# Ricci iteration for coupled K\"ahler-Einstein metrics

## TL;DR

This paper introduces the coupled Ricci iteration, a new dynamical system for studying coupled K"ahler-Einstein metrics, proving convergence in certain cases and linking coercivity of functionals to existence.

## Contribution

It develops the coupled Ricci iteration method and establishes convergence results, connecting functional coercivity to the existence of coupled K"ahler-Einstein metrics.

## Key findings

- Proves smooth convergence of the iteration for negative first Chern class.
- Shows equivalence between Ding functional coercivity and existence of CKE metrics.
- Demonstrates convergence on CKE Fano manifolds with discrete automorphism group.

## Abstract

In this paper, we introduce the "coupled Ricci iteration", a dynamical system related to the Ricci operator and twisted K\"ahler-Einstein metrics as an approach to the study of coupled K\"ahler-Einstein (CKE) metrics. For negative first Chern class, we prove the smooth convergence of the iteration. For positive first Chern class, we also provide a notion of coercivity of the Ding functional, and show its equivalence to existence of CKE metrics. As an application, we prove the smooth convergence of the iteration on CKE Fano manifolds assuming that the automorphism group is discrete.