Coupled continuity equations for constant scalar curvature K\"ahler metrics

TL;DR

This paper introduces a coupled system of elliptic equations for K"ahler metrics and forms, proving convergence to constant scalar curvature K"ahler metrics and recovering classical results for K"ahler-Einstein metrics.

Contribution

It develops a new coupled elliptic system approach for cscK metrics, establishing higher order estimates and convergence results, including classical K"ahler-Einstein cases.

Findings

Proves smooth convergence to cscK metrics under uniform estimates.

Recovers existence of K"ahler-Einstein metrics for negative first Chern class.

Shows convergence to K"ahler-Einstein metrics on Riemann surfaces with genus ≥ 2.

Abstract

Inspired by a parabolic system of Li-Yuan-Zhang and the continuity equation of La Nave-Tian, we study a system of elliptic equations for a K\"ahler metric $\omega$ and a closed $(1, 1)$-form $\alpha$. Assuming a uniform estimate for $\omega$, we prove higher order estimates and smooth convergence to a cscK metric coupled to a harmonic $(1, 1)$-form. A simplification of the system is used to recover existence results for K\"ahler-Einstein metrics when $c_1(X) < 0$. On Riemann surfaces with genus at least $2$, we show smooth convergence to the unique K\"ahler-Einstein metric from a large class of initial data.