The Ricci iteration towards cscK metrics

TL;DR

This paper studies a Ricci iteration sequence related to finding constant scalar curvature Kähler metrics, proving its convergence to such metrics when they exist, thus confirming a conjecture and extending previous results.

Contribution

It demonstrates the global existence and convergence of Rubinstein's Ricci iteration sequence to cscK metrics in arbitrary Kähler classes, confirming Rubinstein's conjecture from 2007.

Findings

The iteration sequence exists for all steps.

The K-energy decreases along the sequence.

Sequence converges smoothly to cscK metrics if they exist.

Abstract

Motivated by the problem of finding constant scalar curvature K\"ahler metrics, we investigate a Ricci iteration sequence of Rubinstein that discretizes the pseudo-Calabi flow. While the long time existence of the flow is still an open question, we show that the iteration sequence does exist for all steps, along which the K-energy decreases. We further show that the iteration sequence, modulo automorphisms, converges smoothly to a constant scalar curvature K\"ahler metric if there is one, thus confirming a conjecture of Rubinstein from 2007 and extending results of Darvas--Rubinstein to arbitrary K\"ahler classes.