# Convergence of the K\"ahler-Ricci iteration

## TL;DR

This paper proves that the Ricci iteration converges to a Kähler-Einstein metric when it exists, confirming a conjecture and providing a new approach to uniformization of the Riemann sphere.

## Contribution

It confirms the conjecture that Ricci iteration converges to Kähler-Einstein metrics, similar to Ricci flow, and introduces a new uniformization method for the Riemann sphere.

## Key findings

- Ricci iteration converges to Kähler-Einstein metrics when they exist
- Provides a new uniformization method for the Riemann sphere
- Confirms the conjectured behavior of Ricci iteration

## Abstract

The Ricci iteration is a discrete analogue of the Ricci flow. According to Perelman, the Ricci flow converges to a Kahler-Einstein metric whenever one exists, and it has been conjectured that the Ricci iteration should behave similarly. This article confirms this conjecture. As a special case, this gives a new method of uniformization of the Riemann sphere.