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The Ricci iteration towards cscK metrics

Kewei Zhang

TL;DR

This work establishes the global existence and monotone behavior of Rubinstein's Ricci iteration as a discrete surrogate for the pseudo-Calabi flow in the search for cscK metrics across arbitrary Kähler classes. By developing the energy-functional framework, the metric completion via the $d_1$-metric, and robust a priori estimates, it proves that the iteration converges smoothly to a cscK metric modulo automorphisms when such a metric exists, thereby confirming Rubinstein's conjecture and extending prior results to general classes. Additionally, the paper extends the analysis to twisted cscK metrics, showing existence and convergence of a twisted iteration under suitable conditions. The results provide both theoretical justification and a potential numerical approach for achieving canonical Kähler metrics beyond Fano or proportional first Chern class settings, with implications for Tian's properness perspective and stability considerations.

Abstract

Motivated by the problem of finding constant scalar curvature Kähler 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 Kähler metric if there is one, thus confirming a conjecture of Rubinstein from 2007 and extending results of Darvas--Rubinstein to arbitrary Kähler classes.

The Ricci iteration towards cscK metrics

TL;DR

This work establishes the global existence and monotone behavior of Rubinstein's Ricci iteration as a discrete surrogate for the pseudo-Calabi flow in the search for cscK metrics across arbitrary Kähler classes. By developing the energy-functional framework, the metric completion via the -metric, and robust a priori estimates, it proves that the iteration converges smoothly to a cscK metric modulo automorphisms when such a metric exists, thereby confirming Rubinstein's conjecture and extending prior results to general classes. Additionally, the paper extends the analysis to twisted cscK metrics, showing existence and convergence of a twisted iteration under suitable conditions. The results provide both theoretical justification and a potential numerical approach for achieving canonical Kähler metrics beyond Fano or proportional first Chern class settings, with implications for Tian's properness perspective and stability considerations.

Abstract

Motivated by the problem of finding constant scalar curvature Kähler 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 Kähler metric if there is one, thus confirming a conjecture of Rubinstein from 2007 and extending results of Darvas--Rubinstein to arbitrary Kähler classes.
Paper Structure (7 sections, 31 theorems, 159 equations)

This paper contains 7 sections, 31 theorems, 159 equations.

Table of Contents

  1. Introduction
  2. Energy functionals
  3. The metric completion
  4. Basic properties of the Ricci iteration
  5. A priori estimates of the Ricci iteration
  6. Smooth convergence of the Ricci iteration
  7. The case of twisted cscK metrics

Key Result

Lemma 1.1

A closed $(1,1)$-form $\theta$ on $(X,\omega)$ satisfies $\mathrm{tr}_\omega\theta=\mathrm{Const.}$ if and only if $\theta$ is harmonic with respect to the Kähler metric $\omega$.

Theorems & Definitions (55)

  • Lemma 1.1
  • Remark 1.2
  • Conjecture 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Definition 2.3
  • ...and 45 more