancient Ricci iterations emerge (in the Gromov--Hausdorff topology) from a collapsed Einstein metric and converge smoothly to a second Einstein metric. In the case of compact homogeneous spaces with maximal isotropy, we prove a relative compactness result that excludes collapsing.